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Econ Theory (2016) 61:515–540DOI 10.1007/s00199-015-0921-8

RESEARCH ARTICLE

On the existence of Nash equilibrium in discontinuousgames

Rabia Nessah1 · Guoqiang Tian2

Received: 12 May 2015 / Accepted: 23 September 2015 / Published online: 7 October 2015© Springer-Verlag Berlin Heidelberg 2015

Abstract This paper offers an equilibrium existence theorem in discontinuous games.We introduce a new notion of very weak continuity, called quasi-weak transfer conti-nuity that guarantees the existence of pure strategy Nash equilibrium in compact andquasiconcave games. We also consider possible extensions and improvements of themain result. We present applications to show that our conditions allow for economi-cally meaningful payoff discontinuities.

Keywords Discontinuous games · Quasi-weak transfer continuity · Various notionsof transfer continuity · Nash equilibrium

JEL Classification C72 · C62

We wish to thank Professors Philip Reny, Nicholas Yannelis, Tarik Tazdaït, Raluca Parvulescu and ananonymous referee for helpful comments and suggestions that significantly improved the exposition of thepaper.Guoqiang Tian Financial support from the National Natural Science Foundation of China(NSFC-71371117) is gratefully acknowledged.

B Rabia [email protected]

Guoqiang [email protected]

1 IESEG School of Management, CNRS-LEM (UMR 9221), 3 rue de la Digue, 59000 Lille, France

2 Department of Economics, Texas A&M University, College Station, TX 77843, USA

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516 R. Nessah, G. Tian

1 Introduction

The concept of Nash equilibrium in Nash (1950, 1951) is probably the most importantsolution concept in game theory. It is immune from unilateral deviations, that is, eachplayer has no incentive to deviate from his/her strategy given that other players donot deviate from theirs. Nash (1951) proved that a finite game has a Nash equilibriumin mixed strategies. Debreu (1952) then showed that games possess a pure strategyNash equilibrium if (1) the strategy spaces are convex and compact, and (2) playershave continuous and quasiconcave payoff functions. However, in many importanteconomic models, such as those in Bertrand (1883), Hotelling (1929), Dasgupta andMaskin (1986), and Jackson (2009), payoffs are particularly discontinuous and/ornonquasiconcave.

Economists then seek weaker conditions that can guarantee the existence of equi-librium. Some seek to weaken the quasiconcavity of payoffs or substitute it with someforms of transitivity/monotonicity of payoffs (cf. McManus 1964; Roberts and Son-nenschein 1977; Nishimura and Friedman 1981; Topkis 1979; Vives 1990; Milgromand Roberts 1990), some seek to weaken the continuity of payoff functions (cf. Das-gupta and Maskin 1986; Simon 1987; Simon and Zame 1990; Tian 1992a, b, c, 2015;Tian andZhou 1992, 1995; Reny 1999, 2009, 2013; Bagh and Jofre 2006;Morgan andScalzo 2007; Carmona 2009, 2011, 2014; Carmona and Podczeck 2015; Prokopovych2011, 2013; Nessah 2011; He and Yannelis 2015a, b), while others seek to weakenboth quasiconcavity and continuity (cf. Yao 1992; Baye et al. 1993; Nessah and Tian2008; Nessah and Tian 2013; Tian 2015;McLennan et al. 2011; Barelli andMeneghel2013; Reny 2013).

This paper investigates the existence of pure strategy Nash equilibria in discontin-uous games. We introduce a new notion of very weak continuity, called quasi-weaktransfer continuity, which holds in a large class of discontinuous games. Roughlyspeaking, a game is quasi-weakly transfer continuous if for every nonequilibriumstrategy x∗, there exists a player i , a neighborhood N and a securing strategy forplayer i such that for every deviation strategy profile z in N , the lower envelope ofagent i’s payoff at securing strategy is strictly above the lower envelope of the agent’spayoff at the local security level even if the others deviate slightly from z.

We establish that a compact, convex, quasiconcave and quasi-weakly transfercontinuous game has a Nash equilibrium. We provide sufficient conditions for quasi-weak transfer continuity such as weak transfer continuity, strong quasi-weak transfercontinuity, quasi-weak upper semicontinuity and payoff security, and transfer lowercontinuity and quasi-upper semicontinuity. These conditions are satisfied in manyeconomic games and are often simple to check. We also provide the existence the-orems for symmetric games, and consider further extensions and improvements ofour main result. We show that our main results are unrelated to Reny (1999, 2009),Carmona (2009, 2011), Nessah (2011), Prokopovych (2011, 2013, 2015), and Barelliand Meneghel (2013).

The remainder of the paper is organized as follows. In Sect. 2, we first introducethe notion of quasi-weak transfer continuity, and then provide the main existenceresult on pure strategy Nash equilibrium. We also provide examples illustrating thetheorems as well as some sufficient conditions for quasi-weak transfer continuity.

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On the existence of Nash equilibrium in discontinuous. . . 517

Section 3 considers the equilibrium existence for symmetric games. Section 4 givessome possible extensions and improvements. Section 5 presents some applications ofinterest to economists that illustrate the usefulness of our results. Section 6 concludesthe paper. All the proofs are presented in the “Appendix.”

2 Existence of Nash equilibria

Consider a game in normal form: G = (Xi , ui )i∈I , where I = {1, . . . , n} is a finiteset of players, Xi is player i’s strategy space that is a nonempty subset of a Hausdorfflocally convex topological vector space, and ui is player i’s payoff function from theset of strategy profiles X = ∏

i∈I Xi to R. For each player i ∈ I , denote by −i allplayers rather than player i . Also denote by X−i = ∏

j �=i X j the set of strategies ofthe players in −i . Product sets are endowed with the product topology.

A game G = (Xi , ui )i∈I is said to be compact if for all i ∈ I , ui is bounded and Xi

is compact. A game G = (Xi , ui )i∈I is said to be quasiconcave if for every i ∈ I , Xi

is convex and the function ui is quasiconcave in xi . A pure strategy Nash equilibriumof G is a strategy profile x∗ ∈ X such that ui (yi , x∗−i ) ≤ ui (x∗) for yi ∈ Xi and alli ∈ I .

The followingweaknotion of continuity,quasi-weak transfer continuity, guaranteesthe existence of equilibrium in compact and quasiconcave games. The function thatcharacterizes quasi-weak transfer continuity is defined as follows: for x ∈ X , letΩ(x)denote the set of all open neighborhoods of x . For all i ∈ I and z ∈ X , define thelower envelope of the player’s payoff function ui (z):

ui (z) = supNz∈Ω(z)

infz′∈Nz

ui (zi , z′−i ).

Definition 1 A game G = (Xi , ui )i∈I is said to be quasi-weakly transfer continuousif whenever x ∈ X is not an equilibrium, there exists a player i , yi ∈ Xi and someneighborhood Nx of x such that for every z ∈ Nx , we have ui (yi , z−i ) > ui (z).

Quasi-weak transfer continuity means that whenever x is not an equilibrium, someplayer i has a strategy yi yielding a strictly large lower envelope of the payoff atthe local security level even if the others play slightly differently than at x .1 Ourexistence result states that a compact quasiconcave game that is quasi-weakly transfercontinuous has a Nash equilibrium.

Theorem 1 If the game G = (Xi , ui )i∈I is compact, quasiconcave, and quasi-weaklytransfer continuous, then it possesses a pure strategy Nash equilibrium.

The proof of the theorem will be presented in the “Appendix.” While we can provethis theorem directly, here we choose to prove the theorem by constructing a suitablegame with an additional player and then applying Theorem 3.4 in Reny (2013). An

1 The local security level at z means the value of the least favorable outcome in a neighborhood of z, givenby ui (z).

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advantage of such a proof is that it nicely connects Theorem 3.4 in Reny (2013) andthis approach is insightful. Indeed, when a game fails to have a pure strategy Nashequilibrium, we construct a new game that includes all of the original players I and anew player i = 0 /∈ I which is point secure with respect to I . Then, we can show thatTheorem 3.4 of Reny (2013) implies that this new game has a Nash equilibrium byquasi-weak transfer continuity and quasiconcavity, and consequently it is also a Nashequilibrium of G by quasi-weak transfer continuity.

Note that, contrary to the results of Reny (1999), Bagh and Jofre (2006), Carmona(2009, 2011), and Prokopovych (2011), which require verifying the closureness of thegraph of the vector payoff function, quasi-weak transfer continuity is relatively easierto verify, requiring no analysis of any closures of high-dimensional objects.

Example 1 Consider the following game with two players and the unit square X1 =X2 = [0, 1]. For player i = 1, 2 and x = (x1, x2) ∈ X = [0, 1]2, let the payofffunctions for the players be given by

ui (x1, x2) =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

xi + 1, if x−i > 12

1, if xi > 12 and x−i = 1

2

−1, if xi ≤ 12 and x−i = 1

2

xi − 1, if x−i < 12 .

