Rationality and Best Response
Frahm, 2015 • Rationality and Best Response
arXiv:1509.08353v1 [cs.GT] 28 Sep 2015
Rationality and Best Response* Gabriel Frahm† Helmut Schmidt University Department of Mathematics/Statistics Chair for Applied Stochastics and Risk Management September 29, 2015
Abstract The Nash equilibrium is the central solution concept in non-cooperative game theory, but it has often been considered insufficient or implausible. It is shown that rational players in a non-cooperative game need not give a best response to each other. Giving a best response in the usual sense of game theory is even impossible. The best-response principle can lead to solutions that are logically inconsistent if all players are rational. Moreover, it is proved that the set of rational solutions of every non-cooperative game is essentially unique. This solves a longstanding problem of non-cooperative game theory, i.e., the possibility of multiple and even Pareto-inefficient solutions. Finally, it is demonstrated that the rational solutions of well-known normal-form games differ from those advocated in traditional game theory. The differences turn out to be essential both from an economic and a social point of view.
Keywords: Best-response principle, dominance principle, Nash equilibrium, normal-form game, Pareto efficiency, rationality. JEL Subject Classification: C72, D81.
* I would like to thank very much Bob Aumann, Steve Brams, Rainer Dyckerhoff, Mathias Risse and Rainer Schüssler
for their helpful comments on the manuscript and our valuable discussions. This paper is dedicated to John Nash and John von Neumann, who influenced generations of scientists with their seminal work. † Phone: +49 40 6541-2791, e-mail: [email protected].
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Frahm, 2015 • Rationality and Best Response
Contents 1. Motivation
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2. Theoretical Background
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2.1. State Space and Private Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The Response Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Strategy Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3. Main Results
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4. Examples
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4.1. 4.2. 4.3. 4.4. 4.5.
Game of Chicken . . . Prisoner’s Dilemma . . The Win-Win Situation Battle of the Sexes . . . Matching Pennies . . .
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5. Conclusion
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The logical roots of game theory are in Bayesian decision theory. Indeed, game theory can be viewed as an extension of decision theory (to the case of two or more decision-makers), or as its essential logical fulfillment. Thus, to understand the fundamental ideas of game theory, one should begin by studying decision theory. Myerson (1991, p. 5)
1. Motivation HE Nash equilibrium (Nash, 1951) is the central solution concept in non-cooperative game theory and represents a cornerstone of modern economics. Over the last decades, the Nash equilibrium has often been considered insufficient or implausible. Colman (2004) points out the possibility of multiple and even Pareto-inefficient solutions (Harsanyi and Selten, 1988). Moreover, it is well-known that Nash equilibria can be subgame imperfect (Selten, 1975) and the idea that the players use a random generator when applying a mixed strategy leads to an obscure and nonsensical description of real-life strategic conflicts (Rubinstein, 1991). Typical arguments supporting the Nash equilibrium are discussed by Risse (2000), who comes to the conclusion that, “All of these arguments either fail entirely or have a very limited scope.” According to Nash (1951),
T
[. . . ] an equilibrium point is an n-tuple [of mixed strategies] such that each player’s mixed strategy maximizes his payoff if the strategies of the others are held fixed. Similarly, Aumann and Brandenburger (1995) describe a Nash equilibrium as,
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Frahm, 2015 • Rationality and Best Response [. . . ] a profile of strategies in which each player’s strategy is optimal for him, given the strategies of the others. Moreover, Aumann and Brandenburger (1995) also observe that, Suppose that each player is rational, knows his own payoff function, and knows the strategy choices of the others. Then the players’ choices constitute a Nash equilibrium in the game being played. Hence, it is claimed that each player gives a best response to the others if the given solution is rational. This assertion has become an integral part of non-cooperative game theory and will be referred to as the best-response principle. As already mentioned, the message that a Nash equilibrium need not be a rational solution is not new. Indeed, one can find a large number of procedures that aim at a refinement of the Nash equilibrium (Govindan and Wilson, 2008, Harsanyi and Selten, 1988, Myerson, 2001). This means additional criteria such as Pareto efficiency, subgame perfectness, stability, etc., have been developed to eliminate all Nash equilibria that are considered implausible. Hence, the Nash equilibrium is assumed to be a valid but insufficient solution concept. Unfortunately, refinement does not address the root of the problem: A rational solution need not be a Nash equilibrium. In more general terms, if all players are rational, they need not give a best response to each other.1 This assertion can be proved by traditional means of decision theory (von Neumann and Morgenstern, 1944, Savage, 1954). The technical framework chosen in this work is very general and reproduces different kinds of strategic interaction that can occur in real life. This covers a wide range of equilibrium models of non-cooperative game theory. Due to the fundamental importance of the Nash equilibrium in economics and many other disciplines such as biology, psychology, and politics, the results presented in this work might be interesting to a broad audience. It is shown that (i) rational players in a normal-form game need not give a best response to each other, (ii) giving a best response, in the usual sense of game theory, is even impossible, (iii) the solutions that are derived by the best-response principle can be logically inconsistent if all players are rational, (iv) the set of rational solutions of every non-cooperative game is essentially unique, and (v) typically, the rational solutions of well-known normal-form games essentially differ from those predicted by the best-response principle. In Section 2, I present the theoretical background. In that section, I discuss how rational players choose an optimal strategy and I show how their strategies are connected to each other. Section 3 contains the main results of this work. In Section 4, I illustrate the given results on the basis of well-known two-player normal games. The last section concludes.
1 This statement essentially depends on the meaning of “response,” which will be discussed in Section 2.3.
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Frahm, 2015 • Rationality and Best Response
2. Theoretical Background Let (Ω, F ) be a measurable space. Each element ω ∈ Ω, i.e., of the sample space, is an outcome of the game. The σ-field F represents the state space and every element of F is said to be a state of the world. Further, every sub-σ-field of F is called an information set. There exist exactly n ∈ N players. Player i is endowed with a private information set Ii ⊆ F (i = 1, 2, . . . , n).2 The tuple ℑ = (I1 , I2 , . . . , In ) is said to be the information structure. Player i applies a strategy s i . This is a real-valued Ii -measurable random variable. The random vector s = (s 1 , s 2 , . . . , s n ) is called a strategy profile. Every realization a i = s i (ω) with ω ∈ Ω is said to be an action. It is assumed that a i ∈ A i ⊆ R, where A i represents the action set of Player i . The Cartesian product A =×ni=1 A i is said to be the action space. Furthermore, the individual preferences of Player i are expressed by an objective function(al) ϕi . This means if Player i is rational, he maximizes ¡ ¢ ϕi s i , s i ∈ R over his strategy s i , where s i denotes the vector of strategies of the other players. The objective functions are bundled in ϕ = (ϕ1 , ϕ2 , . . . , ϕn ), i.e., the joint value of the profile ¡ ¢ s amounts to ϕ(s) = ϕ1 (s), ϕ2 (s), . . . , ϕn (s) ∈ Rn . It is not assumed that the players know the private information sets and objective functions of each other. Now, an n-person normal-form game is specified by Γ = (Ω, F , ℑ, ϕ). For example, we could assume that Player i has a utility function u i : A → R, so that the profile s leads to the utility u i (s). If there exists an objective probability measure P, Player i maximizes ¢ ¡ his expected utility ϕi (s) = E u i (s) , where the expectation is based upon P (von Neumann and Morgenstern, 1944) or a monotone transformation of P (Quiggin, 1993). Otherwise, his expected utility is based upon a subjective probability measure (Fishburn, 1981, Ramsey, 1931, Savage, 1954) or a Choquet capacity (Schmeidler, 1989).3 A nice overview of expected-utility theories is given by Dyckerhoff (1994). Nonetheless, we are not restricted to the concept of cardinal utility. This framework allows both for decisions under risk and under uncertainty (Knight, 1921).4 The reader is free to pursue his favorite model, provided it is based on the general framework described above. The strategies of the players may depend on each other in an arbitrary way. Hence, we are able to explain different forms of strategic interaction that go far beyond the classical models, e.g., the Nash equilibrium and the correlated equilibrium (Aumann, 1974, 1987). In any way, an optimal strategy of Player i is given by ¡ ¢ s i∗ ∈ arg max ϕi s i , s i . s i ∈Ii
Here it is implicitly assumed that s i (ω) ∈ A i for all ω ∈ Ω and that s i is attainable in Γ. The latter will be elaborated in the following sections. Moreover, the set of optimal strategies is supposed to be nonempty. Three points are worth emphasizing: (i) The set of attainable strategies depends on the private information, (ii) the vector s i is an implicit function of s i , and 2 In the following, I omit the enumeration “i = 1, 2, . . . , n” whenever it is clear from the context that the corresponding
statement applies to every player. 3 In this case it may happen that a player assigns probability zero to a state F ∈ F , whereas another player assigns a
positive probability to F . The same holds for capacities. Nonetheless, the players always share the same state space. 4 The term “decision” is understood in the broad sense: It may be an action of a single player in a game against nature
or a strategy of a player against other players in a non-cooperative game.
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Frahm, 2015 • Rationality and Best Response Joe
Joe I2
I3
I1
I1
Plan 2
I2
Plan 1
I3
Plan 3
Plan 3
Plan 1
I2
I3
I1
Plan 1
I2
Plan 3
Mary
Mary
I1
Plan 2
Plan 3
Plan 2
Plan 1
Figure 1: Mary can have more (left hand) or less (right hand) information. (iii) the players make a coherent choice. These points shall be illustrated in the following. Typically, I consider a two-player normal-form game. This is only done for the sake of simplicity, but without loss of generality.
2.1. State Space and Private Information The distinction between normal-form and extensive form games is not understood in the generic sense. This means I do not put any non-cooperative game into the category “normal-form game” or “extensive-form game.” In fact, each non-cooperative game can be represented both in normal form and in extensive form. For example, the game of chess is usually represented in extensive form. Nonetheless, in the general framework established above, it could be also represented without any problem in normal form. Throughout this work, I always refer to the normal form, Γ, of non-cooperative games. The action of a chess player represents an exhaustive plan of moves and countermoves.5 Suppose that Mary and Joe play chess. Each player knows his own plan, but keeps his plan secret from the other player. Hence, the plan of Mary appears to be random to Joe and vice versa. Mary knows whether the outcome of the game, ω ∈ Ω, belongs to any atom I ∈ I1 of her private information set or not.6 This means she has the information I 3 ω, but in general the plan of Joe is still unknown to Mary. More precisely, Joe applies his strategy on the basis of his private information set I2 . His plan remains secret from Mary unless σ(s 2 ), i.e., the σ-field generated by Joe’s strategy, is a sub-σ-field of I1 . The same holds mutatis mutandis for Mary’s plan. Now, what constitutes a strategy under these circumstances? Suppose that Mary and Joe can choose between Plan 1, 2, and 3. In this very simple case, the corresponding action sets of the players are given by A 1 = A 2 = {1, 2, 3}. A strategy is a random variable that is measurable with respect to the private information set of the corresponding player. This means it assigns an action to each atom of the private information set. From Joe’s perspective, each atom of Mary represents a state of the world and vice versa. To avoid unnecessary complications, I assume that the state space, F , looks like in Figure 1. Here, the private information set of Joe contains 5 In decision theory, such a plan is often called a strategy. Unfortunately, an abuse of terminology cannot be completely
avoided when switching between game and decision theory. 6 An event I ∈ I is said to be an atom of I if and only if there exists no element H ⊂ I with H ∈ I . i i i
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Frahm, 2015 • Rationality and Best Response three atoms, whereas Mary’s private information set either contains three atoms (left-hand side) or two atoms (right-hand side). This means she can have more or less information. On the righthand side of Figure 1, Mary has not the same flexibility to adapt her behavior to the (potential) behavior of Joe as on the left-hand side of Figure 1. For this reason, the set of strategies that can be attained by Mary depends on her private information and the more information she has, the more opportunities are available to her. A strategy may always be considered mixed in our general framework, but this should not be interpreted as meaning that the players use a random generator. Of course, Joe knows his own action and so it is deterministic to himself,7 but it is random to Mary, unless her private information set contains the strategy of Joe. Hence, the action of Joe is random to Mary, simply because her information is imperfect, but not because Joe is throwing a dice or spinning a wheel. More details on that topic can be found in Aumann and Brandenburger (1995), Morris (2006), and Rubinstein (1991).
