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Generalized weak transfer continuity and the Nash equilibrium
Journal of Mathematical Economics 47 (2011) 659–662
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Generalized weak transfer continuity and the Nash equilibrium Rabia Nessah IESEG School of Management (CNRS-LEM), 3 rue de la Digue, 59000 Lille, France
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Article history: Received 26 January 2011 Received in revised form 12 August 2011 Accepted 23 August 2011 Available online 31 August 2011
abstract This paper introduces the notion of generalized weak transfer continuity and establishes that a bounded, compact locally convex metric quasiconcave and generalized weak transfer continuous game has a Nash equilibrium. Our equilibrium existence result neither implies nor is implied by the existing results in the literature such as those in [Carmona, G., 2011. Understanding some recent existence results for discontinuous games. Economic Theory 48, 31–45], [Prokopovych, P., 2011. On equilibrium existence in payoff secure games. Economic Theory 48, 5–16], [Carmona, G., 2009. An existence result for discontinuous games. Journal of Economic Theory 144, 1333–1340], and [Reny, P.J., 1999. On the existence of pure and mixed strategy Nash equilibria in discontinuous games, Econometrica 67, 1029–1056]. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Reny (1999) established the existence of a Nash equilibrium in compact and quasiconcave games where the game is better-reply secure, which is a weak notion of continuity and holding in a large class of discontinuous games. Reny (1999) also showed that betterreply security can be imposed separately as reciprocal upper semicontinuity and payoff security. Bagh and Jofre (2006) further weakened reciprocal upper semicontinuity to weak reciprocal upper semicontinuity and showed that, together with payoff security, it implies better-reply security. Prokopovych (2011) introduced the transfer reciprocal upper semicontinuity and established the existence of a Nash equilibrium in compact and quasiconcave games where the game is payoff secure and transfer reciprocal upper semicontinuous. Carmona (2011) introduced the weak better-reply security. He showed that a bounded, convex, compact, quasiconcave game and weakly better-reply secure has a Nash equilibrium and also proved that when players’ action spaces are metric and locally convex. This implies the existence results of Reny (1999) and Carmona (2009) and it is equivalent to the result of Barelli and Soza (2009). McLennan et al. (2010) fully characterized the existence of a Nash equilibrium in compact and convex games and established a single condition, called MRsecure, that is both necessary and sufficient for the existence of equilibrium in games. This paper investigates the existence of a pure strategy Nash equilibrium in discontinuous games and introduces a new notion of weak continuity, called generalized weak transfer continuity. Roughly speaking, a game is generalized weakly transfer continuous if for every nonequilibrium strategy x∗ , some player i
has a well-behaved correspondence yielding a strictly better off payoff even if all players deviate slightly from x∗ . We establish that a bounded, compact locally convex metric quasiconcave and generalized weak transfer continuous game has a Nash equilibrium and show that it is unrelated to the papers by Carmona (2011), Prokopovych (2011), Carmona (2009), Barelli and Soza (2009) and Reny (1999). The remainder of the paper is organized as follows. Section 2 describes the notation, and provides a number of preliminary definitions. Section 3.1 introduces the notions of generalized weak transfer continuity, and Section 3.2 provides the main result on the existence of a pure strategy Nash equilibrium. Section 3.3 describes related results. 2. Preliminaries Consider the following non-cooperative game in a normal form: G = (Xi , ui )i∈I where I = {1, . . . , n} is a finite set of players, Xi is player i’s strategy space which is a nonempty subset of a locally convex metric vector space, ∏and ui : X → R is the payoff function of player i. Denote by X = i∈I Xi the set of strategy profiles of the game. For each player i ∈ I, denote ∏ by −i all players rather than player i. Also denote by X−i = j̸=i Xj the set of strategies of the players −i. We say that a game G = (Xi , ui )i∈I is compact, convex and bounded, respectively if, for all i ∈ I , Xi is compact, convex and ui is bounded, respectively. We say that a game G = (Xi , ui )i∈I is quasiconcave if, for every i ∈ I, Xi is convex and the function ui is quasiconcave in xi . We say that a strategy profile x∗ ∈ X is a Nash equilibrium of game G if, ui (yi , x∗−i ) ≤ ui (x∗ ) ∀i ∈ I , ∀yi ∈ Xi .
E-mail address: [email protected]. 0304-4068/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2011.08.006
The graph of the game is Γ = {(x, u) ∈ X × Rn : ui (x) = ui , ∀i ∈ I }. The closure of Γ in X × Rn is denoted by cl(Γ ). The
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R. Nessah / Journal of Mathematical Economics 47 (2011) 659–662
frontier of Γ is the set of points that are in cl(Γ ) but not in Γ is denoted by Fr(Γ ). Theorem 3.1 in Reny (1999) showed that a game G = (Xi , ui )i∈I possesses a Nash equilibrium if it is compact, bounded, quasiconcave and better-reply secure. Reny (1999) found that a game G = (Xi , ui )i∈I is better-reply secure if it is payoff secure and reciprocally upper semicontinuous. Bagh and Jofre (2006) further proved that G = (Xi , ui )i∈I is better-reply secure if it is payoff secure and weakly reciprocal upper semicontinuous. Corollary 2 in Carmona (2009) showed that a G = (Xi , ui )i∈I possesses a Nash equilibrium if it is compact, quasiconcave, weakly upper semicontinuous and weakly payoff secure. Prokopovych (2011) proved that if a game G = (Xi , ui )i∈I is payoff secure then G is better-reply secure if and only if it is transfer reciprocal upper semicontinuous. Barelli and Soza (2009) defined the notions of generalized payoff secure and generalized better-reply secure as follows: Definition 2.1 (Barelli and Soza, 2009). A game G is generalized payoff secure if for all i ∈ I , x ∈ X and ϵ > 0, there exist a neighborhood V (x−i ) of x−i and a well-behaved correspondence ϕi : V (x−i ) ⇒ Xi such that ui (x′ ) ≥ ui (x) − ϵ , for all x′ ∈ Graph(ϕi ). Definition 2.2 (Barelli and Soza, 2009). A game G is generalized better-reply secure if whenever (x∗ , u∗ ) ∈ cl(Γ ) and x∗ is not a Nash equilibrium, there exist a player i, an open neighborhood V (x∗−i ) of x∗−i , a well-behaved correspondence ϕi : V (x∗−i ) ⇒ Xi and a number αi > u∗i such that ui (x′ ) ≥ αi , for all x′ ∈ Graph(ϕi ). Barelli and Soza (2009) showed that if G = (Xi , ui )i∈I is compact, quasiconcave and generalized better-reply secure, then it has a Nash equilibrium. Theorem 1 in Carmona (2011) proved that a game G = (Xi , ui )i∈I possesses a Nash equilibrium if it is compact, quasiconcave, weakly better-reply secure. To use the result in Reny (1999), Bagh and Jofre (2006) and Carmona (2009), one must analyze the closure of the graph of the vector payoff function. Such an analysis involves a high dimension and is hard to check. A similar remark also applies to Barelli and Soza (2009), Prokopovych (2011) and Carmona (2011).