It can be verified that the game is not (generalized) better-reply secure so thatProposition 2.4 of Barelli and Meneghel (2013), Theorem 1 of Carmona (2011) andTheorem 3.1 in Reny (1999) cannot be applied. It is not generalized weakly transfercontinuous so that Theorem 3.1 of Nessah (2011) cannot be used. It is neither weaklyreciprocal upper semicontinuous. As such, Theorem 4 in Prokopovych (2011) andCorollary 2 in Carmona (2009) cannot be applied.

However, it is quasi-weakly transfer continuous. Indeed, let x = (x1, x2) be anonequilibrium strategy profile. If (x1, x2) �= ( 12 ,

12 ), then it is easy to show that either

player 1 or 2 can secure (x1, x2). If (x1, x2) = ( 12 ,12 ), then by nonequilibrium of x

and continuity of payoffs at x , there exists a player i , yi ∈ Xi and some neighborhoodN of x such that for all z ∈ N , we have

ui (yi , z−i ) > ui (z).

Let ε > 0 be sufficiently small, neighborhood N ⊆ ( 12 − ε, 12 + ε)2 of ( 12 ,

12 ), and

yi = 1. Fix any z ∈ N , then we obtain

(1) If z−i > 12 , then ui (yi , z−i ) = 2 > 1 + zi ≥ ui (z).

(2) If z−i ≤ 12 , then ui (yi , z−i ) = 0 > zi − 1 = ui (z).

Since the game is compact and quasiconcave, by Theorem 1, it possesses a purestrategy Nash equilibrium.

Prokopovych (2013) introduced the notion of weak single-deviation property thatgeneralizes better-reply security of Reny (1999), weak transfer quasi-continuity of

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Nessah and Tian (2008) and single deviation property of Reny (2009) as follows: AgameG = (Xi , ui )i∈I has theweak single-deviation property ifwhenever x ∈ X is notan equilibrium, there exists an open neighborhoodN of x , a set of players I (x) ⊆ I anda collection of deviation strategies {yi (x) ∈ Xi : i ∈ I (x)} such that for every z ∈ Nnonequilibrium, there exists a player j ∈ I (x) satisfying u j (y j (x), z− j ) > u j (z).He then provided a theorem (Theorem 2) that shows under the weak single-deviationproperty and a condition (Condition (ii) in Theorem 2), there is a pure strategy Nashequilibrium in games with compact and convex strategy spaces. Notice that Condition(ii) is unrelated to quasiconcavity. Indeed, Reny (2009) showed with the aid of athree-person game that a game which is compact, quasiconcave, and has the single-deviation propertymay not have a pure strategyNash equilibrium2 (also see Example 2in Prokopovych (2013)).3 Moreover, the following example shows that the quasi-weaktransfer continuity does not imply the weak single deviation property either. As such,Theorem 1 is unrelated to Theorem 2 in Prokopovych (2013).

Example 2 Consider the following concession game with two players and the unitsquare X1 = X2 = [0, 1]. For player i = 1, 2 and x = (x1, x2) ∈ X = [0, 1]2, let thepayoff functions for the players be given by

ui (x1, x2) =

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

1, if xi = 0 and x−i > 0

x−i − xi + 1, if xi < x−i and xi > 0

3xi , if xi = x−i < 12

0, if xi = x−i ≥ 12

x−i − xi − 1, if xi > x−i .

It can be verified that the game does not have the weak single deviation property.Indeed, let x = ( 12 ,

12 ) be a nonequilibrium. For an open neighborhoodN of ( 12 ,

12 ), a

set of players I ( 12 ,12 ) ⊆ I and a collection of deviation strategies {yi (x) ∈ Xi : i ∈

I (x)}, we can find a nonequilibrium strategy z in N so that u j (y j (x), z− j ) ≤ u j (z),for each j ∈ I (x). To see this, consider two cases:

(1) I ( 12 ,12 ) = {i}, i = 1, 2. Let z ∈ N so that zi = z−i = t �= yi , t < 1

2 andt > 1

2 − yi2 if yi > 0. Then

ui (yi , z−i ) =

⎧⎪⎨

⎪⎩

1, if yi = 0

t − yi + 1, if yi < t and yi > 0

t − yi − 1, if yi > t.≤ 3t = ui (z).

2 However, for two-person games where the strategy sets are subsets of the real line, Prokopovych (2013)showed that the conclusion holds. Also, the single-deviation property was called weak transfer quasi-continuity in Nessah and Tian (2008) and will be defined below.3 Thus, strong transfer quasiconcavity defined below is also unrelated to and cannot be replaced by qua-siconcavity. Indeed, as shown in Theorem 4.2, transfer quasi-continuity, together with strong transferquasiconcavity, guarantees the existence of Nash equilibrium in any compact games, but Reny (2009)’scounterexample shows the nonexistence ofNash equilibriumwhen strong transfer quasiconcavity is replacedby quasiconcavity.

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(2) I ( 12 ,12 ) = I = {1, 2}. Let z ∈ N so that zi = z−i = t �= yi , for each i = 1, 2,

t < 12 and t > 1

2 − yi2 if yi > 0, i = 1, 2. Then u j (z) = 3t for j = 1, 2.

(i) If y1 = y2 = 0, then u j (y j , z− j ) = 1 < 3t = u j (z), for j = 1, 2.(ii) If yi = 0 and y−i > 0 for i = 1, 2, then ui (yi , z−i ) = 1 < 3t = ui (z) and

u−i (y−i , zi ) ={t − y−i + 1, if y−i < t

t − y−i − 1, if y−i > t.≤ 3t = u−i (z).

(iii) If y1 > 0 and y2 > 0, then for j = 1, 2 we have

u j (y j , z− j ) ={t − y j + 1, if y j < t

t − y j − 1, if y j > t.≤ 3t = u j (z).

However, it is quasi-weakly transfer continuous. Indeed, x �= ( 12 ,12 ) is obviously

quasi-weakly transfer continuous. Suppose that x = ( 12 ,12 ). Let ε > 0 be sufficiently

small (ε < 14 ), N ⊂ ( 12 − ε, 1

2 + ε)2 and yi = ε, for some i = 1, 2. Fix any z ∈ Nand Vz ⊂ N . Then ui (yi , z−i ) ≥ 1 + z−i − ε > 1 + z−i − zi + ε ≥ ui (z).

While it is somewhat simple to verify quasi-weak transfer continuity, it is sometimeseven simpler to verify other conditions leading to it.

Definition 2 A game G = (Xi , ui )i∈I is said to be strongly quasi-weakly transfercontinuous if whenever x ∈ X is not an equilibrium, there exists a player i , yi ∈Xi , ε > 0, and some neighborhood Nx of x such that for every z ∈ Nx , we haveui (yi , z−i ) > ui (z) + ε.

Definition 3 A game G = (Xi , ui )i∈I is said to be quasi upper semicontinuous(QUSC) if for all i ∈ I , x ∈ X , and ε > 0, there exists a neighborhood N of x suchthat ui (x) ≥ ui (z) − ε, for every z ∈ N .

Definition 4 A game G = (Xi , ui )i∈I is said to be quasi-weakly upper semicontin-uous (QWUSC) if whenever x ∈ X is not an equilibrium, there exists a player i ,xi ∈ Xi , ε > 0, and some neighborhood Nx of x such that ui (xi , x−i ) > ui (z) + ε,for every z ∈ Nx .

Definition 5 A game G = (Xi , ui )i∈I is said to be weakly transfer lower semicon-tinuous (WTLSC) if whenever x is not a Nash equilibrium, there exists a player i ,yi ∈ Xi , ε > 0 and some neighborhoodNx of x such that ui (yi , z−i ) > ui (x)+ ε forall z ∈ Nx .

It is obvious that (1) upper semicontinuity implies quasi upper semicontinuity, whichin turn implies quasi-weak upper semicontinuity, and (2) lower semicontinuity impliespayoff security, which in turn implies weak transfer lower semicontinuity. Also, quasi-weak upper semicontinuity and transfer lower semicontinuity, when combined with

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payoff security4 and quasi upper semicontinuity, respectively, imply quasi-weak trans-fer continuity. We then have the following proposition.

Proposition 1 Suppose that a game G satisfies any of the following conditions:

(a) it is strongly quasi-weakly transfer continuous;(b) it is quasi-weakly upper semicontinuous and payoff secure;(c) it is weakly transfer lower semicontinuous and quasi upper semicontinuous.

Then it is quasi-weakly transfer continuous, and consequently, there exists a purestrategy Nash equilibrium provided that it is also compact and quasiconcave.

Example 3 Consider the two-player game with the following payoff functions definedon [0, 1] × [0, 1]:

ui (x1, x2) ={xi + 1 if x−i > 1

2

xi − 1 if x−i ≤ 12 .

This game is not (generalized) better-reply secure nor (weakly) reciprocal upper semi-continuous. As such, Corollary 3.3, Corollary 3.4 of Reny (1999), Proposition 1 ofBagh and Jofre (2006), and Theorem 4 in Prokopovych (2011) cannot be applied.

