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Variational Convergence: Approximation and Existence of Equilibria in Discontinuous Games

Adib Bagh Departments of Economics and Mathematics University of Kentucky

abstract. We introduce a notion of variational convergence for sequences of games and we show that the Nash equilibrium map is upper semi-continuous with respect to variationally converging sequences. We then show that for a game G with discontinuous payoff, some of the most important existence results of Dasgupta and Maskin, Simon, and Reny are based on constructing approximating sequences of games that variationally converge to G. In fact, this notion of convergence will help simplify these results and make their proofs more transparent. Finally, we use our notion of convergence to establish the existence of a Nash equilibrium for Bertrand-Edgeworth games with very general forms of tie-breaking and residual demand rules.

Keywords: convergence of games, discontinuous games, equilibrium map, Bertrand-Edgeworth games. JEL Classification: C72, C62, D43

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1. Introduction Suppose P and X are respectively the parameters set and the strategies set of a game G. The equilibrium map is a set-valued map EQ from P to X that takes a particular vector of parameters p to the set of Nash equilibria of the game corresponding to these parameters. The continuity properties of the equilibrium map are important for performing comparative statics and stability analysis for the game. In most cases, however, the equilibrium map is not continuous, and the analysis must focus on establishing the upper semi-continuity of this map. The uppersemi continuity of the equilibrium map, moreover, can be used to establish the existence of an equilibrium for this game. It is possible that EQ is empty-valued for some point p0 in P . However, the upper semi-continuity of EQ implies that if EQ(p) is non-empty for points that are arbitrarily close to p0 ∈ P , then EQ(p0 ) is also non-empty. Therefore, when the equilibrium map is upper semi-continuous, showing that a particular game G has an equilibrium can be achieved by showing that an equilibrium exists for games that are “arbitrarily close” to G. For games with discontinuous payoffs, there is little hope in showing the full upper semicontinuity of the equilibrium map. Therefore, we follow an approach that was suggested by Simon [24], and we focus on showing the upper semi-continuity of the equilibrium map with respect to a particular type of sequences. More specifically, we introduce variational convergence, a notion of convergence that can be used with games with possibly discontinuous payoffs. We then show that the Nash equilibrium map is upper semi-continuous with respect to variationally converging sequences. This means that, given a sequence Gn of games converging variationally to some game G, any cluster (an accumulation) point of equilibria of Gn is also an equilibrium of G. We further show that variationally converging sequences arise naturally and under very weak conditions. In particular, such sequences arise when the strategy set of a game is approximated with simpler set of strategies (finite sets, sets with smoother boundaries) as well as when badly behaved payoff functions (discontinuous, non-smooth) are approximated with nicer (continuous, smooth, piece-wise linear) payoff functions. In fact, variational convergence allows for the simultaneous approximation of the payoffs and the strategy sets. This makes this notion of convergence a valuable tool for those interested in developing algorithms for computing equilibria. Furthermore, we use variational convergence to provide a unifying approach to various existence

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results regarding games with discontinuous payoffs. More specifically, we show that for a game G with discontinuous payoffs, some of the classic existence results of Dasgupta and Maskin [6], Simon [24], and Reny [22] are based on constructing approximating sequences of games that converge variationally to G. The unifying approach simplifies the proofs of these earlier results, makes the underlying principal of these proofs more transparent, and in one particular case, actually helps us correct the proof. There are many, and there will be many more, existence results that cannot be viewed in terms of variational convergence. However, the connection with the classis existence results that we discuss in this paper attests to the ubiquity of variationally convergent sequences and to the relevance of this notion of convergence. In some sense, any context where these classic results can be applied is a context where variational convergence is relevant. Finally, we demonstrate that variational convergence can be used to obtain new existence results. We establish the existence of equlibria in a large class of games that includes asymmetric BertrandEdgeworth price competitions with non-constant marginal costs. This class of discontinuous games was investigated by Maskin [19], Dixon [9], and Yoshida [31] among others. Most of the current existence results for these games are based on Theorem 5 in Dasgupta and Maskin [6], henceforth DM [6], where the upper semi-continuity of the sum of the payoffs is required.1 In order to meet this requirement, the cost functions are assumed to be symmetric (identical for all firms) and convex.2 Moreover, proving the upper semi-continuity of the total cost requires different arguments for different rationing and tie-breaking rules (see for example, Dixon [9] and Maskin [19]). In second half of this paper, we use variational convergence to establish the existence of equilibria in games where the sum of the payoffs is not upper semi-continuous. We then apply this result to Bertrand-Edgeworth games (henceforth BE games) between two firms when the cost functions are not identical. Our existence result is robust with respect to specifications of tie-braking and residual demand rules, and can be used to find an equilibrium as the limit of a sequence of equilibria of finite games that can be easily computed. 1 Theorem 5b in DM [6] can also be used to establish the existence of an equilibrium without requiring the upper semi-continuity of the sum of payoffs. Unfortunately, the proof of this theorem seems to be incorrect. See Appendix A for details. 2 Existence results for Bertrand-Edgworth games with symmetric and constant marginal costs were also obtained by Kreps and Sheinkman [16], Osborn and Pitchik [21]. There are some existence results for the asymmetric case but only in the case of constant marginal costs. See Deneckere and Kovenock [8] for more details.

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Many results regarding the continuity of equilibrium maps, as well as various notions of convergence of sequences of games, already exist in the literature. For games with a continuum of players, Hildenbrand and Mertens [13], and Green [14] investigated the upper semi-continuity properties of the equilibrium map in pure exchange economies. Carmona [5] considered a similar question in a more general class of games. For games with finite number of players, the upper-semi continuity of the equilibrium map was studied by Walker [29], Lucchetti and Patrone [18], Jofre and Wets [15], and others. Fundenberg and Levine [10] developed a notion of convergence of games that imply continuity properties for the equilibrium map that are stronger than upper semi-continuity. However, all these results are based on notions of convergence that are not suitable (too strong to be verified) for sequences of games with discontinuous payoffs. This makes them inappropriate for the applications we have in mind in this paper.3 The outline of the paper is as follows. In Section 2, we review the basic definitions and some of the earlier theorems that we will need for our results. In section 3, we introduce the notion of variational convergence and show that it can be used to reproduce (and simplify) earlier existence results for games with discontinuous payoffs. In section 4, we use variational convergence to establish new existence results. Finally, in Section 5 we provide applications to the results of section 4 to BE games. 2. Preliminaries We consider a game G with I players. Each player i = 1, · · · , I has a bounded payoff function Ui : S −→ IR, where S = Πi∈N Si and Si ⊂ Xi is a compact subset of a topological vector space Xi . We denote such game by (Si , Ui , I), and we consider two cases. The first case is when the set S = X is a set of pure strategies and X is a subset of some metric space. The second case is when S is a set of mixed strategies S = M(X) where M(X) =

Q

i Mi (Xi )

and Mi (Xi ) is the

set of regular probability measure defined on Xi that is a compact subset of some separable metric space. In the case of mixed strategies, Ui =

R

ui dµ represents the expected payoff given some von

Neuman-Morgenstern function ui and some probability measure µ ∈ M(X). We use the standard 3 Fundenberg and Levine [10] did introduce a topology that can be used to study the continuity properties of the equilibria of games with discontinuous payoffs. However, as we point out at the end of section 3 of this paper, this approach is not suitable for obtaining existence results.

