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1.
Hadi Charkhgard Martin Savelsbergh Masoud Talebian 《4OR: A Quarterly Journal of Operations Research》2018,16(2):151-171
We study the connection between biobjective mixed integer linear programming and normal form games with two players. We first investigate computing Nash equilibria of normal form games with two players using single-objective mixed integer linear programming. Then, we define the concept of efficient (Pareto optimal) Nash equilibria. This concept is precisely equivalent to the concept of efficient solutions in multi-objective optimization, where the solutions are Nash equilibria. We prove that the set of all points in the payoff (or objective) space of a normal form game with two players corresponding to the utilities of players in an efficient Nash equilibrium, the so-called nondominated Nash points, is finite. We demonstrate that biobjective mixed integer linear programming, where the utility of each player is an objective function, can be used to compute the set of nondominated Nash points. Finally, we illustrate how the nondominated Nash points can be used to determine the disagreement point of a bargaining problem. 相似文献
2.
Elżbieta Z. Ferenstein 《Mathematical Methods of Operations Research》2007,66(3):531-544
We study nonzero-sum stopping games with randomized stopping strategies. The existence of Nash equilibrium and ɛ-equilibrium
strategies are discussed under various assumptions on players random payoffs and utility functions dependent on the observed
discrete time Markov process. Then we will present a model of a market game in which randomized stopping times are involved.
The model is a mixture of a stochastic game and stopping game.
Research supported by grant PBZ-KBN-016/P03/99. 相似文献
3.
We present a method for the derivation of feedback Nash equi- libria in discrete-time finite-horizon nonstationary dynamic
games. A partic- ular motivation for such games stems from environmental economics, where problems of seasonal competition
for water levels occur frequently among heterogeneous economic agents. These agents are coupled through a state variable,
which is the water level. Actions are strategically chosen to max- imize the agents individual season-dependent utility functions.
We observe that, although a feedback Nash equilibrium exists, it does not satisfy the (exogenous) environmental watchdog expectations.
We devise an incentive scheme to help meeting those expectations and calculate a feedback Nash equilibrium for the new game
that uses the scheme. This solution is more environmentally friendly than the previous one. The water allocation game solutions
help us to draw some conclusions regarding the agents behavior and also about the existence of feedback Nash equilibria in
dynamic games.
The paper draws from Refs.1–2. Its earlier version was presented at the Victoria International Conference 2004, Victoria University
of Wellington, Wellington, New Zealand, February 9–13, 2004.
We thank the anonymous referee and Christophe Deissenberg for insightful comments, which have helped us to clarify its message.
We also thank our colleagues Sophie Thoyer, Robert Lifran, Odile Pourtalier, and Vladimir Petkov for helpful discussions on
the model and techniques used in this Paper.
Gratitude is expressed to the Kyoto Institute for Economic Research, Kyoto University, for this author's support in the final
stages of the paper preparation 相似文献
4.
5.
M. S. Nikol’skii 《Proceedings of the Steklov Institute of Mathematics》2015,291(1):232-236
In modern game theory, a lot of attention is paid to the concept of Nash equilibrium. The paper is devoted to the study of some properties of the set A of Nash equilibrium points in two-person games. In particular, the character of possible complexity of the set A is investigated, and the stability of the set A under small perturbations of payoff functions is analyzed. 相似文献
6.
A directed network game of imperfect strategic substitutes with heterogeneous players is analyzed. We consider concave additive separable utility functions that encompass the quasi-linear ones. It is found that pure strategy Nash equilibria verify a non-linear complementarity problem. By requiring appropriate concavity in the utility functions, the existence of an equilibrium point is shown and equilibrium uniqueness is established with a P-matrix. For this reason, it appears that previous findings on network structure and sparsity hold for many more games. 相似文献
7.
