game theory, complexity and simplicity
Game Theory, Complexity, and Simplicity Part I: A Tutorial Martin Shubik
SFI WORKING PAPER: 1998-04-027
SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. ©NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. www.santafe.edu
SANTA FE INSTITUTE
1 July 8, 1997
GAME THEORY, COMPLEXITY AND SIMPLICITY
Part I: A Tutorial The theory of games is now over 50 years old. Its applications and misapplications abound. It is now "respectable". Bright young economic theorists bucking for promotion need to be aware of agency theory, reputation theory, the new industrial organization, auction theory, refinements of equilibria, perfect equilibria, trembling hand equilibria, strong equilibria, Bayesian equilibria, problems with common knowledge and to be aware of game theoretic applications to biology, political science, law and even anthropology. The public is aware of phrases such as the "zero sum game". Many individuals who know next to no game theory have heard of the "Prisoners' Dilemma". The word "strategy" has been given a scientific twist. Newspaper columnists are already using or misusing the terminology. All in all, game theory has arrived. This note is divided into three parts aimed at being a tutorial, retrospective essay and critique. The tutorial is called for to help inform those who are uninformed. Possibly even more important, it aims to help inform those who are misinformed and may be somewhat misled by the academic fashion show. In particular the noncooperative equilibrium is explained and the use and limitations of its applications are discussed. The meaning of cooperative game theory is explained and its uses and limitations are noted. The reasons for the gross inadequacy of a dynamic theory of games are noted and the need for a considerably different approach is suggested1The key to understanding the uses and limitations of game theory is to appreciate that the game theoretic approaches are not monolithic and depending on which one the individual follows the world view obtained is highly different. In particular the four most important distinctions are:
1
Those who wish to delve further into details will find the three volume handbook editied by Aumann and Hart (1992, 1994 and 1997 forthcoming) to be of considerable assistance.
2 1. 2. 3. 4.
Game theory as mathematics; Game theory as science; Game theory as philosophy; Game theory as advocacy.
Some brief observations are made here on these distinctions. They will be expanded further in a subsequent note. After these are discussed, some basic exposition is given. 1. Game theory as mathematics Prior to the theory of games the prime emphasis in mathematical techniques applied to economics was to differential calculus, differential and difference equations. Combinatorics was hardly considered. The stress in von Neumann and Morgenstern was on choices among finite sets2. Not only did they stress finite combinatorics in individual choice. They then proceeded to develop a combinatoric theory of cooperative social choice where the stress was on the powers of all coalitions which could be formed among the players. In their development of the theory of the two-person zero sum game the stress was on finite choices and how to calculate optimal strategies. The method of calculation called for was, in essence, linear programming. There are many purely mathematical problems which are posed in game theoretic investigations and it is possible to regard oneself as a game theorist with no empirical concerns whatsoever, limiting oneself to the new mathematical problems which are posed by the game theory models. 2. Game theory as science It was the original intention of von Neumann and Morgenstern to apply game theory to problems in economics. However the range of scientific applications has been far broader, including all of the social sciences as well as biology. It has also served to provide the underlying structure for experimental gaming, providing the models which help to show the limits of individual rational behavior. The scientific applications can be split into pure and applied. Examples of pure science based on game theory would be the gaming investigations of O'Neill (1987), Roth (1987) and others (see Shubik, 1987 for a survey) and the current work in evolutionary game theory and biology. ( See Weibull, 1996 for a survey.) Game theory as applied science and operation research has been used in voting design (see Shapley and Shubik 1954 and Banzhaf 1965); in designing auction mechanisms , in setting aircraft landing fees in deciding on the assignment of overhead costs (Shubik, 1962) and utilized by the military in weapons design.
2
These could be huge sets, such as the set of strategies in chess. The contemplation of the size of such sets has raised problems concerning how individuals explore large numbers of alternatives.
3 3. Game theory as philosophy Von Neumann and Morgenstern placed extremely stringent conditions on their description of a game. It was assumed that all players were completely informed about all of the rules, including each other's preferences and assets. Deep philosophical problems are posed if the assumption of common knowledge is abandoned. One has to deal with assumptions about A’s knowledge of what B knows and B’s knowledge of what A knows about B and so forth , in an infinite regression. There is a virutual subindustry in Game Theory dealing with how to treat lack of common knowledge (see Aumann, 1976) . Von Neumann and Morgenstern, in their investigation of non-zero sum games showed that the ordinary concept of individual rational behavior did not generalize in a unique natural simple way as soon as there is more than one than one individual in a nonzero sum game. Harsanyi and Selten (1988) have proposed the existence of a unique noncooperative equilibrium as the normative solution to a game; others have argued that the various normative features one might require for a solution (such as symmetry, efficiency and equity) may be in conflict. The meaning of “rational behavior” in an n-person game, even with common knowledge is still open to debate. Furthermore both a normative and descriptive viewpoint to game theoretic solutions are supported with differing emphasis by the professional game theorists. It can be argued that one “should” play in a twoperson zero sum game according to the Minimax strategy and a strong case can be made that the concept of the individual rational decision-maker extends to zero sum games. There is no consensus as to what constitutes the “right” solution for nonconstant sum games. 4. Game theory as advocacy With the growth of the acceptance of game theoretic concepts and ways of formulating the analysis of public problems such as national defense or regulation of industrial organization the use of game theory as metaphor has grown. Possibly the earliest examples are given in the works of Schelling (1960), Ellsberg (1956) and Rapoport (1960). Game theoretic thought mixed in with psychology, social psychology and international relations flavored the works of Wohlstetter (1975) and Herman Kahn (1960). I term this type of use as “conversational game theory”. This is not meant to be a pejorative term. The use of conversational game theoretic argument as portrayed by the writings noted above probably had considerable influence on nuclear deterrence policy and cold war strategy. Von Neumann’s stress was on cooperative game theory and he was adamantly opposed to “one point solutions,” i.e., to using game theory for specific prediction. Thus it is unlikely that his strong views on defense were based on his game theoretic thought.
4 Von Neumann & Morgenstern's great book The seminal book in game theory was published in 1944. It is large, innovative and impressive. But it is such that very few individuals appear to have read it in toto. A reason why many individuals have read some parts of Game Theory and Economic Behavior, but few have read it all the way through is because it may be regarded as four books in one. It deals with four topics all of which can be treated independently. They are: 1. 2. 3. 4.
Preferences and utility theory. The extensive form of a game. The description of, and solution to a 2-person zero sum game The description of, and stable set solution to an n person non-zero sum game played cooperatively.