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notation S−i = ΠN k6=i Sk and s−i = (s1 , · · · , si−1 , si+1 , · · · , sN ) ∈ S−i . The graph Γ of the game G is the set Γ = {(s, α) ∈ S × IRI |Ui (s) = αi , ∀i}. ¯ The (possibly empty) set EQ(G), denotes the set of The closure of Γ in S × IRn is denoted by Γ. all Nash equilibria of the game G. We also define EQi (G) as the projection of EQ(G) on Si (the equilibrium strategies of player i). For a sequence S n of subsets X , we define the following notions of limits: ls S n = {s ∈ X |∃ a subsequence snk such that snk ∈ S nk and snk −→ s}. We also define li S n = {s ∈ X |∃ a sequence sn such that sn ∈ S n and sn −→ s}. In other words, the set li S n is the collection of points in X that are limits of sequences in S n whereas ls S n is the collection of all limits points of all subsequences in S n . Clearly, we always have li Sn ⊂ ls Sn . Furthermore, if the sets Sn are eventually contained in the same compact set, ls Sn will be non-empty. The set li Sn , on the other hand, may or may not be empty. In this paper, we will use the following notion of set convergence. Definition 2.1. A sequence of sets S n converges to a set S, and we write S n → S, if li S n = ls S n = S. In general, this notion of set convergence is weaker than convergence with respect to the Hausdorff metric. However, the two notions are equivalent when S is compact. When S n → S, for each n, S n is finite, and S n ⊂ S n+1 , we say that the sequence Sn densely approximates the set S. → Y from points in X to subsets in Y, there are many When considering a set-valued map F : X → ways to define upper semi-continuity.4 In this paper, we will use the following definition, → Y from points in X to subsets in Y is upper semiDefinition 2.2. A set-valued map F : X → continuous, if ∀s ∈ X and all sn −→ s, we have ls F (sn ) ⊂ F (s). 4

Different notions of convergence of subsets in Y can be used to define different notions of upper semi-continuity. The existence of alternative notions such as upper hemi-continuity and outer semi-continuity reflects the extent of possible variations on the same theme.

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Note that if, for every s ∈ X , F (s) is non-empty and compact, then the above definition implies that the graph of F is closed in X × Y. For a given real valued function, we define the graph and the epigraph of this function Definition 2.3. For f : X −→ IR, Let gph f = {(s, α) ∈ X × IR| α = f (s)}, epi f = {(s, α) ∈ X × IR| α ≥ f (s)}, Intuitively speaking the epigraph is the set above the graph of the function. It can be easily shown that a function f : X −→ IR is lsc, if and only if epi f is closed subset in X × IR. This leads to following definition Definition 2.4. Let f : X −→ IR. The lower semi-continuous envelope f : X −→ IR is the function whose epigraph is the closure (in the product topology) of the epigraph of f in X × IR. The lower semi-continuous envelope of a function was used in an economic setting first by Simon [24] and later by Reny [22]. It also has been used extensively as a standard tool in the mathematical literature investigating the stability of optimization problems. We now review the definitions of various types of continuity that we will need later: Definition 2.5. (DM [6]). For a game G = (Xi , Ui , I), where Xi ⊂ IR, the payoff Ui of player i is weakly lower semi-continuous, if for any x ¯−i ∈ X−i , ∃λ ∈ [0, 1] such that for all x ¯ i ∈ Xi ,

λ lim inf Ui (xi , x ¯−i ) + (1 − λ) lim inf Ui (xi , x ¯−i ) ≥ Ui (¯ xi , x ¯i ). − x + x xi → ¯i xi → ¯i Note that for a given x ¯−i , the same λ has to work for all x ¯i ∈ Xi . Therefore, for a given x ¯−i , this notion of lower semi-continuity, despite its name, is not weaker than the standard notion of lower semi-continuity of the function Ui (·, x ¯−i ). Following DM [6], the above definition is stated only for the case pure strategies S = X ⊂ IR. It is, however, possible to extend it to a more general setting. Definition 2.6. (Reny [22]). A game (Si , Ui , I) is payoff secure, if the following holds for every player i: for all s ∈ S, ∀ ε > 0, ∃ˆ si ∈ Si , ∃V (s−i ), a neighborhood of s−i , such that for all s0−i ∈ V (s−i ) ui (ˆ si , s0−i ) ≥ ui (s) − ε. 6

For a given si , let Ui (si , ·) be the lower envelope of Ui (si , ·). Combining the definitions of the lower envelope and payoff security, we immediately obtain the following lemma Lemma 2.7. If Ui is payoff secure, then ∀(si , s−i ) ∈ S, and ∀ε > 0, there exists sˆi such Ui (ˆ si , s−i ) ≥ Ui (si , s−i ) − ε.

The next two definitions generalize the standard requirement that the sum of the payoffs in the game is usc. Definition 2.8. (Reny [22]). A game is reciprocally upper semi-continuous (rusc) if, whenever (s, α) is in the closure of Γ and Ui (s) ≤ αi for all i, then Ui (s) = αi for all i. Definition 2.9. (Bagh and Jofre [3]). A game is weakly reciprocal upper semi-continuous (wrusc), ¯ \ Γ, there is a player i and sˆi ∈ Si such that Ui (ˆ if for any (s, α) ∈ Γ si , s−i ) > αi . ¯ \ Γ, the collection of points in Γ ¯ but in not in Γ, is often called the frontier of Γ. Unlike The set Γ rusc, the above condition allows for the payoff functions of all the players to “jump down” at some point s as long as the payoff of some player “jumps up”, relative to his old payoff at s, somewhere in S. In some sense, the relation between wrusc and rusc is similar to the relation between payoff security and lower semi-continuity. Furthermore, reciprocal upper semi-continuity implies weak reciprocal upper semi-continuity. When S = X in definitions 2.6, 2.8, and 2.9, we simply say that the game is respectively rusc, payoff secure, and wrusc. When S = M(X), we say that the game is rusc, payoff secure, and wrusc in mixed strategies. As Reny (1999) noted, reciprocal upper semicontinuity in mixed strategies implies reciprocal upper semi-continuity. The converse however, is not true. If the sum of payoffs is usc in pure strategies, then the game is rusc in mixed strategies, and hence in pure strategies. On the other hand, payoff security in mixed strategies neither implies nor is implied by payoff security. Finally, any real-valued lsc function can approximated from below by an increasing sequence of continuous functions. The proof can be found in Rockafellar and Wets [23] and Reny [22]. Lemma 2.10. If f : S −→ IR is lsc, then there exists a sequence of continuous functions f n such that (i) ∀ sn −→ s, lim inf f n (sn ) ≥ f (s), 7

(ii) ∀s ∈ S, ∀n, f n (s) ≤ f n+1 (s) < f (s). 3. Variational Convergence of Games We start this section by giving a precise definition of the notion of variational convergence. Consider first a sequence Gn = (Sin , Uin , I) converging in some sense to a game G = (Si , Ui , I). For the equilibrium map to be upper semi-continuous with respect to this sequence, the following must hold: if s˜n is a (sub)sequence of equilibria points of Gn , and if s˜n −→ s˜, then s˜ is an equilibrium of G. This will certainly hold if the following conditions are satisfied: a) ∀i, and ∀si ∈ Si , lim Uin (˜ sn ) ≥ Ui (si , s˜−i ) b) ∀i, lim Uin (˜ sn ) = Ui (˜ s) The second condition is a continuity condition, along equilibria sequences, of the payoff of each player. The first condition requires lim Uin (˜ sn ) to be an upper bound on the best reply function of player i when the move s˜−i is played by the other players. The combination of these two conditions insures that, in some sense, the equilibrium properties of each s˜n is passed along to the limit point s˜. The notion of variational convergence that we will define will simply consist of two “minimal” conditions that will insure that (a) and (b) above hold. These conditions can be viewed as generalizations of the earlier concepts of payoff security and weak reciprocal upper semi-continuity. In addition to maintaining the upper semi-continuity of the equilibrium map, the main features of variational convergence will be that it is weak enough to hold for sequences of games with discontinuous payoffs. Definition 3.1. A sequence of games Gn = (Sin , Uin , I) converges variationally to a game G = v G, if the following two conditions hold: (Si , Ui , I), and we write Gn →