Systems that involve more than one decision maker are often optimized using the theory of games. In the traditional game theory, it is assumed that each player has a well-defined quantitative utility function over a set of the player decision space. Each player attempts to maximize/minimize his/her own expected utility and each is assumed to know the extensive game in full. At present, it cannot be claimed that the first assumption has been shown to be true in a wide variety of situations involving complex problems in economics, engineering, social and political sciences due to the difficulty inherent in defining an adequate utility function for each player in these types of problems. On the other hand, in many of such complex problems, each player has a heuristic knowledge of the desires of the other players and a heuristic knowledge of the control choices that they will make in order to meet their ends.In this paper, we utilize fuzzy set theory in order to incorporate the players' heuristic knowledge of decision making into the framework of conventional game theory or ordinal game theory. We define a new approach to N-person static fuzzy noncooperative games and develop a solution concept such as Nash for these types of games. We show that this general formulation of fuzzy noncooperative games can be applied to solve multidecision-making problems where no objective function is specified. The computational procedure is illustrated via application to a multiagent optimization problem dealing with the design and operation of future military operations. 相似文献
8.
本文考虑具有损失厌恶行为与破裂风险的Rubinstein谈判博弈。首先构建子博弈完美均衡,并证明子博弈完美均衡的存在性及唯一性。然后分析子博弈完美均衡的性质,结果表明:参与人受益于对手的损失厌恶行为,而因自身具有损失厌恶行为遭受损失;谈判破裂概率对均衡结果的影响取决于贴现因子与参与人的损失厌恶系数;当谈判破裂的概率趋于零时,极限均衡结果收敛于经典的Rubinstein谈判博弈结果。最后建立了与非对称Nash谈判解的关系,其中参与人的议价能力与自身的损失厌恶水平呈负相关性,与对手的损失厌恶水平呈正相关性;参与人的议价能力依赖于谈判破裂概率与出价时间间隔的比值。 相似文献
9.
This paper formally introduces and studies a non-cooperative multi-agent game under uncertainty. The well-known Nash equilibrium is employed as the solution concept of the game. While there are several formulations of a stochastic Nash equilibrium problem, we focus mainly on a two-stage setting of the game wherein each agent is risk-averse and solves a rival-parameterized stochastic program with quadratic recourse. In such a game, each agent takes deterministic actions in the first stage and recourse decisions in the second stage after the uncertainty is realized. Each agent’s overall objective consists of a deterministic first-stage component plus a second-stage mean-risk component defined by a coherent risk measure describing the agent’s risk aversion. We direct our analysis towards a broad class of quantile-based risk measures and linear-quadratic recourse functions. For this class of non-cooperative games under uncertainty, the agents’ objective functions can be shown to be convex in their own decision variables, provided that the deterministic component of these functions have the same convexity property. Nevertheless, due to the non-differentiability of the recourse functions, the agents’ objective functions are at best directionally differentiable. Such non-differentiability creates multiple challenges for the analysis and solution of the game, two principal ones being: (1) a stochastic multi-valued variational inequality is needed to characterize a Nash equilibrium, provided that the players’ optimization problems are convex; (2) one needs to be careful in the design of algorithms that require differentiability of the objectives. Moreover, the resulting (multi-valued) variational formulation cannot be expected to be of the monotone type in general. The main contributions of this paper are as follows: (a) Prior to addressing the main problem of the paper, we summarize several approaches that have existed in the literature to deal with uncertainty in a non-cooperative game. (b) We introduce a unified formulation of the two-stage SNEP with risk-averse players and convex quadratic recourse functions and highlight the technical challenges in dealing with this game. (c) To handle the lack of smoothness, we propose smoothing schemes and regularization that lead to differentiable approximations. (d) To deal with non-monotonicity, we impose a generalized diagonal dominance condition on the players’ smoothed objective functions that facilitates the application and ensures the convergence of an iterative best-response scheme. (e) To handle the expectation operator, we rely on known methods in stochastic programming that include sampling and approximation. (f) We provide convergence results for various versions of the best-response scheme, particularly for the case of private recourse functions. Overall, this paper lays the foundation for future research into the class of SNEPs that provides a constructive paradigm for modeling and solving competitive decision making problems with risk-averse players facing uncertainty; this paradigm is very much at an infancy stage of research and requires extensive treatment in order to meet its broad applications in many engineering and economics domains. 相似文献
10.