Definition 3.2. A game G = (Xi , ui )i∈I is said to be generalized weakly transfer continuous if whenever x∗ ∈ X is not an equilibrium, there exist a player i, a neighborhood V (x∗ ) of x∗ and a well-behaved correspondence ϕi : V (x∗ ) ⇒ Xi such that inf(x′ ,y′ )∈Graph(ϕi ) {ui (y′i , x′−i ) − ui (x′ )} > 0. i
Generalized weak transfer continuity means that whenever x∗ is not an equilibrium, some player i has a well-behaved correspondence that yields a strictly better off payoff even if all players deviate slightly from x∗ . The following example shows that generalized weakly transfer continuous is strictly weaker than the weakly transfer continuous. Example 3.1. Consider the two-player game with the following payoff functions defined on [0, 1] × [0, 1] studied by Barelli and Soza (2009). ui (x1 , x2 ) =
1 0
if x1 = x2 otherwise.
This game is not weakly transfer continuous. Indeed, let x = (1, 0). Clearly x is not a Nash equilibrium. Therefore, for each player i, yi ∈ Xi and each neighborhood V (x) of (x), there exists x′ ∈ V (x) with x′−i ̸= yi such that ui (yi , x′−i ) = 0 ≤ ui (x′ ). Thus, this game is not weakly transfer continuous, so that Theorem 3.2 of Nessah and Tian (2009) cannot be applied. However, it is generalized weakly transfer continuous. Let x ∈ X be not an equilibrium, then x1 ̸= x2 . Thus, there exist a player i, a neighborhood V (x) of x with x′1 ̸= x′2 for each x′ ∈ V (x) and a wellbehaved correspondence ϕi : V (x) ⇒ Xi defined by ϕi (x′ ) = {x′−i } such that for each (x′ , y′i ) ∈ Graph(ϕi ), we have ui (y′i , x′−i ) = ui (x′−i , x′−i ) = 1 and ui (x′ ) = 0. Thus, inf(x′ ,y′ )∈Graph(ϕi ) {ui (y′i , x′−i ) − i
ui (x′ )} = 1 > 0.
3.2. Existence result In this section we investigate the existence of pure strategy Nash equilibrium in discontinuous games. The following definition generalizes the notion of lower semicontinuity of a function.
3. Generalized weak transfer continuity and existence result In this section, we introduce the notion of generalized weak transfer continuity. We show that the generalized weak transfer continuity is unrelated to the better reply security of Reny (1999), the generalized better-reply security of Barelli and Soza (2009), the weak payoff security and weak lower semicontinuity of Carmona (2009), the payoff security and transfer reciprocal upper semicontinuity of Prokopovych (2011) and the weak better-reply security of Carmona (2011). 3.1. Generalized weak transfer continuity Let us recall the definition of weak transfer continuity introduced by Nessah and Tian (2009). Definition 3.1 (Nessah and Tian, 2009). A game G = (Xi , ui )i∈I is said to be weakly transfer continuous if whenever x∗ ∈ X is not an equilibrium, there exist player i, yi ∈ Xi , and a neighborhood V (x∗ ) of x∗ such that infx′ ∈V (x∗ ) {ui (yi , x′−i ) − ui (x′ )} > 0. Theorem 3.2 in Nessah and Tian (2009) shows that a G = (Xi , ui )i∈I possesses a Nash equilibrium if it is bounded, compact, quasiconcave and weakly transfer continuous. Let us consider a correspondence C : X ⇒ Y . C is said to be a well-behaved correspondence if it is upper hemicontinuous with nonempty, closed and convex values.
Definition 3.3. A function f : X × Y → R is said to be generalized lower semicontinuous on X relative to Y if for all x ∈ X and ϵ > 0, there exist a neighborhood V (x) of x and a well-behaved correspondence ϕx : V (x) ⇒ Y such that f (x′ , y′ ) ≥ f (x, y) − ϵ , for all (x′ , y′ ) ∈ Graph(ϕx ). By Definition 2.2, we obtain that a game G is generalized payoff secure if and only if the function ui is generalized lower semicontinuous on X−i relative to Xi , for all i ∈ I. For each player i ∈ I and every (x, yi ) ∈ X × Xi , let us consider the following function ai (x, yi ) = sup
sup
inf
[ui (ti , z−i ) − ui (z )]
V ∈Ω (x) ϕi ∈WV (x,yi ) (z ,ti )∈Graph(ϕi )
where Ω (x) is the set of all open neighborhoods of x, and WV (x, yi ) is the set of all well-behaved correspondence ϕi : V ∈ Ω (x) ⇒ Xi that satisfy (x, yi ) ∈ Graph(ϕi ). Remark 3.1. For all x ∈ X , we have ai (x, xi ) ≤ 0. By Lemma 4 of Carmona (2011), we deduce that for all i ∈ I, the function ai is generalized lower semicontinuous on X relative to Xi , and ai (x, .) is quasiconcave on Xi , for all x ∈ X if the game G is quasiconcave. With the same arguments as in the proof of Carmona’s Lemma 1 (Carmona, 2011), we can show that for each upper semicontinuous function fi (x) < supyi ∈Xi ai (x, yi ), for each x ∈ X , for all i ∈ I, the following system {∃x ∈ X / fi (x) < ai (x, xi ) ≤
R. Nessah / Journal of Mathematical Economics 47 (2011) 659–662
0; ∀i ∈ I } has a solution in X , i.e. there exists x ∈ X such that fi (x) < ai (x, xi ) ≤ 0, for all i ∈ I. Our existence result shows that bounded, compact, quasiconcave games that are generalized weakly transfer continuous have a Nash equilibrium.