2.2. The Response Function Suppose that an investor can choose between two alternatives: (a) A corporate bond and (b) a stock. The annual returns on investment are shown in the following decision matrix: State Action
A
B
Bond Stock
1% -10%
4% 8%
The investor has to make a rational investment decision based on his private information. In a measure-theoretic sense, his action (“bond” or “stock”) is part of his private information set and thus it is not random to himself.8 Each action determines another state variable: If he decides to choose “bond,” he obtains the state variable “bond return,” but if he picks “stock,” he is exposed to the state variable “stock return.” Our theory of rational choice implies that this association is known to the decision maker. The problem is that he does not know whether State A or B occurs. This means he has to make a decision under risk or under uncertainty, depending whether he is able to assign an objective or a subjective probability to each state of the world. In any case, it is assumed that the potential returns on investment are known for each action. This is the way how decision theory deals with economic or natural risk and uncertainty. The same principle applies to strategic uncertainty (Aumann and Dreze, 2009): [. . . ] games against nature and strategic games are in principle quite similar, and can – perhaps should – be treated similarly. Specifically, a player in a strategic game should be able to form subjective probabilities over the strategies of the other players, and his own strategy choice should yield him maximal expected utility with respect to these subjective probabilities. Of course, one could argue that the decision-theoretic approach is unrealistic. What if a player is not able to assign the potential consequences of his strategy to each state of the world? 7 This holds even if he throws a dice. In fact, his strategy is measurable with respect to his private information set and
so the number rolled on the dice is still deterministic to himself in a pure measure-theoretic sense. 8 In fact, as Levi (1997) points out, “Deliberation crowds out prediction.”
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Frahm, 2015 • Rationality and Best Response 1
p Solution set
q
0
Figure 2: A response diagram and the set of solutions.
Well, in this case we could try to translate his strategic uncertainty into our measure-theoretic framework by cutting the state space, F , into sufficiently small pieces. If this is possible, we are right back in our familiar environment. Otherwise, we have to leave the measure-theoretic framework. According to Aumann and Dreze (2009), I presume that rational choice, irrespective of whether it has to be made by a single decision maker against nature or by a player against other players in a non-cooperative game, can be expressed in the usual way of decision theory.9 All players in an n-person normal-form game are decision makers and apply a strategy. For each player j 6= i , the strategy of Player i represents a state variable, whereas Player i considers his own strategy a control variable. According to our basic principle of rational choice, the state variables are known to each player. This means Player i knows that the other players choose s i if he chooses s i . If a player were uncertain about the choice of the others, we would have to express his uncertainty in terms of measure theory, i.e., by choosing an appropriate state space and information structure. Hence, if Γ is properly specified, the vector s i is an implicit function of s i , i.e., ρ i : s i 7→ s i , which is only an analytical description of the logical statement: “s i ⇒ s i .” In the following, ρ i is referred to as the response function of Player i . © ª T Let S i = s i , ρ i (s i ) be the graph of ρ i . The solution set of Γ is given by S = ni=1 S i . This is the set of all profiles that are attainable in the game and each element of S is said to be a solution of Γ. Put another way, each profile s that is not part of S can be ignored, since this combination of strategies is simply impossible for logical reasons.10 This is illustrated in Figure 2. The red line indicates the response function of Mary, whereas the black line is the response function of Joe. If Mary chooses p = 0.5, Joe chooses q = 0, which implies that Mary chooses p = 0. Hence, Mary cannot choose p = 0.5. The only combination of strategies that is possible is p = 0 and q = 0. This singularity represents the solution set for the given response functions.
2.3. Strategy Choice We have seen that the game of chess can be nicely represented in normal form. It is always possible to represent a non-cooperative game in its normal form and thus we do not have to restrict ourselves to one-shot games. In the normal form of a non-cooperative game, there exists no time dimension. This leads to the following question: “Is it possible to choose an 9 In principle, we could readily extend Γ by an additional “Player 0” without objective function, i.e., the nature. This
would allow us to combine games against nature and strategic games into a hybrid model of rational choice. 10 It is implicitly assumed that each player knows his set of attainable strategies and thus the whole set of solutions.
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Frahm, 2015 • Rationality and Best Response
Joe Mary
Left
Right
Up Down
(1, 1) (2, 0)
(0, 2) (−1, −1)
Table 1: The game of chicken.
action or even a strategy after the other players have made their choices?” The answer is “No!” Otherwise we would have had to formulate the game in extensive form before translating it into its normal form. Thus Joe cannot “respond” to Mary in the chronological sense and the same argument applies to Mary. More generally, no player in a normal-form game is able to modify his choice a posteriori, i.e., after knowing the choice of the other players.11 Although no player can change his strategy a posteriori, we could simply assume that the other players hold their strategies fixed if any player changes his strategy. Every player is able to respond to the others in such a pure hypothetical sense. This hypothetical interpretation of “response” seems obvious, but in our general framework of rational choice, no player holds his strategy fixed if any other changes his strategy. As already described in the preceding section, the strategies of the players are associated with each other: If Mary chooses Strategy A then Joe chooses Strategy B and vice versa, i.e., A ⇔ B . More generally, in every n-player normal-form game, the chosen strategies must constitute a solution, i.e., s ∈ S . This means the players make a coherent choice and so the strategy of each player can always be viewed as a “response” to the strategies of the others in a logical sense. Now, we have found (i) a chronological, (ii) a hypothetical, and (iii) a logical interpretation of “response.” Now, the next question arises: “Is it possible to give a best response?” This essentially depends on the specific meaning of “response:” (i) If it is understood in the chronological sense, i.e., “Joe gives a best response to Mary after she has made her choice,” we already know that the answer is “No.” (ii) If it is understood in the hypothetical sense, i.e., “Joe’s strategy can be considered a best response to Mary’s strategy by assuming that Mary holds her strategy fixed if Joe alternates his strategy,” the answer is “Yes,” although the hypothesis itself is incorrect. (iii) Finally, if it is understood in the logical sense, i.e., “Joe applies a strategy that is optimal among all strategies that are associated with Mary’s strategies,” the answer is still “Yes.” Game theory typically pursues the chronological (i) or hypothetical (ii) interpretation of “response.” The chronological interpretation prevails in the context of extensive-form games, whereas the hypothetical interpretation runs through the whole literature on normal-form games. It is precisely this interpretation which underlies the best-response principle. To the best of my knowledge, the logical interpretation (iii) has not yet been considered in game theory, although it turns out to be natural from the perspective of decision theory. 11 It is tempting to conclude that the players choose their strategies simultaneously, but due to the lack of the time
dimension, this conclusion is wrong.
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Frahm, 2015 • Rationality and Best Response 1
1
p
p
q
0
q
0
Figure 3: Best-response diagram (left) and solution set (right) of the game of chicken. The assumption that a player holds his strategy fixed if another player changes his strategy, has no theoretical grounds and leads to fundamental contradictions. For example, consider the wellknown game of chicken in Table 1. Suppose that there exists an objective probability measure. Mary and Joe try to maximize their expected utilities and their strategies are stochastically independent. The best-response diagram of this game is given on the left-hand side of Figure 3. Here p represents the probability that Mary goes “Up,” whereas q is the probability that Joe goes “Left.” In the following, I use the shorthand notation “(p, q)” to indicate a profile, where Mary and Joe choose the strategies p and q, respectively.12 The red line indicates that Joe will always give a best response to Mary, whereas the black line suggests that Mary will always give a best response to Joe.13 Is this really possible from a logical perspective? For example, Joe chooses q = 0 if Mary chooses p = 0.75 and Mary chooses p = 1 if Joe chooses q = 0. To sum up, we have the following implication: p = 0.75 ⇒ q = 0 ⇒ p = 1. This is a contradiction. It cannot happen that Mary chooses p = 0.75 and p = 1 together and thus it is impossible for Mary to choose p = 0.75. The same holds for any other strategy that does not belong to a Nash equilibrium. We conclude that the only strategy profiles that are possible in the game of chicken, are given by the green points on the left-hand side of Figure 3, i.e., the three Nash equilibria (0, 1), (1, 0), and ¡1 1¢ 2 , 2 . Thus we have created a game where the players can only choose among Nash equilibria. Now, an interesting question arises: “Can it be true that Mary holds her strategy fixed, if Joe moves from q = 12 to another possible probability?” This cannot be true. For example, if Joe moves to q = 0, Mary must inevitably move to p = 1. As we can see, the strategies of Mary and Joe are associated with each other so that no player holds his strategy fixed if the other changes his strategy. Similar arguments can be found in the literature. For example, Brams (1994) points out that the players need to consider “the consequences of a series of moves and countermoves from the resulting outcome,” which can be a Nash equilibrium, and they should “think ahead in deciding whether or not to move.” Moreover, Risse (2000) uses the transparency of reason (Bacharach, 1987) and argues like this: One agent judges deviating reasonable if the other one deviates as well. The other one figures this out. Suppose mutual deviation is profitable for him as well. Since 12 The terminology might be somewhat misleading. Actually, the players do not choose any probabilities. They choose
strategies that lead to the corresponding probabilities p for “Up” and q for “Left,” respectively. 13 The best-response diagram involves correspondences rather than functions, but this is only a semantic sophistry,
which can be safely ignored.
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Frahm, 2015 • Rationality and Best Response he knows that the first agent would deviate if he did, he deviates. In this way, two agents transparent to each other could abandon an equilibrium. The point is that a Nash Equilibrium, by definition, only discourages uni-lateral deviation.
3. Main Results Theorem 1. Suppose that S 6= ; and let s, s 0 ∈ S with s 6= s 0 be two solutions of Γ. Then we have that s i 6= s i0 for i = 1, 2, . . . , n. Proof: We have that s i ⇒ s i ⇒ s j for all i 6= j and thus s j ⇒ s i for all j 6= i , i.e., s i ⇔ s j . This means if one player changes his strategy, every other player must change his strategy, too. Q.E.D. Theorem 1 implies that no player can deviate uni-laterally. This renders the hypothetical interpretation of “response” inappropriate and for this reason, the best-response principle fails. The core idea of this argument can be illustrated this way: Suppose that we want to maximize some arbitrary function φ: R2 → R over its first argument. We could search for some point x ∈ R such that φ(x, y) is maximal given that y ∈ R is fixed, but this is not a good idea if y is an implicit ¡ ¢ function of x, i.e., y = f (x). In this case maximizing φ requires us to maximize φ x, f (x) over x. Here the two arguments x and y can be interpreted as strategies in a two-person normal-form game. The function φ represents the objective function of Player 1 and f gives us the response of Player 2 to the strategy of Player 1, i.e., f is the response function of Player 1. Let ψ be the objective function of Player 2 and g his response function. Then Player 2 has to maximize ¡ ¢ ψ g (y), y over y, whereas Player 1 maximizes φ(x, f (x)) over x. Now, a rational solution must be a point (x ∗ , y ∗ ) ∈ R2 that maximizes the objective functions of both players. Definition 1 (Rational solution). Let S 6= ; be the solution set of Γ. A profile s ∗ is said to be a © ª rational solution of Γ if and only if s ∗ ∈ S and ϕi (s ∗ ) ≥ ϕi (s) for all s ∈ S and i ∈ 1, 2, . . . , n . ¡ ¢ If s i , s i ∈ S is any solution, I say that s i is the response of Player i to s i , i.e., to the strategies of the other players. This is the logical interpretation of “response.” Theorem 1 implies that a best response in the usual sense of game theory, i.e., such that the players hold their strategies fixed if one player moves from one strategy to another, is impossible. By contrast, a best response in the logical sense, where each player chooses an optimal strategy given the associated strategies of the others, is clearly possible. A rational solution is nothing else than a profile of strategies in which each player gives a best response to the others in the logical sense. This can be considered a strategic equilibrium of Γ. In the following, S ∗ denotes the set of rational solutions of the game. The next theorem states that the solution of Γ must belong to S ∗ if all players are rational.
Theorem 2. Suppose that S 6= ; and all players are rational. Then we have that s ∈ S ∗ . Proof: Since each player is rational, he chooses a strategy that maximizes his objective function. By definition, the resulting profile of strategies must be a rational solution. Q.E.D. It is not necessary to explain the way how the solution set of a non-cooperative game takes place. Nonetheless, due to the rationality of all players, we are able to localize the potential sets of solutions of Γ. This is guaranteed by the following lemma and demonstrated in Section 4. Lemma 1. Suppose that S 6= ; and all players are rational. Then we have that S ∗ 6= ;.