Preferences and utility theory In their need to define precisely the goal of each player, von Neumann and Morgenstern built upon the usual (highly approximate) assumptions of microeconomic theory that individuals have completely ordered preferences. By providing an axiomatic treatment of preferences including axioms on risk behavior they were able to show that preferences could be represented by a utility function completely specified up to a linear transformation. This result, which von Neumann and Morgenstern needed in order to set up and study the two person zero sum game was the founding seminal contribution to the abstract treatment of the study of many different types of preference structure. Their work led to the investigation of problems concerning individual ability to develop a complete ordering of preferences for items among which they have to choose, and individual ability to gauge risk preferences. It also raised questions concerning the possibility for interpersonal comparison of welfare; and how closely efficient transfer of utility can be achieved by the availability of a money like commodity to be used as a means for making side payments. The formal axiomatic approach of von Neumann and Morgenstern to the measurement of utility led to the investigation of many other preference structures, such as partially ordered preferences. The extensive form of a game. Until the development of the theory of games a language and notation to describe a formal structure for conscious decision making had hardly been developed. The extensive form of game delineated by a game tree (in a one person version it is referred to as a decision tree) provided a means to make precise the concepts of move, choice, information strategy, play, outcome and payoff . The authors’ choice of the title “theory of games” was based on utilizing their insights into the formal structure of games such as chess, Go, or Bridge. Figure 1 shows a game tree for a
5 two person game where Player (or decision maker) 1 chooses his move first among three possible moves. Player 2 is informed if Player 1 selects his first move, but is not fully informed if the choice of Player 1 is move 2 or 3. Player 2 then makes his choice. He has to choose between two moves. The game reaches a terminal node of the tree . At each terminal node there is an outcome. Each player assigns a value or utility to the outcome. For example The game tree Figure 1 in chess an outcome at a terminal point in the tree might be that Black checkmates White. Black evaluates this outcome at a value of 1 and White evaluates it at 0. In Figure 1 the nodes of the tree (except the terminal nodes) represent choice points. They are points at which some player must select a move. The branches emanating from each choice point are the moves. They are numbered so that they can be identified. The choice points are circled by a closed curve which may contain one or more points. These curves are information sets . Each information set is labeled with the name of the player to whom it belongs. In Figure 1 Player 2 has two information sets labeled by P2. The left information set contains a single point. This indicates that if the game proceeds to that node, when Player 2 is called upon to move he knows exactly where he is. This contrasts with the information set on the right. If Player 1 had selected either his second or third moves, when Player 2 is called upon to move he does not know which move Player 1 has selected. If Player 2 selects his first move because of his lack of information he does not know which one of the two possible outcomes will emerge. An information set could be more descriptively called a lack of information set as it describes the level of ignorance of the player. The initial node of the game tree bears the additional label of O which indicates where the game starts. The terminal nodes in Figure 1 are labeled with the outcomes win, draw or lose. Underneath the label at each terminal two numbers appear. The first is the value attached to the outcome by Player 1 and the second, the value attached to the outcome by Player 2. Given this language we can now describe the basic features of any game. A path down the game tree starting from the initial node and ending at some terminal node describes a play of the game. It is a more general form of the type of description of a chess game reported in the newspapers. It indicates where the game began and indicates the selection of moves by each of the players down to the final outcome. The numbers at the terminal nodes indicate the payoffs or the worth of the final outcome to each player. In theory, with this notation we could draw a full game tree for a game such as chess. But a little contemplation of that game is sufficient to indicate that we would exhaust the world’s paper supply if we tried to draw this game tree on the same scale as Figure 1.
6 The notation developed for the description of the game theory enables us to give a precise comprehensive definition of the meaning of strategy. A strategy is an all encompassing plan which contains instructions concerning what a player should do under all contingencies. It can be easily understood and illustrated by considering the game shown in Figure 1 and displaying a few strategies. Player 1 has three strategies which coincide with his moves. This coincidence occurs because he has no contingencies for which he has to plan. Suppose Player 1 were going out of town before he was due to play and that he wanted to leave a complete book of instructions for the individual who was going to play for him. He could leave any one of three extremely simple books of instructions . The first book of instructions would simply say: “Select move 1.” This is somewhat unimpressive, but matters change when we suppose that it is Player 2 who is out of town and he is leaving instructions to someone who is going to play for him. One such book of instructions is of the following form: “If Player 1 selects move 1 select 1; if he selects moves 2 or 3 then select 2.” We could abbreviate this description to (1,2). A little work shows that there are four different books of instruction which could have been left by Player 2. They are (1,2), (2,2), (1,1) or (2,1). Thus in this game Player 1 has three strategies and Player 2 has 4. Rather than analyze the game in extensive form we can obliterate the details of description in terms of the game tree complete with details concerning information and replace it with a matrix which illustrates the interaction between all of the strategies of one player and all of the strategies of the other. P2 (1,2)
(2,2)
(1,1)
(2,1)
1
1,0
1/2,1/2
1,0
1/2,1/2
2
1,0
0,1
0,1
1,0
3
0,1
1,0
1,0
0,1
Table 1 A simple example shows how the entries in the cells are calculated. Suppose that Player 1 selects his first strategy and player 2 selects his strategy denoted by (2,2). This tells us that Player 2’s reply is to select move 2 for the payoffs of (1/2, 1/2).
7 The description of, and solution to a 2-person zero sum game A zero sum game is a game in which the interests of the players are diametrically opposed. What one player wins, the other loses3. Chess, Go and Checkers provide examples of two person zero sum games. A five person Poker game is a five person zero sum game. Von Neumann and Morgenstern devised a special theory to analyze two person zero sum games. In order to do so they made use of the strategic form of a game They confined their interest to a game description only in terms of strategies and payoffs. The detail provided by the extensive form is suppressed. Thus there are many game trees or games in extensive form which give rise to the same game in extensive form. A great loss of detailed information has taken place in going to the strategic form. The sort of structure on which one might wish to base an analysis of dynamics has been replaced by a structure better suited to study equilibrium and statics. A simple example serves to illustrate the relationship between the extensive form and the strategic form. Figure 2a and b show a game in which both Player 1 and 2 each must select between two moves in complete ignorance of the other’s actions. The only difference between the two game trees are that in Figure 2a Player 1 moves first; in 2b Player 2 moves first. As it is assumed that neither player has the opportunity to peek, the games are strategically equivalent. For each player in this simple game a strategy and a move are equivalent. Each must select move one or two. Figures 2a,b The strategic form representation of the game is given in Table 2. Player 1 selects a row and Player 2 selects a column. The first number in each cell represents the payoff to Player 1 and the second number, the payoff to Player 2. Thus, for example. if Player 1 uses his second strategy and Player 2 his first, Player 1 loses 10 and Player 2 gains 10. If Player 1 selects his first P2 1
2
1
5,-5
0,0
2
-10,10
5,-5
Table 2 More generally rather than zero sum we could talk of constant sum games where an arbitrary constant (say a gift from Uncle Harry) is added to each of the payoffs to a player. Third part of his payoff is independent of the strategies employed (See Shubik, 1982 Ch 8) 3
8 move with a probability of 3/4 and his second move with a probability of 1/4 he can guarantee that no matter what Player 2 does his expected gain will be 5/4. This illustrates the Maxmin solution for a two person zero sum game. The Row player can guarantee that his expectation is no less than a certain amount and the column player can prevent the row player from obtaining more than this amount by selecting his first move with a probability of 1/4 and his second move with probability of 3/4. Here we note that the players do not select a single row or column with certainty, but select both rows or columns with a calculated probability for each row or column. Such a strategy is referred to as a mixed strategy. Player 1 is guaranteed his maxmin. The Maxmin solution suggested by von Neumann for the two person zero sum game can be regarded as a normative solution. If the player wishes to maximize his expected gain he should use his maxmin strategy. Except for parlor games and some military problems there are few actual situations which model as pure zero sum games. Experimental evidence on actual performance of players playing a 2 x 2 or a 3 x 3 matrix game is mixed. When the matrix is bigger than 3 x 3 players display information processing and computational limitations. Solutions to an n person non-zero sum game played cooperatively. Von Neumann and Morgenstern developed three different representations of a game of strategy. They are: 1. The extensive form; 2. The strategic form and 3. The cooperative or coalitional form The extensive form can be utilized to provide a complete process description of the game. We discuss the extensive form below after considering the noncooperative equilibrium solution. The strategic form, as illustrated above serves to enable us to analyze the game in terms of strategies. The characteristic function or coalitional form provides a basis from which the combinatorics of potential collaboration can be studied. Table 3 shows a simple 2 x 2 nonconstant sum game in matrix form4. If each player uses his first strategy each obtains a payoff of 5. If Player 1 uses his first strategy and Player 2 uses his second strategy, Player 1 obtains -10 and Player 2 obtains 15. If both use their second strategy each obtains 0. It is easy to observe that jointly, by cooperating, each using their first strategy, the players could obtain 10 together.
4
More accurately, in bimatrix form as there are two entries in each cell.