u1 (p1 , p2 , p3 ) =
Theorem 3.1. If G = (Xi , ui )i∈I is bounded, compact, quasiconcave and generalized weakly transfer continuous, then it has a Nash equilibrium.
u3 (p1 , p2 , p3 ) =
Proof. By Lemma 4 of Carmona (2011), we have that for all i ∈ I, the function ai is generalized lower semicontinuous on X relative to Xi . Then by Lemma 2 of Carmona (2011), the function bi (x) = supyi ∈Xi ai (x, yi ) is lower semicontinuous on X . It follows by Exercise 22, page 60 of Rudin (1987) (or Lemma 3.5 of Reny, 1999) that there exists a sequence {bki }k of continuous real-valued functions on X so that bki (x) ≤ bi (x), and lim infk bki (xk ) ≥ bi (x) for each i ∈ I , k, x ∈ X and each sequence {xk }k converges to x. Let cik (x) = bki (x) − 1k . Therefore cik (x) < bi (x) = supyi ∈Xi ai (x, yi ) for all k, i ∈ I, x ∈ X . cik is continuous, then there exists xk ∈ X such that cik (xk ) < ai (xk , xki ) ≤ 0, for all i ∈ I. Since X is compact, we suppose that the sequence {xk }k converges. Let x = limk xk . Thus, sup ai (x, yi ) = bi (x) ≤ lim inf bki (xk ) = lim inf cik (xk ) k
yi ∈Xi
k
≤ lim inf ai (x , ) ≤ 0. k
k
xki
(This is because we know that bi (x) ≤ lim infk bki (xk ), cik (xk ) ≤ ai (xk , xki ) and by Remark 3.1, ai (xk , xki ) ≤ 0.) Therefore ai (x, yi ) ≤ 0,
for all i ∈ Iand for each yi ∈ Xi .
(3.1)
If x is not a Nash equilibrium and since the game G is generalized weakly transfer continuous, then there exist a player i, a neighborhood V (x) of x and a well-behaved correspondence ϕxi : V (x) ⇒ Xi such that inf(x′ ,y′ )∈Graph(ϕ i ) {ui (y′i , x′−i ) − ui (x′ )} > 0. Let yi ∈ ϕxi (x), i
x
then ai (x, yi ) > 0, which contradicts (3.1). Therefore, x is a Nash equilibrium.
By Theorem 3.1, then Theorem 3.2 in Nessah and Tian (2009) can be obtained immediately. Corollary 3.1 (Nessah and Tian, 2009). If G = (Xi , ui )i∈I is bounded, compact, quasiconcave and weakly transfer continuous, then it has a Nash equilibrium. The following example shows that Theorem 3.1 strictly generalizes Corollary 3.1 (Theorem 3.2 of Nessah and Tian, 2009). Example 3.2. Recall Example 3.1. Consider the two-player game with the following payoff functions defined on [0, 1] × [0, 1]. ui (x1 , x2 ) =
1 0
if x1 = x2 otherwise.
This game is not weakly transfer continuous. Then Theorem 3.2 of Nessah and Tian (2009) cannot be applied. However, it is bounded, compact, quasiconcave and generalized weakly transfer continuous so that by Theorem 3.1 this game has a Nash equilibrium. 3.3. Related results The following examples show that Theorem 3.1 is unrelated to Theorem 1 of Carmona (2011), Theorem 4 of Prokopovych (2011), Corollary 4.5 of Barelli and Soza (2009), and Theorem 3.1 of Reny (1999). Example 3.3. Consider a three-player game with the following payoff functions defined on [0, 1] × [0, 1] × [0, 1].
u2 (p1 , p2 , p3 ) =
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p1 + 2, p1 + p2 + p3 ,
if p2 ≥ p3 if p2 < p3 .
p2 + 2, p2 + p1 + p3 ,
if p1 ≤ p3 if p1 > p3 .
p3 + 2, p3 + p1 + p2 ,
if p1 ≥ p2 if p1 < p2 .