However, the game is payoff secure and quasi-weakly upper semicontinuous. Tosee this, let i ∈ I , ε > 0, and x ∈ X . If x−i �= 1

2 , then it is clear that there exists astrategy yi = 1 and some neighborhood V of x−i such that ui (yi , z−i ) ≥ ui (x)−ε foreach z−i ∈ V . If x−i = 1

2 , then there exists a strategy yi = 1 and some neighborhoodV of x−i such that ui (x) − ε = xi − 1− ε ≤ 0 ≤ ui (yi , z−i ) for each z−i ∈ V . Thus,the game is payoff secure.

Also, let x = (x1, x2)be a nonequilibriumstrategyprofile. Then there exists a playeri such that xi < 1. Let xi +2ε < 1 for some ε > 0. If x−i �= 1

2 , then it is clear that thereexists a strategy yi = 1 and some neighborhood V ⊆ (xi −ε, xi +ε)×[0, 1] of x suchthat for every z ∈ V and every neighborhood Vz of z, ui (yi , x−i ) > ui (zi , z′−i )+ ε for

some z′ ∈ Vz . If x−i = 12 , then there exists a strategy yi = 1 and some neighborhood

V ⊆ (xi − ε, xi + ε) × [0, 1] of x such that for each z ∈ V and every neighborhoodVz of z, there exists z′ ∈ Vz with z′−i = 1

2 so that ui (zi , z′−i ) + ε = zi − 1 + ε ≤xi + 2ε − 1 < ui (yi , x−i ) = 0. Thus, it is quasi-weakly upper semicontinuous. Sincethe game is also compact and quasiconcave, then by Proposition 1(b), it possesses aNash equilibrium.

Example 4 Consider the two-player game with the following payoff functions definedon [−1, 1] × [−1, 1] by

ui (x1, x2) =

⎧⎪⎨

⎪⎩

xi + 1 if x−i > 0

xi if x−i = 0

xi − 12 if x−i < 0.

4 A game is payoff secure if for every x ∈ X , every ε > 0, and every player i , respectively, there existsxi ∈ Xi such that ui (xi , z−i ) ≥ ui (x) − ε for all z−i in some open neighborhood of x−i .

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This game is not generalized better-reply secure. However, it is clearly weakly transferlower continuous. To see that it is also quasi upper semicontinuous, consider a playeri , a strategy x and ε > 0. If x−i �= 0, it is obvious that the game is quasi uppersemicontinuous. If x−i = 0, then there exists a neighborhoodN ⊆ (xi − δ, xi + δ] ×(−δ, δ) of x (with δ < 1

2 ) such that for each z ∈ N and eachNz as a neighborhood of z,there exists z′ ∈ Nz with z′−i < 0 so as ui (x) = xi ≥ (xi+δ)− 1

2−ε ≥ ui (zi , z′−i )−ε,which means it is also quasi upper semicontinuous at x−i = 0. Since the game is alsocompact and quasiconcave, then by Proposition 1(c), it possesses a Nash equilibrium.

3 Pure strategy symmetric Nash equilibrium

In this section, it is assumed that G = (Xi , ui )i∈I is a quasi-symmetric game, i.e.,Z = X1 = · · · = Xn and u1(x, y, . . . , y) = u2(y, x, y, . . . , y) = un(y, . . . , y, x)for all x, y ∈ Z . Recall that a Nash equilibrium (x1, . . . , xn) is symmetric if x1 =x2 = · · · = xn .

Definition 6 A symmetric game G = (Xi , ui )i∈I is said to be diagonally quasi-weaktransfer continuous if whenever (x∗, . . . , x∗) ∈ Xn is not an equilibrium, there existsa player i , a strategy y ∈ Z , and a neighborhood N of x∗ such that for every z ∈ N ,we have ui (z, . . . , z, y, z, . . . , z) > ui (z, . . . , z).

We then have the following existence theorem for quasi-symmetric games.

Theorem 2 If the game G = (Xi , ui )i∈I is quasi-symmetric, compact, quasiconcave,and diagonally quasi-weak transfer continuous, then it has a symmetric pure strategyNash equilibrium.

The following example illustrates Theorem 2.

Example 5 Consider a timing game between two players on the unit square X1 =X2 = [0, 1] studied by Prokopovych (2013). For player i = 1, 2 and x = (x1, x2) ∈X = [0, 1]2, let the payoff functions for the players be given by

ui (x1, x2) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

2, if xi < x−i

2, if xi = x−i < 12

0, if xi = x−i ≥ 12

−2, if xi > x−i .

It can be verified that the game is not diagonally better-reply secure so that The-orem 4.1 in Reny (1999) cannot be applied. This game is not generalized weaklytransfer continuous nor weakly reciprocal upper semicontinuous so that the results inNessah (2011), Prokopovych (2011) and Carmona (2009) cannot be applied.

However, it is diagonally quasi-weak transfer continuous. Indeed, let (x, x) be anonequilibrium strategy profile. By nonequilibrium of (x, x), we have 1

2 ≤ x ≤ 1.Then, there exists a player i = 1, some neighborhood N ⊂ (x − ε, x + ε) (ε issufficiently small) of x and y = 0 such that for all z ∈ N and Vz ⊂ N , we have

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ui (y, t) = 2 > −2 = ui (z, z). Since the game is also quasi-symmetric, compact, andquasiconcave, by Theorem 2, it possesses a symmetric Nash equilibrium.

Quasiconcavity is still a strong assumption formany economic games. For instance,the classicBertrandmodel typically results in nonquasiconcave anddiscontinuous pay-offs. Thus, a general existence result for nonquasiconcave and discontinuous gamesis called for. In the following, we provide an existence result for general nonquasi-concave and discontinuous games. First, recall the following definition of diagonaltransfer quasiconcavity introduced by Baye et al. (1993).

Definition 7 Asymmetric gameG = (Xi , ui )i∈I is said to bediagonally transfer qua-siconcave if X is convex, and for every player i , any finite subset of {y1, y2, . . . , ym} ⊆X , there is a corresponding finite subset {x1, . . . , xm} ⊆ X such that for any subset Jof {1, . . . ,m} and every x ∈ co{x j , j ∈ J }, we have

ui (x, . . . , x) ≥ mink∈J

ui (x, . . . , x, yk, x, . . . , x).

While diagonal transfer quasiconcavity is weaker than diagonal quasiconcavity,5 andconsequentlyweaker than quasiconcavity, the following notion of diagonalweak trans-fer continuity is stronger than the diagonal quasi-weak transfer continuity.

Definition 8 A symmetric game G = (Xi , ui )i∈I is said to be diagonally weak trans-fer continuous if whenever (x∗, . . . , x∗) ∈ Xn is not an equilibrium, there exists aplayer i , strategy y ∈ X and neighborhoodN of x∗ such that ui (z, . . . z, y, z, . . . , z) >

ui (z, . . . , z) for all z ∈ N .

We now state an existence result for nonquasiconcave and discontinuous games.

Theorem 3 Suppose that G = (Xi , ui )i∈I is quasi-symmetric, compact, diagonallytransfer quasiconcave, and diagonally weak transfer continuous. Then, it has a sym-metric pure strategy Nash equilibrium.

Remark 1 Theorem 3 strictly generalizes Proposition 5.2 in Reny (2013).

As an illustration, we will use Bertrand model to show the usefulness of Theorem 3in Sect. 5. Similar to the previous section, we can provide some sets of sufficientconditions for diagonal quasi-weak transfer continuity by introducing various notionsof diagonal upper/lower semicontinuity, such as diagonal quasi-weak upper semi-continuity, diagonal quasi upper semicontinuity, and diagonal weak transfer lowersemicontinuity.6

4 Further extensions and improvements

We can further improve our main result in two directions. One is to weaken continuity,and the other is to weaken quasiconcavity.

5 A game G = (Xi , ui )i∈I is said to be diagonally quasiconcave if X is convex, and for every player i ,all x1, . . . , xm ∈ X and all x ∈ co{x1, . . . , xm }, ui (x, . . . , x) ≥ mink=1,...,m ui (x, . . . , x, x

k , x, . . . , x).6 See Nessah and Tian (2008).

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4.1 Weakening quasiconcavities

For each set Z , denote by coZ the convex hull of Z . Let I = {1, . . . , n} be the set offinite players, Xi the set of pure strategies for player i that is a nonempty subset of aHausdorff locally convex topological vector space Ei , X = ∏

i∈I Xi the set of purestrategies profiles, andi a binary relation on X of player i . A game G = (Xi ,i )i∈Iis compact and convex if, for all i ∈ I , Xi is compact and convex. We say that astrategy profile x ∈ X is a pure strategy Nash equilibrium of a game G if, for eachi ∈ I , and yi ∈ Xi , (xi , x−i ) i (yi , x−i ).

