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.     

Two-person game forms guaranteeing the stability against commitment and delaying tactics

The notions of the struggle for leadership or for followership, introduced by H. Moulin for two-person games, are considered for game forms. Necessary and sufficient conditions for a game form never to generate a game with the struggle of either kind are derived. Connections between these properties and the existence of a Nash equilibrium for any preference profile as well as the possibility to select a Nash equilibrium in an incentive compatible way are established.I thank V. Gurvich for a fruitful discussion, H. Moulin for his comment on a draft version of the paper and several helpful suggestions, and an anonymous referee, whose recommendations helped me to enhance the quality of presentation. I have benefited from the hospitality of Universidad de Alicante and Instituto Valenciano de Investigaciones Económicas. Some financial assistance from the Cultural Initiative Foundation and Academy of Natural Sciences is also acknowledged.     

Characterization of the existence of a pure-strategy Nash equilibrium

In this work, we provide a necessary and sufficient condition for the existence of a pure-strategy Nash equilibrium for non-cooperative games in topological spaces.     

On finding curb sets in extensive games

We characterize strategy sets that are closed under rational behavior (curb) in extensive games of perfect information and finite horizon. It is shown that any such game possesses only one minimal curb set, which necessarily includes all its subgame perfect Nash equilibria. Applications of this result are twofold. First, it lessens computational burden while computing minimal curb sets. Second, it implies that the profile of subgame perfect equilibrium strategies is always stochastically stable in a certain class of games.I am grateful to J. Kamphorst, G. van der Laan and X. Tieman, who commented on the earlier versions of the paper. I also thank an anonymous referee and an associate editor for their helpful remarks. The usual disclaimer applies.     

Strategic bidding in continuous electricity auctions: an application to the Spanish electricity market

In this paper we introduce an asymmetric model of continuous electricity auctions with limited production capacity and bounded supply functions. The strategic bidding is studied with this model by means of an electricity market game. We prove that for every electricity market game with continuous cost functions a mixed-strategy Nash equilibrium always exists. In particular, we focus on the behavior of producers in the Spanish electricity market. We consider a very simple form for the Spanish electricity market: an oligopoly consisting just of independent hydro-electric power production units in a single wet period. We show that a pure-strategy Nash equilibrium for the Spanish electricity market game always exists.     

一类投资价值为变量的研究与开发项目的Nash解问题

本文考虑一类完全信息下研究与开发 (R& D)项目中的纯策略问题 .设有一个 R& D项目 ,两位风险中性的投资人竞争 .项目需要的投资量至少是 I,投资较多的一方取胜 .项目成功后的价值是确定的 ,但是其价值大小对这两位投资人不一样 (品牌知名度 ,管理水平 ,营销能力等方面的差异导致的价格和销量不同 ) ,我们使用博弈论方法对此进行研究 ,得到的结果显示 :如果他们所实现的项目价值是常量 ,则不存在均衡解 ;而当项目价值是一个关于投资总量的单增函数时 ,则在两位投资人的价值函数的差异较大时 ,存在一个同时还是社会占优的 Nash均衡解 ,在两位投资人的价值函数差异不太大时 ,不存在均衡解 ,此时的竞争会导致社会效益下降 .从而提示政府应当通过一定的方式 (如招投标法 ,审批法 )对投资活动予以控制 .     

Weak equilibrium in a spatial model

Spatial models of two-player competition in spaces with more than one dimension almost never have pure-strategy Nash equilibria, and the study of the equilibrium positions, if they exist, yields a disappointing result: the two players must choose the same position to achieve equilibrium. In this work, a discrete game is proposed in which the existence of Nash equilibria is studied using a geometric argument. This includes a definition of equilibrium which is weaker than the classical one to avoid the uniqueness of the equilibrium position. As a result, a “region of equilibrium” appears, which can be located by geometric methods. In this area, the players can move around in an “almost-equilibrium” situation and do not necessarily have to adopt the same position.     

Warm-glow giving in networks with multiple public goods

This paper explores a voluntary contribution game in the presence of warm-glow effects. There are many public goods and each public good benefits a different group of players. The structure of the game induces a bipartite network structure, where players are listed on one side and the public good groups they form are listed on the other side. The main result of the paper shows the existence and uniqueness of a Nash equilibrium. The unique Nash equilibrium is also shown to be asymptotically stable. Then the paper provides some comparative statics analysis regarding pure redistribution, taxation and subsidies. It appears that small redistributions of wealth may sometimes be neutral, but generally, the effects of redistributive policies depend on how public good groups are related in the contribution network structure.     

Mean-field type forward-backward doubly stochastic differential equations and related stochastic differential games

We study a kind of partial information non-zero sum differential games of mean-field backward doubly stochastic differential equations, in which the coefficient contains not only the state process but also its marginal distribution, and the cost functional is also of mean-field type. It is required that the control is adapted to a sub-filtration of the filtration generated by the underlying Brownian motions. We establish a necessary condition in the form of maximum principle and a verification theorem, which is a sufficient condition for Nash equilibrium point. We use the theoretical results to deal with a partial information linear-quadratic (LQ) game, and obtain the unique Nash equilibrium point for our LQ game problem by virtue of the unique solvability of mean-field forward-backward doubly stochastic differential equation.     

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.     

Directed graphical structure,Nash equilibrium,and potential games

This paper considers the directed graphical structure of a game, called influence structure, where a directed edge from player $i$ to player $j$ indicates that player $i$ may be able to affect $j$’s payoff via his unilateral change of strategies. We give a necessary and sufficient condition for the existence of pure-strategy Nash equilibrium of games having a directed graph in terms of the structure of that graph. We also discuss the relationship between the structure of graphs and potential games.     