The game considered in this example is not generalized better-reply secure. Indeed, consider p = (0, 0, 0) ∈ X and u = (2, 2, 2). Then, (p, u) is in the closure of the graph of its vector function, and p is not an equilibrium. We show that player i cannot obtain a payoff strictly above ui = 2. Indeed, player 1 cannot obtain a payoff strictly above u1 = 2, for any neighborhood V (p2 , p3 ) ⊂ [0, 1] × [0, 1] of (p2 , p3 ), choosing (p′2 , p′3 ) ∈ V (p2 , p3 ) with p′2 < p′3 < 12 and for all well-behaved correspondences ϕ1 : V (p2 , p3 ) ⇒ X1 = [0, 1], we then have u1 (p1 , p′2 , p′3 ) = p1 + p′2 + p′3 ≤ u1 , for each p1 ∈ ϕ1 (p′2 , p′3 ). Player 2 cannot obtain a payoff strictly above u2 = 2, for any neighborhood V (p1 , p3 ) ⊂ [0, 1]×[0, 1] of (p1 , p3 ), choosing (p′1 , p′3 ) ∈ V (p1 , p3 ) with 12 > p′1 > p′3 and for all wellbehaved correspondences ϕ2 : V (p1 , p3 ) ⇒ X2 = [0, 1], we then have u2 (p′1 , p2 , p′3 ) = p2 + p′1 + p′3 ≤ u2 , for each p2 ∈ ϕ2 (p′1 , p′3 ). Player 3 cannot obtain a payoff strictly above u3 = 2, for any neighborhood V (p1 , p2 ) ⊂ [0, 1] × [0, 1] of (p1 , p2 ), choosing (p′1 , p′2 ) ∈ V (p1 , p2 ) with p′1 < p′2 < 12 and for all well-behaved correspondences ϕ3 : V (p1 , p2 ) ⇒ X3 = [0, 1], we then have u3 (p′1 , p′2 , p3 ) = p3 + p′2 + p′3 ≤ u3 , for each p3 ∈ ϕ2 (p′1 , p′2 ). Thus, this game is not generalized better-reply secure, so that Corollary 4.5 of Barelli and Soza (2009) cannot be applied. By Theorems 2–3 of Carmona (2011), a game is weakly betterreply secure if and only if it is generalized better-reply secure, then Theorem 1 of Carmona (2011) cannot be applied. Since a game better-reply secure is also generalized better-reply secure, then Theorem 3.1 of Reny (1999) cannot be applied. The game G is not weakly payoff secure. Indeed, consider p = (0, 0, 0) ∈ X and u(p) = (2, 2, 2). Player 1 cannot obtain a payoff strictly above u1 (p) = 2 − ϵ for some ϵ > 0. Let ϵ = 14 , then for each neighborhood V (p2 , p3 ) ⊂ [0, 1]×[0, 1] of (p2 , p3 ), choosing (p′2 , p′3 ) ∈ V (p2 , p3 ) with p′2 = 0 < p′3 < 41 , we then have u1 (p1 , p′2 , p′3 ) = p1 + p′2 + p′3 ≤ 45 < 74 = u1 (p) − ϵ . Thus, this game is not weakly payoff secure, so that Corollary 2 of Carmona (2009) cannot be applied. Since G is not weakly payoff secure, then it is not payoff secure and hence Theorem 4 of Prokopovych (2011) cannot be applied. However, this game is generalized weakly transfer continuous. Indeed, let p = (p1 , p2 , p3 ) be a nonequilibrium strategy profile with at least one non-one coordinate. Then, there exists i ∈ I with pi < 1. Therefore, there exist a neighborhood V (p) of p and ϵ > 0 with p′i + ϵ < 1 for all p′ ∈ V (p) and a well-behaved correspondence ϕi : V (p) ⇒ Xi defined by ϕi (p′ ) = {1}, for each p′ ∈ V (p) such that inf(p′ ,q′ )∈Graph(ϕi ) {ui (q′i , p′−i ) − ui (p′ )} ≥ i ϵ > 0. Since the game is also bounded, compact and quasiconcave, it follows from by Theorem 3.1 that the game possesses a Nash equilibrium. Example 3.4. Let us consider the two-player game with the following payoff functions defined on [0, 1] × [0, 1] studied by Reny (1999) and Bagh and Jofre (2006). li (ti ) = 10, ωi (t ), mi (t−i ) = −10,
ui (t1 , t2 ) =
if ti < t−i if ti = t−i = t if ti > t−i ,
and
ωi (ti ) =
1,
if ti = t−i and ti <
0,
if ti = t−i and ti ≥
1 2 1 2
.
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R. Nessah / Journal of Mathematical Economics 47 (2011) 659–662
Bagh and Jofre (2006) show that this game is better-reply secure. Since the game is also bounded, compact and quasiconcave, then by Theorem 3.1 of Reny (1999), Corollary 4.5 of Barelli and Soza (2009), Theorems 2–3 of Carmona (2011), Theorem 1 of Carmona (2011), Corollary 2 of Carmona (2009), and Theorem 4 of Prokopovych (2011), the considered game possesses a Nash equilibrium. However, it is not generalized weakly transfer continuous. Indeed, let t = 12 , 12 be a nonequilibrium strategy profile. We show that player i cannot obtain a payoff strictly above ui in the neighborhood of t. Indeed, for any neighborhood V 12 , 12 ⊂
[0, 1] × [0, 1] of 21 , 21 , and for any well-behaved correspondence 1 1 ϕi : V 2 , 2 ⇒ Xi , choosing (ti′ , t−′ i ) ∈ V 12 , 21 with ti′ < t−′ i , ′ ′ ′ we then have ui (ti , t− i ) ≤ 10 = ui (t ), for some ti ∈ ϕi (t ).
Thus, this game is not generalized weakly transfer continuous, so Theorem 3.1 cannot be applied. Remark 3.2. The notion of generalized weak transfer continuity can be easily extended to generalized games, quasi-(symmetric) games and games with mixed strategies. Moreover, every game that is generalized weakly transfer continuous in the extended sense has a mixed strategy Nash equilibrium, even if the payoff functions are not quasiconcave.
Acknowledgments We thank Philip Reny, Guilherme Carmona, Pavlo Prokopovych, Guoqiang Tian, Louis Eeckhoudt, Robert Joliet and the referee for helpful discussions, comments, and suggestions References Bagh, A., Jofre, A., 2006. Reciprocal upper semicontinuous and better reply secure games: a comment. Econometrica 74, 1715–1721. Barelli, P., Soza, I., 2009. On the existence of Nash equilibria in discontinuous and qualitative games. Rochester University, working paper. 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, 31–45. McLennan, A., Monteiro, P.K., Tourky, R., 2010. Games with discontinuous payoffs: a strengthening of Reny’s existence theorem. Queensland University, working paper. Nessah, R., Tian, G., 2009. Existence of equilibrium in discontinuous games. IESEG working paper. Prokopovych, P., 2011. On equilibrium existence in payoff secure games. Economic Theory 48, 5–16. Reny, P.J., 1999. On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67, 1029–1056. Rudin, W., 1987. Real and Complex Analysis, third ed. In: International Series in Pure and Applied Mathematics, McGraw-Hill.
