Mean field game of controls and an application to trade crowding

In this paper we formulate the now classical problem of optimal liquidation (or optimal trading) inside a mean field game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader facing a “background noise” (or “mean field”). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. In this paper the trader faces the uncertainty of fair price changes too but not only. He also has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of “extended MFG”, we hence provide generic results to address these “MFG of controls”, before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of “heterogenous preferences” (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can “learn” it day after day, observing others’ behaviors.     

Stationary focusing mean-field games

We consider stationary viscous mean-field games (MFG) systems in the case of local, decreasing and unbounded coupling. These systems arise in ergodic MFG theory and describe Nash equilibria of games with a large number of agents aiming at aggregation. We show how the dimension of the state space, the behavior of the coupling, and the Hamiltonian at infinity affect the existence and nonexistence of regular solutions. Our approach relies on the study of Sobolev regularity of the invariant measure and a blow-up procedure that is calibrated on the scaling properties of the system. In very special cases, we observe uniqueness of solutions. Finally, we apply our methods to obtain new existence results for MFG systems with competition, namely, when the coupling is local and increasing.     

Performance versus informativeness in linear-quadratic Gaussian noncooperative games

In this paper, we study the impact of informativeness on the performance of linear quadratic Gaussian Nash and Stackelberg games. We first show that, in two-person static Nash games, if one of the players acquires more information, then this extra information is beneficial to him, provided that it is orthogonal to both players' information. A special case is that when one of the players is informationally stronger than the other, then any new information is beneficial to him. We then show that a similar result holds for dynamic Nash games. In the dynamic games, the players use strategies that are linear functions of the current estimates of the state, generated by two Kalman filters. The same properties are proved to hold in static and feedback Stackelberg games as well.This work was partially supported by the US Air Force Office of Scientific Research under Grant No. AFOSR-82-0174.     

A unifying model of strategic network formation

We provide a model that merges two basic models of strategic network formation and incorporates them as extreme cases: Jackson and Wolinsky’s connections model based on bilateral formation of links, and Bala and Goyal’s two-way flow model, where links can be unilaterally formed. In our model a link can be created unilaterally, but when it is only supported by one of the two players the flow through it suffers some friction or decay, but more than when it is supported by both players. When the friction in singly-supported links is maximal (i.e. there is no flow) we have Jackson and Wolinsky’s connections model, while when flow in singly-supported links is as good as in doubly-supported links we have Bala and Goyal’s two-way flow model. In this setting, a joint generalization of the results relative to efficiency and stability in both seminal papers is achieved, and the robustness in both models is tested with positive results.     

Evolutionary insights on the willingness to communicate

While in previous models of pre-play communication players are forced to communicate, we investigate what happens if players can choose not to participate in this cheap talk. Outcomes are predicted by analyzing evolutionary stability in a population of a priori identical players. If the game following the communication rewards players who choose the same action then an efficient outcome is only guaranteed when participation in the pre-play communication is voluntary. If however players aim to coordinate on choosing different actions in the underlying game and there are sufficiently many messages then the highest payoff is selected when players are forced to talk to each other before playing the game. Received: September 2000/Revised: March 2003     

带策略约束的区间数双矩阵博弈的双线性规划求解方法

传统区间数双矩阵博弈理论研究局中人支付值为区间数的策略选择问题,但没有考虑局中人策略选择可能受到各种约束.创建一种求解局中人策略选择受约束且支付值为区间数的双矩阵博弈（简称带策略约束的区间数双矩阵博弈）的简单、有效的双线性规划求解方法.首先,将局中人的博弈支付看作支付值区间中数值的函数.通过证明这种函数具有单调性,据此利用支付值区间的上、下界,构造了一对辅助双线性规划模型,可分别用于显式地计算任意带策略约束的区间数双矩阵博弈中局中人区间数博弈支付的上、下界及其相应的最优策略.最后,利用考虑策略约束条件下企业和政府针对发展低碳经济策略问题的算例,通过比较其与不考虑策略约束情形下的结果,说明了提出的模型和方法的有效性、优越性及可应用性.     

Limiting distributions of the number of pure strategy Nash equilibria in n-person games

We study the number of pure strategy Nash equilibria in a “random” n-person non-cooperative game in which all players have a countable number of strategies. We consider both the cases where all players have strictly and weakly ordinal preferences over their outcomes. For both cases, we show that the distribution of the number of pure strategy Nash equilibria approaches the Poisson distribution with mean 1 as the numbers of strategies of two or more players go to infinity. We also find, for each case, the distribution of the number of pure strategy Nash equilibria when the number of strategies of one player goes to infinity, while those of the other players remain finite.     