9 P2 1
2
1
5, 5
-10, 15
2
15, -10
0, 0
Table 3 Each player acting independently cannot guarantee for himself more than 0. The characteristic function of a game with n players illustrates the amounts that any of the 2n coalitions can obtain if the members of the coalition cooperate5. For the simple game illustrated in strategic form in Table 3 the characteristic function is as is shown in Table 4.
v({1}) = 0, v({2}) = 0 v({1,2}) = 10 Table 4 The reason why we use a double set of brackets around the numbers is to indicate that for example v({1}) stands for the value of the coalition consisting of the set containing only Player 1. The curly brackets {} indicate a set. The coalition consisting either of Player 1 or 2 cannot guarantee more than zero. The coalition of both 1 and 2 together can obtain 10. If one is presented with just the characteristic function of the game, the details concerning the strategic structure for individual are suppressed. A way of looking at the information conveyed by the characteristic function is that it at the level of aggregation designed for a diplomat or negotiator. It tells him nothing about the strategic choices facing the generals only the values of coalitions. The extensive form is better designed for use of individuals concerned with the details of information and moves. Bits of it must be considered by those concerned with tactics and the specifics of play rather than with overall strategy. Figure 5 illustrates a three person game in characteristic function form. It tells us that Player 3 is more desirable than 1 or 2 in any two person coalition and that Player 2 is more desirable than 1. Here we have not bothered to illustrate either the extensive or strategic forms which would give rise to this characteristic function. There are many different games which would produce this characteristic function; but if we are only interested in the sort of negotiations which For formal mathematical rigor one usually attaches a value of zero to the coalition consisting of nobody. 5
10 would have the three players decide on how to split the four units they can earn by cooperation we do not need the other detail. v({1}) = v({2}) = v({3}) = 0 v({1,2}) = 1, v({1,3}) = 2, v({2,3}) = 3 v({1,2,3}) = 4. Figure 5 Von Neumann and Morgenstern defined an imputation to be a vector of n numbers whose sum adds up to the amount that the group as a whole could obtain by cooperation. Referring to Table 5 the vector (.9, 1.2, 1.9) is an imputation. It says that Player 1 receives .9; Player 2 receives 1.2 and Player 3 receives 1.9. These three numbers add to 4 which is the amount that the coalition of all could obtain. The key operation they considered was that of domination. An imputation is said to dominate and imputation if some set of players which we call S are better off in than in and by going into business for themselves they could obtain at least as much as. Rather than leave us with an abstract exercise using Greek letters a simple example utilizing the game illustrated in Figure 5 can make this concept clear. Suppose that the imputation were (1, 2, 1) and were (.8, 2.3, .9) then dominates via the set S={1,3} because Players 1 and 3 acting together without Player 2 can obtain 2 thus they have the power to select over. The theory of n-person games developed by von Neumann and Morgenstern was both static and cooperative. The authors were at pains to stress that they regarded the development of a useful dynamics as beyond their scope. They indicated however that much of value could be constructed by concentrating on the statics. Their approach towards the many person problem was in part normative and in part positive in the sense that they felt that the society as a whole should try to make the cake as big as possible, but that groups would bargain and negotiate over their shares of the cake. There are four solution concepts of note which have been developed and applied to cooperative games represented by a characteristic function. They are the stable set solution proposed by von Neumann and Morgenstern and the solutions known as the core, the value and the nucleolus of an n-person game. Without going into mathematical detail it is not possible to do full justice to the specific structure of each of these solutions. However given the concepts of imputation and domination, some intuitive insights can be provided concerning the basic concepts underlying each of the solutions.
11 The stable set solution The stable set solution reflects the possibility for the distribution of resources to depend on a variety of complex social structures. The stress is on extreme nonuniqueness of outcomes. In essence, a stable set solution cannot be used to make a unique prediction, all that it does is to delineate a subset of the outcomes. Many imputations of wealth may make up a single solution. When all of the imputations in a solution are considered together it is the relationship between all of them that imposes a social stability on the whole.6 There can be many different stable sets rewarding different groups considerably different gains which are all nevertheless consistent with the same resource structure The core. The core of an n-person is the set of imputations which are undominated, i.e the set of ways of dividing the overall pie in a manner such that no group could do better if it tried to go into business for itself. Referring to the game illustrated in Table 5 the imputation (.5, 1, 2.5) is in the core as there is no coalition whose members can all do better by themselves. The core can be viewed as the set of outcomes impervious to countervailing power. It reflects a stability against all coalitions. It can be proved that the competitive price system of an economy lies in the core ( for discussion and references see Shubik, 1984). The value The value solution to an n-person game is a solution devised by Shapley (1953) which evaluates an a priori worth of each individual to the overall game. In essence, it averages over the marginal contribution that each individual makes to all the coalitions into which he could enter. A special example of a four person game may help to illustrate the nature of the value solution. We may consider a special type of game which is represented by an extremely simple chartacteristic function. It is called a simple game because all of the payoffs obtained by the coalitions are either 0 or 1. A coalition with a joint payoff of 0 is called a losing coalition. A coalition with joint payoff of 1 is called a losing coalition. The simple game can be used to study the intrinsic assignment of power in a voting structure as a function of the number of votes controlled by each individual. Suppose that there is a corporation with 4 stockholders. The first stockholder has 2 votes and the other three stockholders have 1 vote each. There is a way in which we can assign an a priori value or measure of power to each stockholder. Suppose that instead of there being four stockholders with five shares split among them, there had been five stockholders, each with one share, a reasonable assignment of power among the five stockholders would be .2 to each of them. The technical statement is that a solution consists of a set of imputations which do not dominate each other but dominate all other imputations. 6
12 Each is equally relevant in being able to turn a losing coalition into a winning coalition. But suppose that there were only three voters with the stock distributed as (3,1,1). If the vote is settled by a simple majority of the stock held then it evident that the stockholder with 3 shares holds all of the power and those with 1 share each hold none of the power7. This elementary exercise tells us that as an individual's holding of shares increases his power must change in a nonlinear manner. Once he has more than half of the shares he has all of the power. The value calculation counts all of the marginal contributions that an individual makes to every coalition and averages over them. When the game is large and has little special structure this calculation is long and messy. In games representing voting situations where there are only winning and losing coalitions8 the marginal contribution of a player is either 0 or 1. It is 0 when he joins a losing coalition and it remains losing, or when he joins a winning coalition and it remains winning.9 It is 1 when the addition of the player changes a losing coalition into a winning coalition. We may imagine coalition formation taking place in every possible order. Call the players A, B, C and D where A has 2 votes and the others 1 each. There are 24 ways the individuals could line up. These are shown below. Starting with the player at the left we can consider a coalition growing as each player is added. Credit is given to the pivotal player who coverts a losing coalition into a winning coalition. This is illustrated by a * next to the pivotal player. AB*CD, AB*DC, AC*BD, AC*DB, AD*BC, AD* CB BA*CD, BA*DC, CA*BD, CA*DB, DA*BC, DA*CB BCA*D, CBA*D, DCA*B, CDA*B, BDA*C, DBA*C BCD*A, BDC*A, CBD*A, CDB*,A DBC*A, DCB*A. We note that in the 24 possibilities A is pivotal 12 times and B, C and D are pivotal 4 times each. The a priori division of power among the voters is (1/2, 1/6, 1/6, 1/6). The phrase a priori is used to indicate that this measure is free of any of the sociological structure that might be present in the set of players. All way in which coalitions might be formed are regarded as equiprobable. In an actual society there may be nets of friends or others whose interests are correlated who act together.