(A) for every player i, ∀ε > 0, ∀si ∈ Si and ∀˜ s ∈ ls EQ(Gn ), for any subsequence s˜n−ik → s˜−i such that s˜n−ik ∈ EQ−i (Gnk ), there exists sˆni k ∈ Sink and k0 , such that for k > k0 , Uink (ˆ sni k , s˜n−ik ) > Ui (si , s˜−i ) − ε. (B) for any (s, α) ∈ ls Γn \ Γ, there is a player i and sˇi ∈ Si such that Ui (ˇ si , s−i ) > αi . 8

When Gn ≡ G, the above conditions imply that set of equilibria points of G is closed. More specifically, for a constant sequence Gn ≡ G, condition A implies that G is payoff secure but only along sequences of equilibria points of G.5 Condition B, again for a constant sequence Gn ≡ G, implies that G is wrusc. It is possible to simplify the notation in the above definition (i.e. avoid the double superscript) by replacing ls EQ(Gn ) and ls Γn \ Γ with li EQ(Gn ) and li Γn \ Γ respectively. However, this change will yield a notion of convergence that implies a weaker notion of upper semi-continuity of the equilibrium map.6 Finally, conditions (A) and (B) are not the weakest conditions that imply the upper semi-continuity of the equilibrium map. They do, however, highlight what is needed in a notion of convergence in order to obtain the upper semi-continuity of the equilibrium map. Variational convergence introduced above should be considered as a prototype of sort. This prototype can be later modified, if needed, to fit a particular setting, and still obtain the desired continuity results for the equilibrium map. The following Theorem is our main result for this section. Theorem 3.2. If a sequence of game Gn variationally converge to a game G, then and the equilibrium map is upper semi-continuous. If sn ∈ EQ(Gn ) and sn −→ s, then s ∈ S and lim Un (sn ) = U (s). Proof. Fix an i and let si ∈ Si . Suppose ls EQ(Gn ) is not empty, and let s˜ ∈ ls EQ(Gn ). Consider a subsequence s˜n−ik → s˜−i such that s˜n−ik ∈ EQ−i (Gnk ). Condition (A) and the definition of EQ−i (Gnk ) imply the existence of a sequence sˆni k ∈ Sink such that lim Ui (˜ snk ) ≥ lim Ui (ˆ sni k , s˜n−ik ) ≥ Ui (si , s˜−i ). k

k

(3.1),

where we are assuming, without loss of generality, that the limits in (3.1) exist. The first inequality in (3.1) holds because s˜nk is an equilibrium for Gnk , and the second inequality holds because of condition (A). Condition (B) rules out the possibility that the inequalities in (3.1) are strict for some i. Thus, combining (3.1) with condition (B), we obtain 5

Condition (A) is closely related to a notion of payoff security that Gatti [11] imposed on the function representing the value of the best reply function for each player. 6 The weaker notion of convergence implies li EQ(Gn ) ⊂ EQ(G) rather than ls EQ(Gn ) ⊂ EQ(G). Moreover, showing that li EQ(Gn ) 6= ∅ might require further assumptions.

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lim Ui (˜ snk ) = Ui (si , s˜−i ). k

(3.2)

Since (3.1) holds for any si , inequalities (3.1) and (3.2), both holding for every i, imply that (˜ si , s˜−i ) is an equilibrium for G and s˜ ∈ EQ(G). Note that if the set ls EQ(Gn ) is empty, then condition (A) vacuously holds. However, in all the games considered in this paper, for every n the set EQ(Gn ) will not be empty, and all the strategy sets of the sequence Gn are contained in the same compact set. Hence, ls EQ(Gn ) is not empty.7 With a non-empty ls EQ(Gn ), Theorem 3.2 can be used to establish the existence of an equilibrium for game G. Note further that assuming only that Xn in Gn = (Xin , Ui , I) densely approximate X in G = (Xi , Ui , I) is not sufficient for obtaining ls EQ(Gn ) ⊂ EQ(G). The following game illustrates this point. Example 3.3: Consider a game on the unit square with the following payoffs:

ui (xi , x−i ) =

 10          

when xi < x−i ,

ϕi (x)

when x1 = x2 = x,

0

when xi > x−i .

where ϕi (x) = 10 when x < 0.5 and ϕi (x) = 0 when x ≥ 0.5. Let zn = (0.5 − n1 , 0.5 − n1 ). Let Gn = (Xin , Ui , 2) be a sequence of games that densely approximate G. Without loss of generality, assume that zn ∈ Xn for every n. Clearly, for every n, zn is a pure strategy equilibrium for Gn . Moreover zn −→ (0.5, 0.5), yet (0.5, 0.5) is not an equilibrium for G. We start with a simple application of variational convergence to pure exchange economies. Example 3.4: Convergence of Walras Equilibira We consider an exchange economy W with m commodities and I players. Player one is an auctioneer and players two through I are consumers. For each of the consumers, we define Xi ⊂ IRm to be a survival set for player i (normally Xi = IRm + ). We assume that Xi is compact, convex, subset of 7

Alternatively, we can make the requirement ls EQ(Gn ) 6= ∅ part of the definition of variational convergence.

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IRm with non-empty interior. Each consumer is initially endowed with a bundle ei ∈ int Xi . Given a price p, the feasible set for consumer i is Bi (p) = {xi ∈ IRm |p · xi ≤ p · ei }. Furthermore, each consumer has a utility function ui : Xi −→ IR that is quasi-concave, usc, and bounded with |ui | < M . The payoff of consumer i is Ui (xi , p) =

  ui (xi )

when xi ∈ Xi ∩ Bi (p),



otherwise.

−M

For the auctioneer, X1 is the simplex ∆ in IRm + , and U1 is defined by U1 (p, x) = p · ΣIi=2 (xi − ei ). We denote the economy described above by W = (Xi , ui , ei , I). Let EQ(W ) be the set of Walras Equilibria for W (the set of pure strategy equilibria for the game W ). Following Jofre and Wets [15], we introduce perturbations to the economy W . Consider a sequence of economies Wn = (Xin , Uin , eni , I) where Xin , Uin , eni , are defined in the same manner Xi , Ui , ei were defined. For all n and i > 1 , |uni | < M and Bin (p) = {xi ∈ IRm |p · xi ≤ eni }, and Uin (xi , p)

=

 n  ui (xi )

when xi ∈ Xin ∩ Bin (p),



otherwise.

−M

For the auctioneer, let U1n = p · ΣIi=2 (xi − eni ). Assume for i > 1, i) Xin is convex, Xin −→ Xi , and en −→ e. ii) for i > 2, uni converges continuously to ui .

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Under the above assumptions, the sequence Wn converges variationally to the game W (see Appendix D). Therefore, we have ls EQ(Wn ) ⊂ EQ(W ).