On noncooperative hybrid stochastic games

The present article models and analyzes a noncooperative hybrid stochastic game of two players. The main phase (prime hybrid mode) of the game is preceded by “unprovoked” hostile actions by one of the players (during antecedent hybrid mode) that at some time transforms into a large scale conflict between two players. The game lasts until one of the players gets ruined. The latter occurs when the cumulative damage to the losing player exceeds a fixed threshold. Both hybrid modes are formalized by marked point stochastic processes and the theory of fluctuations is utilized as one of the chief techniques to arrive at a closed form functional describing the status of both players at the ruin time.     

Independent mistakes in large games

Economic models usually assume that agents play precise best responses to others' actions. It is sometimes argued that this is a good approximation when there are many agents in the game, because if their mistakes are independent, aggregate uncertainty is small. We study a class of games in which players' payoffs depend solely on their individual actions and on the aggregate of all players' actions. We investigate whether their equilibria are affected by mistakes when the number of players becomes large. Indeed, in generic games with continuous payoff functions, independent mistakes wash out in the limit. This may not be the case if payoffs are discontinuous. As a counter-example we present the n players Nash bargaining game, as well as a large class of “free-rider games.” Received: November 1997/Final version: December 1999     

Commitments to threat strategies in Nash bargaining

Nash's two-person bargaining model consists of two stages: threat strategies and utility demands are chosen in the first and second stages respectively. Here players commit to play the threat strategies chosen in the first stage for the case where disagreement occurs in the second stage. Whether a player commits or not to play a threat strategy, however, is voluntary in principle. This leads to some possible extensions of Nash's model so that players' commitment choices are taken into account. In this paper, we consider three extensions. In the two person case, these three extensions give essentially the same result as that given by Nash. This is not the case for more than two players; the result depends upon an extension. In one extension, Nash's result always holds for more than two players. In the other two extensions, however, we give a three person example where not all players choose commitments in equilibrium.     

Farsighted stable sets in Hotelling’s location games

We apply the farsighted stable set to two versions of Hotelling’s location games: one with a linear market and another with a circular market. It is shown that there always exists a farsighted stable set in both games, which consists of location profiles that yield equal payoff to all players. This stable set contains location profiles that reflect minimum differentiation as well as those profiles that reflect local monopoly. These results are in contrast to those obtained in the literature that use some variant of Nash equilibrium. While this stable set is unique when the number of players is two, uniqueness no longer holds for both models when the number of players is at least three.     

Inducing cooperation by reciprocative strategy in non-zero-sum games

In this paper, we investigate the use of reciprocative strategy to induce cooperative behavior in non-zero-sum games. Reciprocative behavior is defined mathematically in the context of a two-person non-zero-sum game in which both the players have a common set of pure strategies. Conditions under which mutual cooperative behavior results when one of the players responds optimally to reciprocative behavior by the other player are described. Also, the desirability of playing the reciprocative strategy is investigated by stating conditions under which reciprocative strategy by one of the players or by both the players leading to mutual cooperative behavior is a Nash equilibrium outcome.     

The restricted core of games on distributive lattices: how to share benefits in a hierarchy

Finding a solution concept is one of the central problems in cooperative game theory, and the notion of core is the most popular solution concept since it is based on some rationality condition. In many real situations, not all possible coalitions can form, so that classical TU-games cannot be used. An interesting case is when possible coalitions are defined through a partial ordering of the players (or hierarchy). Then feasible coalitions correspond to teams of players, that is, one or several players with all their subordinates. In these situations, the core in its usual formulation may be unbounded, making its use difficult in practice. We propose a new notion of core, called the restricted core, which imposes efficiency of the allocation at each level of the hierarchy, is always bounded, and answers the problem of sharing benefits in a hierarchy. We show that the core we defined has properties very close to the classical case, with respect to marginal vectors, the Weber set, and balancedness.     

Rankings and values for team games

In this article we attack several problems that arise when a group of individuals is organized in several teams with equal number of players in each one (e.g., for company work, in sports leagues, etc). We define a team game as a cooperative game v that can have non-zero values only on coalitions of a given cardinality; it is further shown that, for such games, there is essentially a unique ranking among the players. We also study the way the ranking changes after one or more players retire. Also, we characterize axiomatically different ways of ranking the players that intervene in a cooperative game.     