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco
Short communication
Generalized weak transfer continuity and the Nash equilibrium Rabia Nessah IESEG School of Management (CNRS-LEM), 3 rue de la Digue, 59000 Lille, France
article
info
Article history: Received 26 January 2011 Received in revised form 12 August 2011 Accepted 23 August 2011 Available online 31 August 2011
abstract This paper introduces the notion of generalized weak transfer continuity and establishes that a bounded, compact locally convex metric quasiconcave and generalized weak transfer continuous game has a Nash equilibrium. Our equilibrium existence result neither implies nor is implied by the existing results in the literature such as those in [Carmona, G., 2011. Understanding some recent existence results for discontinuous games. Economic Theory 48, 31–45], [Prokopovych, P., 2011. On equilibrium existence in payoff secure games. Economic Theory 48, 5–16], [Carmona, G., 2009. An existence result for discontinuous games. Journal of Economic Theory 144, 1333–1340], and [Reny, P.J., 1999. On the existence of pure and mixed strategy Nash equilibria in discontinuous games, Econometrica 67, 1029–1056]. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Reny (1999) established the existence of a Nash equilibrium in compact and quasiconcave games where the game is better-reply secure, which is a weak notion of continuity and holding in a large class of discontinuous games. Reny (1999) also showed that betterreply security can be imposed separately as reciprocal upper semicontinuity and payoff security. Bagh and Jofre (2006) further weakened reciprocal upper semicontinuity to weak reciprocal upper semicontinuity and showed that, together with payoff security, it implies better-reply security. Prokopovych (2011) introduced the transfer reciprocal upper semicontinuity and established the existence of a Nash equilibrium in compact and quasiconcave games where the game is payoff secure and transfer reciprocal upper semicontinuous. Carmona (2011) introduced the weak better-reply security. He showed that a bounded, convex, compact, quasiconcave game and weakly better-reply secure has a Nash equilibrium and also proved that when players’ action spaces are metric and locally convex. This implies the existence results of Reny (1999) and Carmona (2009) and it is equivalent to the result of Barelli and Soza (2009). McLennan et al. (2010) fully characterized the existence of a Nash equilibrium in compact and convex games and established a single condition, called MRsecure, that is both necessary and sufficient for the existence of equilibrium in games. This paper investigates the existence of a pure strategy Nash equilibrium in discontinuous games and introduces a new notion of weak continuity, called generalized weak transfer continuity. Roughly speaking, a game is generalized weakly transfer continuous if for every nonequilibrium strategy x∗ , some player i
has a well-behaved correspondence yielding a strictly better off payoff even if all players deviate slightly from x∗ . We establish that a bounded, compact locally convex metric quasiconcave and generalized weak transfer continuous game has a Nash equilibrium and show that it is unrelated to the papers by Carmona (2011), Prokopovych (2011), Carmona (2009), Barelli and Soza (2009) and Reny (1999). The remainder of the paper is organized as follows. Section 2 describes the notation, and provides a number of preliminary definitions. Section 3.1 introduces the notions of generalized weak transfer continuity, and Section 3.2 provides the main result on the existence of a pure strategy Nash equilibrium. Section 3.3 describes related results. 2. Preliminaries Consider the following non-cooperative game in a normal form: G = (Xi , ui )i∈I where I = {1, . . . , n} is a finite set of players, Xi is player i’s strategy space which is a nonempty subset of a locally convex metric vector space, ∏and ui : X → R is the payoff function of player i. Denote by X = i∈I Xi the set of strategy profiles of the game. For each player i ∈ I, denote ∏ by −i all players rather than player i. Also denote by X−i = j̸=i Xj the set of strategies of the players −i. We say that a game G = (Xi , ui )i∈I is compact, convex and bounded, respectively if, for all i ∈ I , Xi is compact, convex and ui is bounded, respectively. We say that a game G = (Xi , ui )i∈I is quasiconcave if, for every i ∈ I, Xi is convex and the function ui is quasiconcave in xi . We say that a strategy profile x∗ ∈ X is a Nash equilibrium of game G if, ui (yi , x∗−i ) ≤ ui (x∗ ) ∀i ∈ I , ∀yi ∈ Xi .
E-mail address: [email protected]. 0304-4068/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmateco.2011.08.006
The graph of the game is Γ = {(x, u) ∈ X × Rn : ui (x) = ui , ∀i ∈ I }. The closure of Γ in X × Rn is denoted by cl(Γ ). The
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R. Nessah / Journal of Mathematical Economics 47 (2011) 659–662
frontier of Γ is the set of points that are in cl(Γ ) but not in Γ is denoted by Fr(Γ ). Theorem 3.1 in Reny (1999) showed that a game G = (Xi , ui )i∈I possesses a Nash equilibrium if it is compact, bounded, quasiconcave and better-reply secure. Reny (1999) found that a game G = (Xi , ui )i∈I is better-reply secure if it is payoff secure and reciprocally upper semicontinuous. Bagh and Jofre (2006) further proved that G = (Xi , ui )i∈I is better-reply secure if it is payoff secure and weakly reciprocal upper semicontinuous. Corollary 2 in Carmona (2009) showed that a G = (Xi , ui )i∈I possesses a Nash equilibrium if it is compact, quasiconcave, weakly upper semicontinuous and weakly payoff secure. Prokopovych (2011) proved that if a game G = (Xi , ui )i∈I is payoff secure then G is better-reply secure if and only if it is transfer reciprocal upper semicontinuous. Barelli and Soza (2009) defined the notions of generalized payoff secure and generalized better-reply secure as follows: Definition 2.1 (Barelli and Soza, 2009). A game G is generalized payoff secure if for all i ∈ I , x ∈ X and ϵ > 0, there exist a neighborhood V (x−i ) of x−i and a well-behaved correspondence ϕi : V (x−i ) ⇒ Xi such that ui (x′ ) ≥ ui (x) − ϵ , for all x′ ∈ Graph(ϕi ). Definition 2.2 (Barelli and Soza, 2009). A game G is generalized better-reply secure if whenever (x∗ , u∗ ) ∈ cl(Γ ) and x∗ is not a Nash equilibrium, there exist a player i, an open neighborhood V (x∗−i ) of x∗−i , a well-behaved correspondence ϕi : V (x∗−i ) ⇒ Xi and a number αi > u∗i such that ui (x′ ) ≥ αi , for all x′ ∈ Graph(ϕi ). Barelli and Soza (2009) showed that if G = (Xi , ui )i∈I is compact, quasiconcave and generalized better-reply secure, then it has a Nash equilibrium. Theorem 1 in Carmona (2011) proved that a game G = (Xi , ui )i∈I possesses a Nash equilibrium if it is compact, quasiconcave, weakly better-reply secure. To use the result in Reny (1999), Bagh and Jofre (2006) and Carmona (2009), one must analyze the closure of the graph of the vector payoff function. Such an analysis involves a high dimension and is hard to check. A similar remark also applies to Barelli and Soza (2009), Prokopovych (2011) and Carmona (2011).
