A second look at minsup points and the existence of pure Nash equilibria Adib Bagh Departments of Economics and Mathematics University of Kentucky. Abstract. We prove the existence of pure strategy Nash equilibria under very weak conditions of lower semi-continuity and quasi-concavity. Our results generalize previous results by Baye, Tian and Zhou as well as more recent results by Kim and Kum, Hou and Chang. We use the new results to establish the existence of pure strategy equilibria for games with discontinuous and non-quasiconcave payoffs such as probabilistic spatial voting games on the real line. Keywords. Transfer continuity, Transfer quasi-concavity, Payoff security, Minsup points, Nash equilibria, Probabilistic spatial voting games. JEL classification: C72,C62,C69 Corresponding Author: Adib Bagh Gatton College of Business and Economics, University of Kentucky, Lexington, KY, 40506 Tel: (859)257-3057; Fax: (859)323-1920 email: aba223@uky.edu 1
1. Introduction The literature on minimax problems and their applications in game theory and economics is vast and expanding. It stretches from the work of Fan and Von Neuman in the fifties to recent papers on decision making under model uncertainty and robust control problems (Hansson and Sargent [8] and Willams [17]). Given a function f : X Y IR and a real number α, the existence of a minsup point ˆx X such that sup y Y f(ˆx, y) = min sup x X y Y f(x, y) α (1) is closely related to the existence of a pure strategy Nash equilibrium for a game whose aggregator function is f (see Tian [16], Baye et al [3], Chen and Zhou [18], Kim and Kum [12], and Chang [6]). 1 More generally, we can consider a collection C of functions from Y to X and look for a function ĉ C that solves sup y Y f(ĉ(y), y) = min sup c C y Y f(c(y), y). (2) This type of problems arises from a sequential game involving nature and a decision maker (an agent). Nature moves first by choosing a value y representing the state of the world. The agent has a set X of available actions and a collection C of implementable policies, where each policy is a function from Y to X that assigns to every possible state of the world y an action x X. The function f : X Y IR describes the loss the agent suffers when he takes action x and the state of the world is y. The problem in (2) can be interpreted as requiring the agent to choose, and to commit to, a specific policy before the realization of the state of the world in a manner that minimizes his maximum possible loss. It is often important to know whether or not the minimizer of (2) is constant over Y (i.e. whether or not the optimal policy requires the agent to commit to the same action x no matter what the state of the world is). One remarkable fact about establishing the existence of a simple (constant) policy solution for (2) is that such a result would imply that min sup x X y Y f(x, y) = min sup c C y Y f(c(y), y) (3) 1 We us the term minsup instead of minimax since the supremum need not be attained on Y. Note that for some authors such as Aubin [1], the definition of a minsup point includes additional requirements. 2
even when X can be strictly embedded within C (i.e. when C strictly contains all the constant functions from Y to X). In this paper, we establish the existence of a minsup point for f and a constant minimizer for (2) under two conditions. The first condition requires f to be majorized, in a very weak sense, by some function g. The second condition requires the function g to satisfy a very weak concavity condition. We generalize most of the current existence results of minimax points, including Theorem 3.1 in Chang [6], the most recent among such results. 2 We then obtain new existence results for pure strategy Nash equilibria in a large class of two-player games with discontinuous payoffs. Unlike the results of Baye et al [3] and Chang [6], our results do not require the construction of the aggregator function of the underlying game. This is important because imposing strong lower semi-continuity and concavity condtions on the individual payoff functions often does not translate into useful conditions on the aggregator function (see Ziad [19] for an example). We provide a number of applications that include probabilistic spatial voting games with two candidates, where the payoffs of the players are neither continuous nor quasi-concave. Our approach will actually allow us to numerically establish some important patterns regarding the equilibria platforms of the candidates. There are a number of alternative ways for tackling a game with discontinuous and non-quasi-concavity payoffs. One possible approach is to verify that the game is super modular, which requires the best reply functions of the game to satisfy certain monotonicity conditions. This, in turn, requires the payoff of every player to be upper semi-continuous (in the player s own strategy as well as in the strategies of the others). 