Negotiation-proof Nash equilibrium

This paper defines “negotiation-proof Nash equilibrium', a notion that applies to environments where players can negotiate openly and directly prior to the play of a noncooperative game. It recognizes the possibility that a group of self-interested players may choose, voluntarily and without binding agreement, to coordinate their choice of strategies and make joint objections; moreover, it takes the perfect foresight of rational players fully into account. The merit of the notion of negotiation-proof Nash equilibrium is twofold: (1) It offers a way to rectify the nestedness assumption and myopia embedded in the notion of coalition-proof Nash equilibrium. (2) The negotiation process is formalized by a “graph”, which serves as a natural extension to the approach that models preplay communication by an extensive game. Received: October 1998/Final version: May 2000     

Refinements of the Nash equilibrium concept

Selten's concept of perfect equilibrium for normal form games is reviewed, and a new concept of proper equilibrium is defined. It is shown that the proper equilibria form a nonempty subset of the perfect equilibria, which in turn form a subset of the Nash equilibria. An example is given to show that these inclusions may be strict.     

A Shared-Constraint Approach to Multi-Leader Multi-Follower Games

Multi-leader multi-follower games are a class of hierarchical games in which a collection of leaders compete in a Nash game constrained by the equilibrium conditions of another Nash game amongst the followers. The resulting equilibrium problem with equilibrium constraints is complicated by nonconvex agent problems and therefore providing tractable conditions for existence of global or even local equilibria has proved challenging. Consequently, much of the extant research on this topic is either model specific or relies on weaker notions of equilibria. We consider a modified formulation in which every leader is cognizant of the equilibrium constraints of all leaders. Equilibria of this modified game contain the equilibria, if any, of the original game. The new formulation has a constraint structure called shared constraints, and our main result shows that if the leader objectives admit a potential function, the global minimizers of the potential function over this shared constraint are equilibria of the modified formulation. We provide another existence result using fixed point theory that does not require potentiality. Additionally, local minima, B-stationary, and strong-stationary points of this minimization problem are shown to be local Nash equilibria, Nash B-stationary, and Nash strong-stationary points of the corresponding multi-leader multi-follower game. We demonstrate the relationship between variational equilibria associated with this modified shared-constraint game and equilibria of the original game from the standpoint of the multiplier sets and show how equilibria of the original formulation may be recovered. We note through several examples that such potential multi-leader multi-follower games capture a breadth of application problems of interest and demonstrate our findings on a multi-leader multi-follower Cournot game.     

On the stability of a class of noncooperative games

The Nash equilibrium of a class of games generated from a market is examined. Demands are assumed linear, and production constraints are imposed. The equilibrium is shown to be solvable as a complementarity problem. If the demand matrix is a positive definite symmetric z-matrix, then the Nash equilibrium is stable. If the demand matrix is not symmetric, an additional condition yielding stability is developed.     

Hedonic coalition formation games: A new stability notion

This paper studies hedonic coalition formation games where each player’s preferences rely only upon the members of her coalition. A new stability notion under free exit-free entry membership rights, referred to as strong Nash stability, is introduced which is stronger than both core and Nash stabilities studied earlier in the literature. Strong Nash stability has an analogue in non-cooperative games and it is the strongest stability notion appropriate to the context of hedonic coalition formation games. The weak top-choice property is introduced and shown to be sufficient for the existence of a strongly Nash stable partition. It is also shown that descending separable preferences guarantee the existence of a strongly Nash stable partition. Strong Nash stability under different membership rights is also studied.     

Equilibrium selection in multi-leader-follower games with vertical information

We consider two-stage multi-leader-follower games, called multi-leader-follower games with vertical information, where leaders in the first stage and followers in the second stage choose simultaneously an action, but those chosen by any leader are observed by only one “exclusive” follower. This partial unobservability leads to extensive form games that have no proper subgames but may have an infinity of Nash equilibria. So it is not possible to refine using the concept of subgame perfect Nash equilibrium and, moreover, the concept of weak perfect Bayesian equilibrium could be not useful since it does not prescribe limitations on the beliefs out of the equilibrium path. This has motivated the introduction of a selection concept for Nash equilibria based on a specific class of beliefs, called passive beliefs, that each follower has about the actions chosen by the leaders rivals of his own leader. In this paper, we illustrate the effectiveness of this concept and we investigate the existence of such a selection for significant classes of problems satisfying generalized concavity properties and conditions of minimal character on possibly discontinuous data.     