In actuality there is a body of law which protects minority stockholders against the complete power of the majority, but for this example we assume that "winner can take all". 8 Ties must also be accounted for, but here for ease in illustration we consider a voting game with an odd number of votes where all vote and hence there are no ties. 9 In actual political practice although a minimal winning coalition is desirable as there are fewer participants to share the payoff, a few extra players may be of worth in order to insure against accidental failure to be present at the vote. 7
13 The nucleolus The nucleolus is a solution concept devised by Schmeidler (1969) which seeks to find an imputation or a means of dividing the cake in such a manner that the maximum complaint by any coalition against the division is minimized. It may be regarded as a sophisticated version of a splitthe-difference approach to fair division. If the game being studied has a core then the nucleolus is the centroid of the core. The Cournot-Nash Noncooperative Equilibrium In contrast with the cooperative, normative game solution approach noted above Augustin Cournot (1838) suggested a solution to two firms in competition in a market which involved each taking the others expected strategic behavior into account. He showed that there would be a noncooperative equilibrium point which had the property that there would exist a pair of expectations such that if each had the appropriate expectation and based his action on this expectation then the resultant out come would be self-fulfilling, i.e. the expectation of each would turn out to be true. John Nash [20] provided the general definition and proof of the existence of the noncooperative equilibrium. Referring to the simple matrix game shown in Table 3 (known as the Prisoner’s Dilemma game) we can see that if Player 2 expected Player 1 to use his second strategy then Player 1's optimal response would be to use his second strategy and viceversa. Thus the pair of strategies (2,2) would be in equilibrium and would result in a pair of payoffs of (0,0). In contrast if both had selected their first strategy and acted cooperatively they would have obtained payoffs of (5,5). The noncooperative equilibrium solution has also been applied to games in extensive form. Unfortunately when such an attempt is made a host of new problems occur which can be more or less ascribed to the unsatisfactory relationship between statics and dynamics. The equilibrium analysis is essentially static and for it to hold in a game with many stages one approach has been to consider a backward induction where the presumption is that at every point in the game the expectations of the players guide them to an equilibrium. The mathematical formulation and proof of existence of at least one, and often, many noncooperative equilibria tells us little about the dynamics encountered in a game where the players start to play at a position away from equilibrium. In essence, the noncooperative equilibrium solution points out the existence of outcomes which satisfy a condition on the consistency of everyone's expectations. Although there is some experimental evidence that the noncooperative equilibrium serves as a reasonable approximation to long term behavior in some games with many players and a single noncooperative equilibrium point, little is known with any generality about actual behavior. The noncooperative equilibrium solution should not be regarded as a normative solution as, in general, jointly optimal outcomes
14 may not be achieved. Further discussion and a critique of the problems encountered in utilizing the noncooperative equilibrium solution will be given in Part II. Dynamics and the infinite horizon The game theory models presented above have a definite beginning and end. The initial conditions in the instance of a game of chess are given precisely by the placing of the pieces on an 8 x 8 board. The end is well defined by the rules including conventions to prevent endless cycling so that the game is finite. When we try to model most human activities the initial position reflects history and the social context which sets the stage for individual action. Unlike a game of chess, much of human behavior depends in an important way on history from an unbounded past and expectations concerning an unbounded future. When formal game models are constructed to reflect ongoing games, the distinction between cooperative and noncooperative solutions becomes blurred. As there is always the possibility of a tomorrow, the opportunity to punish doublecrossers and to specify threats which can be used to enforce cooperation becomes a reality. Structure and Behavior A brief review of the formalization and different types of representation of a game of strategy has been presented, followed by an exposition of different solution concepts. The stress has been on the development of a language to describe conscious individual decision making in situations involving one or more individuals with well-defined goals and ability to calculate. Von Neumann and Morgenstern provided a seminal idea in devising a means to describe the full structure of a game, such as chess. In human society there are games within games. Although a particular game, in the short run, may take place in a specific context, the actions of the individuals, may in the long run, influence the environment in which they behave. This type of feedback is not represented in the formulations presented here. The provision of a way to make precise "the rules of the game" and to describe in careful detail the set of all possible ways a game might be played does not provide us with a prescription as to how the game should be played, nor with the evidence as to how the game is actually played by humans. Game theorists have approached the problems involving human motivation and behavior by suggesting a mathematical description of individual preferences, attrbiuting individual optimization as the motivation and devising normative or other solution concepts to account for the interactions of many individuals all in pursuit of their own welfare. Each of the five solution concepts suggested has attractive features. The four cooperative solutions are explicit in indicating the different norms they fulfill and useful applications have been found for the value, core and nucleolus. The noncooperative equilibrium solution is justified more for descriptive reasons and fits into the spirit of recent finance and price system economics where the idea of the
15 noncooperative equilibrium is translated into rational expectations. It has also been applied to biology with a somewhat different interpretation of the concept of a mixed strategy equilibrium. In the subsequent discussion, it is suggested that the influence of game theory has been substantial and the applications at several levels have been of great value; but its very success has served to highlight the limitations of the rationalistic models of economic man in coming to grips with the many socio-economic and politico-economic problems which are faced by any large society. The game theoretic language has provided a means to build process models in which the institutions and other aspects of the environment are the carriers of process. These are implicit in the rules of the game9. The game theoretic solutions are of considerable power in studying some problems of interest, but as models of human behavior, they are lacking and the basic apparatus of game theory drives us to abandon the model of the decision maker as completely informed about the rules of the game with common knowledge about what all others know or do not know. The precision and analytical power of game theory forces us to understand the model of utilitarian man and its direct descendent homo oeconomicus instead of being an idealization of purposeful human decision makers are poor approximations to the far more complicated perception and computationaly-limited human being driven by passions, in part ruled by habit and in part controlled by instinct, whose decisions are set in the context of the society he or she belongs to. The mathematical analysis underlying much of game theory and economic theory was made feasible by the gross simplification of the model of the individual. This may have been fine for some of the original purposes at hand. With the advent of the computer and the possibilities of computation and simulation, the stage is set to enrich the study of models of human behavior using the framework provided by game theory as a starting point from which we may investigate multiperson interaction.
16 REFERENCES Aumann, R.J. 1976. Agreeing to disagree. Annals of Statistics. 4: 1236-39. Aumann, R. J. and S. Hart (eds) 1992, 1994, Handbook of Game Theory, vols I and II and III forthcoming 1997. Banzhaf, J.F., 1965. Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review 19: 317-343. Ellsberg, D. 1956. Theory of the Reluctant Duelist. American Economic Review 46: 909-923. Harsanyi, J. and R. Selten, 1988. A General Theory of Equilibrium Selection in Games Cambridge: M.I.T. Press. Kahn, H. 1960. On Thermonuclear War Princeton: Princeton University Press. Luce, D. and H. Raiffa. 1959 Games and Decisions Wiley: New York. Nash, J.F., Jr., 1951. "Noncooperative Games." Annals of Mathematics 54: 286-295. Nash, J.F. Jr., 1950. "Equilibrium Points In N-Person Games." Proceedings of the National Academy of Sciences, USA 36: 48-49. O'Neill, B., 1987 “Nonmetric tesat of the minimax thewory of two-person zerosum games”, Proceedings of the National Academy of Sciences, U.S.A., 84: 2106-2109. Rapoport, A. 1960, Fights, Games and Debates Ann Arbor: University of Michigan Press. Roth, A.E., 1987.. Laboratory Experimentation in Economics. Cambridge: Cambridge University Press. Schelling, T., 1960 The Strategy of Conflict. Bambridge Mass. Harvard University Press. Schmeidler, D., 1969. "The Nucleolus Of A Characteristic Function Game." SIAM Journal on Applied Mathematics 17 (6): 1163-1170. Shapley, L.S., 1953. "A Value For N-Person Games". In Annals of Mathematical Studies, R.D. Luce and A.W. Tucker, eds. Princeton, NJ: Princeton University Press, pp. 307-317. Shapley, L.S., 1975, "Cardinal Utility from Intensity Comparisons", RAND Corporation, Publication R-1683-PR. Shapley, L.S. & M. Shubik., 1954 A Method for Evaluating the Distribution of Power in a Committee System. American Political Science Review 48: 787-792. Shapley, L.S. and Shubik, M., 1969a. "On the Core of an Economic System with Externalities," American Economic Review, 10: 337-362. Shapley, L.S. and Shubik, M., 1969b. "Pure Competition, Coalitional Power and Fair Division," International Economic Review, 10: 337-362.
17 Shubik, M. 1962."Incentives, Decentralized Control, The Assignment Of Joint Costs And Internal Pricing," Management Science, 8, 2: 325-343. Shubik, M.., "Edgeworth Market Games," In Annals of Mathematical Studies, Volume 40, R.D. Luce and A.W. Tucker, eds., Princeton University Press, Princeton NJ, pp. 267-278. Shubik, M., 1959b. Strategy and Market Structure, New York: Wiley. Shubik, M., 1962, "Some Experimental Non Zero Games with Lack of Information about the Rules," Management Science, 8: 215-234. Shubik, M., 1982. Game Theory in the Social Sciences, Vol. 1, MIT Press, Cambridge, MA. Shubik, M., 1984. Game Theory in the Social Sciences, Vol. 2: A Game Theoretic Approach to Political Economy. MIT Press, Cambridge, MA. Shubik, M. "Game Theory and Experimental Gaming," To appear in Handbook of Game Theory, vol III R. J. Aumann and S. Hart (ed) forthcoming 1997. Von Neumann, J. and O. Morgenstern, 1944. Theory of Games and Economic Behavior, Princeton University Press, Princeton NJ. Weibull, J. W. 1996. Evolutionary Game Theory Cambridge, Mass M.I.T Press. Wohlstetter, A. 1975 Optimal Ways To Confuse Ourselves. Foreign Policy 20: 170-198.
