Discontinuous stable games and efficient Nash equilibria

Economics Letters 115 (2012) 387–389

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Discontinuous stable games and efficient Nash equilibria Vincenzo Scalzo ∗ Dipartimento di Matematica e Statistica, Università di Napoli Federico II, via Cinthia (Monte S. Angelo), 80126 Napoli, Italy

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Article history: Received 16 February 2011 Received in revised form 30 November 2011 Accepted 19 December 2011 Available online 29 December 2011

abstract In the recent paper by the author [Scalzo, V., 2010. Pareto efficient Nash equilibria in discontinuous games. Economics Letters 107, 364–365], a class of discontinuous games where efficient Nash equilibria exist has been defined. In the present paper, we complete the previous investigation and recognize a class of discontinuous games where the efficient Nash equilibria are stable with respect to perturbations of the characteristics of players. © 2011 Elsevier B.V. All rights reserved.

JEL classification: C72 Keywords: Efficient Nash equilibria Stability of equilibria Discontinuous games

In recent years, the study of games where the payoffs of players are not continuous functions (shortly: discontinuous games) has captured the attention of many authors and several papers have been devoted to existence and stability properties (under perturbations on characteristics of players) of Nash equilibria: among the wide literature, see: Dasgupta and Maskin (1986), Tian and Zhou (1995), Reny (1999), Morgan and Scalzo (2004), Yu et al. (2007), Morgan and Scalzo (2007, 2008), Carmona (2009), Scalzo (2009b) and Carmona (2011) and references therein. More recently, in Scalzo (2010) it has been recognized a class of discontinuous games which admit efficient Nash equilibria, that are Nash equilibria which maximize the sum of payoffs— such equilibria are also Pareto efficient. Let us underline that the efficiency of Nash equilibria concerns the robustness of noncooperative equilibria under a decision making process where players can cooperate in order to refine inside the set of Nash equilibria; see Auman (1959) and Bernheim et al. (1987). Note that a Nash equilibrium is not necessarily efficient and a profile of strategies which maximizes the sum of payoffs is not necessarily a Nash equilibrium; see the well known Prisoner’s Dilemma. The aim of this note is to complete the investigation started in Scalzo (2010) on the efficient Nash equilibria in the framework of discontinuous games; so, we study a stability property under perturbations on the characteristics of players. More precisely, we define a class of discontinuous games Y, strictly included in the class introduced in Scalzo (2010), where any game has efficient

∗

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0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.12.089

Nash equilibria and is stable in the sense of the following property: if (Gα )α is a net of games converging to G in the space Y and if xα is an efficient Nash equilibrium of Gα for any α , then (xα )α has a subnet converging to an efficient Nash equilibrium of G—the convergence of games is meant in the sense of sup-norms. Such a stability property, also known as Hadamard well-posedness, has been obtained in previous papers for other classes of discontinuous games with respect to (non-efficient) Nash equilibria; see Yu et al. (2007) and Scalzo (2009b). Now, let us recall the setting. An n-person strategic form game – shortly: game – is a set of data G = {X1 , . . . , Xn , f1 , . . . , fn } where, for any i, Xi is the (non-empty) set of strategies of player i and fi is the real-valued function which gives n to i the wins that depend on the choices of all players: fi : X = i=1 Xi −→ R. The elements of X are said strategy profiles and the generic strategy  profile x is also denoted by x = (xi , x−i ), where xi ∈ Xi and x−i ∈ j̸=i Xj . A Nash equilibrium – (Nash, 1950) – of G is a strategy profile x∗ such that, for any player i, fi (x∗i , x∗−i ) ≥ fi (xi , x∗−i ) for any xi ∈ Xi ; a strategy profile if there are no strategy profiles x such xˆn is said to be efficient n that i=1 fi (x) > x). i=1 fi (ˆ The existence of Nash equilibria which are also efficient (efficient Nash equilibria) has been obtained in Scalzo (2010, Theorem 1) for discontinuous games G such that the following assumptions hold: (A1) X is a compact and convex subset of a topological vector space; n (A2) the function F : x ∈ X −→ i=1 fi (x) is transfer weakly upper continuous;

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V. Scalzo / Economics Letters 115 (2012) 387–389

(A3) the function U (·, z ) is strictly quasi-concave for any z ∈ X , that is: for any x1 , x2 , z ∈ X with x1 ̸= x2 we have: U xλ , z > min U x1 , z , U x2 , z





 

λ







∀ λ ∈]0, 1[

where x = (1 − λ)x + λx ; (A4) if x is a maximizer of F and if z ∈ X − {x}, there exists λ0 ∈]0, 1[ such that U (x, x) ≥ U z λ0 , x , with z λ0 = (1 − λ0 )z + λ0 x; 1