Juche versus Sadae: Game-theoretic approach

We argue that to some degree Juche is represented by the concept of Nash equilibrium, and Sadae by Thompson and Faith's truly perfect information equilibrium. We characterize the latter, and show that for a Pareto optimal Nash equilibrium, Juche is as good as, or better than Sadae. This includes the game of brinkmanship.     

Characterizing solution concepts in terms of common knowledge of rationality

Characterizations of Nash equilibrium, correlated equilibrium, and rationalizability in terms of common knowledge of rationality are well known. Analogous characterizations of sequential equilibrium, (trembling hand) perfect equilibrium, and quasi-perfect equilibrium in n-player games are obtained here, using earlier results of Halpern characterizing these solution concepts using non-Archimedean fields.     

Dynamic consistency in incomplete information games under ambiguity

We develop a general framework of incomplete information games under ambiguity which extends the traditional framework of Bayesian games to the context of Ellsberg-type ambiguity. We then propose new solution concepts called ex ante and interim Γ-maximin equilibrium for solving such games. We show that, unlike the standard notion of Bayesian Nash equilibrium, these concepts may lead to rather different recommendations for the same game under ambiguity. This phenomenon is often referred to as dynamic inconsistency. Moreover, we characterize the sufficient condition under which dynamic consistency is assured in this generalized framework.     

Generalising diagonal strict concavity property for uniqueness of Nash equilibrium

In this paper, we extend the notion of diagonally strictly concave functions and use it to provide a sufficient condition for uniqueness of Nash equilibrium in some concave games. We then provide an alternative proof of the existence and uniqueness of Nash equilibrium for a network resource allocation game arising from the so-called Kelly mechanism by verifying the new sufficient condition. We then establish that the equilibrium resulting from the differential pricing in the Kelly mechanism is related to a normalised Nash equilibrium of a game with coupled strategy space.     

Definitions of equilibrium in network formation games

We examine a variety of stability and equilibrium definitions that have been used to study the formation of social networks among a group of players. In particular we compare variations on three types of definitions: those based on a pairwise stability notion, those based on the Nash equilibria of a link formation game, and those based on equilibria of a link formation game where transfers are possible.Bloch is also affiliated with the University of Warwick.     

Detecting all evolutionarily stable strategies

In evolutionary game theory, the central solution concept is the evolutionarily stable state, which also can be interpreted as an evolutionarily stable population strategy (ESS). As such, this notion is a refinement of the Nash equilibrium concept in that it requires an additional stability property. In the present paper, an algorithm for detectingall ESSs of a given evolutionary game consisting of pairwise conflicts is presented which both is efficient and complete, since it involves a procedure avoiding the search for unstable equilibria to a considerable extent, and also has a finite, exact routine to check evolutionary stability of a given equilibrium. The article also contains the generalization of these results to the playing-the-field setting, where the payoff is nonlinear.     

A minimally altruistic refinement of Nash equilibrium

We introduce a minimal notion of altruism and use it to refine Nash equilibria in normal form games. We provide three independent existence proofs, relate minimally altruistic Nash equilibrium to other equilibrium concepts, conduct an in-depth sensitivity analysis, and provide examples where minimally altruistic Nash equilibrium leads to improved predictions.     

On stability of perfect equilibrium points

We consider stability of Selten's perfect equilibrium point against slight imperfections of rationality of players. As its stability is not sufficient, we strengthen the perfectness concept and define astrictly perfect equilibrium point. We provide sufficient conditions for this equilibrium point.     