SFI WORKING PAPER: 1998-04-027
SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. ©NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. www.santafe.edu
SANTA FE INSTITUTE
1 July 8, 1997
GAME THEORY, COMPLEXITY AND SIMPLICITY
Part I: A Tutorial The theory of games is now over 50 years old. Its applications and misapplications abound. It is now "respectable". Bright young economic theorists bucking for promotion need to be aware of agency theory, reputation theory, the new industrial organization, auction theory, refinements of equilibria, perfect equilibria, trembling hand equilibria, strong equilibria, Bayesian equilibria, problems with common knowledge and to be aware of game theoretic applications to biology, political science, law and even anthropology. The public is aware of phrases such as the "zero sum game". Many individuals who know next to no game theory have heard of the "Prisoners' Dilemma". The word "strategy" has been given a scientific twist. Newspaper columnists are already using or misusing the terminology. All in all, game theory has arrived. This note is divided into three parts aimed at being a tutorial, retrospective essay and critique. The tutorial is called for to help inform those who are uninformed. Possibly even more important, it aims to help inform those who are misinformed and may be somewhat misled by the academic fashion show. In particular the noncooperative equilibrium is explained and the use and limitations of its applications are discussed. The meaning of cooperative game theory is explained and its uses and limitations are noted. The reasons for the gross inadequacy of a dynamic theory of games are noted and the need for a considerably different approach is suggested1The key to understanding the uses and limitations of game theory is to appreciate that the game theoretic approaches are not monolithic and depending on which one the individual follows the world view obtained is highly different. In particular the four most important distinctions are:
1
Those who wish to delve further into details will find the three volume handbook editied by Aumann and Hart (1992, 1994 and 1997 forthcoming) to be of considerable assistance.
2 1. 2. 3. 4.
Game theory as mathematics; Game theory as science; Game theory as philosophy; Game theory as advocacy.
Some brief observations are made here on these distinctions. They will be expanded further in a subsequent note. After these are discussed, some basic exposition is given. 1. Game theory as mathematics Prior to the theory of games the prime emphasis in mathematical techniques applied to economics was to differential calculus, differential and difference equations. Combinatorics was hardly considered. The stress in von Neumann and Morgenstern was on choices among finite sets2. Not only did they stress finite combinatorics in individual choice. They then proceeded to develop a combinatoric theory of cooperative social choice where the stress was on the powers of all coalitions which could be formed among the players. In their development of the theory of the two-person zero sum game the stress was on finite choices and how to calculate optimal strategies. The method of calculation called for was, in essence, linear programming. There are many purely mathematical problems which are posed in game theoretic investigations and it is possible to regard oneself as a game theorist with no empirical concerns whatsoever, limiting oneself to the new mathematical problems which are posed by the game theory models. 2. Game theory as science It was the original intention of von Neumann and Morgenstern to apply game theory to problems in economics. However the range of scientific applications has been far broader, including all of the social sciences as well as biology. It has also served to provide the underlying structure for experimental gaming, providing the models which help to show the limits of individual rational behavior. The scientific applications can be split into pure and applied. Examples of pure science based on game theory would be the gaming investigations of O'Neill (1987), Roth (1987) and others (see Shubik, 1987 for a survey) and the current work in evolutionary game theory and biology. ( See Weibull, 1996 for a survey.) Game theory as applied science and operation research has been used in voting design (see Shapley and Shubik 1954 and Banzhaf 1965); in designing auction mechanisms , in setting aircraft landing fees in deciding on the assignment of overhead costs (Shubik, 1962) and utilized by the military in weapons design.
2
These could be huge sets, such as the set of strategies in chess. The contemplation of the size of such sets has raised problems concerning how individuals explore large numbers of alternatives.
3 3. Game theory as philosophy Von Neumann and Morgenstern placed extremely stringent conditions on their description of a game. It was assumed that all players were completely informed about all of the rules, including each other's preferences and assets. Deep philosophical problems are posed if the assumption of common knowledge is abandoned. One has to deal with assumptions about A’s knowledge of what B knows and B’s knowledge of what A knows about B and so forth , in an infinite regression. There is a virutual subindustry in Game Theory dealing with how to treat lack of common knowledge (see Aumann, 1976) . Von Neumann and Morgenstern, in their investigation of non-zero sum games showed that the ordinary concept of individual rational behavior did not generalize in a unique natural simple way as soon as there is more than one than one individual in a nonzero sum game. Harsanyi and Selten (1988) have proposed the existence of a unique noncooperative equilibrium as the normative solution to a game; others have argued that the various normative features one might require for a solution (such as symmetry, efficiency and equity) may be in conflict. The meaning of “rational behavior” in an n-person game, even with common knowledge is still open to debate. Furthermore both a normative and descriptive viewpoint to game theoretic solutions are supported with differing emphasis by the professional game theorists. It can be argued that one “should” play in a twoperson zero sum game according to the Minimax strategy and a strong case can be made that the concept of the individual rational decision-maker extends to zero sum games. There is no consensus as to what constitutes the “right” solution for nonconstant sum games. 4. Game theory as advocacy With the growth of the acceptance of game theoretic concepts and ways of formulating the analysis of public problems such as national defense or regulation of industrial organization the use of game theory as metaphor has grown. Possibly the earliest examples are given in the works of Schelling (1960), Ellsberg (1956) and Rapoport (1960). Game theoretic thought mixed in with psychology, social psychology and international relations flavored the works of Wohlstetter (1975) and Herman Kahn (1960). I term this type of use as “conversational game theory”. This is not meant to be a pejorative term. The use of conversational game theoretic argument as portrayed by the writings noted above probably had considerable influence on nuclear deterrence policy and cold war strategy. Von Neumann’s stress was on cooperative game theory and he was adamantly opposed to “one point solutions,” i.e., to using game theory for specific prediction. Thus it is unlikely that his strong views on defense were based on his game theoretic thought.
4 Von Neumann & Morgenstern's great book The seminal book in game theory was published in 1944. It is large, innovative and impressive. But it is such that very few individuals appear to have read it in toto. A reason why many individuals have read some parts of Game Theory and Economic Behavior, but few have read it all the way through is because it may be regarded as four books in one. It deals with four topics all of which can be treated independently. They are: 1. 2. 3. 4.
Preferences and utility theory. The extensive form of a game. The description of, and solution to a 2-person zero sum game The description of, and stable set solution to an n person non-zero sum game played cooperatively.
Preferences and utility theory In their need to define precisely the goal of each player, von Neumann and Morgenstern built upon the usual (highly approximate) assumptions of microeconomic theory that individuals have completely ordered preferences. By providing an axiomatic treatment of preferences including axioms on risk behavior they were able to show that preferences could be represented by a utility function completely specified up to a linear transformation. This result, which von Neumann and Morgenstern needed in order to set up and study the two person zero sum game was the founding seminal contribution to the abstract treatment of the study of many different types of preference structure. Their work led to the investigation of problems concerning individual ability to develop a complete ordering of preferences for items among which they have to choose, and individual ability to gauge risk preferences. It also raised questions concerning the possibility for interpersonal comparison of welfare; and how closely efficient transfer of utility can be achieved by the availability of a money like commodity to be used as a means for making side payments. The formal axiomatic approach of von Neumann and Morgenstern to the measurement of utility led to the investigation of many other preference structures, such as partially ordered preferences. The extensive form of a game. Until the development of the theory of games a language and notation to describe a formal structure for conscious decision making had hardly been developed. The extensive form of game delineated by a game tree (in a one person version it is referred to as a decision tree) provided a means to make precise the concepts of move, choice, information strategy, play, outcome and payoff . The authors’ choice of the title “theory of games” was based on utilizing their insights into the formal structure of games such as chess, Go, or Bridge. Figure 1 shows a game tree for a
5 two person game where Player (or decision maker) 1 chooses his move first among three possible moves. Player 2 is informed if Player 1 selects his first move, but is not fully informed if the choice of Player 1 is move 2 or 3. Player 2 then makes his choice. He has to choose between two moves. The game reaches a terminal node of the tree . At each terminal node there is an outcome. Each player assigns a value or utility to the outcome. For example The game tree Figure 1 in chess an outcome at a terminal point in the tree might be that Black checkmates White. Black evaluates this outcome at a value of 1 and White evaluates it at 0. In Figure 1 the nodes of the tree (except the terminal nodes) represent choice points. They are points at which some player must select a move. The branches emanating from each choice point are the moves. They are numbered so that they can be identified. The choice points are circled by a closed curve which may contain one or more points. These curves are information sets . Each information set is labeled with the name of the player to whom it belongs. In Figure 1 Player 2 has two information sets labeled by P2. The left information set contains a single point. This indicates that if the game proceeds to that node, when Player 2 is called upon to move he knows exactly where he is. This contrasts with the information set on the right. If Player 1 had selected either his second or third moves, when Player 2 is called upon to move he does not know which move Player 1 has selected. If Player 2 selects his first move because of his lack of information he does not know which one of the two possible outcomes will emerge. An information set could be more descriptively called a lack of information set as it describes the level of ignorance of the player. The initial node of the game tree bears the additional label of O which indicates where the game starts. The terminal nodes in Figure 1 are labeled with the outcomes win, draw or lose. Underneath the label at each terminal two numbers appear. The first is the value attached to the outcome by Player 1 and the second, the value attached to the outcome by Player 2. Given this language we can now describe the basic features of any game. A path down the game tree starting from the initial node and ending at some terminal node describes a play of the game. It is a more general form of the type of description of a chess game reported in the newspapers. It indicates where the game began and indicates the selection of moves by each of the players down to the final outcome. The numbers at the terminal nodes indicate the payoffs or the worth of the final outcome to each player. In theory, with this notation we could draw a full game tree for a game such as chess. But a little contemplation of that game is sufficient to indicate that we would exhaust the world’s paper supply if we tried to draw this game tree on the same scale as Figure 1.