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Frahm, 2015 • Rationality and Best Response Prisoner’s dilemma
Win-win situation
Joe Mary Up Down
Joe
Left
Right
Mary
Left
Right
(−1, −1)
(−3, 0)
Up
(1, 1)
(0, 0)
(0, −3)
(−2, −2)
Down
(0, 0)
(0, 0)
Joe
Joe
Mary
Left
Right
Mary
Up
(2, 3)
(0, 0)
Down
(1, 1)
(3, 2)
Battle of the sexes
Left
Right
Up
(1, −1)
(−1, 1)
Down
(−1, 1)
(1, −1)
Matching pennies
Table 2: Non-cooperative games in normal form. Proof: From Theorem 2 it follows that s ∈ S ∗ and thus S ∗ is nonempty.
Q.E.D.
This means if the solution set is nonempty, but we cannot find any rational solution in S , the players cannot be rational. Hence, only those solution sets that contain at least one rational solution make sense in a game where all players are rational. As I will show in the next section, this rules out many solutions that are advocated in traditional game theory. Colman (2004) points out that one of the most serious problems of the Nash equilibrium is the possibility of multiple and even Pareto-inefficient solutions. This contradicts our common understanding of rationality and makes it difficult to predict the results of strategic conflicts. This longstanding problem evaporates in our decision-theoretic framework. Definition 2 (Essential uniqueness). Let S 6= ; be the solution set of Γ. A subset A ⊆ S is said to be essentially unique if and only if ϕ(s) = ϕ(s 0 ) for all s, s 0 ∈ A . Theorem 3. Suppose that S 6= ; and all players are rational. Then S ∗ is essentially unique. Proof: Suppose that there exist two rational solutions s, s 0 ∈ S ∗ with ϕ(s) 6= ϕ(s 0 ). Then either © ª ϕi (s) < ϕi (s 0 ) or ϕi (s) > ϕi (s 0 ) for a player i ∈ 1, 2, . . . , n . This means Player i can increase the value of his objective function by moving from s i to s i0 or from s i0 to s i , respectively. Hence, either s or s 0 is no rational solution of Γ and so the initial assumption cannot be satisfied. Q.E.D.
4. Examples The following examples shall demonstrate the previous findings. This will be done on the basis of well-known two-player normal-form games. I consider a probability space (Ω, F , P), where Mary and Joe try to maximize their expected utilities, and I assume that their strategies are stochastically independent. The chosen assumptions are only made for the sake of simplicity, but without loss of generality. I do not have to assume complete information, i.e., that each player knows the action set and objective function of the other. Moreover, it is not necessary to assume mutual or even common knowledge of rationality (Aumann and Brandenburger, 1995).
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Frahm, 2015 • Rationality and Best Response
4.1. Game of Chicken Consider once again the game of chicken in Table 1. According to the best-response principle, the solution set of this game is supposed to be given by the green points in Figure 3. Is this really possible if Mary and Joe are rational? On the one hand, Mary knows that Joe goes “Left” if she goes “Down.” Obviously, this is her optimal strategy. On the other hand, Joe knows that Mary goes “Up” if he goes “Right,” which turns out to be his optimal strategy. This means the set of rational solutions, S ∗ , is empty and according to Lemma 1, the players cannot be rational. This can be illustrated as follows: Suppose that Mary and Joe are rational under the given set of solutions. Then she commits herself to “Down” and he commits himself to “Right.” Hence, we obtain two resulting profile distributions, i.e., " # " # 0 0 0 1 and . 1 0 0 0 If any assertion A is true, it cannot imply B as well as its complement ¬B . This means the game cannot end up in two mutually exclusive situations. We conclude that the solution set provided by the best-response principle cannot take place if Mary and Joe are rational and thus we have to search for another set of solutions. Following the above arguments, we can easily see that S cannot run from the lower left to the upper right of the response diagram. By contrast, as is shown on the right-hand side of Figure 3, the solution set may be a straight line from the lower right to the upper left of the response diagram.14 The unique rational solution of the game of chicken turns out to be (1, 1) (see the blue point in Figure 3). This can be seen as follows: We have that p = q and thus Mary’s expected utility amounts to p 2 · 1 + p (1 − p) · 0 + (1 − p) p · 2 + (1 − p)2 · (−1) = −2p 2 + 4p − 1 . Now, we obtain her optimal strategy p ∗ = 1 by setting the first derivative, −4p + 4, to zero. It is clear that Joe has the same optimal strategy, i.e., q ∗ = p ∗ = 1. This contradicts traditional game theory, which advocates one of the three Nash equilibria depicted in Figure 3. A typical argument in game theory is that the solution “Up”/“Left” cannot be rational, since if Mary chooses “Up,” Joe moves to “Right,” etc. As already discussed in Section 2, this kind of argumentation lacks any logic. According to Theorem 1, Joe cannot move to another strategy without accepting that Mary will change her strategy, too. Another inconvenience is that the best-response principle leaves us alone with ambiguity. Which of the three Nash equilibria will be the resulting solution? One could argue that the pure-strategy Nash equilibria of the game of chicken, i.e., (0, 1) and (1, 0), are unsatisfactory, since each of them favors the one or the other player (Rapoport and Chammah, 1966). Hence, the aforementioned authors support the ¡ ¢ mixed-strategy Nash equilibrium 12 , 21 . Since Theorem 1 does not distinguish between pure and mixed strategies, this is not a convincing solution either. We conclude that a rational solution need not be a Nash equilibrium. Since Mary and Joe are rational, they decide to chicken out. If Mary was likely to dare, Joe would take the same opportunity into consideration, but Mary is rational, i.e., she knows this. Thus it makes no sense for her to dare. The same argument applies to Joe and so the message is: “Rational people do 14 We could think also about more complicated shapes of the solution set, but this would not affect the result, essentially.
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Frahm, 2015 • Rationality and Best Response
A
Sun
B
Rain
A
Sun
Sun
Stay
Stay/Fly
A Rain
Sun
Fly
B B
Rain
Rain
Stay/Fly
Figure 4: Decision trees of John’s first decision problem.
not kill each other.” On closer inspection, this result seems intuitive. This demonstrates that peace and rationality do not exclude each other, even if the individual interests of the players are diametrically opposed.
4.2. Prisoner’s Dilemma Now, consider the prisoner’s dilemma on the upper left of Table 2. The best-response diagram of this game can be seen in Figure 2. According to this diagram, the set of solutions is a singularity and contains only the solution (0, 0). Hence, any other profile is unattainable for Mary and Joe. Since the players are restricted to the pure strategies p = 0 and q = 0, we can simply ignore any other solution. The same argument can be applied to every other singular set of solutions and so we could justify any other solution in the prisoner’s dilemma. Indeed, this would reduce game theory to absurdity. Hence, forcing S to be a singularity is not very helpful. We may assume for the sake of simplicity that the solution set consists only of two points. For example, we could think about (0, 1), (1, 0) ∈ S .15 From Lemma 1 it follows that such a set of solutions cannot occur if Mary and Joe are rational, but a solution set with (0, 0), (1, 1) ∈ S is conceivable. In this case we obtain a unique rational solution, i.e., (1, 1). This is the same rational solution as in the game of chicken (see the blue point in Figure 3). Since this result seemingly contradicts the dominance principle, it shall be clarified in the following. Consider a single decision maker, John, who has a flight ticket to Hawaii and thinks about flying off or staying at home. Suppose that John has the following decision matrix: State Action Stay Fly
A
B
Sun Sun
Rain Rain
In the first row, “Sun” and “Rain” refer to the weather in John’s home town, whereas in the second row, the same labels refer to Aloha State. In principle, John could have been faced with two different state variables, depending of whether he decides to stay or fly. Nonetheless, the given decision matrix just indicates that the two state variables are identical, i.e., in any case the 15 This means “Down”/“Left” and “Up”/“Right.”
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Frahm, 2015 • Rationality and Best Response
A
Sun
B
Sun
A
Rain
B
Rain
Sun
Stay
Rain
Fly
Sun
Stay
Rain
Fly
A
Stay
B
Fly
Figure 5: Decision trees of John’s second decision problem.
weather in John’s home town will be the same as the weather on Hawaii. This is also illustrated on the left-hand side of Figure 4.16 Suppose that the corresponding utility matrix looks like this: State Action
A
B
Stay Fly
1 2
−1 0
The action “Fly” dominates the action “Stay.” According to the dominance principle, John will choose the strategy “Fly.” The dominance principle is a fundamental part of our theory of rational choice and thus it should also apply to game theory. In the context of game theory, Mother Nature represents a player. This means she applies a strategy. More precisely, she assigns either “Sun” or “Rain” to each state of the world, i.e., A and B. Indeed, in this very simple example, the state space consists only of two atoms. Can it then happen that nature assigns “Sun” to State A and “Rain” to State B, both if John decides to stay and if he decides to fly? The answer is “No!” John’s state variable “weather” is her control variable, whereas John’s behavior represents her state variable. Obviously, John’s decision is measurable with respect to the “private” information set of Mother Nature. This means her state variable is deterministic. Suppose that John would stay and fly, if nature chooses the strategy “‘Sun’ in State A and ‘Rain’ in State B,” as indicated by the red lines on the right-hand side of Figure 4. In this case, she is not able to determine John’s response to her strategy. Since our general framework of rational choice presumes that all decision makers are able to determine their relevant state variables, the decision matrix outlined above cannot occur in a two-player normal-form game. Now, consider the following decision problem:17 State Action Stay Fly
State
A
B
Sun Rain
Sun Rain
Action
A
B
Stay Fly
1 0
1 0
16 Here, the decision trees do not have any temporal meaning. 17 The decision matrix is given on the left hand, whereas the utility matrix is placed on the right hand.
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Frahm, 2015 • Rationality and Best Response 1
1
p
p
q
0
q
0
Figure 6: Best-response diagram (left) and solution set (right) of the win-win situation.
Now, John is faced with two different state variables. This is also illustrated on the left-hand side of Figure 5. From the perspective of nature, John decides to stay if the weather is sunny, whereas he flies off if it is rainy. Mother Nature’s strategy “Sunny weather” is indicated by the red lines on the right-hand side of Figure 5, whereas the blue lines represent her strategy “Rainy weather.” Contrary to John’s first decision problem, his response now turns out to be unique. As we can see from the utility matrix, he will not fly at all, since the dominance principle leads to the strategy “Stay.” Hence, in the second decision problem, John comes to the opposite conclusion. We see that the dominance principle still holds true, but it has to be applied with care when predicting the result of a non-cooperative game. A two-person normal-form game is typically represented by a bimatrix like in Table 2. Each row and column of a bimatrix represents an action, but not a state of the world, i.e., it is not an element of the state space F .18 While this representation might be useful when deriving the best-response function of each player, it is inadequate if one is interested in their state spaces. On the one hand, we have seen that the best-response functions are not very meaningful. On the other hand, it is not possible to apply the dominance principle without specifying the state space. Hence, from the upper left of Table 2, we must neither conclude that “Up” is dominated by “Down” nor that “Left” is dominated by “Right.” Otherwise, we would confuse actions with states and such a naive application of the dominance principle can be misleading.19 As already mentioned above, the solution (1, 1), i.e., “Cooperate”/“Cooperate,” is a rational solution. This is simply because Mary is rational. Thus she knows that Joe will not defect if she cooperates and vice versa. Otherwise we would have had to consider a set of solutions that runs from the lower right to the upper left of the response diagram.20 This would not alter the rational solution: Suppose that (1, 1) ∈ S is not a rational solution. Then either Mary or Joe can find a better position in S . This means the corresponding player can obtain an expected value that is greater than −1. In this case, the other player obtains an expected value that is lower than −1 and so this cannot be an optimal solution for him. Hence, if Mary and Joe are rational, the result of the game is (1, 1). We conclude that cooperation is a logical consequence of rationality. Like in the game of chicken (see Section 4.1), the presented result is in direct contrast to common belief. Nonetheless, it explains cooperation in a world of selfish but rational people. The fact that some players do not cooperate in real-life situations being similar to the prisoner’s 18 Hence, a bimatrix should never be misunderstood as a bilateral utility matrix. 19 Unfortunately, this mistake is visible throughout the whole literature on the prisoner’s dilemma. 20 As already indicated at the beginning of this section, S cannot run from the lower left to the upper right.