(3.3)

Suppose that, in addition to assumptions (i) through (iii) above, uni ≡ ui (i.e. we only perturb Xi and ei ) and that ui strictly quasi-concave for i ≥ 2. In this case, we have EQ(Wn ) −→ EQ(W ). This follows from (3.3) and the fact that EQ(Wn ) and EQ(W ) are now all singletons. We remark here that Jofre and Wets [15] assumed that, for i > 2, ui is concave whereas we only assume that ui is quasi-concave. Therefore, the conclusion of our example as expressed by (3.3) strictly generalizes Theorem 15 in [15]. Under the assumptions of this example, the fact that EQ(W ) is not empty can be derived directly from standard results in general equilibrium or by following the technique of Example 3.2 in Reny [22]. Therefore, this example is not a terribly interesting application of Theorem 3.2 when viewed only as an existence result. However, when ui are not differentiable, or when Xi is described as the intersection of non-differentiable functions, this example suggests an approximation procedure that can be useful in computing equilibria. Using standard numerical methods, each non-differentiable ui can be approximated by a sequence of smooth functions uni satisfying condition (ii). Moreover, a set Xi with boundaries defined via non-differentiable functions can be approximated with a sequence Xin of sets with smooth boundaries such that condition (i) holds. Similarly, convex strategy sets can be approximated by sequences of convex polytopes. Once the approximating sequence Wn is constructed, the equilibria of the Wn can be found using the standard combination of first order conditions and fixed point theorems. The continuity of the equilibrium map, as implied by (3.3), guarantees that our procedure (the equilibria of Wn ) will eventually converge to the equilibrium of the original problem. We now show that variational convergence provides a unified framework for most of the existence results in discontinuous games that fall under the “topological approach”.8 8

We use the term “topological approach” when the underlying fixed point in the proof is topological in nature as

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Existence results via finite approximation Consider a game G = (M (Xi ), Ui , I) where Xi is a compact subset of a complete metric space and the function Ui is the standard expected utility of a von Neuman-Morgenstern function ui : X −→ IR. This means that for any µ ∈ M(X), Ui (µ) =

R

ui dµ. The seminal paper of DM [6] established

the existence of an equilibria for a large class of games when Xi ⊂ IR, each ui has a discontinuity set with a zero Lesbegue measure, each payoff function is weakly lower semi-continuous, and the sum of the payoffs is upper semi-continuous. Simon [24] generalized the results of [6] to games defined on a metric space under conditions that are strictly weaker than the conditions in [6]. In particular, Simon assumed the following (i) the game G is rusc in mixed strategies (ii) there is a sequence Xn that densely approximates X such that for each i, the function Hi : M(X−i ) −→ IR defined by Z

Hi (µ−i ) = sup

ui (xi , x−i )dµ−i ,

(3.4)

xi ∈Ri

where Ri =

S

n n Xi

∪ Xi , is lsc in µ−i .

Assuming that G satisfies Simon’s conditions (hence satisfies the conditions of DM [6]), we show that the existence of a possibly mixed strategy equilibrium for such a game can follow directly from Theorem 3.2. Consider the sequence Gn = (M (Xin ), Ui , I), where Xn is the sequence specified in condition (ii) of Simon. Note that we are keeping Ui the same. Let Γ(Gn ) and Γ(G) be respectively ¯ the (mixed strategy) graphs of Gn and G. First, ls Γ(Gn ) ⊂ Γ(G) and Simon’s condition (ii) imply that G is wrusc. Hence, condition (B) of the definition of variational convergence holds. Second, condition i of Simon implies that for any µ ∈ M(X), Hi (µ−i ) ≥

Z

ui dµi × dµ−i .

Combining the fact that Hi is a supremum over Ri with the fact that it is lsc in µ−i , we can conclude that condition (A) of the definition of variational convergence holds. Therefore, Gn = opposed to “algebraic” Tarski-type fixed point theorems. Super modularity and other techniques that use Tarski’s fixed point theorem impose monotonicity conditions on the best reply functions of the players. In the application considered in this paper, the best reply functions may not be well defined when the payoff of a player is not usc in the player’s own strategy. Therefore, the algebraic approach is not appropriate for such applications.

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(M (Xin ), Uin , I) converge variationally to G = (M (Xi ), Ui , I). For each n, EQ(Gn ) ⊂ M(X) is non-empty. Hence, ls EQ(Gn ) is not empty, and so is EQ(G). Discretizing a game G = (M (Xi ), Ui , I) via discretizing Xi may introduce equilibria points for the games Gn that are not equilibria for G (see Al´os-Ferrer [2] for examples). However, according to Theorem 3.2, these additional equilibria will eventually converge to equilibria of the original game. Note that it remains possible for G to have an equilibrium that cannot be obtained as a limit of any discretization of Xi (see Example 4 in the appendix of Simon [24]. We finally remark that Fundenberg and Levine [10] introduced a topology on S, which they call the m-topology, that can be used to study the continuity properties of the ε-equilibria points of a game G. In particular, given a game G = (Si , Ui , I), if the sequence S n ⊂ S satisfies a number of conditions with respect to the m-topology, then the ε-equilibria of the game Gn = (Sin , Ui , I) will approximate the ε-equilibria of the game G.9 However, there is no clear set of conditions that can be imposed on discontinuous games that guarantees that a particular discretization will actually satisfy the requirements for convergence in the m-topology. This, in turn, limits the usefulness of this approach in establishing existence results in the setting of discontinuous games (see the discussion on page 275-276 in Fundenberg and Levine [10] for more details). Existence of Equilibria via combining payoff security with reciprocal upper semicontinuity Again consider a game G = (M(Xi ), U, I) where Xi is a compact subset of a separable metric space. For each player i, Ui : M(X) −→ IR is the standard expected utility function Ui (µ) =

R

ui dµ, where

µ ∈ M(X) and ui is the vNM utility of player i. Reny [22] proved that a game G that is both payoff secure and rusc in mixed strategies must have a Nash equilibrium (Corollary 5.3 in Reny [22]).10 We now show that Reny’s above mentioned result can be seen in the context of game convergence 9

S n is required to be “ dense” in S and to “approximate” S in a specific sense. Both the density and the approximation requirement are defined in terms of the m-topology. 10 This corollary is consequence of a more general result by Reny that shows that every game that is better reply secure in mixed strategies has an equilibrium. Better reply security in mixed strategies is a more permissive concept than the combination of payoff security and rusc in mixed strategies.

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in the following way: Assume that G is both payoff secure and rusc in mixed strategies. Let Uin be defined and related to Ui in the same manner f n are related to f as in Lemma 2.10. For each n, let µn be an equilibrium for the game Gn = (M (Xi ), U n , I) (such equilibria exist by classic existence results for continuous games). Now by Lemmas 2.10 and 2.7, ∀(µi , µ−i ) and for all µn−i −→ µ−i , there exist µ ˆni such lim inf Uin (ˆ µni , µn−i ) ≥ Ui (µi , µ−i ),

(3.5)

and condition (A) of variational convergence holds. Let Γn and Γ be respectively the graphs of Gn and G. Note that these graphs are subsets of M(X) × IRI . For any (µ, α) ∈ ls Γn \Γ, there exists a subsequence (µnk , U nk (µnk )) such that (µnk , U nk (µnk )) −→ (µ, α). The definition of ls Γn and the fact that Uin and Ui satisfy (i) of Lemma 2.10 imply that, for all i, α ≥ Ui (µ). Furthermore, at least one these inequalities must be strict since (µ, α) is not in Γ. Moreover, Uink (µnk ) ≤ Ui (µnk ) (condition (ii) of Lemma 2.10), this ¯ and α in turn implies that there exists α ˜ ∈ IRI such (µ, α ˜ ) ∈ Γ\Γ ˜ i ≥ αi , for all i.11 Since G is rusc, it is also wrusc. Therefore, for some player i0 , ∃ˇ µi0 ∈ M(Xi0 ) such that Ui0 (ˇ µi0 , µ−i0 ) > α ˜ i0 ≥ αi0 . Hence, condition (B) of the definition of variational convergence is also satisfied. v G, and both ls EQ(G ) and EQ(G) are not empty. Therefore, Gn → n

We finally note that Reny’s result is in fact constructive. Given a game G that is payoff secure and rusc in mixed strategies, we can combine densely discretizing Xi with approximating the payoff functions using an increasing sequence of continuous functions. Note in particular that given a payoff secure function Ui , a sequence Uin satisfying Lemma 2.10 can be explicitly constructed using Theorem 1.25 of Rockafellar and Wets [23].12 Thus, we can explicitly construct a sequence of games Gn = (Xin , Uin , I) with finite strategies that variationally converges to G. An equilibrium of G can then be found as a limit point of any sequence in EQ(Gn ). 11 ¯