Vitaly Pruzhansky 《International Journal of Game Theory》2011,40(2):351-365
Since the seminal paper of Nash (1950) game theoretic literature has focused mostly on equilibrium and not on maximin (minimax)
strategies. We study the properties of these strategies in non-zero-sum strategic games that possess (completely) mixed Nash
equilibria. We find that under certain conditions maximin strategies have several interesting properties, some of which extend
beyond 2-person strategic games. In particular, for n-person games we specify necessary and sufficient conditions for maximin strategies to yield the same expected payoffs as
Nash equilibrium strategies. We also show how maximin strategies may facilitate payoff comparison across Nash equilibria as
well as refine some Nash equilibrium strategies. 相似文献
11.
Michail Anthropelos 《Mathematics and Financial Economics》2017,11(3):323-368
The paper studies an oligopolistic equilibrium model of financial agents who aim to share their random endowments. The risk-sharing securities and their prices are endogenously determined as the outcome of a strategic game played among all the participating agents. In the complete-market setting, each agent’s set of strategic choices consists of the security payoffs and the pricing kernel that are consistent with the optimal-sharing rules; while in the incomplete setting, agents respond via demand functions on a vector of given tradeable securities. It is shown that at the (Nash) risk-sharing equilibrium, the sharing securities are suboptimal, since agents submit for sharing different risk exposures than their true endowments. On the other hand, the Nash equilibrium prices stay unaffected by the game only in the special case of agents with the same risk aversion. In addition, agents with sufficiently lower risk aversion act as predatory traders, since they absorb utility surplus from the high risk averse agents and reduce the efficiency of sharing. The main results of the paper also hold under the generalized models that allow the presence of noise traders and heterogeneity in agents’ beliefs. 相似文献
12.
Proper rationalizability and backward induction 总被引:1,自引:0,他引:1
Frank Schuhmacher 《International Journal of Game Theory》1999,28(4):599-615
This paper introduces a new normal form rationalizability concept, which in reduced normal form games corresponding to generic
finite extensive games of perfect information yields the unique backward induction outcome. The basic assumption is that every
player trembles “more or less rationally” as in the definition of a ε-proper equilibrium by Myerson (1978). In the same way
that proper equilibrium refines Nash and perfect equilibrium, our model strengthens the normal form rationalizability concepts
by Bernheim (1984), B?rgers (1994) and Pearce (1984). Common knowledge of trembling implies the iterated elimination of strategies
that are strictly dominated at an information set. The elimination process starts at the end of the game tree and goes backwards
to the beginning.
Received: October 1996/Final version: May 1999 相似文献
13.
On Coincidence of Feedback Nash Equilibria and Stackelberg Equilibria in Economic Applications of Differential Games 总被引:1,自引:0,他引:1
S. J. Rubio 《Journal of Optimization Theory and Applications》2006,128(1):203-220
The scope of the applicability of the feedback Stackelberg equilibrium concept in differential games is investigated. First,
conditions for obtaining the coincidence between the stationary feedback Nash equilibrium and the stationary feedback Stackelberg
equilibrium are given in terms of the instantaneous payoff functions of the players and the state equations of the game. Second,
a class of differential games representing the underlying structure of a good number of economic applications of differential
games is defined; for this class of differential games, it is shown that the stationary feedback Stackelberg equilibrium coincides
with the stationary feedback Nash equilibrium. The conclusion is that the feedback Stackelberg solution is generally not useful
to investigate leadership in the framework of a differential game, at least for a good number of economic applications
This paper was presented at the 8th Viennese Workshop on Optimal Control, Dynamic Games, and Nonlinear Dynamics: Theory and
Applications in Economics and OR/MS, Vienna, Austria, May 14–16, 2003, at the Seminar of the Instituto Complutense de Analisis
Economico, Madrid, Spain, June 20, 2003, and at the Sevilla Workshop on Dynamic Economics and the Environment, Sevilla, Spain,
July 2–3, 2003. The author is grateful to the participants in these sessions, in particular F.J. Andre and J. Ruiz, for their
comments. Five referees provided particularly helpful suggestions. Financial support from the Ministerio de Ciencia y Tecnologia
under Grant BEC2000-1432 is gratefully acknowledged. 相似文献
14.