6 The notation developed for the description of the game theory enables us to give a precise comprehensive definition of the meaning of strategy. A strategy is an all encompassing plan which contains instructions concerning what a player should do under all contingencies. It can be easily understood and illustrated by considering the game shown in Figure 1 and displaying a few strategies. Player 1 has three strategies which coincide with his moves. This coincidence occurs because he has no contingencies for which he has to plan. Suppose Player 1 were going out of town before he was due to play and that he wanted to leave a complete book of instructions for the individual who was going to play for him. He could leave any one of three extremely simple books of instructions . The first book of instructions would simply say: “Select move 1.” This is somewhat unimpressive, but matters change when we suppose that it is Player 2 who is out of town and he is leaving instructions to someone who is going to play for him. One such book of instructions is of the following form: “If Player 1 selects move 1 select 1; if he selects moves 2 or 3 then select 2.” We could abbreviate this description to (1,2). A little work shows that there are four different books of instruction which could have been left by Player 2. They are (1,2), (2,2), (1,1) or (2,1). Thus in this game Player 1 has three strategies and Player 2 has 4. Rather than analyze the game in extensive form we can obliterate the details of description in terms of the game tree complete with details concerning information and replace it with a matrix which illustrates the interaction between all of the strategies of one player and all of the strategies of the other. P2 (1,2)
(2,2)
(1,1)
(2,1)
1
1,0
1/2,1/2
1,0
1/2,1/2
2
1,0
0,1
0,1
1,0
3
0,1
1,0
1,0
0,1
Table 1 A simple example shows how the entries in the cells are calculated. Suppose that Player 1 selects his first strategy and player 2 selects his strategy denoted by (2,2). This tells us that Player 2’s reply is to select move 2 for the payoffs of (1/2, 1/2).
7 The description of, and solution to a 2-person zero sum game A zero sum game is a game in which the interests of the players are diametrically opposed. What one player wins, the other loses3. Chess, Go and Checkers provide examples of two person zero sum games. A five person Poker game is a five person zero sum game. Von Neumann and Morgenstern devised a special theory to analyze two person zero sum games. In order to do so they made use of the strategic form of a game They confined their interest to a game description only in terms of strategies and payoffs. The detail provided by the extensive form is suppressed. Thus there are many game trees or games in extensive form which give rise to the same game in extensive form. A great loss of detailed information has taken place in going to the strategic form. The sort of structure on which one might wish to base an analysis of dynamics has been replaced by a structure better suited to study equilibrium and statics. A simple example serves to illustrate the relationship between the extensive form and the strategic form. Figure 2a and b show a game in which both Player 1 and 2 each must select between two moves in complete ignorance of the other’s actions. The only difference between the two game trees are that in Figure 2a Player 1 moves first; in 2b Player 2 moves first. As it is assumed that neither player has the opportunity to peek, the games are strategically equivalent. For each player in this simple game a strategy and a move are equivalent. Each must select move one or two. Figures 2a,b The strategic form representation of the game is given in Table 2. Player 1 selects a row and Player 2 selects a column. The first number in each cell represents the payoff to Player 1 and the second number, the payoff to Player 2. Thus, for example. if Player 1 uses his second strategy and Player 2 his first, Player 1 loses 10 and Player 2 gains 10. If Player 1 selects his first P2 1
2
1
5,-5
0,0
2
-10,10
5,-5
Table 2 More generally rather than zero sum we could talk of constant sum games where an arbitrary constant (say a gift from Uncle Harry) is added to each of the payoffs to a player. Third part of his payoff is independent of the strategies employed (See Shubik, 1982 Ch 8) 3
8 move with a probability of 3/4 and his second move with a probability of 1/4 he can guarantee that no matter what Player 2 does his expected gain will be 5/4. This illustrates the Maxmin solution for a two person zero sum game. The Row player can guarantee that his expectation is no less than a certain amount and the column player can prevent the row player from obtaining more than this amount by selecting his first move with a probability of 1/4 and his second move with probability of 3/4. Here we note that the players do not select a single row or column with certainty, but select both rows or columns with a calculated probability for each row or column. Such a strategy is referred to as a mixed strategy. Player 1 is guaranteed his maxmin. The Maxmin solution suggested by von Neumann for the two person zero sum game can be regarded as a normative solution. If the player wishes to maximize his expected gain he should use his maxmin strategy. Except for parlor games and some military problems there are few actual situations which model as pure zero sum games. Experimental evidence on actual performance of players playing a 2 x 2 or a 3 x 3 matrix game is mixed. When the matrix is bigger than 3 x 3 players display information processing and computational limitations. Solutions to an n person non-zero sum game played cooperatively. Von Neumann and Morgenstern developed three different representations of a game of strategy. They are: 1. The extensive form; 2. The strategic form and 3. The cooperative or coalitional form The extensive form can be utilized to provide a complete process description of the game. We discuss the extensive form below after considering the noncooperative equilibrium solution. The strategic form, as illustrated above serves to enable us to analyze the game in terms of strategies. The characteristic function or coalitional form provides a basis from which the combinatorics of potential collaboration can be studied. Table 3 shows a simple 2 x 2 nonconstant sum game in matrix form4. If each player uses his first strategy each obtains a payoff of 5. If Player 1 uses his first strategy and Player 2 uses his second strategy, Player 1 obtains -10 and Player 2 obtains 15. If both use their second strategy each obtains 0. It is easy to observe that jointly, by cooperating, each using their first strategy, the players could obtain 10 together.
4
More accurately, in bimatrix form as there are two entries in each cell.