Exploring the VCG mechanism in combinatorial auctions: The threshold revenue and the threshold-price rule

We explore interesting potential extensions of the Vickrey–Clarke–Groves (VCG) rule under the assumption of players with independent and private valuations and no budget constraints. First, we apply the VCG rule to a coalition of bidders in order to compute the second price of the coalition. Then, we introduce and formulate the problem of determining that partition of players into coalitions which maximize the auctioneer’s revenue in the case whereby such coalitions take part to a VCG auction each one as a single agent; in particular, we provide an integer linear formulation of this problem. We also generalize this issue by allowing players to simultaneously belong to distinct coalitions in the case that players’ valuation functions are separable. Finally, we propose some applications of these theoretical results. For instance, we exploit them to provide a class of new payment rules and to decide which bids should be defined as the highest losing ones in combinatorial auctions.     

基于全支付拍卖模型的众包激励策略优化分析

考虑协作式众包与众包竞赛在发包方期望收益上存在差异,本文对两种众包模式下的激励策略优化问题进行了研究。基于全支付拍卖理论建立了协作式众包模式中全部接包方贡献总收益和众包竞赛模式中接包方贡献最高质量收益的数学模型。接下来,针对两种模式下接包方收益的区别,分析了各自的最优激励策略。最后,采用比较静态均衡法分析了接包方数量和能力在两种众包模式中对激励策略的不同影响,并采用了数值算例验证了相关结论。结果显示:在不设置奖励限制时,协作式众包模式中存在一个最优的奖励分配个数使得接包方贡献总收益取得最大值;而众包竞赛模式中存在一个最优的奖励分配个数使得接包方贡献最高质量收益取得最大值。因此,若接包方参与数量及能力水平提高时,发包方应增加奖励分配个数,以促进接包方贡献总收益与接包方贡献最高质量收益的最大化。     

Fictitious play with incomplete learning

In this paper, we consider a case that a game is played repeatedly in an incomplete learning process where each player updates his belief only in the learning periods rather than all the stages. For fictitious play process with incomplete learning, we discuss the absorbability of Nash equilibriums and the consistency of utilities in a finite game and discuss the convergence in a 2×2 game with an identical learning-period set. The main results for incomplete learning models are that, if it is uniformly played, a strict Nash equilibrium is absorbing in a fictitious play process; a fictitious play has the property of utility consistency if it exhibits infrequent switches and players learn frequently enough; a 2×2 game with an identical learning-period set has fictitious play property that any fictitious process for the game converges to equilibrium provided that players learn frequently enough.     

Collusive game solutions via optimization

A Nash-based collusive game among a finite set of players is one in which the players coordinate in order for each to gain higher payoffs than those prescribed by the Nash equilibrium solution. In this paper, we study the optimization problem of such a collusive game in which the players collectively maximize the Nash bargaining objective subject to a set of incentive compatibility constraints. We present a smooth reformulation of this optimization problem in terms of a nonlinear complementarity problem. We establish the convexity of the optimization problem in the case where each player's strategy set is unidimensional. In the multivariate case, we propose upper and lower bounding procedures for the collusive optimization problem and establish convergence properties of these procedures. Computational results with these procedures for solving some test problems are reported. It is with great honor that we dedicate this paper to Professor Terry Rockafellar on the occasion of his 70th birthday. Our work provides another example showing how Terry's fundamental contributions to convex and variational analysis have impacted the computational solution of applied game problems. This author's research was partially supported by the National Science Foundation under grant ECS-0080577. This author's research was partially supported by the National Science Foundation under grant CCR-0098013.     

Global convergence of a kinetic model of chemotaxis to a perturbed Keller–Segel model

We consider a class of kinetic models of chemotaxis with two positive non-dimensional parameters coupled to a parabolic equation of the chemo-attractant. If both parameters are set equal zero, we have the classical Keller–Segel model for chemotaxis. We prove global existence of solutions of this two-parameters kinetic model and prove convergence of this model to models of chemotaxis with global existence when one of these two parameters is set equal zero. In one case, we find as a limit model a kinetic model of chemotaxis while in the other case we find a perturbed Keller–Segel model with global existence of solutions.     

Emergence and nonemergence of alternating offers in bilateral bargaining

We study in what circumstance players alternate offers in bilateral bargaining. To examine this question, we suppose that players choose whether to take the initiative in each period. The player who tries to take the initiative is able to make an offer only when the other player does not. The probability that a player tries to take the initiative is referred to as the frequency of initiative taking. We assume that this is conditioned on mutually observable states and is, once chosen, unchangeable. When players make their frequency of initiative taking dependent on the identity of the latest proposer, the players alternate their offers (possibly with some stochastic delay). In contrast, when players always use the same frequency of initiative taking, or when players only distinguish odd-numbered from even-numbered periods for the frequency of initiative taking, both players constantly try to take the initiative. Consequently, an impasse arises.