2

where U is the function defined on X × X by U (x, x′ ) =  n ′ i=1 fi (xi , x−i ). We recall that a function F : X −→ R is said to

be transfer weakly upper continuous – see Tian and Zhou (1995) – if F (x′ ) > F (x) implies that there exists a neighborhood Nx of x and x′′ ∈ X such that F (x′′ ) ≥ F (z ) for any z ∈ Nx . The assumptions above guarantee that any maximizer of F is a Nash equilibrium of G. On the other hand, maximizers of F exist since a real-valued function has maximizers over a compact set if and only if it is transfer weakly upper continuous—see Tian and Zhou (1995, Theorem 1). Here we are interested in a stability property related to efficient Nash equilibria of discontinuous games when the characteristics of players change. As usual, the evaluation of changes is given by means of the metric ρ induced by the sup-norms of payoffs recalled below, where G and G′ are games with bounded payoffs defined on X : n

ρ(G, G′ ) =



sup fi (x) − fi′ (x) .





i=1 x∈X

We consider n-person games having a common set X of strategy profiles and bounded payoffs. Now, a game G is said to be stable if, given a net of games (Gα )α such that ρ(G, Gα ) −→ 0 and a net of strategy profiles (xα )α such that xα is an efficient Nash equilibrium of Gα for any α , there exists a subnet of (xα ) which converges to an efficient Nash equilibrium of G. First, we remark that one does not have to expect that this stability property be verified in the class of discontinuous games satisfying the assumptions A1–A4 since, in general, the maximum points of transfer weakly upper continuous functions are not stable: in fact, let F and Fk – k ∈ N and k ≥ 2—be the transfer weakly upper continuous functions defined below: F (x) =

Fk (x) =



x x−1

if x ∈ [0, 1[ if x ∈ [1, 2]

and



 kx     k   −1

if x ∈ 0, 1 −

 1     

if x ∈ 1 −

x−1



1 k

1



k



,1

if x ∈ [1, 2].

One has: δ(F , Fk ) = supx∈X |F (x) − Fk (x)| −→ 0; xk = 1 − 1k is a maximum point of Fk for each k ≥ 2; the sequence (xk )k converges to 1 which is not a maximizer of F . Hence we need to consider games where the sum of payoffs has a property stronger than transfer weakly upper continuity. We use the upper pseudocontinuity: F : X −→ R is upper pseudocontinuous – (Morgan and Scalzo, 2004, 2007; Scalzo, 2009a) – if 1 F (x) < F (x′ ) H⇒ lim sup F (xα ) < F (x′ )

for any net (xα )α converging to x.

As pointed out by examples in Morgan and Scalzo (2004, 2007), any upper semicontinuous function is also upper pseudocontinuous but the converse is not true.2

1 lim sup F (xα ) = inf α α αo ∈A supα≥αo F (x ) where A is the index set of the net (x )α . 2 Similarly, the lower pseudocontinuity is defined and a function which is both upper and lower pseudocontinuous is said to be pseudocontinuous; we

The upper pseudocontinuity guarantees the stability of maximum points, as stated by the following result. First, we recall that a set-valued function T : Y ⇒ X is said to be closed if its graph Γ (T ) = {(y, x) : x ∈ T (y)} is a closed set. Proposition 1. Let Y be the set of bounded and upper pseudocontinuous function F defined on a compact topological space X and let T be the set-valued function defined on Y by T (F ) = arg max(F , X ).3 Then, T is closed with respect to the topology on Y induced by the metric δ . Moreover, given a net of functions (Fα )α such that δ(F , Fα ) −→ 0 and a net (xα )α such that xα ∈ T (Fα ) for any α , there exists a subnet of (xα ) which converges to a maximum point of F . Proof. By contradiction, assume that T is not closed. So, there exist:

• F ∈ Y and x ∈ X such that x ̸∈ T (F ); • a net (Fα )α converging to F in the metric δ ; • a net (xα )α converging to x such that xα ∈ T (Fα ) for any α . Since x is not a maximizer of F , we get F (x) < F (x′ ) for some x′ ∈ X and, in light of the upper pseudocontinuity of the function F , we have: lim sup F (xα ) < t < F (x′ )