3 A second approach, developed by Bich [4], requires the strategy set of the players to be convex and imposes conditions on the quasiconcave lower envelops of the payoff functions. Of course, there are games where the strategy sets are not convex, the payoffs are not upper semi-continuous, or the quasi-concave lower envelops of payoff functions cannot be easily calculated. For such games, the approach we follow in this paper will be more appropriate. We start the paper by reviewing basic definitions in Section 2. We present our main results in section 3. Finally, in Sections 4 and 5, we provide some concrete applications. 2 An exception would be the minimax existence results that do not require the set Y to have a linear structure as in Kalmoun and Riahi [11]. 3 The same is true for the approach of Nishimura and Friedman [13]. 3
2. Preliminaries In this section, we introduce the notation and the definitions that will be used throughout the paper. Henceforth, let X be a topological space and let Y be a subset of a topological vector space. 4 We denote by n the standard unit simplex in IR n, and for every λ n, we define I(λ) = {k {1,, n} λ k 0}, which is simply the collection of indices where λ has a non-zero component. We remind the reader of the following definition of the lower level set of a function. Definition 1 Given a function h from a set U to IR and given an α IR, the α-lower level sets of h is the set lev α h = {z U h(z) α}. The lower semi-continuous closure in x of a function f : X Y IR is defined in the following manner. Definition 2 Let W(x) be the collection of all open sets containing x X. The lower closure of f : X Y IR in x is defined by f x (x, y) = sup W W(x) inf x W f(x, y). It is clear from the definition that for any y Y, f x (, y) is lower semi-continuous in x, and that f x (x, y) f(x, y) for any (x, y) X Y. Thus, we now have the following lemma whose proof follows immediately from the lower semi-continuity of f x Lemma 1 For every α and every y Y, the set lev α f x (, y) = {x X f x (x, y) α is closed in X. The following notion of lower semi-continuity was introduced in Tian [16]. Definition 3 A function f : X Y IR is transfer lower continuous in x at α IR, if for any (x, y) X Y, f(x, y) > α implies the existence of ŷ Y and the existence of a neighborhood W of x such that f(x, ŷ) > α for all x W. The most important feature of the above notion is the connection between the lower level sets of f and the lower level sets of f x. Lemma 2 f is transfer lower continuous in x at α, then for any y Y, there exists ŷ Y such that lev α f x (, ŷ) lev α f(, y). 4 When X is a subset of topological space U, we can simply equip X with the relative topology it inherits from U. 4
Proof. Suppose x lev α f(, y). This implies that f(x, y) > α. Hence, by transfer lower continuity, there exists ŷ and neighborhood W (x) of x such that f(x, ŷ) > α for all x W (x), and therefore f x (x, ŷ) > α and x lev α f x (, ŷ). Payoff security is another notion of lower semi-continuity that was introduced by Reny [14] in a game theoretic setting. Definition 4 The function f : X Y IR is payoff secure in x, if for every (x, y) X Y and every ε > 0, there exists ŷ Y and a neighborhood W of x such that f(x, ŷ) f(x, y) ε, for all x W. Payoff security and transfer lower continuity are closely related via the following Lemma. Lemma 3 The function f : X Y IR is payoff secure in x, if and only if f is transfer lower continuous in x at any α. Proof. Suppose f is transfer lower continuous in x for all α. Then, for any (x, y) X Y and any ε > 0, f is transfer continuous at in x at f(x, y) ε, and therefore the existence of neighborhood W (x) and y Y such that f(x, ŷ) > f(x, y) ε for all x W (x). Hence, f is payoff secure. Now suppose f is payoff secure and f(x, y) > α. Let ε be such that f(x, y) ε > α. The payoff security of f implies the existence of neighborhood W (x) and y Y such that f(x, ŷ) > f(x, y) ε > α for all x W (x). Hence, f is transfer lower continuous at α. We also have the following property of payoff secure functions in terms of their lower level sets. Lemma 4 If f is payoff secure in x, then for every α, y Y levα f x (, y) = y Y levα f(, y). Proof. The fact that f x f implies that, for every α and every y, lev α f(, y) lev α f x (, y). The payoff security of f implies (via Lemmas 2 and 3) that for every α and every y, there exists ŷ Y such lev α f x (, ŷ) lev α f(, y), which implies that lev α f x (, y) lev α f(, y). y Y The conclusion of the Lemma follows immediately. The following notion of concavity is a very minor variation on the notion of 0-pair-concavity introduced by Chang [6]. 5