Γ is the closure of the graph of G in the weak topology of M(X) × IRI . Theorem 1.25 in Rockafellar and Wets [23] is stated for functions over IRn but it can be easily generalized to functions over any metric space. 12

15

4. New Existence Results In this section, we use variational convergence to establish the existence of equilibria under very weak conditions. Our results in this section will later allow us to establish the existence of equilibria in a large class of Bertrand-Edgeworth games. We start with a simple lemma. Lemma 4.1. Consider a sequence of games Gn = (M(Xin ), Ui , I) where Xin densely approximates Xi and Xi is the pure strategies set of the game G = (M(Xi ), Ui , I). If G is wrusc with respect to mixed strategies, then Gn and G satisfy condition (B) of the definition of variational convergence. Proof. Let Γn , Γ be respectively the graphs of Gn and G. The conclusion of the lemma is ¯ and Definition 2.9. immediate from the fact that ls Γn ⊂ Γ

Consider a game G = (M(Xi ), Ui , I) with Ui (µ) =

R

ui dµ with possibly discontinuous ui . We

→ Xi : define, for every player i, the discontinuity map Ai : Xi → Ai (xi ) = {x−i ∈ X−i |ui is discontinuous at (xi , x−i )}. The following theorem, whose proof is in Appendix B, is our main result in this section Theorem 4.2. Consider a sequence of games Gn = (M(Xin ), Ui , I) where Xin densely approximate the set Xi in the game G = (M(Xi ), Ui , I). Assume (i) For all i, ∀xi ∈ Xi , ∀˜ µi ∈ lsEQ(Gn ), ∀ε > 0, ∃ˆ xi ∈ Xi such that µ ˜−i (Ai ((ˆ xi ))) = 0 and Z

Z

ui (ˆ xi , x−i ) d˜ µ−i >

ui (xi , x−i ) d˜ µ−i − ε.

(ii) G is wrusc in mixed strategies. v G, ls EQ(G ) ⊂ EQ(G), and the game G has an equilibrium. Then, Gn → n

The precise economic interpretation of condition (i) depends on the source of discontinuities in the game. If the discontinuities in the games arise because of tie-breaking rules, then condition (i) says that, from the perspective of player i and given a move µ ˜−i by the other players, any action that leads to a tie with positive probability is epsilon-dominated by an action that avoids a tie for sure. 16

Moreover, Condition (i) is strictly weaker the uniform payoff security (see Monteiro and Page [20], and it neither implies nor is implied by payoff security in mixed strategies. However, in the context of the examples that we will investigate in this paper checking condition (i) will be much easier than checking payoff security in mixed strategies. In many applications, condition (i) above can be replaced with a stronger condition: (i’) For all i, for any µ−i ∈ M(X−i ), ∀xi ∈ Xi , ∀ε > 0, ∃ˆ xi ∈ Xi such that µ−i (Ai (ˆ xi )) = 0 and Z

ui (ˆ xi , x−i )dµ−i ≥

Z

ui (xi , x−i )dµ−i − ε.

Since (i) implies (i’), we immediately obtain Corollary 4.3. If G satisfies conditions (i’) above and condition (ii) in Theorem 4.2, then G has an equilibrium. As a direct application to Theorem 4.2, we prove the existence of equilibria in a class of rentseeking games introduced by Tullock [28]. Example 4.4: Consider a game with I players competing for a “rent” V . The actions space of each player (the set of pure strategies) is Xi = [0, 1]. The payoff of player i is ui (xi , x−i ) =

  fi (xi , x−i ) · V − Ci (xi )

when (xi , x−i ) 6= (0, 0),



when (xi , x−i ) = (0, 0) .

(4.1)

bi

where bi ≥ 0, V ≥ ΣI bi , and ΣI fi = 1. The function fi represents the probability of player i wining V giving the actions (efforts) of all the players. bi represents the fraction of V player i gets when none of the players exerts any effort. The effort of player i comes at a cost given by the function Ci . Assume that Ci is continuous with Ci (0) = 0, fi is bounded and continuous except possibly at the origin (0, 0), and limxi −→0 ui (xi , 0) = V .13 ˜ where the payoff of player i is We introduce a modified game G u ˜i (xi , x−i ) =

  fi (xi , x−i )V − Ci (xi )

when (xi , x−i ) 6= (0, 0),

˜

when (xi , x−i ) = (0, 0) .

bi

(4.2)

where 0 < ˜bi < Vi and ΣI ˜bi = V (for example, take ˜bi =

V n ).

Clearly, ΣI u ˜i is usc. The point (0, 0)

is the only point of payoff discontinuity. Moreover, limxi −→0 u ˜i (xi , 0) = V , and therefore for any 13

All the payoff functions suggests by Tullock and others satisfy all these assumptions.

17

µ−i ∈ M(X−i ) and any ε > 0, there is x ˆi arbitrarily close to 0 such that u ˜i (ˆ xi , 0) > u ˜i (0, 0),

(4.3)

and Z

Z

u ˜i (ˆ xi , x−i ) dµ−i >

u ˜i (0, x−i ) dµ−i − ε,

(4.4)

˜ satisfies condition (i’) of Corollary 4.3. Since which implies that G ˜ has an equilibrium µ∗ . Finally, note that µ∗ cannot assign a Hence, by Corollary 4.3., the game G non-zero weight to (0, 0) due to (4.4). This, combined with the fact u ˜i = ui except at the origin, imply that µ∗ is also an equilibrium for the original game G. The conclusion of this example generalizes a similar result by Yang [30]. We now list a number of immediate corollaries for Theorem 4.2. Recall that a game is symmetric, if the strategy sets for all the players are the same and the payoffs depend only on the strategies employed, not on who is playing them. A symmetric equilibrium is an equilibrium where all the players play the same strategy. Corollary 4.5. A symmetric game G satisfying the assumptions of Theorem 4.2 has a symmetric equilibrium. Proof. Every finite symmetric game has a symmetric equilibrium. Therefore, Gn has symmetric equilibrium µ∗n . Let µ∗ be an accumulation point of the sequence mu∗n . By Theorem 4.2, µ∗ is an equilibrium for the game G. Since the limit, in the weak topology on M(X), of a sequence of pure strategies is also a pure strategy, we also obtain the following corollaries. Corollary 4.6. Consider a sequence of games Gn that variationally converge to G. Suppose that, for all n > n0 for some n0 , Gn has a pure Nash equilibrium, then G has a pure Nash equilibrium. Recall that a function f : X × Y −→ IR has increasing differences, if for any y2 ≥ y1 , the function f (·, y2 ) − f (·, y1 ) is increasing. Corollary 4.7. Consider a game G with I players and a (pure) strategy set X = [0, 1]I . Assume the following: (i) G satisfies the conditions Theorem 4.2, (ii) the payoff of each player has increasing 18

differences, then G has an equilibrium in pure strategies. Proof. Consider a sequence Gn of games that densely approximate G. Without loss of generality, assume that for each n the strategy set Xn of Gn is a finite lattice. For each n, the payoffs of Gn have increasing differences. Hence, by Theorem 4.2.1 in Topkis [27], each game Gn is super modular and has an equilibrium in pure strategies. Corollary 4.5 now implies that G has an equilibrium in pure strategies. Note that the assumptions of the corollary neither require nor imply that G is super modular (despite the fact that each Gn is super modular). In fact, the best reply functions can be well defined in each Gn even when they are not well defined for the original game (see Al´os Ferrer [2] for more details). Note finally that unlike the standard results for super modular games on IRn , condition (ii) of Theorem 4.2 does not require the payoff of each player in G to be usc in the player’s own strategy. Next, we prove a theorem that can used to establish the existence of equilibria for a large class of Bertrand-Edgeworth games and other types of “timing duels” between two players. The similarity between the proof of our theorem and the proof of Theorem 5b in DM [6] will be evident. However, our result will be more general, and it will avoid the technical difficulties in the proof of Theorem 5b that we mentioned earlier (see Appendix A for details). 5. Applications: Existence result for timing games between two players Consider a game between two players i ∈ {1, 2} where the pure strategy set of player i is the interval Xi = [ai , bi ]. We denote the strategy set of the whole game by B = [a1 , b1 ] × [a2 , b2 ]. Rather than restricting the discontinuity sets of the payoff to the diagonal of B, we will allow the discontinuity points to be a subset of a more general curve. To make the description of such points more precise, we introduce some notation. Let f : IR −→ IR be a strictly increasing function. Let O denote the point where the function f enters the box B. More specifically, O ∈ B is the intersection of the graph of f and the line x2 = a2 (see figure 1 below).