Definition 3.2. A game G = (Xi , ui )i∈I is said to be generalized weakly transfer continuous if whenever x∗ ∈ X is not an equilibrium, there exist a player i, a neighborhood V (x∗ ) of x∗ and a well-behaved correspondence ϕi : V (x∗ ) ⇒ Xi such that inf(x′ ,y′ )∈Graph(ϕi ) {ui (y′i , x′−i ) − ui (x′ )} > 0. i
Generalized weak transfer continuity means that whenever x∗ is not an equilibrium, some player i has a well-behaved correspondence that yields a strictly better off payoff even if all players deviate slightly from x∗ . The following example shows that generalized weakly transfer continuous is strictly weaker than the weakly transfer continuous. Example 3.1. Consider the two-player game with the following payoff functions defined on [0, 1] × [0, 1] studied by Barelli and Soza (2009). ui (x1 , x2 ) =
1 0
if x1 = x2 otherwise.
This game is not weakly transfer continuous. Indeed, let x = (1, 0). Clearly x is not a Nash equilibrium. Therefore, for each player i, yi ∈ Xi and each neighborhood V (x) of (x), there exists x′ ∈ V (x) with x′−i ̸= yi such that ui (yi , x′−i ) = 0 ≤ ui (x′ ). Thus, this game is not weakly transfer continuous, so that Theorem 3.2 of Nessah and Tian (2009) cannot be applied. However, it is generalized weakly transfer continuous. Let x ∈ X be not an equilibrium, then x1 ̸= x2 . Thus, there exist a player i, a neighborhood V (x) of x with x′1 ̸= x′2 for each x′ ∈ V (x) and a wellbehaved correspondence ϕi : V (x) ⇒ Xi defined by ϕi (x′ ) = {x′−i } such that for each (x′ , y′i ) ∈ Graph(ϕi ), we have ui (y′i , x′−i ) = ui (x′−i , x′−i ) = 1 and ui (x′ ) = 0. Thus, inf(x′ ,y′ )∈Graph(ϕi ) {ui (y′i , x′−i ) − i
ui (x′ )} = 1 > 0.
3.2. Existence result In this section we investigate the existence of pure strategy Nash equilibrium in discontinuous games. The following definition generalizes the notion of lower semicontinuity of a function.
3. Generalized weak transfer continuity and existence result In this section, we introduce the notion of generalized weak transfer continuity. We show that the generalized weak transfer continuity is unrelated to the better reply security of Reny (1999), the generalized better-reply security of Barelli and Soza (2009), the weak payoff security and weak lower semicontinuity of Carmona (2009), the payoff security and transfer reciprocal upper semicontinuity of Prokopovych (2011) and the weak better-reply security of Carmona (2011). 3.1. Generalized weak transfer continuity Let us recall the definition of weak transfer continuity introduced by Nessah and Tian (2009). Definition 3.1 (Nessah and Tian, 2009). A game G = (Xi , ui )i∈I is said to be weakly transfer continuous if whenever x∗ ∈ X is not an equilibrium, there exist player i, yi ∈ Xi , and a neighborhood V (x∗ ) of x∗ such that infx′ ∈V (x∗ ) {ui (yi , x′−i ) − ui (x′ )} > 0. Theorem 3.2 in Nessah and Tian (2009) shows that a G = (Xi , ui )i∈I possesses a Nash equilibrium if it is bounded, compact, quasiconcave and weakly transfer continuous. Let us consider a correspondence C : X ⇒ Y . C is said to be a well-behaved correspondence if it is upper hemicontinuous with nonempty, closed and convex values.
Definition 3.3. A function f : X × Y → R is said to be generalized lower semicontinuous on X relative to Y if for all x ∈ X and ϵ > 0, there exist a neighborhood V (x) of x and a well-behaved correspondence ϕx : V (x) ⇒ Y such that f (x′ , y′ ) ≥ f (x, y) − ϵ , for all (x′ , y′ ) ∈ Graph(ϕx ). By Definition 2.2, we obtain that a game G is generalized payoff secure if and only if the function ui is generalized lower semicontinuous on X−i relative to Xi , for all i ∈ I. For each player i ∈ I and every (x, yi ) ∈ X × Xi , let us consider the following function ai (x, yi ) = sup
sup
inf
[ui (ti , z−i ) − ui (z )]
V ∈Ω (x) ϕi ∈WV (x,yi ) (z ,ti )∈Graph(ϕi )
where Ω (x) is the set of all open neighborhoods of x, and WV (x, yi ) is the set of all well-behaved correspondence ϕi : V ∈ Ω (x) ⇒ Xi that satisfy (x, yi ) ∈ Graph(ϕi ). Remark 3.1. For all x ∈ X , we have ai (x, xi ) ≤ 0. By Lemma 4 of Carmona (2011), we deduce that for all i ∈ I, the function ai is generalized lower semicontinuous on X relative to Xi , and ai (x, .) is quasiconcave on Xi , for all x ∈ X if the game G is quasiconcave. With the same arguments as in the proof of Carmona’s Lemma 1 (Carmona, 2011), we can show that for each upper semicontinuous function fi (x) < supyi ∈Xi ai (x, yi ), for each x ∈ X , for all i ∈ I, the following system {∃x ∈ X / fi (x) < ai (x, xi ) ≤
R. Nessah / Journal of Mathematical Economics 47 (2011) 659–662
0; ∀i ∈ I } has a solution in X , i.e. there exists x ∈ X such that fi (x) < ai (x, xi ) ≤ 0, for all i ∈ I. Our existence result shows that bounded, compact, quasiconcave games that are generalized weakly transfer continuous have a Nash equilibrium.