Definition 5 A function g : X Y IR is α-pair-concave in y, if for every finite collection {y 1,, y n } in Y, there exists a continuous function ϕ from n to X such that for any λ n, we have ϕ(λ) i I(λ) levα g(, y i ). 3. Main results As we claimed in the introduction, our first theorem generalizes the current existence results of minimax points. Theorem 1 Consider a function f : X Y IR and a fixed α IR. Assume the following: (i) there exists a function g : X Y IR such that for every y Y, there exists ŷ Y and a closed set F (y) X such that lev α g(, ŷ) F (y) lev α f(, y) (ii) there exists a finite collection of points K 0 Y such that y K 0 F (y) is compact Assume further that for every finite collection of point {y 1,, y n } in Y, there exists a function ϕ : n X such that (iii) for every i {1,, n}, the set G i = {λ n ϕ(λ) F (y i )} is closed in n (iv) for any λ n, we have ϕ(λ) lev α g(, y i ). i I(λ) Note that unlike similar results in Chang [6], Baye et al [3], and Tian [16], the above theorem does not impose an explicit continuity conditions on f. Instead, condition (i) requires f to be majorized, in a very weak sense, by some function g. In fact, as a result of Lemmas 2 and 3, condition (i) is weaker than requiring f to be transfer lower continuous in x at α, and hence it also weaker than requiring f to payoff secure in x. Condition (iv) by itself is weaker than α- pair-concavity in y since condition (iv) above does not require ϕ to be continuous. As we will see 6
from the proof of Theorem 1, the combination of (iii) and (iv) provides, in essence, the weakest conditions that will guarantee that the set y K levα g(, y) for any finite subset K of Y. We will also demonstrate in a subsequent corollary that the above theorem generalizes the results of Chang [6], and therefore it also generalizes similar results of Hou [10], Baye et al [3], and Tian [16]. It is possible to introduce conditions (iii) and (iv) as a separate and new notion of transfer quasiconcavity. However, we choose not do so since the literature is already cluttered with such notions (diagonal quasi-concavity, α-diagonal quasi-concavity, C-quasi-concavity, 0-pair-concavity). Proof of Theorem 1. Clearly, y Y F (y) implies y Y levα f(, y), and hence there exists ˆx X such that sup y Y f(ˆx, y) α. Therefore, we only need to show that y Y We define, for every y Y, the following set A 0 = F (y), y K 0 F (y). where K 0 is the finite set in assumption (ii). We also define, for every y Y, the set A(y) = F (y) A 0. For every y, A(y) is a closed subset of A 0. We now show that for every finite set K Y, y K F (y), which in turn implies, via the finite intersection property of closed subsets of a compact set, that y Y F (y) and we are done. Let K be some finite collection of points {y 1,, y n } in Y. By assumption (iii), for every i {1,, n}, there exists a function ϕ such that is closed in n. G i = {λ n ϕ(λ) F (y i )} Furthermore, assumptions (i) and (iv) of implies that for any indexing set {i 1,, i J } that is subset of {1,, n}, and for any λ n such that λ con{e i1,..., e ij }, we have and hence ϕ(λ) lev α g(, ŷ i1 ) lev α g(, ŷ ij ), ϕ(λ) G i1 G ij. (6) 7
We now have demonstrated that the collection of sets G 1,, G n satisfies all the assumptions of the KKM theorem (see Appendix A), and therefore n i=1 G i. Hence, y K F (y). Since K was an arbitrary finite subset of Y, this completes the proof. Corollary 1 Let f : X Y IR be transfer lower continuous in x at α. Assume that there is finite subset K 0 Y such that y K 0 lev α f(, y) is compact. Assume further that for every finite collection {y 1,, y n } in Y, there exists a function ϕ : n X such that (i) for every i, the set G i = {λ n f x (ϕ(λ), y i ) α} is closed. (ii) for any λ n, we have ϕ(λ) lev α f(, y i ) i I(λ) Then, there exists ˆx X such that sup y Y f(ˆx, y) α Proof. Take g = f x, F (y) = lev α f x (, y), and