A correspondence C : X × X ⇒ X × X is said to be inherited if it satisfies:

(a) (Reflexivity:) for all x, y ∈ X , (x, y) ∈ C(x, y).(b) (Transitivity:) for all x1, x2, x3, y1, y2, y3 ∈ X , (x1, y1) ∈ C(x2, y2) and

(x2, y2) ∈ C(x3, y3) imply (x1, y1) ∈ C(x3, y3).(c) (Completeness:) for all x1, x2, y1, y2 ∈ X , either (x1, y1) ∈ C(x2, y2) or

(x2, y2) ∈ C(x1, y1).(d) (Smallest Finite Element:) for all x, x1, . . . , xm, y1, . . . , ym, z1, . . . , zm ∈ X ,

whenever (x, yh) ∈ C(xh, zh), h = 1, . . . ,m, there is h0 = 1, . . . ,m such that(x, yh) ∈ C(xh0 , zh0) for h = 1, . . . ,m.

Let us consider the following definition.

Definition 9 A game G = (Xi ,i )i∈I is said to be P-correspondence secure (CS)if for every i ∈ I , there exists an inherited correspondence P

i : X × X ⇒ X × Xsuch that whenever x ∈ X is not an equilibrium of G, there exists a neighborhoodV of x and a well-behaved correspondence φx : V ⇒ X satisfying that for everynonequilibrium z ∈ V and for some player j , we have

z /∈ co{t ∈ X such that (z, t) ∈ Pj (x, y)} holds for each (x, y) ∈ Graph(φx ).

Theorem 4 If the gameG = (Xi ,i )i∈I is compact, convex, and P-correspondencesecure, then it has a pure strategy Nash equilibrium.

Remark 2 If for each x, y, z ∈ X , the set {t ∈ X such that (z, t) ∈ Pj (x, y)} is

convex, then the condition z /∈ co{t ∈ X such that (z, t) ∈ Pj (x, y)} becomes

(z, z) /∈ Pj (x, y).

Example 6 Consider the two-player game on the square [0, 1] × [0, 1].

ui (p) ={4pi if p−i ≥ 1

2

pi otherwise.

It is clear that this game is bounded, compact, and quasiconcave, but it is not contin-uously secure. To see this, consider p = ( 12 ,

12 ) ∈ X . Then p is not an equilibrium.

For every neighborhood V of p, correspondence ϕ : V ⇒ X , and α ∈ Rn , if αi ≤ 1,

i = 1, 2, choosing z ∈ V with zi > 12 , we have ui (z) = 4zi ≥ 2 > αi ; if α j > 1 for

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some j , choosing z ∈ V with zi < 12 , i = 1, 2, we have u j (t j , z− j ) = t j ≤ 1 < α j

for all t j ∈ ϕ j (z). Thus, this game is not continuously secure, so Theorem 2.2 andProposition 2.4 in Barelli and Meneghel (2013) cannot be applied. Since the contin-uous security condition is weaker than the C-security, Proposition 2.7 of McLennanet al. (2011) cannot be applied either.

This gamedoes not have the generalized deviation property (it is not correspondencesecure when i can be represented by a utility function ui ). Indeed, consider p =( 12 ,

12 ) ∈ X . Then, p is not an equilibrium. For every neighborhood V of p and

correspondence ϕ ∈ ∏i∈I WV (pi , p−i ), choosing z ∈ V with z = ( 12 ,

12 ), and z

′ ∈ Vwith z

′−i < 1

2 , i ∈ I , we have uϕi (y

′i , z

′−i ) ≤ ui (y

′i , z

′−i ) = y

′i ≤ 2 = ui (z) for

all y′i ∈ ϕi (z

′−i ). Hence, Theorem 5.6 of Reny (2013) and Theorem 5.8 of Bich and

Laraki (2012) cannot be applied.This game does not have the single lower deviation property (it is not corre-

spondence secure when i can be represented by a utility function ui ).7 Indeed,consider p = ( 12 ,

12 ) ∈ X , then p is not an equilibrium. For any neighborhood

V of p and all y ∈ X , choosing z ∈ V with z j > 12 for j = 1, 2 and t ∈ V

with t−i < 12 for i ∈ I , then there exists a neighborhood Vz ⊂ ( 12 , 1]2 satisfying

ui (z) ≥ inf z′ ∈Vzui (zi , z

′−i ) = 4zi > 2 > yi = ui (yi , t−i ). Thus, this game does not

have the single lower deviation property. As such, Theorem 2.2 in Reny (2009) cannotbe applied.

This game is not (generalized) better-reply secure. Indeed, consider p = ( 12 ,12 ) ∈ X

and u = (2, 2). Then, (p, u) is in the closure of the graph of its vector function, andp is not an equilibrium. Each player i cannot obtain a payoff strictly above ui = 2.For any neighborhood V ⊂ [0, 1] of p−i and for all well-behaved correspondenceϕi : V ⇒ Xi = [0, 1], choosing p

′−i ∈ V with p

′−i < 1

2 , we then have ui (qi , p′−i ) =

qi ≤ 2 = ui for all qi ∈ ϕi (p′−i ). Thus, this game is not (generalized) better-

reply secure. Therefore, Corollary 4.5 of Barelli and Meneghel (2013), Theorem 1 ofCarmona (2011) and Theorem 3.1 in Reny (1999) cannot be applied.

This game is not generalized payoff secure. Indeed, let i = 1, x = (1, 12 ), ε = 1

2 .Then for any neighborhood V ⊂ [0, 1] of 1

2 and for all well-behaved correspondence

ϕi : V ⇒ [0, 1], choosing p′2 ∈ V with p

′2 < 1

2 , we have u1(q1, p′2) = q1 ≤ 1 <

4 − ε, for each q1 ∈ ϕ1(p′2). Thus, this game is not generalized payoff secure, and

consequently Theorem 4 in Prokopovych (2011) and Corollary 2 in Carmona (2009)cannot be applied.

However, it is P-correspondence securewhere the correspondence Pi : X×X ⇒

X × X is defined as

(z, t) ∈ Pi (x, y) iff ui (ti , z−i ) − ui (z) ≥ ui (yi , x−i ) − ui (x).

It is clear that for each i = 1, 2, Pi is inherited. Let p = (p1, p2) be a nonequilibrium

strategy profile with at least one non-one coordinate. Then, there exists i ∈ I with

7 A game G = (Xi , ui )i∈I has the single lower deviation property if whenever x∗ is not an equilibrium,there is a player, a neighborhood V of x∗ and a strategy y ∈ X such that for each z ∈ V , there is a player jso as u j (y j , t− j ) > u j (z) for all t ∈ V .

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pi < 1. Therefore, there exists a neighborhood V of p, ε > 0 with p′i + ε < 1 for all

p′ ∈ V and awell-behaved correspondenceφp : V ⇒ X defined byφp(p

′) = {(1, 1)},

for each p′ ∈ V such that for each z ∈ V , there exists a player j = i so as for each

(t, y) ∈ Graph(φp), we have ui (yi , t−i ) ={4, if t−i ≥ 1

2

1, otherwiseand ui (t) =

{4ti , if t−i ≥ 1

2

ti , otherwise.Thus, for each (t, y) ∈ Graph(φp) we have u j (y j , t− j )− u j (t) > 0 = u j (z j , z− j )−u j (z). As the game is compact and convex, by Theorem 4, the considered gamepossesses a Nash equilibrium.

Remark 3 The considered game in Example 2 is P-correspondence secure and con-sequently by Theorem 4 it has a Nash equilibrium.

The following definition extends Definition 1 to a nonquasiconcave game.

Definition 10 A game G = (Xi , ui )i∈I is said to be correspondence quasi-weaklytransfer continuous if whenever x ∈ X is not an equilibrium, there exists a neighbor-hoodN of x and a well-behaved correspondence φ : N ⇒ X satisfying that for everynonequilibrium z ∈ N and for some player j , we have

z j /∈ co{t j ∈ X j such that u j (t j , z− j ) − u j (z) ≥ u j (y j , x− j ) − u j (x)}

holds for each (x, y) ∈ Graph(φ).

We then have the following corollary which is a strict generalization of Theorem 1.

Corollary 1 If G = (Xi , ui )i∈I is compact, convex, and correspondence quasi-weakly transfer continuous, then it has a pure strategy Nash equilibrium.

Proof It is sufficient to consider in Theorem 4 the following inherited correspondencePi : X × X ⇒ X × X ( j = 1, . . . , n) defined by:

Pj (x, y) = {(t, z) ∈ X × X : u j (t j , z− j ) − u j (z) ≥ u j (y j , x− j ) − u j (x)}.

��The weak transfer continuity was first introduced in our previously circulated Nessahand Tian (2008), which is the same as the single player deviation property introducedin Prokopovych (2013).

Definition 11 A game G = (Xi , ui )i∈I is said to be weakly transfer continuous ifwhenever x ∈ X is not an equilibrium, there exists a player i , yi ∈ Xi and someneighborhood Nx of x such that ui (yi , z−i ) > ui (z) for all z ∈ Nx .

The following definition extends Definition 11 to a nonquasiconcave game.

Definition 12 A game G = (Xi , ui )i∈I is said to be correspondence weakly transfercontinuous if whenever x ∈ X is not an equilibrium, there exists a neighborhood

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N of x and a well-behaved correspondence φ : N ⇒ X satisfying that for everynonequilibrium z ∈ N and for some player j , we have

z j /∈ co{t j ∈ X j such that u j (t j , z− j ) − u j (z) ≥ u j (y j , x− j ) − u j (x)}

holds for each (x, y) ∈ Graph(φ).