u1 (p1 , p2 , p3 ) =
Theorem 3.1. If G = (Xi , ui )i∈I is bounded, compact, quasiconcave and generalized weakly transfer continuous, then it has a Nash equilibrium.
u3 (p1 , p2 , p3 ) =
Proof. By Lemma 4 of Carmona (2011), we have that for all i ∈ I, the function ai is generalized lower semicontinuous on X relative to Xi . Then by Lemma 2 of Carmona (2011), the function bi (x) = supyi ∈Xi ai (x, yi ) is lower semicontinuous on X . It follows by Exercise 22, page 60 of Rudin (1987) (or Lemma 3.5 of Reny, 1999) that there exists a sequence {bki }k of continuous real-valued functions on X so that bki (x) ≤ bi (x), and lim infk bki (xk ) ≥ bi (x) for each i ∈ I , k, x ∈ X and each sequence {xk }k converges to x. Let cik (x) = bki (x) − 1k . Therefore cik (x) < bi (x) = supyi ∈Xi ai (x, yi ) for all k, i ∈ I, x ∈ X . cik is continuous, then there exists xk ∈ X such that cik (xk ) < ai (xk , xki ) ≤ 0, for all i ∈ I. Since X is compact, we suppose that the sequence {xk }k converges. Let x = limk xk . Thus, sup ai (x, yi ) = bi (x) ≤ lim inf bki (xk ) = lim inf cik (xk ) k
yi ∈Xi
k
≤ lim inf ai (x , ) ≤ 0. k
k
xki
(This is because we know that bi (x) ≤ lim infk bki (xk ), cik (xk ) ≤ ai (xk , xki ) and by Remark 3.1, ai (xk , xki ) ≤ 0.) Therefore ai (x, yi ) ≤ 0,
for all i ∈ Iand for each yi ∈ Xi .
(3.1)
If x is not a Nash equilibrium and since the game G is generalized weakly transfer continuous, then there exist a player i, a neighborhood V (x) of x and a well-behaved correspondence ϕxi : V (x) ⇒ Xi such that inf(x′ ,y′ )∈Graph(ϕ i ) {ui (y′i , x′−i ) − ui (x′ )} > 0. Let yi ∈ ϕxi (x), i
x
then ai (x, yi ) > 0, which contradicts (3.1). Therefore, x is a Nash equilibrium.
By Theorem 3.1, then Theorem 3.2 in Nessah and Tian (2009) can be obtained immediately. Corollary 3.1 (Nessah and Tian, 2009). If G = (Xi , ui )i∈I is bounded, compact, quasiconcave and weakly transfer continuous, then it has a Nash equilibrium. The following example shows that Theorem 3.1 strictly generalizes Corollary 3.1 (Theorem 3.2 of Nessah and Tian, 2009). Example 3.2. Recall Example 3.1. Consider the two-player game with the following payoff functions defined on [0, 1] × [0, 1]. ui (x1 , x2 ) =
1 0
if x1 = x2 otherwise.
This game is not weakly transfer continuous. Then Theorem 3.2 of Nessah and Tian (2009) cannot be applied. However, it is bounded, compact, quasiconcave and generalized weakly transfer continuous so that by Theorem 3.1 this game has a Nash equilibrium. 3.3. Related results The following examples show that Theorem 3.1 is unrelated to Theorem 1 of Carmona (2011), Theorem 4 of Prokopovych (2011), Corollary 4.5 of Barelli and Soza (2009), and Theorem 3.1 of Reny (1999). Example 3.3. Consider a three-player game with the following payoff functions defined on [0, 1] × [0, 1] × [0, 1].
u2 (p1 , p2 , p3 ) =
661
p1 + 2, p1 + p2 + p3 ,
if p2 ≥ p3 if p2 < p3 .
p2 + 2, p2 + p1 + p3 ,
if p1 ≤ p3 if p1 > p3 .
p3 + 2, p3 + p1 + p2 ,
if p1 ≥ p2 if p1 < p2 .