apply Lemma 2 and Theorem 1. Note that Lemma 2 also implies that if there is a finite K 0 such that y K 0 lev α f(, y) is compact, then there is a finite K such that y K 0 lev α f(, y) is also compact. As we indicated earlier, we also obtain Theorem 3.1 in Chang [6] a corollary of Theorem 1. Corollary 2 Let f : X Y IR be transfer lower continuous in x at α. Assume there is a finite collection of point K Y such that y K levα f(, ŷ) is compact. Then, there exists ˆx X such that sup y Y f(ˆx, y) α, if and only f is α-pair-concave in y. Proof. The α-pair-concave in y, the lower semi-continuity of f x in x, and the continuity of ϕ imply, for any {y 1,, y n }, the existence of ϕ satisfying conditions (i) and (ii) of Corollary 1. Hence, one direction of the claim follows from Corollary 1. The other direction of the claim is obtained by simply taking ϕ(λ) ˆx for all λ n and using the fact that f x f. Combining Corollary 1 with Lemma 3, we obtain a corollary which will be useful in establishing the existence of a constant solution satisfying (3). Corollary 3 Consider a function payoff secure f : X Y IR in x and a fixed α IR. Assume there is a finite set K 0 Y such that y Y levα f(, y) is compact. Assume further that for any 8
collection {y 1,, y n } of points in Y, there exists a function ϕ satisfying conditions (i) and (ii) of Corollary 1 at α. Then, there exists ˆx X such that sup y Y f(ˆx, y) α. Proof. Apply Corollary 1 to f x to obtain the existence of a point ˆx X such that sup y Y f x (ˆx, y) α. By Lemma 4, this implies that sup y Y f(ˆx, y) α. Our next theorem provides conditions that guarantee the existence of a constant function that is a minimizer of problem (2) in the introduction. Theorem 2 Consider a collection C of functions from Y to a compact space X. For each c C, let α c = sup y Y f(c(y), y), and let α = inf c C α c. Assume (a) f is payoff secure in x (b) for every c C, such that for every finite collection {y 1,, y n } in Y, there exists a function ϕ : n IR such that conditions (i) and (ii) of Corollary 1 hold at α = f(c(y c ), y c ) for some y c Y. Then, there exists ˆx X such that sup f(ˆx, y) α. (7) y Y Proof. Consider a sequence of functions c n and corresponding sequence of {α cn } of decreasing real numbers such that α < α cn+1 < α cn, and α cn α. For each y Y and each α cn, consider the set lev αcn f x (, y). Due to the lower semi-continuity of f x in x, and for every y Y and every α cn, lev αcn f x (, y) is a closed and compact subset of X. The function f is payoff secure in x, and by Lemma 3, it is transfer lower continuous at α cn. Condition (b), implies that conditions (i) of (ii) of Corollary 1 hold at f(c n (y cn ), y cn ), and hence they also hold at α cn. Therefore, Corollary 1 implies that, for every n 1, that the set y Y levαcn f x (, y) is not empty. Furthermore, these sets are compact and nested. In particular, for every n 1 we have Therefore, lev αcn f x (, y) lev αc n+1 f x (, y). y Y y Y ( lev αcn f x (, y)), n 1 y Y and there exists ˆx X such that sup y Y f x (ˆx, y) α, and by Lemma 4, sup y Y f(ˆx, y) α. 9
In the above proof, if the collection C contains the constant function ĉ(y) ˆx (or if it contains all constant functions), then clearly ĉ satisfies (2). Example 1: Let X = Y = [0, 1] and consider the function f : [0, 1] Y IR defined by 0 when ψ(x) < y, f(x, y) = 5 when ψ(x) = y and when y = 0, 10 when ψ(x) > y and y 0, where ψ is a function from [0, 1] to [0, 1] with ψ(x) = 0 for some x [0, 1]. The function f is payoff secure. Furthermore, for any function c : Y [0, 1] and for any {y 1,, y n } in Y, f satisfies condition (b) of Theorem 2 (take ϕ(λ) x). Therefore, Theorem 2 implies that ˆx such that sup f(ˆx, y) = inf y sup c C y where C is the collection of all functions from Y onto [0, 1]. 4. Applications f(c(y), y), Theorem 1 immediately leads to yet another characterization of the existence of pure strategy equilibria à al Baye et al, Hou, and Chang. Consider a game G with I players each with strategy set X i U where U is a Hausdorff topological vector space. Let X = I i=1 X i be the strategy set of the game, and let u i : X IR be the payoff of player i. Let x i denote the moves of all players other than player i. We define an aggregator function U : X X IR as U(x, y) = I i=1 [u i(y i, x i ) u i (x)]. By the construction of U, a point x X is minsup of U, if and only if it is a pure Nash equilibrium of G. Therefore, by applying Theorem 1 to U, we obtain Theorem 3 Let U : X X IR be the aggregator function of a game defined above, and assume there exists a finite set K 0 Y such y K 0 lev 0 U(, y) is compact. Assume that there exists a function V : X X such that for every y Y, there exists ŷ Y and closed set F (y) such that lev 0 V (, ŷ) F (y) lev 0 U(, y). Assume further that for every {y 1,, y n }, there exists a function ϕ : n X such (i) for every i {1,, n}, the set G i = {λ n ϕ(λ) F (y i )} 10