19

f

x2

x2

b2

m2

l1 φ1 m1 a2

o a1

u1

b1

φ2

x1

o

u2

l2 x1

Fig. 1

The payoff of the players are

u1 (x1 , x2 ) =

 l1 (x1 , x2 )            

when x2 > f (x1 )

ϕ1 (x1 )

when x2 = f (x1 )

m1 (x1 , x2 )

when x2 < f (x1 )

(5.1)

where xi ∈ [ai , bi ] and ϕ1 : IR −→ IR. Similarly,

u2 (x2 , x1 ) =

 l2 (x2 , x1 )            

when x2 < f (x1 ),

ϕ2 (x1 )

when x2 = f (x1 ),

m2 (x2 , x1 )

when x2 > f (x1 ).

20

(5.2)

where xi ∈ [ai , bi ] and ϕ2 : IR −→ IR. Note that the function f is used to describe (parameterize) the set of potential points of discontinuities whereas the function ϕi describes the payoff of player i over this set. We henceforth refer to the above game as the the standard game on the rectangle. In many applications involving standard games on the rectangle, condition (i) of Theorem 4.2 can be easily verified but condition (ii) of the same theorem does not hold. In some cases, however, the original game can be modified in such a way that the new game satisfies both conditions, and hence the new game has an equilibrium. Furthermore, the equilibrium of the new game can be shown to be an equilibrium of the original game. The following theorem provides sufficient conditions for such procedure to work. Theorem 5.1 Consider a standard game on the rectangle G. Assume the following (i) For all i, li , mi are continuous and bounded (ii) ϕ1 and ϕ2 are rusc on the real line with only “jump ” discontinuities where at any point of discontinuity limits from the right and from the left exist but are not necessarily equal. (iii) For all i, ui is continuous at the point O. (iv) For any point (xi , x−i ) ∈ B such that x2 = f (x1 ), we have mi (xi , x−i ) ≤ ϕi (x1 ) ≤ li (xi , x−i ) (v) For any point (xi , x−i ) ∈ B where x2 = f (x1 ) and the payoff of some player is discontinuous, there exists i such that li (xi , x−i ) > ϕi (x1 ) and m−i (xi , x−i ) > ϕ−i (x1 ). The game G has an equilibrium. The proof is in Appendix C. The above conditions imply that the set of discontinuities for both players is the same, and that it is a subset of the intersection of gph fi with B. Condition (iii) will be naturally satisfied by all the games that we have in mind. In particular, in our examples of BE games, the point O is the point 21

(0, 0), and (ii) will be satisfied since profits go to zero when prices go to zero as long as there are no fixed costs. Condition (iv) means that the payoff of a tie cannot be better than winning and cannot be worse than losing. Modifying the game at points of discontinuity such that the sum of modified payoffs is usc is easy. The problematic step in the proof is to insure that the modified game continues to satisfy condition (i) of Theorem 4.2. Condition (ii), (iv), and (v) together insure that this critical step can be carried out. For more on the role of these conditions see Appendix A and Appendix B. The strategy sets of both players are not assumed to be the same and the points of discontinuity are subset of a “generalized diagonal” of sort. Therefore, the setting of the above theorem is more general than the setting of Theorem 5b in DM [6]. Bertrand-Edgeworth games Recall that BE games are competitions between various producers of a homogenous good where the prices are the only strategic variables available to the firms. Furthermore, and unlike the classic Bertrand games, firms are not required to meet the market demand at the posted prices. Firms announce prices simultaneously, and productions decisions are made after the realization of the demand for each firm. We will assume that, for i ∈ {1, 2}, each firm has an increasing, strictly convex, and continuous cost function Ci (possibly different for different firms) with Ci (0) = 0. We further assume that the market demand function D(p) is continuous, and ∃¯ p such that D(¯ p) = 0 and ∀p > p¯, D(p) ≤ 0. We don’t require the differentiability of D or Ci . Note, however, that the continuity of Ci is crucial as there exist standard examples of non-existence of equilibria when C is discontinuous. For a firm with capacity Ki , the firm production problem is

M axz∈[0,Ki ] πi (z) = p · z − Ci (z).

(5.3)

Let the solution to the above problem (supply function of the firm) be Si (p) = argmaxz∈[0,Ki ] πi .

(5.4)

Without loss of generality, we assume that Si is single valued and continuous for every p ∈ [0, p¯].14 14

The continuity of Si in p can be shown under the assumptions that Ci is lsc and Ki is compact.

22

The demand function facing seller i is

d(pi , p−i ) =

 D(pi )            

when pi < p−i ,

tbi (p)

when pi = p−i = p,

max[0, rdi (pi , p−i )]

when pi > p−i .

(5.4)

where tbi and rdi are the tie breaking rule and the rationing rule (rule for residual demand) respectively. Ideally, we like to only require rdi (p, p) ≤ tbi (p) ≤ Di (p).

(5.5)

This will allow a variety of tie breaking rules including the rules used by Dixon [9], Dasgupta and Maskin [7], and Deneckere and Kovenock [8]. This setting also allows for very general rationing rules such as the combined rationing rule suggested by Tasn´adi [26].15 Now consider a price competition game given a level of capacity Ki . Assume there is unique price pi,m ≤ p¯ such that pi,m maximizes the profit of a firm i when it is the only firm in the market. Such game can be expressed as a standard game on the rectangle [0, p1,m ] × [0, p2,m ] with payoffs define in the following manner: Let xi (pi , p−i ) = min{Si (p), di (pi , p−i )}.

(5.6)

ui (pi , p−i ) = pi xi (pi , p−i ) − Ci (xi (pi , p−i )).

(5.7)

and the payoff of player i is

Assume that, for every i and every p ∈ [0, pi,m ], the function πi (z) = pz − Ci (z) satisfy the following properties: a) πi has unique maximizer denoted by Si (p), b) πi is non-decreasing to the left of Si (p). Remark 1: The strict convexity of Ci is a sufficient but not a necessary condition for the above requirements to hold. In fact, conditions (a) and (b) above allow for “S” shaped cost functions. However, (a) and the continuity of Si (p) together imply the strict convexity of Ci .16 15 16

The proportional and efficient rational rules for residual demand are both special cases of this rule. This fact was pointed out by a anonymous referee.

