On the nonemptiness of the α-core of discontinuous games: Transferable and nontransferable utilities

doi:10.1016/j.jet.2015.04.005

Journal of Economic Theory

Volume 158, Part A, July 2015, Pages 213-231

https://doi.org/10.1016/j.jet.2015.04.005 Get rights and content

Abstract

The nonemptiness of the α-core of games with continuous payoff functions was proved by Scarf (1971) for nontransferable utilities and by Zhao (1999) for transferable utilities. In this paper we present generalizations of their results to games with possibly discontinuous payoff functions. Our handling of discontinuity is based on Reny's (1999) better-reply-security concept. We present examples to show that our generalizations are nonvacuous.

Introduction

Two major solution concepts for normal form games are the Nash equilibrium and the core. A Nash equilibrium is a noncooperative solution in which the joint interest of groups of players is not explicitly considered, whereas the core is a cooperative solution involving the behavior of coalitions of players. Motivated by economic problems that are suitably modeled by games with discontinuous payoff functions, Nash's existence result has been considerably extended following the seminal works of Dasgupta and Maskin (1986) and Reny (1999).¹ The existence of a cooperative solution for such games has yet to be provided. This note fulfills this objective.

An action profile belongs to the core of a game if no group of players has an incentive to form a coalition in which each of its members are made better-off, i.e. the action profile cannot be blocked by any coalition. In a (normal form) game, the actions of the complementary coalition affect the payoff of the members of a coalition, and therefore the definition of “blocking” hinges crucially on what the complementary coalition does. Among various blocking concepts defined in the literature, the α-core due to Aumann (1961) has attracted a significant attention.² In this paper we study the existence of this cooperative solution for games with possibly discontinuous payoff functions without transferable utilities (α-core) and with transferable utilities ( $α^{T}$ -core).

An action profile is in the α-core of a game if no coalition has an alternative action which makes all of its members better off, independently of the actions of the complementary coalition. Hence it is a pessimistic solution concept regarding the actions of the complementary coalition. And a pair of an action profile and a payoff profile is in the $α^{T}$ -core of a game if the action profile maximizes the grand coalition's aggregate payoff, and no coalition has an alternative action which guarantees a higher aggregate payoff, independently of the actions of the complementary coalition. Therefore, the α-core allows the coordination of the actions among the members of the coalitions, whereas the $α^{T}$ -core allows payoff transfers within the coalitions in addition to the coordination of actions. Scarf (1971) and Zhao (1999) proved the following theorems.

Theorem Scarf

Let $G = {(X_{i}, u_{i})}_{i \in N}$ be a game such that for each player i,

(i)
$X_{i}$ is a nonempty, compact and convex subset of a finite dimensional Euclidean space,
(ii)
$u_{i}$ is a quasiconcave and continuous function on $X = \prod_{i \in N} X_{i}$ .

Then G has a nonempty α-core.

Theorem Zhao

Let $G = {(X_{i}, u_{i})}_{i \in N}$ be a game such that for each player i,

(i)
$X_{i}$ is a nonempty, compact and convex subset of a finite dimensional Euclidean space,
(ii)
$u_{i}$ is a concave and continuous function on $X = \prod_{i \in N} X_{i}$ ,
(iii)
G is weakly separable.

Then G has a nonempty

α^{T}

-core.

In this paper we generalize these results to games where the continuity assumptions are weakened. And in line with the recent literature on discontinuous games pioneered by Reny (1999), the notions of coalitionally C-secure, coalitionally $C^{T}$ -secure and coalitionally $C_{N}^{T}$ -secure games are presented, and the existence of an imputation in the α-core and $α^{T}$ -core are shown for them.

The paper is organized as follows. Section 2 defines the basic concepts and states the results, Section 3 illustrates examples, Section 4 provides proofs of the results, and Section 5 concludes.

Section snippets

The model and the results

A (normal form) game is a list $G = {(X_{i}, u_{i})}_{i \in N}$ where

(i)
$N = {1, \dots, n}$ is the finite set of players,
(ii)
$X_{i}$ is the nonempty set of actions of player $i \in N$ ,
(iii)
$u_{i} : X \to R$ is the utility function of player $i \in N$ defined on $X = \prod_{i \in N} X_{i}$ .

A quasiconcave (concave) game is a game

G = {(X_{i}, u_{i})}_{i \in N}

such that for each player i,

X_{i}

is a nonempty, compact and convex subset of a finite dimensional Euclidean space and

u_{i}

is a quasiconcave (concave) function on X.

In his pioneering work, Reny (1999) proved the existence of a Nash equilibrium of

Examples

The first example illustrates a game which does not have a (pure strategy) Nash equilibrium, but has a nonempty α-core and $α^{T}$ -core. The game satisfies all the assumptions of Proposition 1, Proposition 2, particularly coalitional C-security and coalitional $C_{N}^{T}$ -security. However, it neither satisfies C-security nor coalitional $C^{T}$ -security.

Example 1

Consider the following Bertrand duopoly game. Each firm's action set is $P_{i} = [0, 10]$ . The market demand function $D : [0, 10] \to R$ is defined as $D (p) = \max {4 - p, 0} .$ And the

Proofs of the results

The proofs of Theorem 1, Theorem 2, Theorem 3 require a delicate construction. In order to give a general overview of the proofs, we first provide a heuristic outline of the proof of Theorem 1. Whereas the details of the proofs of Theorem 2, Theorem 3 are different, the construction in the proof is quite similar. Our proof of Theorem 1 is by contradiction. We assume the α-core of the game is empty. By using the compactness of the action sets and the coalitional C-security of the game, we obtain

Concluding remarks

This paper provides sufficient conditions for the nonemptiness of the TU and NTU α-cores of games with possibly discontinuous payoff functions. We end this paper with three remarks. First, although the α-core is widely applied cooperative solution concept for normal form games, a number of different solution concepts, such as β-core, strong equilibrium and hybrid solution of Zhao (1999) are also of interest for analyzing specific problems. It will be of interest to discuss the existence of such

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^☆: The results reported here constitute part of the author's Ph.D. dissertation being written under the supervision of Professors Hülya Eraslan and M. Ali Khan. I am grateful to them and to Richard P. McLean for their careful reading of the preliminary drafts. I also thank Ying Chen, Idione Meneghel, Burçin Kısacıkoğlu, Sezer Yaşar and the seminar participants at the University of Queensland and at the Sunday Theory Workshops at Johns Hopkins University for stimulating discussion. Finally, I am grateful to an anonymous referee of this journal for his/her report and for bringing Zhao (1999) to my attention.

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NotesOn the nonemptiness of the α-core of discontinuous games: Transferable and nontransferable utilities☆

Abstract

Introduction

Section snippets

The model and the results

Examples

Proofs of the results

Concluding remarks

Econ. Lett.

J. Econ. Theory

J. Econ. Theory

J. Econ. Theory

The core of a cooperative game without side payments

Trans. Am. Math. Soc.

A note on the equilibrium existence problem in discontinuous games

Econometrica

The theory of the core in an n-person game

Vestn. Leningr. Univ.

Symposium on: existence of Nash equilibria in discontinuous games

Econ. Theory

The existence of equilibrium in discontinuous economic games, I: theory, II: applications

Rev. Econ. Stud.

Notes
On the nonemptiness of the α-core of discontinuous games: Transferable and nontransferable utilities☆