is closed in n (ii) for λ n, we have ϕ(λ) lev α V (, y i ). i I(λ) Then, the game G has a pure strategy equilibrium. We can also use Theorem 1 to provide conditions for the existence of pure Nash equilibria that can be imposed directly on the individual payoff functions rather than on the aggregator function of the game. As we mentioned in the introduction, this is important since properties such as transfer lower continuity and transfer quasi-concavity might hold for each payoff function, and yet fail to hold for the aggregator function. Theorem 4 Consider a two-player game defined on X 1 X 2. For every player i, assume there exists a function g i : X i X i IR and a real number α i such that for every x i, exist a closed set F (x i ) and a point ˆx i such that (i) lev α i g i (ˆx i, ) F (x i ) lev α i u i (x i, ) (ii) there exists a finite collection of points K 0 X i such that x K 0 F (x) is compact and nonempty Moreover, assume that for each i {1, 2}, for every finite collection {y 1,, y n } in X i, there exists a function ϕ i : X i such that (iii) for every k {1,, n}, the set G k = {λ n ϕ i (λ) F (y k )} is closed in n (iv) for any λ n, we have Assume further that ϕ i (λ) lev α g i (y k, ). k I(λ) (v) u 1 (x 1, x 2 ) + u 2 (x 2, x 1 ) α 1 + α 2 for any (x 1, x 2 ) X 1 X 2. Then, the game G has a pure strategy Nash equilibrium (x 1, x 2 ) which solves the system of equations u 1 (x 1, x 2 ) = α 1 and u 2 (x 1, x 2 ) = α 2. Proof. By Theorem 1, there exists x 1 and x 2 such that sup u 1 (x 1, x 2) α 1 and sup u 2 (x 1, x 2 ) α 2, x 1 X 1 x 2 X 2 11
Hence, by assumption (v), we have and Hence, which implies and therefore, Thus sup x 2 X 2 [α 1 + α 2 u 1 (x 1, x 2 )] α 2 α 1 + α 2 + sup x 2 X 2 [ u 1 (x 1, x 2 )] α 2 α 1 α 2 + inf u 1 (x 1, x 2 ) α 2, x 2 X 2 inf x 2 X 2 u 1 (x 1, x 2 ) α 1, u 1 (x 1, x 2) = sup u 1 (x 1, x 2) = inf u 1 (x 1, x 2 ). x 1 X 1 x 2 X 2 u 1 (x 1, x 2) = sup x 1 X 1 u 1 (x 1, x 2) The same exact argument can be used to show that u 2 (x 2, x 1) = sup x 2 X 2 u 2 (x 2, x 1 ), and (x 1, x 2 ) is a pure strategy Nash equilibrium. Moreover, u 1 (ˆx 1, x 2) α 1 and u 2 (x 2, x 1) α 2 and u 1 (x 1, x 2) + u 2 (x 2, x 1) α 1 + α 2. The last three inequalities imply that u 1 (x 1, x 2 ) = α 1 and u 2 (x 2, x 1 ) = α 2. Note that Theorems 3 and 4 impose no convexity or compactness conditions on the strategy sets. Note further, that Theorem 4 does not require the game to be a zero-sum game. 5 In the next section, we apply Theorem 4 to voting games. 5 More generally, Theorem 4 does not require the game to be reciprocally upper semi-continuous (Reny [14]). 12
5. Pure strategy Equilibrium in PSVM Probabilistic Spatial Voting Models (PSVM) are essentially Hotelling models of spatial competition between candidates where the voters ideal platforms are randomly distributed over a subset of the real line. As an application to the results of the previous section, we consider the following PSVM model that leads to a game with discontinuous and non-quasi-concave payoffs. Consider two candidates RED and BLUE. Both candidates simultaneously announce their platforms as two points in the interval [0, 1]. We assume full commitment to the announced policies. The probability that candidate RED wins the elections, when RED and BLUE respectively announce the platforms (r, b), is given by the function P (a, b), and thus the probability of BLUE winning is 1 P (r, b). We further assume that the candidates can derive the function P given the information they have about the voters. For example, suppose there exists a pivotal voter (a median voter for example) such that a candidate wins the election, if and only if this candidate gets the vote of this pivotal voter. The function P can then represent a situation where the candidates view the location of the ideal platform of the pivotal voter as a random variable with values in [0, 1]. Three minimal assumptions are commonly imposed on P (see Ball [2], Hansson and Stuart [9]): A1) 0 < P (r, b) < 1 A2) P (r, b) = 1 P (b, r) A3) For r < b, P (r, b) is