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Frahm, 2015 • Rationality and Best Response
1
Ambiguous
1
Unattainable
p
p
0
q
q
0
Figure 7: Best-response diagram (left) and solution set (right) of the Pareto game.
dilemma could be attributed to a lack of “rationality” in our strict sense. This means some players defect, because our basic model of rational choice is not (fully) satisfied in real-life situations, but not because the players are rational in the traditional sense of decision theory.
4.3. The Win-Win Situation The win-win situation is given on the upper right of Table 2 and its best-response diagram can be found on the left-hand side of Figure 6. The corresponding solution set consists of two points, i.e., on the lower right and upper left corner, which are depicted on the right-hand side of Figure 6. Only the upper left corner, i.e., the blue point, represents a rational solution. In fact, due to Theorem 3, the rational solution is essentially unique and thus we do not have to rule out the Pareto-inefficient solution on the lower right by refinement (Harsanyi and Selten, 1988). While the two preceding examples demonstrate that a rational solution need not be a Nash equilibrium, this example shows that a Nash equilibrium need not be a rational solution. The fact that the Nash equilibrium can lead to multiple and even Pareto-inefficient solutions is often considered its most serious deficiency (Colman, 2004). Another drawback of the Nash equilibrium is the fact that it can produce subgame-imperfect equilibrium points (Selten, 1965, 1975).21 It is argued that such solutions are implausible, since an arbitrarily small uncertainty regarding the action of one player destroys the Nash equilibrium. As already mentioned, one can find a tremendous number of refinement procedures, which aim at eliminating Nash equilibria that are considered in any sense implausible. The potential pitfalls of refinement and perfectness shall be illustrated by the following “Pareto game:” Joe Mary
Left
Right
Up Down
(6, 9) (3, 7)
(5, 9) (4, 8)
A typical argument is that Mary goes “Up,” since “Down” appears to be strictly dominated by “Up.” If Joe has complete information and knows that Mary is rational, he is indifferent about “Left” and “Right.” This game has two pure-strategy Nash equilibria, i.e., “Up”/“Left” 21 In our context, this terminology is somewhat misleading, since we do not consider games in extensive form. For this
reason, I will drop the prefix “subgame.”
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Frahm, 2015 • Rationality and Best Response 1
1
p
p
q
0
q
0
Figure 8: Best-response diagram (left) and solution set (right) of the battle of the sexes.
and “Up”/“Right,” and a multitude of mixed-strategy Nash equilibria. More precisely, Joe can choose “Left” with probability q and “Right” with probability 1−q for all 0 < q < 1. Only the purestrategy Nash equilibrium “Up”/“Left” is Pareto efficient. All Pareto-inefficient Nash equilibria can be considered implausible and we end up with the unique rational solution “Up”/“Left.” Nonetheless, the strategy “Left” is weakly dominated by “Right” and so the Pareto-efficient Nash equilibrium “Up”/“Left” is imperfect. For example, Joe could doubt that Mary is rational or think that she might go “Down” by accident. If he has the slightest fear that Mary goes “Down,” we could expect that he goes “Right.” In this case we would end up with any Pareto-inefficient solution. This demonstrates that refinement does not guarantee a unique and plausible solution. The above arguments are based on a naive application of the dominance principle and the hypothetical interpretation of “best response.” The best-response diagram of the Pareto game is given on the left-hand side of Figure 7. It contains two areas: (i) The right border indicates the best responses of Joe to the strategies of Mary. (ii) The upper border represents the set of all Nash equilibria. All profiles on the right border (except for the upper right corner) are unattainable. Hence, Mary cannot apply a “trembling-hand strategy” for logical reasons.22 Moreover, the upper border is ambiguous and thus it cannot be used as a response function for Mary. Our theory of rational choice stipulates that it is always possible to express Mary’s uncertainty by one and only one state variable. This means Joe’s response to Mary’s strategy must be uniquely determined, but this goal cannot be achieved by the best-response diagram. Hence, the best-response principle does not help much if we are searching for a unique rational solution. By contrast, consider the solution set on the right-hand side of Figure 7. The reader might convince himself that this set of solutions is possible in the Pareto game. Under these circumstances, the blue point on the upper left turns out to be the rational solution. Interestingly, although this is the Pareto-efficient Nash equilibrium, this result cannot be derived by the best-response principle.
4.4. Battle of the Sexes The battle of the sexes can be found on the lower left of Table 2 and its best-response diagram is given on the left-hand side of Figure 8. The set of solutions, S , apparently consists of the 22 For example, we have that p = 0.5 ⇒ q = 0 ⇒ p = 1 and so Mary cannot choose p = 0.5.
17
Frahm, 2015 • Rationality and Best Response ¡ ¢ three Nash equilibria (0, 0), (1, 1), and 14 , 34 (see the green points in Figure 8). This solution set does not contain a rational solution and thus it cannot take place if Mary and Joe are rational. Otherwise, Mary would choose “Down” to force Joe to go “Right,” whereas Joe would choose “Left” to force Mary to go “Up,” which leads to a contradiction. Indeed, every solution set that runs from the lower right to the upper left is impossible in the battle of the sexes. By contrast, consider the solution set on the right-hand side of Figure 8. ¡ ¢ This set contains a unique rational solution, i.e., 83 , 58 , which can be seen as follows: For each 0 ≤ p ≤ 1 we have that q = 1 − p and thus Mary’s expected utility amounts to
p (1 − p) · 2 + p 2 · 0 + (1 − p)2 · 1 + (1 − p) p · 3 = −4p 2 + 3p + 1 . Setting the first derivative, −8p + 3, to zero leads to Mary’s optimal strategy p ∗ = 38 . It can be easily seen that the optimal strategy of Joe corresponds to q ∗ = (1 − p ∗ ) = 58 . This is not a corner solution. The reason is that Mary and Joe can improve their situation by moving away from the lower left corner. To show this, let us analyze the battle of the sexes in its generic form: Joe Mary
Left
Right
Up Down
(x, y) (w, w)
(v, v) (y, x)
Here we have that v ≤ w < x < y with x + y > 2w. Then the solution set in Figure 8 leads to the rational solution µ ¶ 1 w −v ∗ ∗ p = 1−q = 1− . 2 x −v −w +y For example, suppose that “Up” and “Left” mean “Football,” whereas “Down” and “Right” stand for “Concert.” Since we have that w − v < x − v − w + y, it is not optimal for Mary to apply the pure strategy “Concert,” since in this case Joe would always go to “Football,” i.e., they never meet each other. She is better off keeping the possibility open to meet Joe at the football match. Of course, the same argument applies to Joe. In the special case w = v, i.e., if “Concert” (“Football”) has no special meaning to Mary (Joe) when they are separated from each other, we ¡ ¢ obtain the rational solution 12 , 21 .
4.5. Matching Pennies Matching pennies is a two-person zero-sum game. It can be seen on the lower right of Table 2. Its best-response diagram is given on the left-hand side of Figure 9. The recommendation from ¡ ¢ traditional game theory is clear: It is the minimax solution 21 , 12 (Kjeldsen, 2001, von Neumann, 1928, von Neumann and Morgenstern, 1944), i.e., the green point in Figure 9. Once again, the best-response principle suggests a singular set of solutions and one might ask if it is possible to justify a more comprehensive solution set. Interestingly, here this is not possible. Suppose that S were not a singularity. For example, consider a solution set that runs from the lower right to the upper left of the response diagram. Then the set of rational solutions is empty: Mary would prefer “Up” and “Down,” whereas Joe would choose “Left” and “Right” with equal probability. Hence, Mary’s optimal strategies are located on the lower right and upper left corner, whereas Joe’s optimal strategy is in the middle of the response diagram. If S runs from
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Frahm, 2015 • Rationality and Best Response 1
1
p
p
q
0
q
0
Figure 9: Best-response diagram (left) and solution set (right) of matching pennies. the lower left to the upper right of the response diagram, Mary prefers the solution in the middle, whereas Joe tries to escape from the middle. In every two-person zero-sum game, i.e., ϕ2 = −ϕ1 , where all players are rational,23 the set of solutions must be essentially unique. This can be seen as follows: According to Lemma 1, the set of rational solutions, S ∗ , is nonempty. Then any rational solution s ∗ = (s 1∗ , s 2∗ ) must belong to ¡ ¢ ¡ ¢ arg max ϕ1 s 1 , s 2 = arg min ϕ2 s 1 , s 2 . (s 1 ,s 2 )∈S
(s 1 ,s 2 )∈S
This means by maximizing her own objective function with s 1∗ , Mary implicitly minimizes the objective function of Joe. Now, S must be essentially unique. Otherwise, Joe could find a strategy that is better than s 2∗ , but then s ∗ cannot be a rational solution. Hence, our theory of rational choice implies that there exists essentially one and only one solution of matching pennies. In principle, we could defend any singular set of solutions, but the minimax solution stands out because it is fair. This means it leads to the same value for both players. Indeed, if we assume that Mary and Joe have the same opportunities, there is no reason ¡ ¢ to suggest another solution than 12 , 12 , i.e., the blue point on the right-hand side of Figure 9. This demonstrates that the minimax solution can be justified by simple arguments of decision theory and so the best-response principle is void.
5. Conclusion The Nash equilibrium allows for multiple and Pareto-inefficient solutions of non-cooperative games. Hence, a Nash equilibrium need not be a rational solution. This contradicts our common understanding of rationality and makes it difficult to predict the results of strategic conflicts. We have seen that also the converse is true: Rational players in a non-cooperative game need not give a best response to each other. This renders the Nash equilibrium and other equilibria that are based on the best-response principle inappropriate as a solution concept for strategic conflicts. This problem is highly relevant, since the Nash equilibrium represents the central solution concept in non-cooperative game theory and is frequently applied in many different scientific disciplines. Our theory of rational choice implies that the strategies of the players are associated with each other. For this reason, giving a best response in the usual sense of game theory is impossible. 23 It is implicitly assumed that the set of solutions, S , is nonempty.
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Frahm, 2015 • Rationality and Best Response Solutions that are based on the best-response principle can be logically inconsistent if all players are rational. This means the best-response principle creates situations which cannot occur for logical reasons. Moreover, it has been shown that every non-cooperative game has essentially one and only one rational solution. Hence, the multiple-solution problem disappears in our general framework of rational choice and thus we do not need to eliminate Pareto-inefficient solutions by refinement. This solves a longstanding problem of non-cooperative game theory. Finally, it has been demonstrated that rational solutions of well-known normal-form games typically differ from those predicted by the best-response principle. The differences turned out to be essential both from an economic and a social point of view.
References R.J. Aumann (1974), ‘Subjectivity and correlation in randomized strategies’, Journal of Mathematical Economics 1, pp. 67–96. R.J. Aumann (1987), ‘Correlated equilibrium as an expression of Bayesian rationality’, Econometrica 55, pp. 1–18. R.J. Aumann and A. Brandenburger (1995), ‘Epistemic conditions for Nash Equilibrium’, Econometrica 63, pp. 1161–1180. R.J. Aumann and J.H. Dreze (2009), ‘Assessing strategic risk’, American Economic Journal 1, pp. 1–16. M. Bacharach (1987), ‘A theory of rational decision in games’, Erkenntnis 27, pp. 17–55. S.J. Brams (1994), Theory of Moves, Cambridge University Press, 3rd Edition. A.M. Colman (2004), ‘Reasoning about strategic interaction: Solution concepts in game theory’, in: K. Manktelow and M.C. Chung, eds., ‘Psychology of Reasoning: Theoretical and Historical Perspectives’, Psychology Press. R. Dyckerhoff (1994), Choquet-Erwartungsnutzen und antizipierter Nutzen, Ph.D. thesis, University of the Federal Armed Forces Hamburg. P. Fishburn (1981), ‘Subjective expected utility theory: An overview of normative theories’, Theory and Decision 13, pp. 139–199. S. Govindan and R.B. Wilson (2008), ‘Refinements of Nash equilibrium’, in: S.N. Durlauf and L.E. Blume, eds., ‘The New Palgrave Dictionary of Economics’, Palgrave Macmillan, 2nd Edition. J.C. Harsanyi and R. Selten (1988), A General Theory of Equilibrium Selection in Games, MIT Press. T.H. Kjeldsen (2001), ‘John von Neumann’s conception of the Minimax Theorem: A journey through different mathematical contexts’, Archive for History of Exact Sciences 56, pp. 39–68. F.H. Knight (1921), Risk, Uncertainty and Profit, Hart, Schaffner and Marx.