Stéphane Le Roux 《Mathematical Logic Quarterly》2014,60(4-5):354-371
Game theory is usually considered applied mathematics, but a few game‐theoretic results, such as Borel determinacy, were developed by mathematicians for mathematics in a broad sense. These results usually state determinacy, i.e., the existence of a winning strategy in games that involve two players and two outcomes saying who wins. In a multi‐outcome setting, the notion of winning strategy is irrelevant yet usually replaced faithfully with the notion of (pure) Nash equilibrium. This article shows that every determinacy result over an arbitrary game structure, e.g., a tree, is transferable into existence of multi‐outcome (pure) Nash equilibrium over the same game structure. The equilibrium‐transfer theorem requires cardinal or order‐theoretic conditions on the strategy sets and the preferences, respectively, whereas counter‐examples show that every requirement is relevant, albeit possibly improvable. When the outcomes are finitely many, the proof provides an algorithm computing a Nash equilibrium without significant complexity loss compared to the two‐outcome case. As examples of application, this article generalises Borel determinacy, positional determinacy of parity games, and finite‐memory determinacy of Muller games. 相似文献
15.
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. 相似文献
16.
基于经典博弈模型的Nash均衡点集的通有稳定性和具有不确定参数的n人非合作博弈均衡点的概念,探讨了具有不确定参数博弈的均衡点集的通有稳定性.参照Nash均衡点集稳定性的统一模式,构造了不确定博弈的问题空间和解空间,并证明了问题空间是一个完备度量空间,解映射是上半连续的,且解集是紧集(即usco(upper semicontinuous and compact-valued)映射),得到不确定参数博弈模型的解集通有稳定性的相关结论. 相似文献
17.
A generalization of the Nash demand game is examined. Agents make simultaneous offers in each period as to how a pie is to be divided. Incompatible offers send the game to the next period, while compatible offers end the game with a split-the-difference trade. The set of perfect equilibria of this game includes any individually rational outcome, including inefficient outcomes and even including the outcome of perpetual disagreement. We suggest a stronger equilibrium concept of universal perfection, which requires robustness against every rather than just one sequence of perturbed games. The set of universally perfect equilibria also includes all individually rational outcomes. The results provide useful insights into both simultaneous-offers bargaining and the nature of the perfect equilibrium and similar concepts (such as stability and hyperstability) in infinite games. 相似文献
18.
In this paper, we consider a class of n-person noncooperative games, where the utility function of every player is given by a homogeneous polynomial defined by the payoff tensor of that player, which is a natural extension of the bimatrix game where the utility function of every player is given by a quadratic form defined by the payoff matrix of that player. We will call such a problem the multilinear game. We reformulate the multilinear game as a tensor complementarity problem, a generalization of the linear complementarity problem; and show that finding a Nash equilibrium point of the multilinear game is equivalent to finding a solution of the resulted tensor complementarity problem. Especially, we present an explicit relationship between the solutions of the multilinear game and the tensor complementarity problem, which builds a bridge between these two classes of problems. We also apply a smoothing-type algorithm to solve the resulted tensor complementarity problem and give some preliminary numerical results for solving the multilinear games. 相似文献
19.
Ryoichi Nishimura Shunsuke Hayashi Masao Fukushima 《Journal of Global Optimization》2012,53(1):107-120
Consider the N-person non-cooperative game in which each player’s cost function and the opponents’ strategies are uncertain. For such an
incomplete information game, the new solution concept called a robust Nash equilibrium has attracted much attention over the
past several years. The robust Nash equilibrium results from each player’s decision-making based on the robust optimization
policy. In this paper, we focus on the robust Nash equilibrium problem in which each player’s cost function is quadratic,
and the uncertainty sets for the opponents’ strategies and the cost matrices are represented by means of Euclidean and Frobenius
norms, respectively. Then, we show that the robust Nash equilibrium problem can be reformulated as a semidefinite complementarity
problem (SDCP), by utilizing the semidefinite programming (SDP) reformulation technique in robust optimization. We also give
some numerical example to illustrate the behavior of robust Nash equilibria. 相似文献
20.
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. 相似文献