non-decreasing in r and non-decreasing in b, and for b < r, P is nonincreasing in r and non-increasing in b The monotonicity assumption A3 says that if a candidate moves his platform closer to his opponent s, then he will not decrease his chances of winning the election. Similarly, if he moves his platform away from the his opponent s strategy, he will not increase his chances for winning the elections. Ball [2] pointed out that a function P satisfying A1-A3 is either constant with value 1 2 over all of [0, 1] [0, 1], or it is discontinuous on the diagonal of [0, 1] [0, 1]. In other words, in any interesting application of this model, the function P is discontinuous on the diagonal. Therefore, we can simply assume that the probability of RED winning is given by the function of the form 13
f(r, b) when r < b, P (r, b) = 1 2 when r = b = z, g(r, b) when r > b. We also assume voters have single-peaked utility functions, and we assume that the location of the ideal point (the peak of the utility function) of the median voter is a random variable with a uniform distribution of the interval [0, 1]. This setting yields a winning probability function for RED of the following form (see section 4 of Ball [2]) r+b 2 when r < b, P (r, b) = 1 2 when r = b = z, 1 r+b 2 when r > b. It is clear that the function P described above satisfies conditions A1 through A3. Assume further that both candidates have mixed motives in the sense that they care about winning the office and about the policy that will be implemented by the winner. More specifically, we assume that in addition to the utilities of winning K R and K B, candidates have respectively utilities of u R (z) and u B (z) when the policy z [0, 1] is implemented regardless to who wins the elections. Therefore, the expected payoffs of the candidates are U R (r, b) = P (r, b)[u R (r) + K R ] + (1 P (r, b))u R (b) and U B (b, r) = (1 P (r, b))[u B (b) + K B ] + P (r, b)u B (r). Or alternatively, U R (r, b) = P (r, b)[u R (r) + K R u R (b)] + u R (b) and U B (b, r) = P (r, b))[u B (r) u B (b) K B ] + u B (b) + K B. Note that when either K A or K B is not zero, the discontinuity in P (r, b) leads to a discontinuity in the payoff functions of the candidates. Finally, let u B (z) = a B z 1, u R (z) = a R z and K R = K B = K. 14
Using the setting of our previous section, the payoffs of RED and BLUE can be expressed as r+b 2 ( r + b + K) + a R b when r < b, U R (r, b) = 1 2 K + a R z when r = b = z, (1 r+b 2 )( r + b + K) + a R b when r > b. (1 r+b 2 )(r b K) + a B + b 1 + K when b < r, U B (b, r) = 1 2 K + a B + z 1 when r = b = z, r+b 2 (r b K) + a B + b 1 + K when b > r. The existence of mixed strategy equilibria for the voting games discussed above was investigated by Ball [2], Saporiti [15], and Duggan [7] among others. In this paper, however, we are only interested in the existence of pure strategy equilibrium for such games. (8) (9) When u 1 = u 2 0, K > 0 and candidates care only about winning the election, it is straight forward to show that assumptions A1-A3 imply that (r, b ) = (0.5, 0.5) is the only pure strategy equilibrium for our game. This is the so-called platforms convergence result where both candidates choose the policy corresponding to the expected location of the median voter (recall in our model 0.5 is the expected location of the median voter). When the candidates don t care about winning per say and they only care about the policy that will be implemented by the winner (i.e. K = 0), the platforms convergence result does not hold (see Hansson and Stuart [9]). Moreover, when K = 0, the payoff of both candidates are in fact continuous everywhere. As the value of K starts to increase above zero, a discontinuity is introduced to the candidates payoffs. Furthermore, given a value of K, and for some values of b and a, the functions U R (, b) and U B (, r) may fail to be quasi-concave. As Saporiti [15] pointed out, this lack of quasi-concavity can potentially lead to the non-existence of pure strategy equilibria despite the fact that the voting games under consideration are in fact better reply secure. More specifically, Theorem 3.1 in Reny [14] and related results in Carmona [5], two important tools in establishing the existence of pure strategy equilibria in better reply secure games, cannot be applied due to the lack of quasi-concavity. Note also that the functions U R and U B are not diagonal quasi-concave à la Baye et al (Definition 2 in Baye et al [3]), and therefore, their results cannot be applied here. Furthermore, the second order conditions 2 U R (r,b) r 2 0 and 2 U B (r,b) b 2 0 hold only almost everywhere on [0, 1]. In fact, they don t hold at points where x = y because these partial derivatives are not defined at such points. Therefore, it is not possible to find an equilibrium for this game 15