Consequently, we obtain the following corollary which is a strict generalization ofNessah and Tian (2008) and Nessah (2011).

Corollary 2 If G = (Xi , ui )i∈I is compact, convex, and correspondence weaklytransfer continuous, then it has a pure strategy Nash equilibrium.

Proof It is sufficient to consider in Theorem 4 the following inherited correspondencePj : X × X ⇒ X × X ( j = 1, . . . , n) defined as follows:

Pj (x, y) = {(t, z) ∈ X × X : u j (t j , z− j ) − u j (z) ≥ u j (y j , x− j ) − u j (x)}.

��Reny (2013) introduced the following notion. A game G = (Xi ,i )i∈I is said tobe correspondence secure if whenever x ∈ X is not an equilibrium, there exists aneighborhood V of x and a well-behaved correspondence φ : V ⇒ X so that for eachz ∈ V , there exists a player j for whom z j /∈ co{t j ∈ X j : (t j , z− j ) j (y j , x− j )}holds for each (x, y) ∈ Graph(φ).

For the game G = (Xi ,i )i∈I , we associate the following correspondence Pi :

X × X ⇒ X × X (i = 1, . . . , n) as follows:

Pi (x, y) = {(t, z) ∈ X × X : (ti , z−i ) i (yi , x−i )}.

Obviously, if the relation i is complete, reflexive and transitive, then Pi is inherited

and consequently we have the following corollaries:

Corollary 3 (Prokopovych 2015) If G = (Xi , ui )i∈I is compact, quasiconcave, andcorrespondence secure, then it has a pure strategy Nash equilibrium.

Corollary 4 (Reny 2013) If G = (Xi , ui )i∈I is compact, convex, complete, reflexive,transitive, and correspondence secure, then it has a pure strategy Nash equilibrium.

Corollary 5 (Barelli andMeneghel 2013) If G = (Xi , ui )i∈I is compact, convex, andcontinuously secure, then it has a pure strategy Nash equilibrium.

Corollary 6 (Barelli and Meneghel 2013) If G = (Xi , ui )i∈I is compact, quasicon-cave and generalized better-reply secure, then it has a pure strategy Nash equilibrium.

Corollary 7 (McLennan et al. 2011) If G = (Xi , ui )i∈I is compact, convex, and C-secure, then it has a pure strategy Nash equilibrium. If the restriction operator X isuniversal, then Theorem 4 is a strict generalization of Theorem 3.4 of McLennan et al.(2011).

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4.2 Further weakening continuity

Definition 13 A game G = (Xi , ui )i∈I is said to be pseudo quasi-weakly transfercontinuous if whenever x∗ ∈ X is not an equilibrium, there exists a neighbor-hood N of x∗, a player i , and a strategy yi ∈ Xi such that inf z′∈N ui (yi , z

′−i ) >

inf z′′∈N ui (zi , z′′−i ), for each z ∈ N .

The difference between pseudo quasi-weak transfer continuity and quasi-weaktransfer continuity is that the former takes the neighborhood Nz equal to Nx so thatquasi-weak transfer continuity implies pseudo quasi-weak transfer continuity.

Definition 14 A game G = (Xi , ui )i∈I is said to be strongly quasiconcave iffor each i ∈ I , two sets N 1 and N 2 with N 1 ∩ N 2 �= ∅ and y1i , y

2i ∈ Xi

with inf z′∈N ui (yji , z′−i ) > inf z′′∈N j ui (zi , z′′−i ) for each z ∈ N j , j = 1, 2,

inf z′∈N ui (yi , z′−i ) > inf z′′∈N ui (zi , z′′−i ) for each z ∈ N = N 1 ∩ N 2 and

yi ∈ co(y1i , y2i ).

We then have the following result.

Theorem 5 If the game G = (Xi , ui )i∈I is compact, convex, strongly quasiconcaveand pseudo quasi-weakly transfer continuous, then it possesses a pure strategy Nashequilibrium.

The following proposition shows that pseudo quasi-weak transfer continuity is alsoweaker than better-reply security, and consequently Theorem 5 extends Theorem 3.1in Reny (1999) by weakening better-reply security.

Proposition 2 If G = (Xi , ui )i∈I is better-reply secure, then it is pseudoquasi-weaklytransfer continuous.

While in terms of continuity, Theorem 5 is more interesting than Theorem 1 as wellas Theorem 3.1 in Reny (1999), the strong quasiconcavity of G is more complicatedto check. A question is then whether strong quasiconcavity of G can be replaced byquasiconcavity of ui (xi , x−i ) in xi . Unfortunately, the answer is negative.

Example 7 Consider, on the unit square, the following game that has no pure strategyNash equilibrium.

u1(x) ={x1, if x2 = 0

2 − x1, otherwise

u2(x) ={x2, if x1 = 1

2 − x2, otherwise

The considered game is pseudo quasi-weakly transfer continuous and quasiconcave,but it is not strongly quasiconcave. Indeed, letN 1 = [0, 1

2 + ε),N 2 = ( 12 − ε, 12 + ε)

(for sufficiently small ε) and y11 = 12 + ε, y21 = 0. Then, we have u1(y

j1 , z2) >

inf z′∈N j u1(z1, z′2), for each z ∈ N j , j = 1, 2. Or if we consider z = ( 12 ,12 ) ∈ N 2 =

N 1 ∩N 2 and y1 = 12 ∈ co(y11 , y

21 ), we obtain u1(y1, z2) = inf z′∈N 2 u1(z1, z′2) = 3

2 .

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Our main result can be further improved by introducing the following notions oftransfer quasi-continuity and strong diagonal transfer quasiconcavity.

Definition 15 AgameG = (Xi , ui )i∈I is said to beweakly transfer quasi-continuousif whenever x ∈ X is not an equilibrium, there exists a y ∈ X and a neighborhoodNx

of x such that for every x ′ ∈ Nx , there exists a player i satisfying ui (yi , x ′−i ) > ui (x ′).

Weak transfer quasi-continuity, which was first introduced in our previously circulatedNessah and Tian (2008) and then called single-deviation property in Reny (2009), onlyrequires that each strategy profile in a neighborhood of x be upset by one, but not allplayers. Thus, it is a very weak notion of continuity so that it is a form of quasi-continuity.

Definition 16 A game G = (Xi , ui )i∈I is said to be strongly diagonal transferquasiconcave if for any finite subset {y1, . . . , ym} ⊆ X , there exists a correspond-ing finite subset {x1, . . . , xm} ⊆ X such that for any subset {xk1 , xk2 , . . . , xks } ⊆{xk1 , xk2 , . . . , xks }, 1 ≤ s ≤ m, and any x ∈ co{xk1 , xk2 , . . . , xks }, there existsy ∈ {yk1 , . . . , yks } so that

ui (yi , x−i ) ≤ ui (x) ∀i ∈ I. (1)

It is clear that a game is diagonally transfer quasiconcave if it is strongly diagonaltransfer quasiconcave.8 We then have the following result that generalizes Corollary1 in Prokopovych (2013).

Theorem 6 Suppose that a gameG = (Xi , ui )i∈I is convex, compact, weakly transferquasi-continuous. Then, the game possesses a pure strategy Nash equilibrium if andonly if it is strongly diagonal transfer quasiconcave.

It may be remarked that weak transfer quasi-continuity and quasi-weak transfer con-tinuity are not implied by nor imply each other. The game considered in Example 2is quasi-weakly transfer continuous, but it does not have the weak single-deviationproperty which in turn does not satisfy the single-deviation property/weak transferquasi-continuity. On the other hand, the game in Example 3.1 in Reny (2009) isweakly transfer quasi-continuous, but it is not quasi-weakly transfer continuous.

While weak transfer quasi-continuity in Theorem 6 is weaker than the better-replysecurity and diagonal transfer continuity, it requires that the game be strongly diagonaltransfer quasiconcave. Can strong diagonal transfer quasiconcavity in Theorem 6 bereplacedby conventional quasiconcavity?Unfortunately, the answer is no.Reny (2009)showed this by giving a counterexample (Example 3.1 in his paper) where a gameG = (Xi , ui )i∈I is compact, quasiconcave, and weakly transfer quasi-continuous, butit may not possess a pure strategy Nash equilibrium.

Thus, Theorems 3 and 6 both show that there is a trade-off between continuitycondition and quasiconcavity condition.

8 Indeed, summing up (1) and denoting U (x, y) = ∑i∈I ui (yi , xi ), we have min1≤l≤sU (x, yk

l) ≤

U (x, x), which is the condition for diagonal transfer quasiconcavity.

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5 Applications

In this section, we show how our main existence results are applied to some importanteconomic games. We provide two applications: one is the shared resource gamesintensively studied by Rothstein (2007), and the other is the classic Bertrand pricecompetition games studied first by Bertrand (1883).