9 P2 1
2
1
5, 5
-10, 15
2
15, -10
0, 0
Table 3 Each player acting independently cannot guarantee for himself more than 0. The characteristic function of a game with n players illustrates the amounts that any of the 2n coalitions can obtain if the members of the coalition cooperate5. For the simple game illustrated in strategic form in Table 3 the characteristic function is as is shown in Table 4.
v({1}) = 0, v({2}) = 0 v({1,2}) = 10 Table 4 The reason why we use a double set of brackets around the numbers is to indicate that for example v({1}) stands for the value of the coalition consisting of the set containing only Player 1. The curly brackets {} indicate a set. The coalition consisting either of Player 1 or 2 cannot guarantee more than zero. The coalition of both 1 and 2 together can obtain 10. If one is presented with just the characteristic function of the game, the details concerning the strategic structure for individual are suppressed. A way of looking at the information conveyed by the characteristic function is that it at the level of aggregation designed for a diplomat or negotiator. It tells him nothing about the strategic choices facing the generals only the values of coalitions. The extensive form is better designed for use of individuals concerned with the details of information and moves. Bits of it must be considered by those concerned with tactics and the specifics of play rather than with overall strategy. Figure 5 illustrates a three person game in characteristic function form. It tells us that Player 3 is more desirable than 1 or 2 in any two person coalition and that Player 2 is more desirable than 1. Here we have not bothered to illustrate either the extensive or strategic forms which would give rise to this characteristic function. There are many different games which would produce this characteristic function; but if we are only interested in the sort of negotiations which For formal mathematical rigor one usually attaches a value of zero to the coalition consisting of nobody. 5
10 would have the three players decide on how to split the four units they can earn by cooperation we do not need the other detail. v({1}) = v({2}) = v({3}) = 0 v({1,2}) = 1, v({1,3}) = 2, v({2,3}) = 3 v({1,2,3}) = 4. Figure 5 Von Neumann and Morgenstern defined an imputation to be a vector of n numbers whose sum adds up to the amount that the group as a whole could obtain by cooperation. Referring to Table 5 the vector (.9, 1.2, 1.9) is an imputation. It says that Player 1 receives .9; Player 2 receives 1.2 and Player 3 receives 1.9. These three numbers add to 4 which is the amount that the coalition of all could obtain. The key operation they considered was that of domination. An imputation is said to dominate and imputation if some set of players which we call S are better off in than in and by going into business for themselves they could obtain at least as much as. Rather than leave us with an abstract exercise using Greek letters a simple example utilizing the game illustrated in Figure 5 can make this concept clear. Suppose that the imputation were (1, 2, 1) and were (.8, 2.3, .9) then dominates via the set S={1,3} because Players 1 and 3 acting together without Player 2 can obtain 2 thus they have the power to select over. The theory of n-person games developed by von Neumann and Morgenstern was both static and cooperative. The authors were at pains to stress that they regarded the development of a useful dynamics as beyond their scope. They indicated however that much of value could be constructed by concentrating on the statics. Their approach towards the many person problem was in part normative and in part positive in the sense that they felt that the society as a whole should try to make the cake as big as possible, but that groups would bargain and negotiate over their shares of the cake. There are four solution concepts of note which have been developed and applied to cooperative games represented by a characteristic function. They are the stable set solution proposed by von Neumann and Morgenstern and the solutions known as the core, the value and the nucleolus of an n-person game. Without going into mathematical detail it is not possible to do full justice to the specific structure of each of these solutions. However given the concepts of imputation and domination, some intuitive insights can be provided concerning the basic concepts underlying each of the solutions.
11 The stable set solution The stable set solution reflects the possibility for the distribution of resources to depend on a variety of complex social structures. The stress is on extreme nonuniqueness of outcomes. In essence, a stable set solution cannot be used to make a unique prediction, all that it does is to delineate a subset of the outcomes. Many imputations of wealth may make up a single solution. When all of the imputations in a solution are considered together it is the relationship between all of them that imposes a social stability on the whole.6 There can be many different stable sets rewarding different groups considerably different gains which are all nevertheless consistent with the same resource structure The core. The core of an n-person is the set of imputations which are undominated, i.e the set of ways of dividing the overall pie in a manner such that no group could do better if it tried to go into business for itself. Referring to the game illustrated in Table 5 the imputation (.5, 1, 2.5) is in the core as there is no coalition whose members can all do better by themselves. The core can be viewed as the set of outcomes impervious to countervailing power. It reflects a stability against all coalitions. It can be proved that the competitive price system of an economy lies in the core ( for discussion and references see Shubik, 1984). The value The value solution to an n-person game is a solution devised by Shapley (1953) which evaluates an a priori worth of each individual to the overall game. In essence, it averages over the marginal contribution that each individual makes to all the coalitions into which he could enter. A special example of a four person game may help to illustrate the nature of the value solution. We may consider a special type of game which is represented by an extremely simple chartacteristic function. It is called a simple game because all of the payoffs obtained by the coalitions are either 0 or 1. A coalition with a joint payoff of 0 is called a losing coalition. A coalition with joint payoff of 1 is called a losing coalition. The simple game can be used to study the intrinsic assignment of power in a voting structure as a function of the number of votes controlled by each individual. Suppose that there is a corporation with 4 stockholders. The first stockholder has 2 votes and the other three stockholders have 1 vote each. There is a way in which we can assign an a priori value or measure of power to each stockholder. Suppose that instead of there being four stockholders with five shares split among them, there had been five stockholders, each with one share, a reasonable assignment of power among the five stockholders would be .2 to each of them. The technical statement is that a solution consists of a set of imputations which do not dominate each other but dominate all other imputations. 6
12 Each is equally relevant in being able to turn a losing coalition into a winning coalition. But suppose that there were only three voters with the stock distributed as (3,1,1). If the vote is settled by a simple majority of the stock held then it evident that the stockholder with 3 shares holds all of the power and those with 1 share each hold none of the power7. This elementary exercise tells us that as an individual's holding of shares increases his power must change in a nonlinear manner. Once he has more than half of the shares he has all of the power. The value calculation counts all of the marginal contributions that an individual makes to every coalition and averages over them. When the game is large and has little special structure this calculation is long and messy. In games representing voting situations where there are only winning and losing coalitions8 the marginal contribution of a player is either 0 or 1. It is 0 when he joins a losing coalition and it remains losing, or when he joins a winning coalition and it remains winning.9 It is 1 when the addition of the player changes a losing coalition into a winning coalition. We may imagine coalition formation taking place in every possible order. Call the players A, B, C and D where A has 2 votes and the others 1 each. There are 24 ways the individuals could line up. These are shown below. Starting with the player at the left we can consider a coalition growing as each player is added. Credit is given to the pivotal player who coverts a losing coalition into a winning coalition. This is illustrated by a * next to the pivotal player. AB*CD, AB*DC, AC*BD, AC*DB, AD*BC, AD* CB BA*CD, BA*DC, CA*BD, CA*DB, DA*BC, DA*CB BCA*D, CBA*D, DCA*B, CDA*B, BDA*C, DBA*C BCD*A, BDC*A, CBD*A, CDB*,A DBC*A, DCB*A. We note that in the 24 possibilities A is pivotal 12 times and B, C and D are pivotal 4 times each. The a priori division of power among the voters is (1/2, 1/6, 1/6, 1/6). The phrase a priori is used to indicate that this measure is free of any of the sociological structure that might be present in the set of players. All way in which coalitions might be formed are regarded as equiprobable. In an actual society there may be nets of friends or others whose interests are correlated who act together.