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Frahm, 2015 • Rationality and Best Response I. Levi (1997), ‘Prediction, deliberation, and correlated equilibrium’, in: ‘The Covenant of Reason’, Chapter 5, Cambridge University Press. S. Morris (2006), ‘Purification’, Technical report, Princeton University. R. Myerson (2001), ‘Refinements of the Nash Equilibrium Concept’, in: Y. Varoufakis, ed., ‘Game Theory: Critical Concepts in the Social Sciences’, pp. 159–166, Routledge. R.B. Myerson (1991), Game Theory: Analysis of Conflict, Harvard University Press. J.F. Nash (1951), ‘Non-cooperative games’, Annals of Mathematics 54, pp. 286–295. J. von Neumann (1928), ‘Zur Theorie der Gesellschaftsspiele’, Mathematische Annalen 100, pp. 295–320. J. von Neumann and O. Morgenstern (1944), Theory of Games and Economic Behavior, Princeton University Press. J. Quiggin (1993), Generalized Expected Utility Theory. The Rank-Dependent Model, Kluwer Academic Publishers. F.P. Ramsey (1931), ‘Truth and probability’, in: R.B. Braithwaite, ed., ‘The Foundations of Mathematics and other Logical Essays’, pp. 156–198, Harcourt, Brace and Company (New York). A. Rapoport and A.M. Chammah (1966), ‘The game of chicken’, The American Behavioral Scientist 10, pp. 10–28. M. Risse (2000), ‘What is rational about Nash equilibria?’, Synthese 124, pp. 361–384. A. Rubinstein (1991), ‘Comments on the interpretation of Game Theory’, Econometrica 59, pp. 909–924. L.J. Savage (1954), The Foundations of Statistics, Wiley. D. Schmeidler (1989), ‘Subjective probability and expected utility without additivity’, Econometrica 57, pp. 571–587. R. Selten (1965), ‘Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit’, Zeitschrift für die gesamte Staatswissenschaft 121, pp. 301–24, 667–89. R. Selten (1975), ‘Re-examination of the perfectness concept for equilibrium points in extensive games’, International Journal of Game Theory 4, pp. 25–55.
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arXiv:1509.08353v1 [cs.GT] 28 Sep 2015
Rationality and Best Response* Gabriel Frahm† Helmut Schmidt University Department of Mathematics/Statistics Chair for Applied Stochastics and Risk Management September 29, 2015
Abstract The Nash equilibrium is the central solution concept in non-cooperative game theory, but it has often been considered insufficient or implausible. It is shown that rational players in a non-cooperative game need not give a best response to each other. Giving a best response in the usual sense of game theory is even impossible. The best-response principle can lead to solutions that are logically inconsistent if all players are rational. Moreover, it is proved that the set of rational solutions of every non-cooperative game is essentially unique. This solves a longstanding problem of non-cooperative game theory, i.e., the possibility of multiple and even Pareto-inefficient solutions. Finally, it is demonstrated that the rational solutions of well-known normal-form games differ from those advocated in traditional game theory. The differences turn out to be essential both from an economic and a social point of view.
Keywords: Best-response principle, dominance principle, Nash equilibrium, normal-form game, Pareto efficiency, rationality. JEL Subject Classification: C72, D81.
* I would like to thank very much Bob Aumann, Steve Brams, Rainer Dyckerhoff, Mathias Risse and Rainer Schüssler
for their helpful comments on the manuscript and our valuable discussions. This paper is dedicated to John Nash and John von Neumann, who influenced generations of scientists with their seminal work. † Phone: +49 40 6541-2791, e-mail: [email protected].
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Frahm, 2015 • Rationality and Best Response
Contents 1. Motivation
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2. Theoretical Background
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2.1. State Space and Private Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The Response Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Strategy Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3. Main Results
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4. Examples
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The logical roots of game theory are in Bayesian decision theory. Indeed, game theory can be viewed as an extension of decision theory (to the case of two or more decision-makers), or as its essential logical fulfillment. Thus, to understand the fundamental ideas of game theory, one should begin by studying decision theory. Myerson (1991, p. 5)
1. Motivation HE Nash equilibrium (Nash, 1951) is the central solution concept in non-cooperative game theory and represents a cornerstone of modern economics. Over the last decades, the Nash equilibrium has often been considered insufficient or implausible. Colman (2004) points out the possibility of multiple and even Pareto-inefficient solutions (Harsanyi and Selten, 1988). Moreover, it is well-known that Nash equilibria can be subgame imperfect (Selten, 1975) and the idea that the players use a random generator when applying a mixed strategy leads to an obscure and nonsensical description of real-life strategic conflicts (Rubinstein, 1991). Typical arguments supporting the Nash equilibrium are discussed by Risse (2000), who comes to the conclusion that, “All of these arguments either fail entirely or have a very limited scope.” According to Nash (1951),
T
[. . . ] an equilibrium point is an n-tuple [of mixed strategies] such that each player’s mixed strategy maximizes his payoff if the strategies of the others are held fixed. Similarly, Aumann and Brandenburger (1995) describe a Nash equilibrium as,
2
Frahm, 2015 • Rationality and Best Response [. . . ] a profile of strategies in which each player’s strategy is optimal for him, given the strategies of the others. Moreover, Aumann and Brandenburger (1995) also observe that, Suppose that each player is rational, knows his own payoff function, and knows the strategy choices of the others. Then the players’ choices constitute a Nash equilibrium in the game being played. Hence, it is claimed that each player gives a best response to the others if the given solution is rational. This assertion has become an integral part of non-cooperative game theory and will be referred to as the best-response principle. As already mentioned, the message that a Nash equilibrium need not be a rational solution is not new. Indeed, one can find a large number of procedures that aim at a refinement of the Nash equilibrium (Govindan and Wilson, 2008, Harsanyi and Selten, 1988, Myerson, 2001). This means additional criteria such as Pareto efficiency, subgame perfectness, stability, etc., have been developed to eliminate all Nash equilibria that are considered implausible. Hence, the Nash equilibrium is assumed to be a valid but insufficient solution concept. Unfortunately, refinement does not address the root of the problem: A rational solution need not be a Nash equilibrium. In more general terms, if all players are rational, they need not give a best response to each other.1 This assertion can be proved by traditional means of decision theory (von Neumann and Morgenstern, 1944, Savage, 1954). The technical framework chosen in this work is very general and reproduces different kinds of strategic interaction that can occur in real life. This covers a wide range of equilibrium models of non-cooperative game theory. Due to the fundamental importance of the Nash equilibrium in economics and many other disciplines such as biology, psychology, and politics, the results presented in this work might be interesting to a broad audience. It is shown that (i) rational players in a normal-form game need not give a best response to each other, (ii) giving a best response, in the usual sense of game theory, is even impossible, (iii) the solutions that are derived by the best-response principle can be logically inconsistent if all players are rational, (iv) the set of rational solutions of every non-cooperative game is essentially unique, and (v) typically, the rational solutions of well-known normal-form games essentially differ from those predicted by the best-response principle. In Section 2, I present the theoretical background. In that section, I discuss how rational players choose an optimal strategy and I show how their strategies are connected to each other. Section 3 contains the main results of this work. In Section 4, I illustrate the given results on the basis of well-known two-player normal games. The last section concludes.
1 This statement essentially depends on the meaning of “response,” which will be discussed in Section 2.3.
3
Frahm, 2015 • Rationality and Best Response
2. Theoretical Background Let (Ω, F ) be a measurable space. Each element ω ∈ Ω, i.e., of the sample space, is an outcome of the game. The σ-field F represents the state space and every element of F is said to be a state of the world. Further, every sub-σ-field of F is called an information set. There exist exactly n ∈ N players. Player i is endowed with a private information set Ii ⊆ F (i = 1, 2, . . . , n).2 The tuple ℑ = (I1 , I2 , . . . , In ) is said to be the information structure. Player i applies a strategy s i . This is a real-valued Ii -measurable random variable. The random vector s = (s 1 , s 2 , . . . , s n ) is called a strategy profile. Every realization a i = s i (ω) with ω ∈ Ω is said to be an action. It is assumed that a i ∈ A i ⊆ R, where A i represents the action set of Player i . The Cartesian product A =×ni=1 A i is said to be the action space. Furthermore, the individual preferences of Player i are expressed by an objective function(al) ϕi . This means if Player i is rational, he maximizes ¡ ¢ ϕi s i , s i ∈ R over his strategy s i , where s i denotes the vector of strategies of the other players. The objective functions are bundled in ϕ = (ϕ1 , ϕ2 , . . . , ϕn ), i.e., the joint value of the profile ¡ ¢ s amounts to ϕ(s) = ϕ1 (s), ϕ2 (s), . . . , ϕn (s) ∈ Rn . It is not assumed that the players know the private information sets and objective functions of each other. Now, an n-person normal-form game is specified by Γ = (Ω, F , ℑ, ϕ). For example, we could assume that Player i has a utility function u i : A → R, so that the profile s leads to the utility u i (s). If there exists an objective probability measure P, Player i maximizes ¢ ¡ his expected utility ϕi (s) = E u i (s) , where the expectation is based upon P (von Neumann and Morgenstern, 1944) or a monotone transformation of P (Quiggin, 1993). Otherwise, his expected utility is based upon a subjective probability measure (Fishburn, 1981, Ramsey, 1931, Savage, 1954) or a Choquet capacity (Schmeidler, 1989).3 A nice overview of expected-utility theories is given by Dyckerhoff (1994). Nonetheless, we are not restricted to the concept of cardinal utility. This framework allows both for decisions under risk and under uncertainty (Knight, 1921).4 The reader is free to pursue his favorite model, provided it is based on the general framework described above. The strategies of the players may depend on each other in an arbitrary way. Hence, we are able to explain different forms of strategic interaction that go far beyond the classical models, e.g., the Nash equilibrium and the correlated equilibrium (Aumann, 1974, 1987). In any way, an optimal strategy of Player i is given by ¡ ¢ s i∗ ∈ arg max ϕi s i , s i . s i ∈Ii
Here it is implicitly assumed that s i (ω) ∈ A i for all ω ∈ Ω and that s i is attainable in Γ. The latter will be elaborated in the following sections. Moreover, the set of optimal strategies is supposed to be nonempty. Three points are worth emphasizing: (i) The set of attainable strategies depends on the private information, (ii) the vector s i is an implicit function of s i , and 2 In the following, I omit the enumeration “i = 1, 2, . . . , n” whenever it is clear from the context that the corresponding
statement applies to every player. 3 In this case it may happen that a player assigns probability zero to a state F ∈ F , whereas another player assigns a
positive probability to F . The same holds for capacities. Nonetheless, the players always share the same state space. 4 The term “decision” is understood in the broad sense: It may be an action of a single player in a game against nature
or a strategy of a player against other players in a non-cooperative game.
4
Frahm, 2015 • Rationality and Best Response Joe
Joe I2
I3
I1
I1
Plan 2
I2
Plan 1
I3
Plan 3
Plan 3
Plan 1
I2
I3
I1
Plan 1
I2
Plan 3
Mary
Mary
I1
Plan 2
Plan 3
Plan 2
Plan 1
Figure 1: Mary can have more (left hand) or less (right hand) information. (iii) the players make a coherent choice. These points shall be illustrated in the following. Typically, I consider a two-player normal-form game. This is only done for the sake of simplicity, but without loss of generality.