Nash equilibrium based fairness

There are several approaches of sharing resources among users. There is a noncooperative approach wherein each user strives to maximize its own utility. The most common optimality notion is then the Nash equilibrium. Nash equilibria are generally Pareto inefficient. On the other hand, we consider a Nash equilibrium to be fair as it is defined in a context of fair competition without coalitions (such as cartels and syndicates). We show a general framework of systems wherein there exists a Pareto optimal allocation that is Pareto superior to an inefficient Nash equilibrium. We consider this Pareto optimum to be ??Nash equilibrium based fair.?? We further define a ??Nash proportionately fair?? Pareto optimum. We then provide conditions for the existence of a Pareto-optimal allocation that is, truly or most closely, proportional to a Nash equilibrium. As examples that fit in the above framework, we consider noncooperative flow-control problems in communication networks, for which we show the conditions on the existence of Nash-proportionately fair Pareto optimal allocations.     

A general characterization of the mean field limit for stochastic differential games

The mean field limit of large-population symmetric stochastic differential games is derived in a general setting, with and without common noise, on a finite time horizon. Minimal assumptions are imposed on equilibrium strategies, which may be asymmetric and based on full information. It is shown that approximate Nash equilibria in the n-player games admit certain weak limits as n tends to infinity, and every limit is a weak solution of the mean field game (MFG). Conversely, every weak MFG solution can be obtained as the limit of a sequence of approximate Nash equilibria in the n-player games. Thus, the MFG precisely characterizes the possible limiting equilibrium behavior of the n-player games. Even in the setting without common noise, the empirical state distributions may admit stochastic limits which cannot be described by the usual notion of MFG solution.     

带有网络外部性的动态寡头垄断定价

给出了带有网络外部性的两阶段寡头垄断定价模型,并用博弈论方法求解.通过与带有网络外部性的完全垄断定价模型的比较,得出重要结论:在网络外部性足够大的情况下,①寡头竞争情况下与完全垄断下情况一样,“科斯假设”将得到克服,均衡定价将呈现先低后高的情况.②对于完全垄断厂商来说,网络外部性k的增加能够增加其利润,但对于寡头竞争的企业来说,正好相反,k的增加将导致其利润的下降.     

Establishing Nash equilibrium of the manufacturer–supplier game in supply chain management

We study a game model of multi-leader and one-follower in supply chain optimization where n suppliers compete to provide a single product for a manufacturer. We regard the selling price of each supplier as a pre-determined parameter and consider the case that suppliers compete on the basis of delivery frequency to the manufacturer. Each supplier's profit depends not only on its own delivery frequency, but also on other suppliers' frequencies through their impact on manufacturer's purchase allocation to the suppliers. We first solve the follower's (manufacturer's) purchase allocation problem by deducing an explicit formula of its solution. We then formulate the n leaders' (suppliers') game as a generalized Nash game with shared constraints, which is theoretically difficult, but in our case could be solved numerically by converting to a regular variational inequality problem. For the special case that the selling prices of all suppliers are identical, we provide a sufficient and necessary condition for the existence and uniqueness of the Nash equilibrium. An explicit formula of the Nash equilibrium is obtained and its local uniqueness property is proved.     

基于再保险人分担理赔费用的保险欺诈博弈模型分析

本文首先建立了基于再保险人分担理赔费用的夸大风险损失的索赔欺诈博弈模型，然后分析了保险双方的博弈策略选择及其可能达到的均衡。研究表明，博弈模型存在某种形式的精炼贝叶斯纳什均衡，但如实告知一般不是保单持有人的精炼贝叶斯纳什均衡策略。而且，博弈模型不存在分离均衡，当费用自负率在一定范围内变化时，局中人将交替选用分离策略与合并策略。     

Hotelling’s location model with negative network externalities

We study a variation of Hotelling’s location model in which consumers choose between firms based on travel distances as well as the number of consumers visiting each firm. The model in which the network externality is the same for all firms was proposed by Kohlberg (Econ Lett 11:211–216, 1983), who claims that no equilibrium exists for more than two firms. We assume the network effects to be linear and, in contrast to the claim in Kohlberg (Econ Lett 11:211–216, 1983), derive a condition under which a subgame perfect Nash equilibrium exists for four and six firms. Moreover, we show that for more than two firms the equilibrium locations of the firms are different from the equilibrium locations in Hotelling’s location model. Our results suggest that a subgame perfect Nash equilibrium exists if and only if the number of firms is even. We also provide examples of subgame perfect equilibria in which the network externality is different for some of the firms.