23

In order to express the game in the format of a standard game on the rectangle, we define the following functions that reflect the different possible production decisions of firm i: xi,l (pi , p−i ) = min{Si (pi ), Di (pi )}, xi,ϕ (p, p) = min{Si (p), tbi (p)}, xi,m (pi , p−i ) = min{Si (pi ), rdi (pi , p−i )}. We also let

li (pi , p−i ) = pi xi,l (pi , p−i ) − Ci (xi,l (pi , p−i )),

(5.8)

ϕi (p) = pi xi,ϕi (p, p) − Ci (xϕi ,l (p, p)),

(5.9)

mi (pi , p−i ) = pi xi,m (pi , p−i ) − Ci (xi,m (pi , p−i )),

(5.10)

and finally let f : IR −→ IR be the identity map. Under assumptions (a) and (b) above, inequality (5.5) leads to similar inequality with profits and payoffs: mi (p, p) ≤ ϕi (p) ≤ li (p, p). Remark 2: The above inequality holds regardless to whether Ci is convex or not. This, together with Remark 1, imply that our assumption of the strict convexity of Ci was only needed to guarantee that Si (p) is continuous and single-valued. This suggests that the same analysis can be used for models where Si (p) is a continuous selection from a (lower semi-continuous) set-valued map that represents the maximizers of (5.3) Since the profit of each firm goes to zero as prices goes to zero, the payoffs are continuous at (0, 0). Finally, tbi , is continuous in all the variations that we discussed earlier, and therefore xi,ϕ is also continuous. We now have the following application of Theorem 5.1: Theorem 5.2 Consider a Bertrand-Edgeworth duopoly defined by equations (5.3) through (5.10) and satisfying conditions (a) and (b). Suppose further that the tie-breaking rules of the game satisfy condition (ii) of Theorem 5.1. Then, this game has an equilibrium.

24

For the BE game discussed above, an equilibrium µ∗ will assign a zero weight to each discontinuity point. Furthermore, µ∗ will also assign zero weight to (0, 0).17 To see this, assume that µ∗ (0, 0) 6= 0. Note ui (0, p−i ) = 0 and πi (ˆ pi , 0) > 0 for some pˆi . Therefore, Z

πi (ˆ pi , p−i )dµ∗i

Z

>

πi (0, p−i ) dµ∗i

which contradicts µ∗i (0, 0) 6= 0. If we further assume that the BE game under consideration is symmetric with payoffs that are discontinuous at every point of the diagonal except for (0, 0), then the above argument shows that µ∗ has to be atom-less.18 Using Theorem 5.2, we can reproduce all the results of Dixon [9], Maskin [19], Yoshida [31]. We can also use Theorem 5.2 to reproduce the existence results for the first stage game of Kreps and Scheinkman [16], Osborne and Pitchik [21], Deneckere and Kovenock [8]. This can be done without convexity or symmetry of costs functions. Note further that unlike the current existence results, we do not need different proofs for different specifications of the tie breaking and residual demand rules as long as these rule satisfy assumption (5.5). As we mentioned earlier, price competitions often represents a second stage game that follows a competitions in a decision variable other than prices. Consider a two-stage game where the players first simultaneously choose their capacities Ki from a compact convex subset of IRn , and then choose their prices. Suppose that the pricing game is a BE competition in the format discussed earlier, and hence the equilibrium for this stage exists. Suppose further that for a fix choice of capacities (K1 , K2 ), the payoff of the pricing game at equilibrium is unique, and denote it by πi∗ (Ki , K− i).19 Consider a sequence Kin −→ Ki . Using Theorem 3.2, we can show that n πi∗ (Kin , K−i ) −→ πi∗ (Ki , K−i ).

(5.8)

Assume now that the payoff of the first stage is given by π ˜i (Ki , K−i ) = πi∗ (Ki , K−i ) − gi (Ki ), 17 This is unlike the classic Bertrand game where requiring firms to meet the market demand at the posted price leads to πi (pi , 0) = 0 for any pi . 18 Consider for example a BE game where at a price tie, the market is split evenly between the two firms. 19 The uniqueness assumption is satisfied by the games of Deneckere and Kovenock [8] and by most models of two-stage BE games.

25

where gi is continuous function denoting the capacity cost of firm i. The continuity of gi and (5.8) imply the continuity of the π ˜i . This, together with convexity and compactness of the strategy sets of capacities, imply that the entire game has a backward induction equilibrium. Furthermore, when πi∗ (·, K−i ) is quasi-concave and gi is convex, the equilibrium is in pure strategies (K1∗ , K2∗ ).

26

Appendix A. Theorem 5b in DM [6] Theorem (5b in [6]). Consider a game between two players on the unit square in IR2 . Assume the payoffs are bounded and continuous except possibly on the diagonal. Suppose that for each x ∈ [0, 1], there exists a player i, j ∈ {1, 2} such that lim

ui (x1 , x2 ) ≥ ui (x, x) ≥

lim

uj (x1 , x2 ) ≤ uj (x, x) ≤

− x, x + x x1 → 2→ − x, x + x x1 → 2→

lim

ui (x1 , x2 )

(a1)

lim

uj (x1 , x2 )

(a2)

+ x, x − x x1 → 2→ + x, x − x x1 → 2→

where the left(right) inequality in (i) is strict, if and only if the right (left) inequality in (ii) is strict. Such game has an equilibrium. The original proof of the above theorem is based on the premise that any game satisfying the assumptions of the above theorem can be modified in such a way that every modified payoff is weakly lower semi-continuous (see Definition 2.5) and the sum of the modified payoffs is usc. Hence using an earlier result in DM [6], the new game has an equilibrium. Furthermore, the modification of the original game can be done such that the equilibrium of the new game is also an equilibrium of the original game. We first go over the modification process that is used in DM [6]. We then use two examples to show that this process, hence the original proof of theorem, may not work.20 A slightly modified version of this theorem and a proof based on the results of this paper appear in appendix C. The outline of the proof of Theorem 5b in DM[6] (page 17 in [6]): The payoffs of the player are modified at their points of discontinuity. The modified payoffs u ˆi must satisfy the following conditions. i) At every point (x, x) such that inequalities (a1) and (a2) are strict, we must have ui (x, x) ≤ u ˆi (x)
x−i .

where ϕ1 (x) = 10 when x < 0.5 and ϕ1 (x) = 0 when x ≥ 0.5. Similarly, ϕ2 (x) = 10 when x < 0.5 and ϕ1 (x) = 0 when x ≥ 0.5. Suppose we modify the payoffs along the diagonal using some functions ϕ˜i instead of ϕi . Then the

28

sum of the payoff is

u ˜1 + u ˜2 =

 10          

when x1 < x2 ,

ϕ˜1 (x) + ϕ˜2

when x1 = x2 = x,

10

when x1 > x2 .

The only way for the sum to be usc is for ϕ˜1 (0.5) + ϕ˜2 (0.5) to be larger or equal to 10. This is possible, only if ϕ˜1 (0.5) ≥ 10 or ϕ˜2 (0.5) ≥ 10. This clearly violates the second set of inequalities in (3a) and (3b) of requirement (i) in the proof of DM [6].

21

Example A.2: Consider a pay-your-bid auction on the unit square with two players. The payoff of player i is

ui (bi , b−i ) =

 1 − bi      1 2

    

when bi > b−i ,

− bi , when bi = b−i ,

−bi ,

when bi < b−i .

It is straightforward to check that his example satisfies the conditions of Theorem 5b in DM [6] and that the payoff function of each player is not wlsc at (1, 1). Furthermore, it is not possible to redefine the payoff on the diagonal such that the modified payoff satisfies requirement (i), (ii) and such that the modified payoffs are wlsc. More specifically, in order for the proof to work, the modified payoff must be such that u ˆi ≥ ui with u ˆi = ui off the diagonal. This, however, will force u ˆi to violate weak lower semi-continuity at the point (1, 1) since limti −→1 u ˆi (ti , 1) = limti −→1 ui (ti , 1) = −1 and u ˆi (1, 1) ≥ ui (1, 1) = −0.5. B. Theorem 4.2. First, we list some important properties of weak convergence of measures in M(X) where X is a subset of some separable space (See chapter 3 in Stroock [25] or Chapters 1 and 2 in Billingsley [4]. 1) µn −→ µ implies

R

f (x)dµn −→

R

f (x)dµ for any f that is bounded on X and continuous

µ-almost everywhere in X. 2) If Z ⊂ X is a dense subset in X, then M(Z) is a dense subset of M(X). 21

If we ignore the requirement that second inequalities in (3a) and (4a) are not strict, then we will not be able to claim that any equilibrium of the modified game is an equilibrium of original game. In our example, (0.5, 0.5) is an equilibrium of the former but not the latter.