2.1. State Space and Private Information The distinction between normal-form and extensive form games is not understood in the generic sense. This means I do not put any non-cooperative game into the category “normal-form game” or “extensive-form game.” In fact, each non-cooperative game can be represented both in normal form and in extensive form. For example, the game of chess is usually represented in extensive form. Nonetheless, in the general framework established above, it could be also represented without any problem in normal form. Throughout this work, I always refer to the normal form, Γ, of non-cooperative games. The action of a chess player represents an exhaustive plan of moves and countermoves.5 Suppose that Mary and Joe play chess. Each player knows his own plan, but keeps his plan secret from the other player. Hence, the plan of Mary appears to be random to Joe and vice versa. Mary knows whether the outcome of the game, ω ∈ Ω, belongs to any atom I ∈ I1 of her private information set or not.6 This means she has the information I 3 ω, but in general the plan of Joe is still unknown to Mary. More precisely, Joe applies his strategy on the basis of his private information set I2 . His plan remains secret from Mary unless σ(s 2 ), i.e., the σ-field generated by Joe’s strategy, is a sub-σ-field of I1 . The same holds mutatis mutandis for Mary’s plan. Now, what constitutes a strategy under these circumstances? Suppose that Mary and Joe can choose between Plan 1, 2, and 3. In this very simple case, the corresponding action sets of the players are given by A 1 = A 2 = {1, 2, 3}. A strategy is a random variable that is measurable with respect to the private information set of the corresponding player. This means it assigns an action to each atom of the private information set. From Joe’s perspective, each atom of Mary represents a state of the world and vice versa. To avoid unnecessary complications, I assume that the state space, F , looks like in Figure 1. Here, the private information set of Joe contains 5 In decision theory, such a plan is often called a strategy. Unfortunately, an abuse of terminology cannot be completely
avoided when switching between game and decision theory. 6 An event I ∈ I is said to be an atom of I if and only if there exists no element H ⊂ I with H ∈ I . i i i
5
Frahm, 2015 • Rationality and Best Response three atoms, whereas Mary’s private information set either contains three atoms (left-hand side) or two atoms (right-hand side). This means she can have more or less information. On the righthand side of Figure 1, Mary has not the same flexibility to adapt her behavior to the (potential) behavior of Joe as on the left-hand side of Figure 1. For this reason, the set of strategies that can be attained by Mary depends on her private information and the more information she has, the more opportunities are available to her. A strategy may always be considered mixed in our general framework, but this should not be interpreted as meaning that the players use a random generator. Of course, Joe knows his own action and so it is deterministic to himself,7 but it is random to Mary, unless her private information set contains the strategy of Joe. Hence, the action of Joe is random to Mary, simply because her information is imperfect, but not because Joe is throwing a dice or spinning a wheel. More details on that topic can be found in Aumann and Brandenburger (1995), Morris (2006), and Rubinstein (1991).
2.2. The Response Function Suppose that an investor can choose between two alternatives: (a) A corporate bond and (b) a stock. The annual returns on investment are shown in the following decision matrix: State Action
A
B
Bond Stock
1% -10%
4% 8%
The investor has to make a rational investment decision based on his private information. In a measure-theoretic sense, his action (“bond” or “stock”) is part of his private information set and thus it is not random to himself.8 Each action determines another state variable: If he decides to choose “bond,” he obtains the state variable “bond return,” but if he picks “stock,” he is exposed to the state variable “stock return.” Our theory of rational choice implies that this association is known to the decision maker. The problem is that he does not know whether State A or B occurs. This means he has to make a decision under risk or under uncertainty, depending whether he is able to assign an objective or a subjective probability to each state of the world. In any case, it is assumed that the potential returns on investment are known for each action. This is the way how decision theory deals with economic or natural risk and uncertainty. The same principle applies to strategic uncertainty (Aumann and Dreze, 2009): [. . . ] games against nature and strategic games are in principle quite similar, and can – perhaps should – be treated similarly. Specifically, a player in a strategic game should be able to form subjective probabilities over the strategies of the other players, and his own strategy choice should yield him maximal expected utility with respect to these subjective probabilities. Of course, one could argue that the decision-theoretic approach is unrealistic. What if a player is not able to assign the potential consequences of his strategy to each state of the world? 7 This holds even if he throws a dice. In fact, his strategy is measurable with respect to his private information set and
so the number rolled on the dice is still deterministic to himself in a pure measure-theoretic sense. 8 In fact, as Levi (1997) points out, “Deliberation crowds out prediction.”
6
Frahm, 2015 • Rationality and Best Response 1
p Solution set
q
0
Figure 2: A response diagram and the set of solutions.
Well, in this case we could try to translate his strategic uncertainty into our measure-theoretic framework by cutting the state space, F , into sufficiently small pieces. If this is possible, we are right back in our familiar environment. Otherwise, we have to leave the measure-theoretic framework. According to Aumann and Dreze (2009), I presume that rational choice, irrespective of whether it has to be made by a single decision maker against nature or by a player against other players in a non-cooperative game, can be expressed in the usual way of decision theory.9 All players in an n-person normal-form game are decision makers and apply a strategy. For each player j 6= i , the strategy of Player i represents a state variable, whereas Player i considers his own strategy a control variable. According to our basic principle of rational choice, the state variables are known to each player. This means Player i knows that the other players choose s i if he chooses s i . If a player were uncertain about the choice of the others, we would have to express his uncertainty in terms of measure theory, i.e., by choosing an appropriate state space and information structure. Hence, if Γ is properly specified, the vector s i is an implicit function of s i , i.e., ρ i : s i 7→ s i , which is only an analytical description of the logical statement: “s i ⇒ s i .” In the following, ρ i is referred to as the response function of Player i . © ª T Let S i = s i , ρ i (s i ) be the graph of ρ i . The solution set of Γ is given by S = ni=1 S i . This is the set of all profiles that are attainable in the game and each element of S is said to be a solution of Γ. Put another way, each profile s that is not part of S can be ignored, since this combination of strategies is simply impossible for logical reasons.10 This is illustrated in Figure 2. The red line indicates the response function of Mary, whereas the black line is the response function of Joe. If Mary chooses p = 0.5, Joe chooses q = 0, which implies that Mary chooses p = 0. Hence, Mary cannot choose p = 0.5. The only combination of strategies that is possible is p = 0 and q = 0. This singularity represents the solution set for the given response functions.
2.3. Strategy Choice We have seen that the game of chess can be nicely represented in normal form. It is always possible to represent a non-cooperative game in its normal form and thus we do not have to restrict ourselves to one-shot games. In the normal form of a non-cooperative game, there exists no time dimension. This leads to the following question: “Is it possible to choose an 9 In principle, we could readily extend Γ by an additional “Player 0” without objective function, i.e., the nature. This
would allow us to combine games against nature and strategic games into a hybrid model of rational choice. 10 It is implicitly assumed that each player knows his set of attainable strategies and thus the whole set of solutions.
7
Frahm, 2015 • Rationality and Best Response
Joe Mary
Left
Right
Up Down
(1, 1) (2, 0)
(0, 2) (−1, −1)
Table 1: The game of chicken.
action or even a strategy after the other players have made their choices?” The answer is “No!” Otherwise we would have had to formulate the game in extensive form before translating it into its normal form. Thus Joe cannot “respond” to Mary in the chronological sense and the same argument applies to Mary. More generally, no player in a normal-form game is able to modify his choice a posteriori, i.e., after knowing the choice of the other players.11 Although no player can change his strategy a posteriori, we could simply assume that the other players hold their strategies fixed if any player changes his strategy. Every player is able to respond to the others in such a pure hypothetical sense. This hypothetical interpretation of “response” seems obvious, but in our general framework of rational choice, no player holds his strategy fixed if any other changes his strategy. As already described in the preceding section, the strategies of the players are associated with each other: If Mary chooses Strategy A then Joe chooses Strategy B and vice versa, i.e., A ⇔ B . More generally, in every n-player normal-form game, the chosen strategies must constitute a solution, i.e., s ∈ S . This means the players make a coherent choice and so the strategy of each player can always be viewed as a “response” to the strategies of the others in a logical sense. Now, we have found (i) a chronological, (ii) a hypothetical, and (iii) a logical interpretation of “response.” Now, the next question arises: “Is it possible to give a best response?” This essentially depends on the specific meaning of “response:” (i) If it is understood in the chronological sense, i.e., “Joe gives a best response to Mary after she has made her choice,” we already know that the answer is “No.” (ii) If it is understood in the hypothetical sense, i.e., “Joe’s strategy can be considered a best response to Mary’s strategy by assuming that Mary holds her strategy fixed if Joe alternates his strategy,” the answer is “Yes,” although the hypothesis itself is incorrect. (iii) Finally, if it is understood in the logical sense, i.e., “Joe applies a strategy that is optimal among all strategies that are associated with Mary’s strategies,” the answer is still “Yes.” Game theory typically pursues the chronological (i) or hypothetical (ii) interpretation of “response.” The chronological interpretation prevails in the context of extensive-form games, whereas the hypothetical interpretation runs through the whole literature on normal-form games. It is precisely this interpretation which underlies the best-response principle. To the best of my knowledge, the logical interpretation (iii) has not yet been considered in game theory, although it turns out to be natural from the perspective of decision theory. 11 It is tempting to conclude that the players choose their strategies simultaneously, but due to the lack of the time
dimension, this conclusion is wrong.
8
Frahm, 2015 • Rationality and Best Response 1
1
p
p
q
0
q
0
Figure 3: Best-response diagram (left) and solution set (right) of the game of chicken. The assumption that a player holds his strategy fixed if another player changes his strategy, has no theoretical grounds and leads to fundamental contradictions. For example, consider the wellknown game of chicken in Table 1. Suppose that there exists an objective probability measure. Mary and Joe try to maximize their expected utilities and their strategies are stochastically independent. The best-response diagram of this game is given on the left-hand side of Figure 3. Here p represents the probability that Mary goes “Up,” whereas q is the probability that Joe goes “Left.” In the following, I use the shorthand notation “(p, q)” to indicate a profile, where Mary and Joe choose the strategies p and q, respectively.12 The red line indicates that Joe will always give a best response to Mary, whereas the black line suggests that Mary will always give a best response to Joe.13 Is this really possible from a logical perspective? For example, Joe chooses q = 0 if Mary chooses p = 0.75 and Mary chooses p = 1 if Joe chooses q = 0. To sum up, we have the following implication: p = 0.75 ⇒ q = 0 ⇒ p = 1. This is a contradiction. It cannot happen that Mary chooses p = 0.75 and p = 1 together and thus it is impossible for Mary to choose p = 0.75. The same holds for any other strategy that does not belong to a Nash equilibrium. We conclude that the only strategy profiles that are possible in the game of chicken, are given by the green points on the left-hand side of Figure 3, i.e., the three Nash equilibria (0, 1), (1, 0), and ¡1 1¢ 2 , 2 . Thus we have created a game where the players can only choose among Nash equilibria. Now, an interesting question arises: “Can it be true that Mary holds her strategy fixed, if Joe moves from q = 12 to another possible probability?” This cannot be true. For example, if Joe moves to q = 0, Mary must inevitably move to p = 1. As we can see, the strategies of Mary and Joe are associated with each other so that no player holds his strategy fixed if the other changes his strategy. Similar arguments can be found in the literature. For example, Brams (1994) points out that the players need to consider “the consequences of a series of moves and countermoves from the resulting outcome,” which can be a Nash equilibrium, and they should “think ahead in deciding whether or not to move.” Moreover, Risse (2000) uses the transparency of reason (Bacharach, 1987) and argues like this: One agent judges deviating reasonable if the other one deviates as well. The other one figures this out. Suppose mutual deviation is profitable for him as well. Since 12 The terminology might be somewhat misleading. Actually, the players do not choose any probabilities. They choose
strategies that lead to the corresponding probabilities p for “Up” and q for “Left,” respectively. 13 The best-response diagram involves correspondences rather than functions, but this is only a semantic sophistry,
which can be safely ignored.
9
Frahm, 2015 • Rationality and Best Response he knows that the first agent would deviate if he did, he deviates. In this way, two agents transparent to each other could abandon an equilibrium. The point is that a Nash Equilibrium, by definition, only discourages uni-lateral deviation.