5.1 The shared resource games

The shared resource games that usually result in discontinuous payoffs include awide class of games such as the canonical game of fiscal competition for mobilecapital. In these games, players compete for a share of a resource that is in fixed totalsupply, except perhaps at certain joint strategies. Each player’s payoff depends onher opponents’ strategies only through the effect those strategies have on the amountof the shared resource that the player obtains. As Rothstein (2007) argued, when advalorem taxes instead of unit taxes are adopted and the aggregate amount of mobilecapital is fixed instead of variable, it will typically result in at least one, and possiblymany, discontinuity points.

Formally, for such a game G = (Xi , ui )i∈I , each player i has a convex andcompact strategy space Xi ⊂ R

l and a payoff function ui that depends on otherplayers’ strategies only through the sharing rule defined by Si : X → [0, s] withs ∈ (0,+∞). That is to say, each player has a payoff function ui : X → R withthe form ui (xi , x−i ) = Fi [xi , Si (xi , x−i )] where Fi : Xi × [0, s] → R and ui isbounded.9

Let Di ⊆ X be the set of joint strategies at which Si is discontinuous and let the setΔ = ⋃

i∈I Di be all of the joint strategies at which one or more of the sharing rulesare discontinuous. The set X\Δ is then all of the joint strategies at which the sharingrules are continuous.

Rothstein (2007) showed a shared resource game possesses a pure strategy Nashequilibrium if the following conditions are satisfied: (1) X is compact and convex; (2)ui is continuous on X and quasiconcave in xi ; (3) Si satisfies: (3.i)

∑ni=1 Si (x) = s

for all x ∈ X \ Δ; (3.ii) there exists s ∈ [0, s] such that∑n

i=1 Si (x) = s for allx ∈ Δ; (3.iii) for all i , (xi , x−i ) ∈ Di and every neighborhood V(xi ) of xi , there existsx ′i ∈ V(xi ) such that (x ′

i , x−i ) ∈ X \ Di ; (3.iv) there exists a constant si satisfyings ≥ si > s/n such that for all i , all (xi , x−i ) ∈ Δ, and all (x ′

i , x−i ) ∈ X \ Di ,Si (x ′

i , x−i ) ≥ si ≥ Si (xi , x−i ); (4) Fi is continuous, nondecreasing in si , and satisfiesmaxxi∈Xi Fi (xi , si ) > maxxi∈Xi Fi (xi , s/n) for any si > s/n.

In the following, we will give an existence result with much simpler conditions andits proof is also much easier:

Assumption 1: The game is compact and quasiconcave.Assumption 2: If (yi , x−i ) ∈ Di and Fi (yi , Si (yi , x−i )) > Fi (xi , Si (x)) for player

i , then there exists some player j ∈ I and y j such that (y j , x− j ) ∈X\Dj and Fj (y j , S j (y j , x− j )) > Fj (x j , S j (x)).

9 For more details on this model, see Rothstein (2007).

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Assumption 3: If (yi , x−i ) ∈ X\Di and Fi (yi , Si (yi , x−i )) > Fi (xi , Si (x)) forplayer i , then there exists a player j ∈ I , a deviation strategy profileyi and a neighborhood Nx of x such that for every z ∈ Nx , we haveF j (y j , S j (y j , z− j )) > F j (z j , S j (z j , z− j )).

Assumption 1 is standard. A well-known sufficient condition for a composite functionui = Fi [xi , Si (xi , x−i )] to be quasiconcave is that Fi is quasiconcave and nondecreas-ing in si , and Si is concave. Assumption 2 means that if x is not an equilibrium andcan be improved at a discontinuous strategy profile (yi , x−i ) when player i uses thedeviation strategy yi , then there exists a player j such that it must also be improvedby a continuous strategy profile (y j , x− j ) when player j uses the deviation strategyy j . Assumption 3 means that if a strategy profile x is not an equilibrium and canbe improved by a continuous strategy profile (yi , x−i ) when player i uses a deviationstrategy yi , then there exists a securing strategy profile y and a neighborhood of x suchthat all points in the neighborhood cannot be equilibria. We then have the followingresult.

Proposition 3 A shared resource game possesses a pure strategy Nash equilibrium ifit satisfies Assumptions 1–3.

5.2 The Bertrand price competition games

It is well known that Bertrand competition typically results in discontinuous andnonquasiconcave payoffs. It is a normal form game in which each of n ≥ 2 firms,i = 1, 2, . . . , n, simultaneously sets a price pi ∈ P = [0, p]. Under the assumptionof profit maximization, the payoff to each firm i is

πi (pi , p−i ) = pi Di (pi , p−i ) − Ci (Di (pi , p−i )),

where p−i denotes the vector of prices charged by all firms other than i , Di (pi , p−i )

represents the total demand forfirm i’s product at prices (pi , p−i ), andCi (Di (pi , p−i ))

is firm i’s total cost of producing the output Di (pi , p−i ). A Bertrand equilibrium is aNash equilibrium of this game.

Let Ai ⊆ Pn be the set of joint strategies at which πi is discontinuous, Δ =⋃i∈I Ai be the set of all of the joint strategies at which one or more of the payoffs are

discontinuous, and X\Δ be the set of all joint strategies at which all of the payoffs arecontinuous.

We make the following assumptions:

Assumption 1′: The game is compact, convex, and strongly diagonal transfer quasi-concave.

Assumption 2′: If (qi , p−i ) ∈ Ai andπi (qi , p−i ) > πi (p) for i ∈ I , then there existsa firm j ∈ I , and q j such that (q j , p−i ) ∈ Pn\A j andπ j (q j , p−i ) >

πi (p).Assumption 3′: If (qi , p−i ) ∈ Pn\Ai and πi (qi , p−i ) > πi (p) for player i , then

there exists a player j ∈ I , a deviation strategy profile q j and

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a neighborhood Np of p such that for every r ∈ Np, we haveπ j (q j , z− j ) > π j (r).

We then have the following result.

Proposition 4 Each Bertrand price competition game has a pure strategy Nash equi-librium if it satisfies Assumptions 1′–3′.

Example 8 Consider a quasi-symmetric two-player Bertrand price competition gameon the square [0, a] × [0, a] with a > 0. Assume that the demand function is discon-tinuous and is defined by

Di (pi , p−i ) =

⎧⎪⎨

⎪⎩

α f (pi ) if pi < p−i

β f (pi ) if pi = p−i

γ f (pi ) if pi > p−i

where f : R+ → R+ is a continuous and nonincreasing function andα > β > γ ≥ 0.Suppose that the total cost of production is zero for each firm. Then, the payoff functionfor each firm i becomes

πi (pi , p−i ) =

⎧⎪⎨

⎪⎩

αpi f (pi ) if pi < p−i

βpi f (pi ) if pi = p−i

γ pi f (pi ) if pi > p−i

.

The game is quasi-symmetric and compact. Since α > β > γ and f is nonincreasing,it is clearly diagonally quasiconcave. Thus, Assumption 1′ is satisfied. Note that the setof discontinuity points is given by A = A1 = A2 = {(p1, p2) : p1 = p2}. Assumption2′ is obviously satisfied. Indeed, if πi (qi , p) > πi (p, p), we must have qi �= p, i.e.,(qi , p) /∈ A.

We show that Assumption 3′ is also satisfied. Let (qi , p) /∈ A and πi (qi , p) >

πi (p, p) + 3ε. Thus, there exists a neighborhood N of p with qi /∈ N such thatπi (qi , p) − ε ≤ πi (qi , p′) for all p′ ∈ N by the continuity of f . We also haveπi (p, p) + ε ≥ πi (p′, p′) for all p′ ∈ N . Therefore, π i (qi , p

′) > πi (p′, p′) ≥π i (p

′, p′) for every p′ ∈ N . Then, by Proposition 4, the considered game possessesa symmetric pure strategy Nash equilibrium.

6 Conclusion

In this paper, we investigate the existence of Nash equilibria in games that maybe discontinuous and/or nonquasiconcave. We offer new existence results on Nashequilibrium for a large class of discontinuous games by introducing new notions ofweak continuity, such as quasi-weak transfer continuity, strong quasi-weak transfercontinuity, pseudo quasi-weak transfer continuity, weak transfer quasi-continuity, P-correspondence security, etc. We prove our main result—Theorem 1 by constructinga suitable game with an additional player and then applying Theorem 3.4 of Reny(2013). We prove Theorem 4 in a similar manner by applying Theorem 5.6 in Reny

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(2013). An advantage of such an approach is that these results are nicely connected tothe results obtained in Reny (2013). Since this literature is becoming filled with manydifferent results and proofs, we should make connections whenever possible so thatthe literature can begin to become somewhat more unified.

Our equilibrium existence results neither imply nor are implied by those results inBaye et al. (1993), Reny (1999, 2009), Carmona (2009, 2011), Prokopovych (2011,2013), Nessah (2011), McLennan et al. (2011), and Barelli and Meneghel (2013).These results permit us to significantly weaken the continuity conditions for the exis-tence of Nash equilibria. We also provide examples and economic applications whereour general results are applicable.