Switching defence for switched systems under malicious attacks: A Stackelberg game approach

The security issue of switched systems is researched from a noncooperative dynamic game-theoretic perspective in this paper. A Stackelberg game is developed for the switched autonomous system suffering malicious attacks, and the Stackelberg equilibrium switching and attack strategies are constructed respectively. Afterwards, the proposed game-theoretic approach is extended to switched control systems. A Stackelberg Nash game is consequently established to characterize the hierarchical decision making processes, where the controller and the attacker are the followers who simultaneously make their own decisions, and a sufficient condition is provided for the construction of the Stackelberg Nash equilibrium. Finally, a continuous stirred tank reactor is exploited to validate the effectiveness and applicability of the proposed results.     

Identification of classes of differential games for which the open loop is a degenerate feedback Nash equilibrium

In general, it is clear that open-loop Nash equilibrium and feedback Nash equilibrium do not coincide. In this paper, we study the structure of differential games and develop a technique using which we can identify classes of games for which the open-loop Nash equilibrium is a degenerate feedback equilibrium. This technique clarifies the relationship between the assumptions made on the structure of the game and the resultant equilibrium.The author would like to thank E. Dockner, A. Mehlmann, and an anonymous referee for helpful comments.     

Stochastic differential game model of a common property fishery

The paper presents a stochastic differential game model of a common-property commercial fishery and determines a feedback Nash equilibrium of the game. Closed-form expressions for the value functions, the equilibrium harvesting strategies, and stationary distributions of the fish stock are derived. Sensitivity analyses with respect tot he model parameters are carried out. The paper also considers equilibrium outcomes under joint maximization and surplus maximization. In the latter case, an optimal market size (i.e., number of firms) is identified.The research of the first author was supported in part by Grant 5.20.31.02 from the Danish Research Council for the Social Sciences. The research of the second author was supported in part by the CRGC Fund of the University of Hong Kong. The authors would like to thank the editor for constructive comments.     

On the incentive properties of multi-item auctions

In a market of indivisible objects where a buyer consumes at most one object, the buyer-optimal auction is a multi-item generalization of Vickrey's second-price auction. If the optimal auction is formulated as a strategic game, it is well-known that it satisfies good incentive properties, i.e., the honest strategy profile is a Nash equilibrium, a unique perfect equilibrium and a dominant strategy equilibrium. For each of the three incentive properties, it is shown that the optimal auction is aunique auction satisfying the property. The uniqueness results are derived in a general setting with budget constraints and non-linear utilities.I would like to thank three anonymous referees for detailed comments and constructive suggestions. I would also like to thank Edward J. Green and John G. Riley for their valuable comments and advice on earlier drafts of this paper. I am solely responsible for any errors.     

A silent duel over a cake

The division of a cake by two players is modelled by means of a silent game of timing. It is shown that this game has a unique Nash equilibrium. The strategies of the Nash equilibrium are explicitly given.I thank Peter Borm, Eric van Damme, Feico Drost, Harold Houba, Jos Potters, Stef Tijs and the referees for their comments.     

竞争环境下考虑顾客惰性行为的易逝品动态定价

消费者购买决策的行为倾向对企业的收益管理有着不可忽视的影响。本文在竞争企业动态定价问题中, 考虑了顾客惰性行为——即顾客购买决策的非理性拖延倾向——的影响。证明了库存充足情况下, 多企业多期博弈各销售期都存在唯一的纯策略纳什均衡。数值模拟表明, 顾客惰性对企业的最优价格和企业最大期望收益均造成负面影响, 且对销售低质量产品的企业的负面影响更大。此外, 顾客惰性宽度对各企业最优价格和最大期望收益的影响边际效应递增, 而顾客惰性深度产生的影响边际效应递减。     

On differential games with a feedback nash equilibrium

This note provides a lemma on differential games which possess a feedback Nash equilibrium (FNE). In particular, it shows that (i) a class of games with a degenerate FNE can be constructucted from every game which has a nondegenerate FNE and (ii) a class of games with a nondegenerate FNE can be constructed from every game which has a degenerate FNE.The author would like to thank an anonymous referee for invaluable comments and suggestions.     

Game Theoretic Analysis of a Distribution System with Customer Market Search

Consider a distribution system with one supplier and two retailers. When a stockout occurs at one retailer customers may go to the other retailer. We study a single period model in which the supplier may have infinite or finite capacity. In the latter case, if the total quantity ordered (claimed) by the retailers exceeds the supplier’s capacity, an allocation policy is involved to assign the limited capacity to the retailers. We analyze the inventory control decisions for the retailers using a game theoretical approach. The necessary and sufficient conditions are derived for the existence of a unique Nash equilibrium. A computational procedure is also proposed to calculate the Nash equilibrium. In case the Nash equilibrium does not exist, we use the concept of Stackelberg game to develop optimal strategies for both the leader and the follower. The work was partially supported by the National Textile Center of the US Department of Commerce under Grant No. I01-S01. The second author is supported in part by NSF under DMI-0196084 and DMI-0200306.     

On variational inequality approach to nash dynamic game of oil stockpiling ∗

Oil market disruption and strategic stockpiling are consideredin this work. A variational inequality approach is developed.The model treated here is more general than previous work.It is established that the Nash equilibrium is characterized by a system of inequalities. A sufficient condition for the existence and uniqueness of the Nash equilibrium is derived and an efficient iterative algorithm is proposed and analyzed