3. Main Results Theorem 1. Suppose that S 6= ; and let s, s 0 ∈ S with s 6= s 0 be two solutions of Γ. Then we have that s i 6= s i0 for i = 1, 2, . . . , n. Proof: We have that s i ⇒ s i ⇒ s j for all i 6= j and thus s j ⇒ s i for all j 6= i , i.e., s i ⇔ s j . This means if one player changes his strategy, every other player must change his strategy, too. Q.E.D. Theorem 1 implies that no player can deviate uni-laterally. This renders the hypothetical interpretation of “response” inappropriate and for this reason, the best-response principle fails. The core idea of this argument can be illustrated this way: Suppose that we want to maximize some arbitrary function φ: R2 → R over its first argument. We could search for some point x ∈ R such that φ(x, y) is maximal given that y ∈ R is fixed, but this is not a good idea if y is an implicit ¡ ¢ function of x, i.e., y = f (x). In this case maximizing φ requires us to maximize φ x, f (x) over x. Here the two arguments x and y can be interpreted as strategies in a two-person normal-form game. The function φ represents the objective function of Player 1 and f gives us the response of Player 2 to the strategy of Player 1, i.e., f is the response function of Player 1. Let ψ be the objective function of Player 2 and g his response function. Then Player 2 has to maximize ¡ ¢ ψ g (y), y over y, whereas Player 1 maximizes φ(x, f (x)) over x. Now, a rational solution must be a point (x ∗ , y ∗ ) ∈ R2 that maximizes the objective functions of both players. Definition 1 (Rational solution). Let S 6= ; be the solution set of Γ. A profile s ∗ is said to be a © ª rational solution of Γ if and only if s ∗ ∈ S and ϕi (s ∗ ) ≥ ϕi (s) for all s ∈ S and i ∈ 1, 2, . . . , n . ¡ ¢ If s i , s i ∈ S is any solution, I say that s i is the response of Player i to s i , i.e., to the strategies of the other players. This is the logical interpretation of “response.” Theorem 1 implies that a best response in the usual sense of game theory, i.e., such that the players hold their strategies fixed if one player moves from one strategy to another, is impossible. By contrast, a best response in the logical sense, where each player chooses an optimal strategy given the associated strategies of the others, is clearly possible. A rational solution is nothing else than a profile of strategies in which each player gives a best response to the others in the logical sense. This can be considered a strategic equilibrium of Γ. In the following, S ∗ denotes the set of rational solutions of the game. The next theorem states that the solution of Γ must belong to S ∗ if all players are rational.
Theorem 2. Suppose that S 6= ; and all players are rational. Then we have that s ∈ S ∗ . Proof: Since each player is rational, he chooses a strategy that maximizes his objective function. By definition, the resulting profile of strategies must be a rational solution. Q.E.D. It is not necessary to explain the way how the solution set of a non-cooperative game takes place. Nonetheless, due to the rationality of all players, we are able to localize the potential sets of solutions of Γ. This is guaranteed by the following lemma and demonstrated in Section 4. Lemma 1. Suppose that S 6= ; and all players are rational. Then we have that S ∗ 6= ;.
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Frahm, 2015 • Rationality and Best Response Prisoner’s dilemma
Win-win situation
Joe Mary Up Down
Joe
Left
Right
Mary
Left
Right
(−1, −1)
(−3, 0)
Up
(1, 1)
(0, 0)
(0, −3)
(−2, −2)
Down
(0, 0)
(0, 0)
Joe
Joe
Mary
Left
Right
Mary
Up
(2, 3)
(0, 0)
Down
(1, 1)
(3, 2)
Battle of the sexes
Left
Right
Up
(1, −1)
(−1, 1)
Down
(−1, 1)
(1, −1)
Matching pennies
Table 2: Non-cooperative games in normal form. Proof: From Theorem 2 it follows that s ∈ S ∗ and thus S ∗ is nonempty.
Q.E.D.
This means if the solution set is nonempty, but we cannot find any rational solution in S , the players cannot be rational. Hence, only those solution sets that contain at least one rational solution make sense in a game where all players are rational. As I will show in the next section, this rules out many solutions that are advocated in traditional game theory. Colman (2004) points out that one of the most serious problems of the Nash equilibrium is the possibility of multiple and even Pareto-inefficient solutions. This contradicts our common understanding of rationality and makes it difficult to predict the results of strategic conflicts. This longstanding problem evaporates in our decision-theoretic framework. Definition 2 (Essential uniqueness). Let S 6= ; be the solution set of Γ. A subset A ⊆ S is said to be essentially unique if and only if ϕ(s) = ϕ(s 0 ) for all s, s 0 ∈ A . Theorem 3. Suppose that S 6= ; and all players are rational. Then S ∗ is essentially unique. Proof: Suppose that there exist two rational solutions s, s 0 ∈ S ∗ with ϕ(s) 6= ϕ(s 0 ). Then either © ª ϕi (s) < ϕi (s 0 ) or ϕi (s) > ϕi (s 0 ) for a player i ∈ 1, 2, . . . , n . This means Player i can increase the value of his objective function by moving from s i to s i0 or from s i0 to s i , respectively. Hence, either s or s 0 is no rational solution of Γ and so the initial assumption cannot be satisfied. Q.E.D.
4. Examples The following examples shall demonstrate the previous findings. This will be done on the basis of well-known two-player normal-form games. I consider a probability space (Ω, F , P), where Mary and Joe try to maximize their expected utilities, and I assume that their strategies are stochastically independent. The chosen assumptions are only made for the sake of simplicity, but without loss of generality. I do not have to assume complete information, i.e., that each player knows the action set and objective function of the other. Moreover, it is not necessary to assume mutual or even common knowledge of rationality (Aumann and Brandenburger, 1995).
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Frahm, 2015 • Rationality and Best Response
4.1. Game of Chicken Consider once again the game of chicken in Table 1. According to the best-response principle, the solution set of this game is supposed to be given by the green points in Figure 3. Is this really possible if Mary and Joe are rational? On the one hand, Mary knows that Joe goes “Left” if she goes “Down.” Obviously, this is her optimal strategy. On the other hand, Joe knows that Mary goes “Up” if he goes “Right,” which turns out to be his optimal strategy. This means the set of rational solutions, S ∗ , is empty and according to Lemma 1, the players cannot be rational. This can be illustrated as follows: Suppose that Mary and Joe are rational under the given set of solutions. Then she commits herself to “Down” and he commits himself to “Right.” Hence, we obtain two resulting profile distributions, i.e., " # " # 0 0 0 1 and . 1 0 0 0 If any assertion A is true, it cannot imply B as well as its complement ¬B . This means the game cannot end up in two mutually exclusive situations. We conclude that the solution set provided by the best-response principle cannot take place if Mary and Joe are rational and thus we have to search for another set of solutions. Following the above arguments, we can easily see that S cannot run from the lower left to the upper right of the response diagram. By contrast, as is shown on the right-hand side of Figure 3, the solution set may be a straight line from the lower right to the upper left of the response diagram.14 The unique rational solution of the game of chicken turns out to be (1, 1) (see the blue point in Figure 3). This can be seen as follows: We have that p = q and thus Mary’s expected utility amounts to p 2 · 1 + p (1 − p) · 0 + (1 − p) p · 2 + (1 − p)2 · (−1) = −2p 2 + 4p − 1 . Now, we obtain her optimal strategy p ∗ = 1 by setting the first derivative, −4p + 4, to zero. It is clear that Joe has the same optimal strategy, i.e., q ∗ = p ∗ = 1. This contradicts traditional game theory, which advocates one of the three Nash equilibria depicted in Figure 3. A typical argument in game theory is that the solution “Up”/“Left” cannot be rational, since if Mary chooses “Up,” Joe moves to “Right,” etc. As already discussed in Section 2, this kind of argumentation lacks any logic. According to Theorem 1, Joe cannot move to another strategy without accepting that Mary will change her strategy, too. Another inconvenience is that the best-response principle leaves us alone with ambiguity. Which of the three Nash equilibria will be the resulting solution? One could argue that the pure-strategy Nash equilibria of the game of chicken, i.e., (0, 1) and (1, 0), are unsatisfactory, since each of them favors the one or the other player (Rapoport and Chammah, 1966). Hence, the aforementioned authors support the ¡ ¢ mixed-strategy Nash equilibrium 12 , 21 . Since Theorem 1 does not distinguish between pure and mixed strategies, this is not a convincing solution either. We conclude that a rational solution need not be a Nash equilibrium. Since Mary and Joe are rational, they decide to chicken out. If Mary was likely to dare, Joe would take the same opportunity into consideration, but Mary is rational, i.e., she knows this. Thus it makes no sense for her to dare. The same argument applies to Joe and so the message is: “Rational people do 14 We could think also about more complicated shapes of the solution set, but this would not affect the result, essentially.
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Frahm, 2015 • Rationality and Best Response
A
Sun
B
Rain
A
Sun
Sun
Stay
Stay/Fly
A Rain
Sun
Fly
B B
Rain
Rain
Stay/Fly
Figure 4: Decision trees of John’s first decision problem.
not kill each other.” On closer inspection, this result seems intuitive. This demonstrates that peace and rationality do not exclude each other, even if the individual interests of the players are diametrically opposed.
4.2. Prisoner’s Dilemma Now, consider the prisoner’s dilemma on the upper left of Table 2. The best-response diagram of this game can be seen in Figure 2. According to this diagram, the set of solutions is a singularity and contains only the solution (0, 0). Hence, any other profile is unattainable for Mary and Joe. Since the players are restricted to the pure strategies p = 0 and q = 0, we can simply ignore any other solution. The same argument can be applied to every other singular set of solutions and so we could justify any other solution in the prisoner’s dilemma. Indeed, this would reduce game theory to absurdity. Hence, forcing S to be a singularity is not very helpful. We may assume for the sake of simplicity that the solution set consists only of two points. For example, we could think about (0, 1), (1, 0) ∈ S .15 From Lemma 1 it follows that such a set of solutions cannot occur if Mary and Joe are rational, but a solution set with (0, 0), (1, 1) ∈ S is conceivable. In this case we obtain a unique rational solution, i.e., (1, 1). This is the same rational solution as in the game of chicken (see the blue point in Figure 3). Since this result seemingly contradicts the dominance principle, it shall be clarified in the following. Consider a single decision maker, John, who has a flight ticket to Hawaii and thinks about flying off or staying at home. Suppose that John has the following decision matrix: State Action Stay Fly
A
B
Sun Sun
Rain Rain
In the first row, “Sun” and “Rain” refer to the weather in John’s home town, whereas in the second row, the same labels refer to Aloha State. In principle, John could have been faced with two different state variables, depending of whether he decides to stay or fly. Nonetheless, the given decision matrix just indicates that the two state variables are identical, i.e., in any case the 15 This means “Down”/“Left” and “Up”/“Right.”
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Frahm, 2015 • Rationality and Best Response
A
Sun
B
Sun
A
Rain
B
Rain
Sun
Stay
Rain
Fly
Sun
Stay
Rain
Fly
A
Stay
B
Fly
Figure 5: Decision trees of John’s second decision problem.
weather in John’s home town will be the same as the weather on Hawaii. This is also illustrated on the left-hand side of Figure 4.16 Suppose that the corresponding utility matrix looks like this: State Action
A
B
Stay Fly
1 2
−1 0
The action “Fly” dominates the action “Stay.” According to the dominance principle, John will choose the strategy “Fly.” The dominance principle is a fundamental part of our theory of rational choice and thus it should also apply to game theory. In the context of game theory, Mother Nature represents a player. This means she applies a strategy. More precisely, she assigns either “Sun” or “Rain” to each state of the world, i.e., A and B. Indeed, in this very simple example, the state space consists only of two atoms. Can it then happen that nature assigns “Sun” to State A and “Rain” to State B, both if John decides to stay and if he decides to fly? The answer is “No!” John’s state variable “weather” is her control variable, whereas John’s behavior represents her state variable. Obviously, John’s decision is measurable with respect to the “private” information set of Mother Nature. This means her state variable is deterministic. Suppose that John would stay and fly, if nature chooses the strategy “‘Sun’ in State A and ‘Rain’ in State B,” as indicated by the red lines on the right-hand side of Figure 4. In this case, she is not able to determine John’s response to her strategy. Since our general framework of rational choice presumes that all decision makers are able to determine their relevant state variables, the decision matrix outlined above cannot occur in a two-player normal-form game. Now, consider the following decision problem:17 State Action Stay Fly
State
A
B
Sun Rain
Sun Rain
Action
A
B
Stay Fly
1 0
1 0
16 Here, the decision trees do not have any temporal meaning. 17 The decision matrix is given on the left hand, whereas the utility matrix is placed on the right hand.
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Frahm, 2015 • Rationality and Best Response 1
1
p
p
q
0
q
0
Figure 6: Best-response diagram (left) and solution set (right) of the win-win situation.