The approach developed in the paper can be similarly used to study the existenceof mixed strategy Nash and Bayesian Nash equilibria in general discontinuous games.For details, see our earlier version of this paper (cf. Nessah and Tian (2008)). Byimposing diagonal transfer continuity defined in Baye et al. (1993) and transfer lowersemicontinuity introduced in Tian (1992a) and applying the KKM lemma properly,Prokopovych and Yannelis (2014) proved the existence of the mixed strategy equi-librium. Our conjecture is that, by replacing diagonal transfer continuity with weaktransfer quasi-continuity, one may also similarly prove the existence of the mixedstrategy equilibrium.

7 Appendix

Proof of Theorem 1 Define a new game G ′ that includes all of the original playersi ∈ I and a new player i = 0 /∈ I . Thus, the new set of players is N = I ∪ {0}. Player0’s strategy set is X and the strategy set of player i ∈ I is Xi . If player 0 choosesz ∈ X and each player i ∈ I chooses xi ∈ Xi , then the payoff to player 0 is

v0(z, x) ={1, if z = x

0, otherwise

and the payoff to player i ∈ I is

vi (z, x) = ui (x) − ui (z).

Suppose that the game G does not have an equilibrium in X . Let x ∈ X be a non-equilibrium of G ′. Then, by quasi-weak transfer continuity, there exists a player i , astrategy yi ∈ Xi and some neighborhood Nx of x such that for every z ∈ Nx , wehave ui (yi , z−i ) > ui (z). Therefore, for each t ∈ Nx , there is a player j = i ∈ I forwhom, we have for each z ∈ Nx ,

v j (z, (y j , z− j )) = ui (yi , z−i ) − ui (z) > 0 = v j (t, t).

Hence G ′ is point secure with respect to I . Consequently, since ui (xi , x−i ) is quasi-concave in xi , then this new game satisfies the hypotheses of Theorem 3.4 in Reny(2013) and therefore possesses a pure strategy Nash equilibrium (z, x). Payoff of

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player 0 implies that z = x . For each player j ∈ I , his payoff implies that x j max-imizes u j (y j , x− j ). By quasi-weak transfer continuity, there exists a player i and astrategy yi ∈ Xi such that ui (yi , x−i ) > ui (x), which contradicts that xi maximizesui (yi , x−i ). ��

Proof of Proposition 1 (a) Suppose that G is strongly quasi-weakly transfer continu-ous. Then, if x ∈ X is not an equilibrium, there exists a player i , yi ∈ Xi , ε > 0, andsome neighborhood V of x such that for every z ∈ V , we have ui (yi , z−i ) ≥ ui (z)+ε.Therefore, for each z ∈ V and Nz ⊂ V as a neighborhood of z, we have

infz′∈Nz

ui (yi , z′−i ) ≥ inf

z′∈Nz

ui (zi , z′−i ) + ε. (2)

Indeed, if (2) is false for some z ∈ V and Nz ⊂ V , then there is z ∈ Nz so that

ui (yi , z−i ) < infz′∈Nz

ui (zi , z′−i ) + ε ≤ ui (zi , z−i ) + ε ≤ ui (yi , z−i ),

which is impossible. Obviously, for each z ∈ X and each neighborhood Nz of z, wehave

infz′∈Nz

ui (ti , z′−i ) = inf

z′∈Nz

ui (ti , z′−i ), for each ti ∈ Xi . (3)

Combining (2) and (3), we obtain for each Nz ⊂ V ,

ui (yi , z−i ) ≥ infz′∈Nz

ui (yi , z′−i ) ≥ inf

z′∈Nz

ui (zi , z′−i ) + ε = inf

z′∈Nz

ui (zi , z′−i ) + ε.

Consequently for each z ∈ V , we have ui (yi , z−i ) ≥ ui (z)+ ε > ui (z), which meansG is quasi-weakly transfer continuous.

(b) Suppose thatG isQWUSC and payoff secure. If x ∈ X is not aNash equilibrium,then by quasi-weak upper semicontinuity, some player i has a strategy xi ∈ Xi , ε > 0and a neighborhood N 1 of x such that for every z ∈ N 1 and every neighborhoodNz ⊆ N 1 of z, ui (xi , x−i ) > ui (zi , z′−i ) + ε for some z′ ∈ Nz . The payoff securityof G implies that there exists a strategy yi and a neighborhood N 2 of x such thatui (yi , z−i ) ≥ ui (xi , x−i ) − ε

2 for all z ∈ N 2. Thus, for every z ∈ N = N 1 ∩N 2 andits neighborhoodNz ⊆ N , there exists z′ ∈ Nz such that ui (yi , z−i ) > ui (zi , z′−i )+ ε

2 .(c) Suppose that G isWTLSC andQUSC. Then, if x ∈ X is not a Nash equilibrium,

by WTLSC, there exists a player i , yi ∈ Xi , ε > 0 and a neighborhood N 1 of x suchthat ui (yi , z−i ) > ui (x) + ε for all z ∈ N 1. The QUSC implies that for i ∈ I , x ∈ X ,and ε > 0, there exists a neighborhoodN 2 of x such that for every z ∈ N 2 and everyneighborhood Nz ⊆ N 2 of z, ui (x) ≥ ui (zi , z′−i ) − ε

2 for some z′ ∈ Nz . Thus, forevery z ∈ N = N 1 ∩ N 2 and its neighborhood Nz ⊆ N , there exists z′ ∈ Nz suchthat ui (yi , z−i ) > ui (zi , z′−i ) + ε

2 .Thus, by Theorem 1, the game possesses a pure strategy Nash equilibrium. ��

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Proof of Theorem 2 Define a new two-player game G ′ by a simple player’s strategyset Z . If player 1 chooses t ∈ Z and player 2 chooses z ∈ Z , player 1’s preferencesare presented by

v1(t, z) ={1, if z = t

0, otherwise

and player 2’s preference relation is defined by v2(t, z) = f (z, t)− f (t, t). The gameG ′ is point secure with respect to I ′ = {2}. Then, by Theorem 3.4 in Reny (2013), G ′possesses a Nash equilibrium (x, x) which is also a symmetric Nash equilibrium ofG (because G is diagonally quasi-weak transfer continuous). ��Proof of Theorem 3 Suppose that the considered game does not have a symmetricNash equilibrium. Then, by diagonal weak transfer continuity, for each x ∈ X , thereexists an open neighborhoodNx and y ∈ X so that f (z, y) > f (z, z), for each z ∈ Nx .Thus, we obtain a collection {(Nx , yx )}x∈X where {Nx }x∈X forms an open cover ofX . Since X is compact, then one can extract a finite subcollection {(Nxk , y

k)}k∈K (Kis a finite set), so as for each k ∈ K and for each z ∈ Nxk , f (z, yk) > f (z, z) foreach z ∈ Nxk . Let {βk}k∈K be a partition of the unity subordinate to {Nxk }k∈K . Bythe diagonal transfer quasiconcavity, for the finite set {yk, k ∈ K } ⊆ X , there is acorresponding finite subset {x k, k ∈ K } such that for any subset J of K and everyx ∈ co{x j , j ∈ J }, we have

f (x, x) ≥ minj∈J

f (x, yk). (4)

Let us consider the following function g : X → X defined by g(z) = ∑k∈K βk(z)x k .

Brouwer fixed point theorem implies that there is x ∈ X so that x = g(x) =∑j∈J β j (x)x j where J = { j ∈ K : β j (x) > 0}. Then, for each j ∈ J , x ∈ Nx j and

consequently f (x, y j ) > f (x, x). Hence f (x, x) ≥ min j∈J f (x, yk) > f (x, x),which is impossible. ��Proof of Theorem 4 Suppose that there is no equilibrium in X . Define the followingtwo-player game by a single player’s strategy set X . If player 1 chooses t ∈ X andplayer 2 chooses z ∈ X , player 1’s preferences are presented by

u1(t, z) ={1, if z = t

0, otherwise

and player 2’s preference relation is defined by (t, z) (t ′, z′) if and only if (t, z) ∈Pi (t ′, z′), for each i ∈ I .The two-player game is correspondence secure with respect to I ′ = {2}. Indeed,

assume that it is not correspondence secure with respect to I ′ = {2} in nonequilibriumx . Then combining it with the P-correspondence security, there exists a neighbor-hood N ⊆ X of x , a well-behaved correspondence ϕ : N ⇒ X , z ∈ N , y ∈ ϕ(x)and a player j such that

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z ∈ co {w : (z, w) (x, y)} (5)

z /∈ co{w ∈ X such that (z, w) ∈ P

j (x, y)}

. (6)

By definition of , we obtain

z ∈ co {w : (z, w) ∈ Pi (x, y), for each i ∈ I }

= co

(⋂

i∈I{w : (z, w) ∈ P

i (x, y)})

.