29

3) µn1 −→ µ1 in M(X1 ) and µn2 −→ µ2 in M(X2 ), if and only if, µn1 ×µn2 −→ µ1 ×µ2 in M(X1 ×X2 ). Proof of Theorem 4.2 In light of Theorem 3.2 and Lemma 4.1, we only need to show that a game satisfying the assumptions of Theorem 4.2 also satisfies assumption (A) in the definition of variational convergence. We start with µ ˜n ∈ EQ(Gn ) and µ ˜n → µ ˜. For any µi ∈ M(Xi ), there is xi ∈ Xi such that Z

ui (xi , x−i ) d˜ µ−i ≥

Z

ui µi × µ ˜−i

(b1)

since µi is a probability measure. If µ ˜−i (Ai (xi )) = 0, then define µ ˆi (xi ) to be the probability measure with weight one on xi . Since Xin densely approximates Xi , M(Xin ) also densely approximates M(Xi ). Therefore, ∃ˆ µni ∈ M(Xin ) such that µ ˆni −→ µ ˆi . Moreover, Z

lim

ui dˆ µn−i × µn−i =

Z

ui (xi , x−i )d˜ µi × µ−i

(b2)

since ui (xi , ·) is continuous on X−i \ Ai (xi ). Combining (b1) and (b2) implies that condition (A) of variational convergence holds. If µ ˜−i (Ai (xi )) 6= 0, then by assumption ∀ ε, ∃ˆ xi ∈ Xi such that µ ˜−i (Ai (ˆ xi )) = 0 and Z

ui (ˆ xi , x−i )d˜ µ−i ≥

Z

ui (xi , x−i )d˜ µ−i − ε

(b3)

We now repeat the previous argument: Let µi (ˆ xi ) be the measure that puts unit weight on the point x ˆi . Note that

µi (ˆ xi ) × µ ˜−i (Ai (ˆ x−i )) = 0 Since M(Xin ) is dense in M(Xi ), there is a µ ˆni → µ ˆi (ˆ xi ). Due to (b4), we have 30

(b4)

Z

lim

ui dˆ µni

×

d˜ µn−i

→

Z

ui dµi (ˆ xi ) × d˜ µ−i

This, combined with (b1), in turn implies that for any ε > 0, there is µ ˆi and n0 such that for all n > n0 , we have Z

lim

ui µ ˆi ×

µ ˜n−i

≥

Z

ui (xi , x−i ) d˜ µ−i − ε

Hence, using (1b), we Z

lim

ui µ ˆi × µ ˜n−i ≥

Z

ui dµi × µ ˜i − ε,

(b5)

which is condition (B) of the definition of variational convergence. C. Theorem 5.1 Proof of Theorem 5.1 As we mentioned earlier, the idea of the proof is that a game satisfying the assumptions of Theorem 5.1 can be modified into a new game that satisfies the conditions of Theorem 4.2, and any equilibrium of the new game is also an equilibrium of the original game. First, we state, without proof, a simple lemma about functions form IR to IR. The lemma states that any two reciprocally upper semi-continuous functions that are “sandwiched” between continuous function can be modified in such a way that the sum of the modified functions is usc and each of the modified functions remains “sandwiched” between the original continuous functions. Lemma C.1. Consider two functions f1 : IR −→ IR and f1 : IR −→ IR. Suppose both functions have only jump discontinuities.22 Moreover, for each i, hi ≤ fi ≤ gi where gi and hi are continuous functions. Assume further that the two functions fi are rusc on IR.23 In this case, both functions P can be modified into new functions f˜i : IR −→ IR such that i f˜i is usc and f˜i < gi whenever

fi < gi and hi < f˜i whenever hi < fi . Proof of Theorem 5.1. Define the following functions on [a1 , b1 ]: For any x and limx0 −→x+ fi (x0 ) and limx0 −→x− fi (x0 ) both exist. Recall that on IR, rusc simply means that if lim inf n fi (xn ) > fi (x) for some i and some xn −→ x, then lim supn f−i (xn ) < f−i (x). 22

23

31

ϕˆ1 = max{l1 (x1 , f (x1 )) − [ϕ2 (x1 ) − m2 (f (x1 ), x1 )], ϕ1 (x1 )} and ϕˆ2 = max{l2 (f (x1 ), x1 ) − [ϕ1 (x1 ) − m1 (x1 , f (x1 ))], ϕ2 (x1 )} ˆ on the square where Define a new game G

u ˆ1 (x1 , x2 ) =

 l1 (x1 , x2 )            

when x2 > f (x1 ),

ϕˆ1 (x1 )

when x2 = f (x1 ),

m1 (x1 , x2 )

when x2 < f (x1 ).

(c1)

where xi ∈ [0, 1]. Similarly, define

u ˆ2 (x2 , x1 ) =

 l (x , x )    2 2 1         

when x2 < f (x1 ),

ϕˆ2 (x1 )

when x2 = f (x1 ),

m2 (x2 , x1 )

when x2 > f (x1 ).

(c2)

Note the following: (ic) Since ϕi and ϕ2 are rusc, ϕˆ1 and ϕˆ2 are also rusc. As a result of Lemma C.1., we can assume that Σi ϕˆi is usc on [0, 1]. Moreover, for any point (x1 , x2 ) ∈ B such that x2 = f (x1 ) and ui is discontinuous, we have ϕˆ1 (x1 ) + ϕˆ2 (x1 ) ≥ max{l1 (x1 , x2 ) + m2 (x2 , x1 ), l2 (x2 , x1 ) + m1 (x1 , x2 )}. This above inequality, the fact that the sum of ϕˆi is usc, and the continuity of li and mi imply that P

ˆi iu

is upper-semi continuous on the rectangle [a1 , b1 ] × [a1 , b1 ].

(iic) For every (x1 , x2 ) such that x2 = f (x1 ) and ui is discontinuous, we have ϕˆi (x1 ) < li (xi , x−i ). ˆ satisfies assumption (i) of Theorem 4.2. Therefore G ˆ has an Furthermore, and because of (iic), G equilibrium µ∗ . Now we show that µ∗ is an equilibrium of the original game. Because of (iic), µ∗ must assign zero weight to any point of discontinuity, and hence

32

R

u ˆi dµ∗ =

R

ui dµ∗ . Therefore, µ∗

is also an equilibrium for G since otherwise ∃i and ∃˜ xi such that Z

u ˆi (˜ xi , x−i )dµ∗−i

≥

Z

ui (˜ xi , x−i )dµ∗−i

Z

>

∗

Z

ui dµ =

u ˆi dµ∗ ,

ˆ contradicting the fact that µ∗ is an equilibrium for G.

D. Variational Convergence in Example 3.4. The proof that W n −→ W : For every i > 1 and for every pn −→ p in the simplex, we have Bin (pn ) −→ Bi (p). Combining this fact, assumption (i) in the example, and the fact that int(Xi ∩ Bi (p)) is not empty, we can apply Theorem 4.32 in [23] to conclude that ∀pn −→ p in the simplex, we have Xin ∩ Bin (pn ) −→ Xi ∩ Bi (p),

(d1)

where the convergence is in the sense of Definition 2.1. Assumptions (i), (ii) of the example and (d1) above imply that condition (A) of the definition of variational convergence holds. Furthermore, the assumptions of the example imply that Ls gph Γ(W n ) ⊂ Γ(W ). Since the sum of the payoffs of W is usc, W is wrusc in mixed strategies. Hence, condition (B) of the definition of variational convergence also holds.

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