by constructing the best reply functions of the players using first order conditions. The problem with such approach is that the first order (necessary) conditions for the maximizers of the function U R (, b) and U B (, r) may no longer be sufficient if the second order conditions fail at a single point (see Section B in the appendix for an example). 6 Therefore, we establish the existence of a pure strategy equilibrium using Theorem 4. Let U R be the lower semi-continuous closure of U R in b. The functions U R and U R are different only at points of discontinuity where r = b = z 0.5. Moreover, from (8), we can see that any such point, we either have lim b z U R (z, b) > U(z, z), or we have lim b z + U R (z, b) > U(z, z). Hence, for every r [0, 1], and for any α, there exists ˆr [0, 1] such that lev α U R (ˆr, ) lev α U R (r, ). (10) Letting F (r) = lev α U R (ˆr, b), we can claim that assumptions (i), and (ii) of Theorem 4 hold for any α. For any collection {r 1,, r n }, there exists ϕ : n [0, 1] such that conditions (iii) and (iv) of Theorem 4 hold for α 1 = a R + 0.5(K 1) (see Appendix C for details). Similarly, we can show that U B also satisfies assumptions (i) through (iv) of Theorem 4 for α 2 = a B + 0.5(K 1). Moreover, straight forward calculations shows that U R + U B α 1 + α 2. (11) Hence, by Theorem 4, the game has a pure strategy Nash equilibrium. Furthermore, Theorem 4 provides us with a systematic method to find these equilibria. For K = 0, α 1 = 0.5 + a R and α 2 = 0.5 + a B. We start by solving the system U R (r, b) = 0.5 + a R U B (b, r) = 0.5 + a B, and among all the solutions that we find, we look for points (r, b ) where sup U R (r, b ) 0.5 + a R and r [0,1] sup U B (b, r ) 0.5 + a B. b [0,1] 6 This suggests that the approach for finding pure Nash equilibria taken in section 3.2 in [15] and the approach taken in [2] for constructing the best reply functions are both incomplete. More specifically, footnote 10 in [2] and footnote 19 in [15] are incorrect because the second derivative inequalities in these footnotes hold almost every where instead of holding everywhere. 16
The leads to the equilibrium (r, b ) = (0, 1) (independent of a R and a B ), and as expected we have an extreme divergence of platforms. Hence, we obtain the same results of Hansson and Stuart [9] but without any differentiability assumptions on the function P. For 0 < K < 1, repeating the same process leads to an equilibrium at (r, b ) = (0.5K, 1 0.5K). For example, when K = 0.5, (r, b ) = (0.25, 0.75) is an equilibrium. For K 1, the point (r, b ) = (0.5, 0.5) is an equilibrium, and the platforms for both candidates are the same. These result provide a concrete demonstration of the-convergence-of-platforms results when candidates office motivation surpasses a certain threshold. We emphasize again that unlike the results of Baye et al [3], Tian [16], and Chang [6], our proof does not require the construction of the aggregator function of the game. Moreover, the same proof works when the strategy set of each candidate is a finite (discrete) subset of the real line instead of the entire interval [0, 1]. Assuming that the location of the median vote is uniformly distributed simplifies our calculations to obtain (8) and (9) but it is not essential for the application of Theorem 4. The same is true for the specific functional forms we used for u R and u B. The only critical assumption is assuming K R = K B = K. When K R K B, we may not be able verify the assumptions of Theorem 4. In fact, Saporiti [15] suggests that when the difference between K R and K B is large enough, pure strategy equilibria will fail to exist. 17