Since co(⋂

i∈I Ai) ⊆ ⋂

i∈I (coAi ), then for each i ∈ I ,

z ∈ co{w ∈ X such that (z, w) ∈ P

i (x, y)},

which contradicts (5). It is straightforward to show that the preference is complete,reflexive, and transitive. Then, by Theorem 5.6 in Reny (2013), this game has anequilibrium (x, x), i.e., for each y ∈ X , we have (x, x) (x, y) which is equivalentto

(x, x) ∈ Pi (x, y), for each i ∈ I. (7)

Or x is not a Nash equilibrium of G. Then by P-correspondence security of G, thereis a neighborhood N ⊆ X of x , a well-behaved correspondence ϕ : N ⇒ X and aplayer j ∈ I so that

x /∈ co{t ∈ X : (x, t) ∈ P

j (x, y)}

holds for each y ∈ ϕ(x), which contradicts (7). ��Proof of Theorem 5 Let Ω(x) be the set of all open neighborhoods N of x . For eachplayer i ∈ I and every x ∈ X , define the following correspondence Ci : X ⇒ Xi by

Ci (x) = {yi ∈ Xi : ∃N ∈ Ω(x), ∀z ∈ N , infz′∈N

ui (yi , z′−i ) > inf

z′′∈Nui (zi , z

′′−i )}.

By the strong quasiconcavity, Ci (x) is convex for all x ∈ X and i ∈ I . The corre-spondence Ci has the lower open section. Now suppose, by way of contradiction,that for each x ∈ X , there exists a player i ∈ I such that Ci (x) �= ∅. Then,by Theorem 3a in Deguire and Lassonde (1995), there exists a point x ∈ X andi ∈ I such that xi ∈ Ci (x), which is impossible. Thus, there exists x ∈ X suchthat for each i ∈ I , we have Ci (x) = ∅. If x is not a Nash equilibrium, then asthe game G is quasi-weakly transfer continuous, there exists a player i , a strategyyi ∈ Xi and some neighborhood Nx of x such that for every z ∈ Nx , we haveinf z′∈N ui (yi , z

′−i ) > inf z′′∈N ui (zi , z′′−i ). Then, yi ∈ Ci (x), which is a contradic-tion to Ci (x) = ∅. ��

123


Proof of Proposition 2 Let G = (Xi , ui )i∈I be better-reply secure. Suppose, by wayof contradiction, that the game is not pseudo quasi-weakly transfer continuous. Then,there exists a nonequilibrium x∗ ∈ X such that for all player j , ε > 0, every neigh-borhood N of x∗, and all y j , there exists x ′ ∈ N satisfying

u j (y j , x′− j ) ≤ u j (x

′j , x

′′− j ) + ε, for all x ′′ ∈ N .

Letting u be the limit of the vector of payoffs corresponding to some sequence ofstrategies converging to x∗, and U∗ be the set of all such points, which is a compactset by the boundedness of payoffs, we have (x∗, u) ∈ cl(Γ ) for all u ∈ U∗. Then,for each (x∗, u) ∈ cl(Γ ) with u ∈ U∗, there exists a player i , a strategy yi , ε > 0and a neighborhood N of x∗ such that ui (yi , x

′−i ) > ui + ε for all x ′ ∈ N . Theninfx ′∈N ui (yi , x

′−i ) ≥ ui + ε. Let U∗i be the projection of U∗ to coordinate i and

u∗i = sup

{

ui ∈ U∗i : inf

x ′∈Nui (yi , x

′−i ) ≥ ui + ε

}

.

Then, for ε/2 > 0, there is a neighborhood N i,∗ of x∗ and a strategy y∗i such that

infx ′∈N i,∗

ui (y∗i , x ′−i ) ≥ (u∗

i + ε) − ε/2 = u∗i + ε/2. (8)

Now, since the game is not pseudo quasi-weakly transfer continuous, then for a directedsystem of neighborhoods {N k}k of x∗, a sequence {εk}k converging to 0, and everyj ∈ I , there exists a sequence {x j,k}k with x j,k ∈ N k so that {x j,k}k converges to x∗and

u j (y∗j , x

j,k− j ) ≤ u j (x

j,kj , x ′− j ) + εk, for each x ′ ∈ N k . (9)

Consider the following sequence: for each k, let x k = (x1,k1 , . . . , xn,kn ). Since for each

j ∈ I , x j,k ∈ N k and {x j,k}k converges to x∗, then x k ∈ N k and the sequence {xk}kconverges to x∗. Therefore, inequality (9) becomes

u j

(y∗j , x

j,k− j

)≤ u j

(x j,kj , x k− j

)= u j (x

k) + εk, for each k, j ∈ I. (10)

Assume that {u(xk)}k converges and u = limk→∞u(xk). Hence, (x∗, u) ∈ cl(Γ )

with u ∈ U∗, then there exists a player i ∈ I such that ui ≤ u∗i . Thus, for ε/3 > 0,

there exists k1 such that whenever k > k1, we have ui (y∗i , xi,k−i ) ≤ u∗

i + ε/3 ≤infx ′∈N i,∗ ui (y∗

i , x ′−i ) − ε/6. Then for k > k1, we obtain

ui (y∗i , xi,k−i ) ≤ ui (y

∗i , x ′−i ) − ε/6, for each x ′ ∈ N i,∗. (11)

Since the sequence {xi,k}k converges to x∗, then for N i,∗, there exists k2 such thatfor k > k2, we have xi,k ∈ N i,∗. Thus, by (11) for k > max(k1, k2), we have

123


ui (y∗i , xi,k−i ) ≤ ui (y∗

i , xi,k−i ) − ε/6, which is impossible. Hence, the game must bepseudo quasi-weakly transfer continuous. ��Proof of Theorem 6 Sufficiency. For each y ∈ X , let

F(y) = {x ∈ X : ui (yi , x−i ) ≤ ui (x), ∀i ∈ I }.

It is clear that G is weakly transfer quasi-continuous if and only if F is transferclosed-valued. For y ∈ X , let F(y) = cl F(y). Then F(y) is closed, and by the strongdiagonal transfer quasiconcavity, it is also transfer FS-convex. By Lemma 1 in Tian(1993), we know that

⋂y∈X F(y) = ⋂

y∈X F(y) �= ∅. Thus, there exists a strategyprofile x ∈ X such that

ui (yi , x−i ) ≤ ui (x), for all y ∈ X and i ∈ I.

Thus x is a pure strategy Nash equilibrium of the game G.Necessity: Suppose the game Γ has a pure strategy Nash equilibrium x∗ ∈ X .

We want to show that it is strongly diagonal transfer quasiconcave in y. Indeed,for any finite subset {y1, . . . , ym} ⊂ X , let the corresponding finite subset Xm ={x1, . . . , xm} = {x∗}. Then, for any subset {xk1 , xk2 , . . . , xks } ⊂ Xm = {x∗},1 ≤ s ≤ m, x ∈ co {xk1 , xk2 , . . . , xks } = {x∗}, and y ∈ {yk1 , yk2 , . . . , yks }, wehave

ui (yi , x−i ) = ui (yi , x∗−i ) ≤ ui (x

∗i , x∗−i ) = ui (xi , x−i ).

Hence U is strongly diagonal transfer quasiconcave in x . ��Proof of Proposition 3 Suppose x is not an equilibrium. Then, some player i has astrategy yi such that ui (yi , x−i ) > ui (x), i.e., Fi (yi , Si (yi , x−i )) > Fi (xi , Si (x)). If(yi , x−i ) ∈ X\Di , then by Assumption 3, there exists a player j ∈ I , a deviationstrategy profile y, and a neighborhood V of x such that for every z ∈ V , we haveF j (y j , S j (y j , z− j )) > F j (z j , S j (z)), i.e., u j (y

′j , z− j ) > u j (z). If (yi , x−i ) ∈ Di ,

then by Assumption 2, there exists a player j ∈ I and y j such that (y j , x− j ) ∈ X\Dj

and Fj (y j , S j (y j , x− j )) > Fj (x j , S j (x)). Thus, by Assumption 3, there exists aplayer k ∈ I , a deviation strategy profile y, and a neighborhood V of x such that forevery z ∈ V , we have Fk(yk, Sk(yk, z−k)) > Fk(zk, Sk(z)), i.e., uk(yk, z−k) > uk(z).Therefore, the game is quasi-weakly transfer continuous. Since it is also compact andquasiconcave, by Theorem 1, it has a pure strategy Nash equilibrium. ��Proof of Proposition 4 Suppose p is not an equilibrium. Then, some player i has astrategy qi such that πi (qi , p−i ) > πi (p). If (qi , p−i ) ∈ Pn\Ai , then by Assumption3, there exists a player j ∈ I , a deviation strategy profile q j and a neighborhoodN of p such that for every r ∈ Np, π j (q j , r−i ) > π j (r). If (qi , p−i ) ∈ Ai , thenby Assumption 2, there exists a firm j ∈ I , and q j such that (q j , p−i ) ∈ Pn\A j

and π j (q j , p−i ) > πi (p). Thus, by Assumption 3, there exists a player j ∈ I , adeviation strategy profile q j and a neighborhood N of p such that for every r ∈ Np,

123


π j (q j , r−i ) > π j (r). Therefore, the game is weakly transfer quasi-continuous. Sincethe game is also compact, convex, and quasiconcave, by Theorem 6, it has a purestrategy Nash equilibrium. ��

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