The game considered in this example is not generalized better-reply secure. Indeed, consider p = (0, 0, 0) ∈ X and u = (2, 2, 2). Then, (p, u) is in the closure of the graph of its vector function, and p is not an equilibrium. We show that player i cannot obtain a payoff strictly above ui = 2. Indeed, player 1 cannot obtain a payoff strictly above u1 = 2, for any neighborhood V (p2 , p3 ) ⊂ [0, 1] × [0, 1] of (p2 , p3 ), choosing (p′2 , p′3 ) ∈ V (p2 , p3 ) with p′2 < p′3 < 12 and for all well-behaved correspondences ϕ1 : V (p2 , p3 ) ⇒ X1 = [0, 1], we then have u1 (p1 , p′2 , p′3 ) = p1 + p′2 + p′3 ≤ u1 , for each p1 ∈ ϕ1 (p′2 , p′3 ). Player 2 cannot obtain a payoff strictly above u2 = 2, for any neighborhood V (p1 , p3 ) ⊂ [0, 1]×[0, 1] of (p1 , p3 ), choosing (p′1 , p′3 ) ∈ V (p1 , p3 ) with 12 > p′1 > p′3 and for all wellbehaved correspondences ϕ2 : V (p1 , p3 ) ⇒ X2 = [0, 1], we then have u2 (p′1 , p2 , p′3 ) = p2 + p′1 + p′3 ≤ u2 , for each p2 ∈ ϕ2 (p′1 , p′3 ). Player 3 cannot obtain a payoff strictly above u3 = 2, for any neighborhood V (p1 , p2 ) ⊂ [0, 1] × [0, 1] of (p1 , p2 ), choosing (p′1 , p′2 ) ∈ V (p1 , p2 ) with p′1 < p′2 < 12 and for all well-behaved correspondences ϕ3 : V (p1 , p2 ) ⇒ X3 = [0, 1], we then have u3 (p′1 , p′2 , p3 ) = p3 + p′2 + p′3 ≤ u3 , for each p3 ∈ ϕ2 (p′1 , p′2 ). Thus, this game is not generalized better-reply secure, so that Corollary 4.5 of Barelli and Soza (2009) cannot be applied. By Theorems 2–3 of Carmona (2011), a game is weakly betterreply secure if and only if it is generalized better-reply secure, then Theorem 1 of Carmona (2011) cannot be applied. Since a game better-reply secure is also generalized better-reply secure, then Theorem 3.1 of Reny (1999) cannot be applied. The game G is not weakly payoff secure. Indeed, consider p = (0, 0, 0) ∈ X and u(p) = (2, 2, 2). Player 1 cannot obtain a payoff strictly above u1 (p) = 2 − ϵ for some ϵ > 0. Let ϵ = 14 , then for each neighborhood V (p2 , p3 ) ⊂ [0, 1]×[0, 1] of (p2 , p3 ), choosing (p′2 , p′3 ) ∈ V (p2 , p3 ) with p′2 = 0 < p′3 < 41 , we then have u1 (p1 , p′2 , p′3 ) = p1 + p′2 + p′3 ≤ 45 < 74 = u1 (p) − ϵ . Thus, this game is not weakly payoff secure, so that Corollary 2 of Carmona (2009) cannot be applied. Since G is not weakly payoff secure, then it is not payoff secure and hence Theorem 4 of Prokopovych (2011) cannot be applied. However, this game is generalized weakly transfer continuous. Indeed, let p = (p1 , p2 , p3 ) be a nonequilibrium strategy profile with at least one non-one coordinate. Then, there exists i ∈ I with pi < 1. Therefore, there exist a neighborhood V (p) of p and ϵ > 0 with p′i + ϵ < 1 for all p′ ∈ V (p) and a well-behaved correspondence ϕi : V (p) ⇒ Xi defined by ϕi (p′ ) = {1}, for each p′ ∈ V (p) such that inf(p′ ,q′ )∈Graph(ϕi ) {ui (q′i , p′−i ) − ui (p′ )} ≥ i ϵ > 0. Since the game is also bounded, compact and quasiconcave, it follows from by Theorem 3.1 that the game possesses a Nash equilibrium. Example 3.4. Let us consider the two-player game with the following payoff functions defined on [0, 1] × [0, 1] studied by Reny (1999) and Bagh and Jofre (2006). li (ti ) = 10, ωi (t ), mi (t−i ) = −10,
ui (t1 , t2 ) =
if ti < t−i if ti = t−i = t if ti > t−i ,
and
ωi (ti ) =
1,
if ti = t−i and ti <
0,
if ti = t−i and ti ≥
1 2 1 2
.
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Bagh and Jofre (2006) show that this game is better-reply secure. Since the game is also bounded, compact and quasiconcave, then by Theorem 3.1 of Reny (1999), Corollary 4.5 of Barelli and Soza (2009), Theorems 2–3 of Carmona (2011), Theorem 1 of Carmona (2011), Corollary 2 of Carmona (2009), and Theorem 4 of Prokopovych (2011), the considered game possesses a Nash equilibrium. However, it is not generalized weakly transfer continuous. Indeed, let t = 12 , 12 be a nonequilibrium strategy profile. We show that player i cannot obtain a payoff strictly above ui in the neighborhood of t. Indeed, for any neighborhood V 12 , 12 ⊂
[0, 1] × [0, 1] of 21 , 21 , and for any well-behaved correspondence 1 1 ϕi : V 2 , 2 ⇒ Xi , choosing (ti′ , t−′ i ) ∈ V 12 , 21 with ti′ < t−′ i , ′ ′ ′ we then have ui (ti , t− i ) ≤ 10 = ui (t ), for some ti ∈ ϕi (t ).
Thus, this game is not generalized weakly transfer continuous, so Theorem 3.1 cannot be applied. Remark 3.2. The notion of generalized weak transfer continuity can be easily extended to generalized games, quasi-(symmetric) games and games with mixed strategies. Moreover, every game that is generalized weakly transfer continuous in the extended sense has a mixed strategy Nash equilibrium, even if the payoff functions are not quasiconcave.
Acknowledgments We thank Philip Reny, Guilherme Carmona, Pavlo Prokopovych, Guoqiang Tian, Louis Eeckhoudt, Robert Joliet and the referee for helpful discussions, comments, and suggestions References Bagh, A., Jofre, A., 2006. Reciprocal upper semicontinuous and better reply secure games: a comment. Econometrica 74, 1715–1721. Barelli, P., Soza, I., 2009. On the existence of Nash equilibria in discontinuous and qualitative games. Rochester University, working paper. 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, 31–45. McLennan, A., Monteiro, P.K., Tourky, R., 2010. Games with discontinuous payoffs: a strengthening of Reny’s existence theorem. Queensland University, working paper. Nessah, R., Tian, G., 2009. Existence of equilibrium in discontinuous games. IESEG working paper. Prokopovych, P., 2011. On equilibrium existence in payoff secure games. Economic Theory 48, 5–16. Reny, P.J., 1999. On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67, 1029–1056. Rudin, W., 1987. Real and Complex Analysis, third ed. In: International Series in Pure and Applied Mathematics, McGraw-Hill.