REFERENCES REFERENCES References [1] Jean-Pierre Aubin. Mathematical Methods of Game and Economic Theory. Dover, 2007. [2] R. Ball. Discontinuity and non-existence of equilibirum in the probabilistic spatial voting model. Social Choice and and Welfare, 16:533 555, 1999. [3] Michael Baye, Guoqiang Tian, and Jianxin Zhou. Characterization of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs. Review of Economic Studies, 60:935 948, 1993. [4] Philip Bich. Existence of pure nash equilibria in discontinuous and non quasiconcave games. International Journal of Game Theory, 38:395 410, 2009. [5] G. Carmona. An existence result for dicontinuous games. Journal of Economic Theory, 144:1333 1340, 2009. [6] Shiow-Yu Chang. Inequalities and nash equilibria. Nonlinear Analysis, 73:2933 2940, 2010. [7] J. Duggan. Equilibrium existence for zero sum games and spatial models of elections. Games and Economic Behavior, 60:52 74, 2007. [8] L.P. Hansen and T. J. Sargent. Robust control and model uncertainty. American Economic Review, 91:60 66, 2001. [9] I. Hansson and C. Stuart. Voting competitions with interested politicians: Paltforms do not converge to the prefrences of the median voter. Public choice, 44:431 441, 1984. [10] Ji-Cheng Hou. Characterization of the existence of pure strategy nash equilibrium. Applied Mathematics Letters, 22:689 692, 2009. [11] E. Kalmoun and H. Riahi. Topological kkm theorems and generalized vector equilibira on g-connvex spaces with applications. 129:1335 1348, 2001. Proceedings of the American Mathematical Soceity, 18
REFERENCES REFERENCES [12] W. K. Kim and S. Kum. Existence of nash equilibria with c-convexity. Nonlinear Analysis, 63:1857 1865, 2005. [13] K. Nishimura and J. Friedman. Existence of nash equilibrium in n person games without quasi-concavity. International Economic Review, 22:637 648, 1981. [14] Philip Reny. On the existence of equilibrium in discontinuous games. Econometrica, 167:1029 1056, 1999. [15] Alejandro Saporiti. Existence and uniquness of nash equilibrium in electoral competition game: The hybrid case. Journal of Public Economic Theory, 10:827 857, 2008. [16] Guoqiang Tian. Generalization of the fkkm theorem and the ky fan minimax inequality, with applications to maximal elements, price equilibrium, and complementarrity. Journal of Mathematical Analysis and Applications, 170:457 471, 1992. [17] N. Williams. Robust control. New Palgrave Dictionary of Economics, 2008. [18] Jian Xin Zhou and Goong Chen. Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities. The Journal of Mathematical Analysis and Applications, 132:213 225, 1988. [19] A. Ziad. A counterexample to 0-diagonal quasiconcavity in a minimax inequality. Journal of Optimization Theory and Applications, 22:457 462, 2001. 19
REFERENCES REFERENCES Appendix A. The KKM Theorem Theorem A1. Let m denote the standard simplex in IR m, and let e k be the vector of m corresponding to the vector in IR m with one in the kth component and zeros everywhere else. Consider m subsets F i of the simplex m such that for (i) each F i is closed (i) for any J m and any face of of m of the form con{e i0, e i1,, e ij } is contained in the union F i0 Fi1 FiJ Then, the set m i=1 F i is not empty. For an excellent reference on the connection between KKM lemma, the classic Key Fan inequality, and the Brouwer and Kakutani fixed point theorems, see the appendix of [1] B. Lack of sufficiency when the second order condition fails at a single point The following example illustrates the difficulty in establishing best reply functions for the game PSVM in Section 4. Example B1: Consider the following function f : IR IR Clearly, the condition d2 f dx 2 f(x) = { x 2 when x < 1 x 2 + 4 when x 1 < 0 holds everywhere except at the point x = 1, and yet the (only) critical point x = 0 obtained via the first order condition df dx = 0, is not a maximum. C. The functions U R and U B satisfy the assumptions of Theorem 1. The first panel in the Figure C.1 below shows a region in [0, 1] [0, 1] where U R (r, b) α 1. In particular, using (8) and simple constrained maximization techniques, we can easily show that when 1 2 b 1, 0 r 1 2, and r + b 1, we have U R(r, b) α 1. Similarly, when 1 2 r 1, 0 b 1 2, and r + b 1, we also have U R (r, b) α 1. Consider the function c : [0, 1] [0, 1] defined as c(z) = 1 z. For any finite collection {r 1,, r n }, and any λ n, let ϕ(λ) = c(λ 1 r 1 + +λ n r n ), then the continuity of ϕ, the lower semi-continuity of U R in b, and Lemma 1, imply that for any 20
REFERENCES REFERENCES k {1,, n}, the set G k = {λ n ϕ(λ) lev α 1 U R (r k, )} is closed in n. Furthermore, the first panel of Figure C1 implies that for any r in the convex hull of {r 1,, r n }, we have which implies that for λ n, min k I(λ) U R(r k, c( r)) α 1, ϕ(λ) k I(λ) lev α 1 U R (r k, ). A similar argument using the second panel in Figure C1 can be used to show that U B satisfies the conditions of Theorem 1 at α 2. b b α α α r 1 2 U R 1 2 U B α r Fig. C1 21