In actuality there is a body of law which protects minority stockholders against the complete power of the majority, but for this example we assume that "winner can take all". 8 Ties must also be accounted for, but here for ease in illustration we consider a voting game with an odd number of votes where all vote and hence there are no ties. 9 In actual political practice although a minimal winning coalition is desirable as there are fewer participants to share the payoff, a few extra players may be of worth in order to insure against accidental failure to be present at the vote. 7
13 The nucleolus The nucleolus is a solution concept devised by Schmeidler (1969) which seeks to find an imputation or a means of dividing the cake in such a manner that the maximum complaint by any coalition against the division is minimized. It may be regarded as a sophisticated version of a splitthe-difference approach to fair division. If the game being studied has a core then the nucleolus is the centroid of the core. The Cournot-Nash Noncooperative Equilibrium In contrast with the cooperative, normative game solution approach noted above Augustin Cournot (1838) suggested a solution to two firms in competition in a market which involved each taking the others expected strategic behavior into account. He showed that there would be a noncooperative equilibrium point which had the property that there would exist a pair of expectations such that if each had the appropriate expectation and based his action on this expectation then the resultant out come would be self-fulfilling, i.e. the expectation of each would turn out to be true. John Nash [20] provided the general definition and proof of the existence of the noncooperative equilibrium. Referring to the simple matrix game shown in Table 3 (known as the Prisoner’s Dilemma game) we can see that if Player 2 expected Player 1 to use his second strategy then Player 1's optimal response would be to use his second strategy and viceversa. Thus the pair of strategies (2,2) would be in equilibrium and would result in a pair of payoffs of (0,0). In contrast if both had selected their first strategy and acted cooperatively they would have obtained payoffs of (5,5). The noncooperative equilibrium solution has also been applied to games in extensive form. Unfortunately when such an attempt is made a host of new problems occur which can be more or less ascribed to the unsatisfactory relationship between statics and dynamics. The equilibrium analysis is essentially static and for it to hold in a game with many stages one approach has been to consider a backward induction where the presumption is that at every point in the game the expectations of the players guide them to an equilibrium. The mathematical formulation and proof of existence of at least one, and often, many noncooperative equilibria tells us little about the dynamics encountered in a game where the players start to play at a position away from equilibrium. In essence, the noncooperative equilibrium solution points out the existence of outcomes which satisfy a condition on the consistency of everyone's expectations. Although there is some experimental evidence that the noncooperative equilibrium serves as a reasonable approximation to long term behavior in some games with many players and a single noncooperative equilibrium point, little is known with any generality about actual behavior. The noncooperative equilibrium solution should not be regarded as a normative solution as, in general, jointly optimal outcomes
14 may not be achieved. Further discussion and a critique of the problems encountered in utilizing the noncooperative equilibrium solution will be given in Part II. Dynamics and the infinite horizon The game theory models presented above have a definite beginning and end. The initial conditions in the instance of a game of chess are given precisely by the placing of the pieces on an 8 x 8 board. The end is well defined by the rules including conventions to prevent endless cycling so that the game is finite. When we try to model most human activities the initial position reflects history and the social context which sets the stage for individual action. Unlike a game of chess, much of human behavior depends in an important way on history from an unbounded past and expectations concerning an unbounded future. When formal game models are constructed to reflect ongoing games, the distinction between cooperative and noncooperative solutions becomes blurred. As there is always the possibility of a tomorrow, the opportunity to punish doublecrossers and to specify threats which can be used to enforce cooperation becomes a reality. Structure and Behavior A brief review of the formalization and different types of representation of a game of strategy has been presented, followed by an exposition of different solution concepts. The stress has been on the development of a language to describe conscious individual decision making in situations involving one or more individuals with well-defined goals and ability to calculate. Von Neumann and Morgenstern provided a seminal idea in devising a means to describe the full structure of a game, such as chess. In human society there are games within games. Although a particular game, in the short run, may take place in a specific context, the actions of the individuals, may in the long run, influence the environment in which they behave. This type of feedback is not represented in the formulations presented here. The provision of a way to make precise "the rules of the game" and to describe in careful detail the set of all possible ways a game might be played does not provide us with a prescription as to how the game should be played, nor with the evidence as to how the game is actually played by humans. Game theorists have approached the problems involving human motivation and behavior by suggesting a mathematical description of individual preferences, attrbiuting individual optimization as the motivation and devising normative or other solution concepts to account for the interactions of many individuals all in pursuit of their own welfare. Each of the five solution concepts suggested has attractive features. The four cooperative solutions are explicit in indicating the different norms they fulfill and useful applications have been found for the value, core and nucleolus. The noncooperative equilibrium solution is justified more for descriptive reasons and fits into the spirit of recent finance and price system economics where the idea of the
15 noncooperative equilibrium is translated into rational expectations. It has also been applied to biology with a somewhat different interpretation of the concept of a mixed strategy equilibrium. In the subsequent discussion, it is suggested that the influence of game theory has been substantial and the applications at several levels have been of great value; but its very success has served to highlight the limitations of the rationalistic models of economic man in coming to grips with the many socio-economic and politico-economic problems which are faced by any large society. The game theoretic language has provided a means to build process models in which the institutions and other aspects of the environment are the carriers of process. These are implicit in the rules of the game9. The game theoretic solutions are of considerable power in studying some problems of interest, but as models of human behavior, they are lacking and the basic apparatus of game theory drives us to abandon the model of the decision maker as completely informed about the rules of the game with common knowledge about what all others know or do not know. The precision and analytical power of game theory forces us to understand the model of utilitarian man and its direct descendent homo oeconomicus instead of being an idealization of purposeful human decision makers are poor approximations to the far more complicated perception and computationaly-limited human being driven by passions, in part ruled by habit and in part controlled by instinct, whose decisions are set in the context of the society he or she belongs to. The mathematical analysis underlying much of game theory and economic theory was made feasible by the gross simplification of the model of the individual. This may have been fine for some of the original purposes at hand. With the advent of the computer and the possibilities of computation and simulation, the stage is set to enrich the study of models of human behavior using the framework provided by game theory as a starting point from which we may investigate multiperson interaction.
16 REFERENCES Aumann, R.J. 1976. Agreeing to disagree. Annals of Statistics. 4: 1236-39. Aumann, R. J. and S. Hart (eds) 1992, 1994, Handbook of Game Theory, vols I and II and III forthcoming 1997. Banzhaf, J.F., 1965. Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review 19: 317-343. Ellsberg, D. 1956. Theory of the Reluctant Duelist. American Economic Review 46: 909-923. Harsanyi, J. and R. Selten, 1988. A General Theory of Equilibrium Selection in Games Cambridge: M.I.T. Press. Kahn, H. 1960. On Thermonuclear War Princeton: Princeton University Press. Luce, D. and H. Raiffa. 1959 Games and Decisions Wiley: New York. Nash, J.F., Jr., 1951. "Noncooperative Games." Annals of Mathematics 54: 286-295. Nash, J.F. Jr., 1950. "Equilibrium Points In N-Person Games." Proceedings of the National Academy of Sciences, USA 36: 48-49. O'Neill, B., 1987 “Nonmetric tesat of the minimax thewory of two-person zerosum games”, Proceedings of the National Academy of Sciences, U.S.A., 84: 2106-2109. Rapoport, A. 1960, Fights, Games and Debates Ann Arbor: University of Michigan Press. Roth, A.E., 1987.. Laboratory Experimentation in Economics. Cambridge: Cambridge University Press. Schelling, T., 1960 The Strategy of Conflict. Bambridge Mass. Harvard University Press. Schmeidler, D., 1969. "The Nucleolus Of A Characteristic Function Game." SIAM Journal on Applied Mathematics 17 (6): 1163-1170. Shapley, L.S., 1953. "A Value For N-Person Games". In Annals of Mathematical Studies, R.D. Luce and A.W. Tucker, eds. Princeton, NJ: Princeton University Press, pp. 307-317. Shapley, L.S., 1975, "Cardinal Utility from Intensity Comparisons", RAND Corporation, Publication R-1683-PR. Shapley, L.S. & M. Shubik., 1954 A Method for Evaluating the Distribution of Power in a Committee System. American Political Science Review 48: 787-792. Shapley, L.S. and Shubik, M., 1969a. "On the Core of an Economic System with Externalities," American Economic Review, 10: 337-362. Shapley, L.S. and Shubik, M., 1969b. "Pure Competition, Coalitional Power and Fair Division," International Economic Review, 10: 337-362.
17 Shubik, M. 1962."Incentives, Decentralized Control, The Assignment Of Joint Costs And Internal Pricing," Management Science, 8, 2: 325-343. Shubik, M.., "Edgeworth Market Games," In Annals of Mathematical Studies, Volume 40, R.D. Luce and A.W. Tucker, eds., Princeton University Press, Princeton NJ, pp. 267-278. Shubik, M., 1959b. Strategy and Market Structure, New York: Wiley. Shubik, M., 1962, "Some Experimental Non Zero Games with Lack of Information about the Rules," Management Science, 8: 215-234. Shubik, M., 1982. Game Theory in the Social Sciences, Vol. 1, MIT Press, Cambridge, MA. Shubik, M., 1984. Game Theory in the Social Sciences, Vol. 2: A Game Theoretic Approach to Political Economy. MIT Press, Cambridge, MA. Shubik, M. "Game Theory and Experimental Gaming," To appear in Handbook of Game Theory, vol III R. J. Aumann and S. Hart (ed) forthcoming 1997. Von Neumann, J. and O. Morgenstern, 1944. Theory of Games and Economic Behavior, Princeton University Press, Princeton NJ. Weibull, J. W. 1996. Evolutionary Game Theory Cambridge, Mass M.I.T Press. Wohlstetter, A. 1975 Optimal Ways To Confuse Ourselves. Foreign Policy 20: 170-198.