(1)

for a suitable real number t. Since δ(F , Fα ) −→ 0, from the first inequality of (1) we obtain F (xα ) + δ(Fα , F ) < t for any index α greater than a suitable αo , which implies F (xα ) + Fα (xα ) − F (xα ) < t. The second inequality of (1) implies that t < Fα (x′ ) for any α greater than an index α1 ≥ αo and we have Fα (xα ) < Fα (x′ ) for any α ≥ α1 , which contradicts xα ∈ T (Fα ). Finally, since X is compact and T is closed, it is easy to show the last sentence.  Now, let Y be the class of discontinuous games satisfying assumptions A1, A3, A4 and having upper pseudocontinuous sums of payoffs. Since δ(F , F ′ ) ≤ ρ(G, G′ ) for games G and G′ , where F and F ′ denote the respective sums of payoffs, by virtue of Proposition 1, we have that our stability property is verified in Y: Proposition 2. Any game belonging to Y is stable. We remark that the upper pseudocontinuity of the sum of payoffs considered in Proposition 2 can be obtained for games where any payoff is less than upper pseudocontinuous; moreover, games having upper pseudocontinuous payoffs could not have an upper pseudocontinuous sum of payoffs; see the examples below. Example 1. Consider the Dirichlet’s functions on [0, 1] defined by: f1 (x) = 0 if x is rational and f1 (x) = 1 otherwise, f2 (x) = 1 if x is rational and f2 (x) = 0 otherwise. These functions are less than upper pseudocontinuous but f1 + f2 is constant. Example 2. Consider the functions f1 and f2 defined on [0, 1] by f1 (x) = 2x − 1 for any x and f 2 ( x) =



1−x −1

if x ∈ [0, 1[ if x = 1.

These functions are upper pseudocontinuous – f1 is even continuous – but the sum is neither upper pseudocontinuous nor transfer weakly upper continuous:

note that pseudocontinuity is the common topological property of the numerical representations of a continuous, total and transitive binary relations; see Morgan and Scalzo (2007, Proposition 2.2). 3 The set of maximum points of F over X , denoted by arg max(F , X ), is non-empty in light of Morgan and Scalzo (2007, Proposition 2.1).

V. Scalzo / Economics Letters 115 (2012) 387–389

(f1 + f2 )(x) =



x 0

if x ∈ [0, 1[ if x = 1.

We conclude observing that a stability-like property for (nonefficient) Nash equilibria in the setting of discontinuous games has been studied in Morgan and Scalzo (2004, 2008) from a different viewpoint: in these previous papers, the payoffs depend on an exogenous parameter, which is a variable inside the payoffs, and the stability of Nash equilibria is meant with respect to changes in values of the parameter; so, results have been obtained by using convergence properties of the payoffs with respect to the parameter. The point of view of the present paper is different from Morgan and Scalzo (2004, 2008) since here we recognize a metric space of discontinuous games described only by the characteristics of players in which a stability property is obtained with efficient Nash equilibria and the evaluation of the changes is given by means of the distance of the payoffs of games. References Auman, J.P., 1959. Acceptable points in general cooperative n-person games. In: Tucker, A.W., Luce, R.D. (Eds.), Contributions to the Theory of Games IV. Princeton University Press, Princeton.

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Bernheim, B.D., Peleg, B., Whinston, M.D., 1987. Coalition-proof Nash equilibria I. Concepts. Journal of Economic Theory 42 (1), 1–12. Carmona, G., 2009. An existence result for discontinuous games. Journal of Economic Theory 144, 1333–1340. Carmona, G., 2011. Understanding some recent existence results for discontinuous games. Economic Theory 48 (1), 31–45. Dasgupta, P., Maskin, E., 1986. The existence of equilibrium in discontinuous economic games, I: Theory. Review of Economic Studies 53, 1–26. Morgan, J., Scalzo, V., 2004. Pseudocontinuity in optimization and non-zero sum games. Journal of Optimization Theory and Applications 120 (1), 181–197. Morgan, J., Scalzo, V., 2007. Pseudocontinuous functions and existence of Nash equilibria. Journal of Mathematical Economics 43 (2), 174–183. Morgan, J., Scalzo, V., 2008. Variational stability of social Nash equilibria. International Game Theory Review 10 (1), 17–24. Nash, J., 1950. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences of the USA 36, 48–49. Reny, P.J., 1999. On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67 (5), 1029–1056. Scalzo, V., 2009a. Pseudocontinuity is necessary and sufficient for order-preserving continuous representations. Real Analysis Exchange 34 (1), 239–248. Scalzo, V., 2009b. Hadamard well-posedness in discontinuous non-cooperative games. Journal of Mathematical Analysis and Applications 360, 697–703. Scalzo, V., 2010. Pareto efficient Nash equilibria in discontinuous games. Economics Letters 107, 364–365. Tian, G., Zhou, J., 1995. Transfer continuities, generalizations of the Weierstrass and maximum theorems: a full characterization. Journal of Mathematical Economics 24, 281–303. Yu, J., Yang, H., Yu, C., 2007. Well-posed Ky Fan’s point, quasi-variational inequality and Nash equilibrium problems. Nonlinear Analysis 66, 777–790.