Now, John is faced with two different state variables. This is also illustrated on the left-hand side of Figure 5. From the perspective of nature, John decides to stay if the weather is sunny, whereas he flies off if it is rainy. Mother Nature’s strategy “Sunny weather” is indicated by the red lines on the right-hand side of Figure 5, whereas the blue lines represent her strategy “Rainy weather.” Contrary to John’s first decision problem, his response now turns out to be unique. As we can see from the utility matrix, he will not fly at all, since the dominance principle leads to the strategy “Stay.” Hence, in the second decision problem, John comes to the opposite conclusion. We see that the dominance principle still holds true, but it has to be applied with care when predicting the result of a non-cooperative game. A two-person normal-form game is typically represented by a bimatrix like in Table 2. Each row and column of a bimatrix represents an action, but not a state of the world, i.e., it is not an element of the state space F .18 While this representation might be useful when deriving the best-response function of each player, it is inadequate if one is interested in their state spaces. On the one hand, we have seen that the best-response functions are not very meaningful. On the other hand, it is not possible to apply the dominance principle without specifying the state space. Hence, from the upper left of Table 2, we must neither conclude that “Up” is dominated by “Down” nor that “Left” is dominated by “Right.” Otherwise, we would confuse actions with states and such a naive application of the dominance principle can be misleading.19 As already mentioned above, the solution (1, 1), i.e., “Cooperate”/“Cooperate,” is a rational solution. This is simply because Mary is rational. Thus she knows that Joe will not defect if she cooperates and vice versa. Otherwise we would have had to consider a set of solutions that runs from the lower right to the upper left of the response diagram.20 This would not alter the rational solution: Suppose that (1, 1) ∈ S is not a rational solution. Then either Mary or Joe can find a better position in S . This means the corresponding player can obtain an expected value that is greater than −1. In this case, the other player obtains an expected value that is lower than −1 and so this cannot be an optimal solution for him. Hence, if Mary and Joe are rational, the result of the game is (1, 1). We conclude that cooperation is a logical consequence of rationality. Like in the game of chicken (see Section 4.1), the presented result is in direct contrast to common belief. Nonetheless, it explains cooperation in a world of selfish but rational people. The fact that some players do not cooperate in real-life situations being similar to the prisoner’s 18 Hence, a bimatrix should never be misunderstood as a bilateral utility matrix. 19 Unfortunately, this mistake is visible throughout the whole literature on the prisoner’s dilemma. 20 As already indicated at the beginning of this section, S cannot run from the lower left to the upper right.
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Frahm, 2015 • Rationality and Best Response
1
Ambiguous
1
Unattainable
p
p
0
q
q
0
Figure 7: Best-response diagram (left) and solution set (right) of the Pareto game.
dilemma could be attributed to a lack of “rationality” in our strict sense. This means some players defect, because our basic model of rational choice is not (fully) satisfied in real-life situations, but not because the players are rational in the traditional sense of decision theory.
4.3. The Win-Win Situation The win-win situation is given on the upper right of Table 2 and its best-response diagram can be found on the left-hand side of Figure 6. The corresponding solution set consists of two points, i.e., on the lower right and upper left corner, which are depicted on the right-hand side of Figure 6. Only the upper left corner, i.e., the blue point, represents a rational solution. In fact, due to Theorem 3, the rational solution is essentially unique and thus we do not have to rule out the Pareto-inefficient solution on the lower right by refinement (Harsanyi and Selten, 1988). While the two preceding examples demonstrate that a rational solution need not be a Nash equilibrium, this example shows that a Nash equilibrium need not be a rational solution. The fact that the Nash equilibrium can lead to multiple and even Pareto-inefficient solutions is often considered its most serious deficiency (Colman, 2004). Another drawback of the Nash equilibrium is the fact that it can produce subgame-imperfect equilibrium points (Selten, 1965, 1975).21 It is argued that such solutions are implausible, since an arbitrarily small uncertainty regarding the action of one player destroys the Nash equilibrium. As already mentioned, one can find a tremendous number of refinement procedures, which aim at eliminating Nash equilibria that are considered in any sense implausible. The potential pitfalls of refinement and perfectness shall be illustrated by the following “Pareto game:” Joe Mary
Left
Right
Up Down
(6, 9) (3, 7)
(5, 9) (4, 8)
A typical argument is that Mary goes “Up,” since “Down” appears to be strictly dominated by “Up.” If Joe has complete information and knows that Mary is rational, he is indifferent about “Left” and “Right.” This game has two pure-strategy Nash equilibria, i.e., “Up”/“Left” 21 In our context, this terminology is somewhat misleading, since we do not consider games in extensive form. For this
reason, I will drop the prefix “subgame.”
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Frahm, 2015 • Rationality and Best Response 1
1
p
p
q
0
q
0
Figure 8: Best-response diagram (left) and solution set (right) of the battle of the sexes.
and “Up”/“Right,” and a multitude of mixed-strategy Nash equilibria. More precisely, Joe can choose “Left” with probability q and “Right” with probability 1−q for all 0 < q < 1. Only the purestrategy Nash equilibrium “Up”/“Left” is Pareto efficient. All Pareto-inefficient Nash equilibria can be considered implausible and we end up with the unique rational solution “Up”/“Left.” Nonetheless, the strategy “Left” is weakly dominated by “Right” and so the Pareto-efficient Nash equilibrium “Up”/“Left” is imperfect. For example, Joe could doubt that Mary is rational or think that she might go “Down” by accident. If he has the slightest fear that Mary goes “Down,” we could expect that he goes “Right.” In this case we would end up with any Pareto-inefficient solution. This demonstrates that refinement does not guarantee a unique and plausible solution. The above arguments are based on a naive application of the dominance principle and the hypothetical interpretation of “best response.” The best-response diagram of the Pareto game is given on the left-hand side of Figure 7. It contains two areas: (i) The right border indicates the best responses of Joe to the strategies of Mary. (ii) The upper border represents the set of all Nash equilibria. All profiles on the right border (except for the upper right corner) are unattainable. Hence, Mary cannot apply a “trembling-hand strategy” for logical reasons.22 Moreover, the upper border is ambiguous and thus it cannot be used as a response function for Mary. Our theory of rational choice stipulates that it is always possible to express Mary’s uncertainty by one and only one state variable. This means Joe’s response to Mary’s strategy must be uniquely determined, but this goal cannot be achieved by the best-response diagram. Hence, the best-response principle does not help much if we are searching for a unique rational solution. By contrast, consider the solution set on the right-hand side of Figure 7. The reader might convince himself that this set of solutions is possible in the Pareto game. Under these circumstances, the blue point on the upper left turns out to be the rational solution. Interestingly, although this is the Pareto-efficient Nash equilibrium, this result cannot be derived by the best-response principle.
4.4. Battle of the Sexes The battle of the sexes can be found on the lower left of Table 2 and its best-response diagram is given on the left-hand side of Figure 8. The set of solutions, S , apparently consists of the 22 For example, we have that p = 0.5 ⇒ q = 0 ⇒ p = 1 and so Mary cannot choose p = 0.5.
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Frahm, 2015 • Rationality and Best Response ¡ ¢ three Nash equilibria (0, 0), (1, 1), and 14 , 34 (see the green points in Figure 8). This solution set does not contain a rational solution and thus it cannot take place if Mary and Joe are rational. Otherwise, Mary would choose “Down” to force Joe to go “Right,” whereas Joe would choose “Left” to force Mary to go “Up,” which leads to a contradiction. Indeed, every solution set that runs from the lower right to the upper left is impossible in the battle of the sexes. By contrast, consider the solution set on the right-hand side of Figure 8. ¡ ¢ This set contains a unique rational solution, i.e., 83 , 58 , which can be seen as follows: For each 0 ≤ p ≤ 1 we have that q = 1 − p and thus Mary’s expected utility amounts to
p (1 − p) · 2 + p 2 · 0 + (1 − p)2 · 1 + (1 − p) p · 3 = −4p 2 + 3p + 1 . Setting the first derivative, −8p + 3, to zero leads to Mary’s optimal strategy p ∗ = 38 . It can be easily seen that the optimal strategy of Joe corresponds to q ∗ = (1 − p ∗ ) = 58 . This is not a corner solution. The reason is that Mary and Joe can improve their situation by moving away from the lower left corner. To show this, let us analyze the battle of the sexes in its generic form: Joe Mary
Left
Right
Up Down
(x, y) (w, w)
(v, v) (y, x)
Here we have that v ≤ w < x < y with x + y > 2w. Then the solution set in Figure 8 leads to the rational solution µ ¶ 1 w −v ∗ ∗ p = 1−q = 1− . 2 x −v −w +y For example, suppose that “Up” and “Left” mean “Football,” whereas “Down” and “Right” stand for “Concert.” Since we have that w − v < x − v − w + y, it is not optimal for Mary to apply the pure strategy “Concert,” since in this case Joe would always go to “Football,” i.e., they never meet each other. She is better off keeping the possibility open to meet Joe at the football match. Of course, the same argument applies to Joe. In the special case w = v, i.e., if “Concert” (“Football”) has no special meaning to Mary (Joe) when they are separated from each other, we ¡ ¢ obtain the rational solution 12 , 21 .
4.5. Matching Pennies Matching pennies is a two-person zero-sum game. It can be seen on the lower right of Table 2. Its best-response diagram is given on the left-hand side of Figure 9. The recommendation from ¡ ¢ traditional game theory is clear: It is the minimax solution 21 , 12 (Kjeldsen, 2001, von Neumann, 1928, von Neumann and Morgenstern, 1944), i.e., the green point in Figure 9. Once again, the best-response principle suggests a singular set of solutions and one might ask if it is possible to justify a more comprehensive solution set. Interestingly, here this is not possible. Suppose that S were not a singularity. For example, consider a solution set that runs from the lower right to the upper left of the response diagram. Then the set of rational solutions is empty: Mary would prefer “Up” and “Down,” whereas Joe would choose “Left” and “Right” with equal probability. Hence, Mary’s optimal strategies are located on the lower right and upper left corner, whereas Joe’s optimal strategy is in the middle of the response diagram. If S runs from
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Frahm, 2015 • Rationality and Best Response 1
1
p
p
q
0
q
0
Figure 9: Best-response diagram (left) and solution set (right) of matching pennies. the lower left to the upper right of the response diagram, Mary prefers the solution in the middle, whereas Joe tries to escape from the middle. In every two-person zero-sum game, i.e., ϕ2 = −ϕ1 , where all players are rational,23 the set of solutions must be essentially unique. This can be seen as follows: According to Lemma 1, the set of rational solutions, S ∗ , is nonempty. Then any rational solution s ∗ = (s 1∗ , s 2∗ ) must belong to ¡ ¢ ¡ ¢ arg max ϕ1 s 1 , s 2 = arg min ϕ2 s 1 , s 2 . (s 1 ,s 2 )∈S
(s 1 ,s 2 )∈S
This means by maximizing her own objective function with s 1∗ , Mary implicitly minimizes the objective function of Joe. Now, S must be essentially unique. Otherwise, Joe could find a strategy that is better than s 2∗ , but then s ∗ cannot be a rational solution. Hence, our theory of rational choice implies that there exists essentially one and only one solution of matching pennies. In principle, we could defend any singular set of solutions, but the minimax solution stands out because it is fair. This means it leads to the same value for both players. Indeed, if we assume that Mary and Joe have the same opportunities, there is no reason ¡ ¢ to suggest another solution than 12 , 12 , i.e., the blue point on the right-hand side of Figure 9. This demonstrates that the minimax solution can be justified by simple arguments of decision theory and so the best-response principle is void.
5. Conclusion The Nash equilibrium allows for multiple and Pareto-inefficient solutions of non-cooperative games. Hence, a Nash equilibrium need not be a rational solution. This contradicts our common understanding of rationality and makes it difficult to predict the results of strategic conflicts. We have seen that also the converse is true: Rational players in a non-cooperative game need not give a best response to each other. This renders the Nash equilibrium and other equilibria that are based on the best-response principle inappropriate as a solution concept for strategic conflicts. This problem is highly relevant, since the Nash equilibrium represents the central solution concept in non-cooperative game theory and is frequently applied in many different scientific disciplines. Our theory of rational choice implies that the strategies of the players are associated with each other. For this reason, giving a best response in the usual sense of game theory is impossible. 23 It is implicitly assumed that the set of solutions, S , is nonempty.
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Frahm, 2015 • Rationality and Best Response Solutions that are based on the best-response principle can be logically inconsistent if all players are rational. This means the best-response principle creates situations which cannot occur for logical reasons. Moreover, it has been shown that every non-cooperative game has essentially one and only one rational solution. Hence, the multiple-solution problem disappears in our general framework of rational choice and thus we do not need to eliminate Pareto-inefficient solutions by refinement. This solves a longstanding problem of non-cooperative game theory. Finally, it has been demonstrated that rational solutions of well-known normal-form games typically differ from those predicted by the best-response principle. The differences turned out to be essential both from an economic and a social point of view.
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