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A note on robust Nash equilibria with uncertainties

Vianney Perchet (2014)

RAIRO - Operations Research - Recherche Opérationnelle

In this short note, we investigate the framework where agents or players have some uncertainties upon their payoffs or losses, the behavior (or the type, number or any other characteristics) of other players. More specifically, we introduce an extension of the concept of Nash equilibria that generalize different solution concepts called by their authors, and depending on the context, either as robust, ambiguous, partially specified or with uncertainty aversion. We provide a simple necessary and...

A symbolic shortest path algorithm for computing subgame-perfect Nash equilibria

Pedro A. Góngora, David A. Rosenblueth (2015)

International Journal of Applied Mathematics and Computer Science

Consider games where players wish to minimize the cost to reach some state. A subgame-perfect Nash equilibrium can be regarded as a collection of optimal paths on such games. Similarly, the well-known state-labeling algorithm used in model checking can be viewed as computing optimal paths on a Kripke structure, where each path has a minimum number of transitions. We exploit these similarities in a common generalization of extensive games and Kripke structures that we name “graph games”. By extending...

Analyse mathématique élémentaire d'un jeu sportif

P. Parlebas (1974)

Mathématiques et Sciences Humaines

Un jeu sportif bien connu, les quatre coins, est l'objet de cette étude. Les règles de ce jeu déterminent une succession de déplacements qui sont organisés selon une structure de groupe. L'ensemble des graphes de déplacement peut être distribué selon trois partitions qui offrent un précieux support à l'étude expérimentale: on obtient alors des classes d'équivalence de type cyclique, de type spatial et de type métrique. Les règles sont porteuses d'une logique qui accorde une grande importance à l'espace...

Approximations of dynamic Nash games with general state and action spaces and ergodic costs for the players

Tomasz Bielecki (1997)

Applicationes Mathematicae

The purpose of this paper is to prove existence of an ε -equilib- rium point in a dynamic Nash game with Borel state space and long-run time average cost criteria for the players. The idea of the proof is first to convert the initial game with ergodic costs to an ``equivalent" game endowed with discounted costs for some appropriately chosen value of the discount factor, and then to approximate the discounted Nash game obtained in the first step with a countable state space game for which existence...

Bilateral sequential bargaining with perfect information and different protocols

Robert Golański (2006)

Banach Center Publications

Most research done in the bargaining literature concentrates on the situations in which players get to be proposers alternately, with the first player being the proposer in the first period, the second player being the proposer in the second period, and so on until the cycle ends and the order of proposers is repeated. However, allowing for only this kind of order is a rather simplifying assumption. This paper looks at the situation in which we allow for much more general kind of protocols. We characterize...

Complementarities and the existence of strong Berge equilibrium

Kerim Keskin, H. Çağrı Sağlam (2014)

RAIRO - Operations Research - Recherche Opérationnelle

This paper studies the existence and the order structure of strong Berge equilibrium, a refinement of Nash equilibrium, for games with strategic complementarities à la strong Berge. It is shown that the equilibrium set is a nonempty complete lattice. Moreover, we provide a monotone comparative statics result such that the greatest and the lowest equilibria are increasing.

Control of a team of mobile robots based on non-cooperative equilibria with partial coordination

Krzysztof Skrzypczyk (2005)

International Journal of Applied Mathematics and Computer Science

In this work we present an application of the concept of non-cooperative game equilibria to the design of a collision free movement of a team of mobile robots in a dynamic environment. We propose the solution to the problem of feasible control synthesis, based on a partially centralized sensory system. The control strategy based on the concept of non-cooperative game equilibria is well known in the literature. It is highly efficient through phases where the solution is unique. However, even in simple...

Convergence method, properties and computational complexity for Lyapunov games

Julio B. Clempner, Alexander S. Poznyak (2011)

International Journal of Applied Mathematics and Computer Science

We introduce the concept of a Lyapunov game as a subclass of strictly dominated games and potential games. The advantage of this approach is that every ergodic system (repeated game) can be represented by a Lyapunov-like function. A direct acyclic graph is associated with a game. The graph structure represents the dependencies existing between the strategy profiles. By definition, a Lyapunov-like function monotonically decreases and converges to a single Lyapunov equilibrium point identified by...

Core solutions and nash equilibria in noncooperative games with a measure space of players

Sjur Didrik Flåm, Andrzej Wieczorek (2006)

Banach Center Publications

The paper deals with noncooperative games in which players constitute a measure space. Strategy profiles that are equal almost everywhere are assumed to have the same interactive effects. Under these circumstances we explore links between core solutions and Nash equilibria. Conditions are given which guarantee that core outcomes must be Nash equilibria and vice versa. The main contribution are results on nonemptieness of the core.

Delegation equilibrium payoffs in integer-splitting games

Sylvain Sorin, Cheng Wan (2013)

RAIRO - Operations Research - Recherche Opérationnelle

This work studies a new strategic game called delegation game. A delegation game is associated to a basic game with a finite number of players where each player has a finite integer weight and her strategy consists in dividing it into several integer parts and assigning each part to one subset of finitely many facilities. In the associated delegation game, a player divides her weight into several integer parts, commits each part to an independent delegate and collects the sum of their payoffs in...

Equilibria in a class of games and topological results implying their existence.

R.S. Simon, S. Spiez, H. Torunczyk (2008)


We survey results related to the problem of the existence of equilibria in some classes of infinitely repeated two-person games of incomplete information on one side, first considered by Aumann, Maschler and Stearns. We generalize this setting to a broader one of principal-agent problems. We also discuss topological results needed, presenting them dually (using cohomology in place of homology) and more systematically than in our earlier papers.

Equilibria in constrained concave bimatrix games

Wojciech Połowczuk, Tadeusz Radzik (2013)

Applicationes Mathematicae

We study a generalization of bimatrix games in which not all pairs of players' pure strategies are admissible. It is shown that under some additional convexity assumptions such games have equilibria of a very simple structure, consisting of two probability distributions with at most two-element supports. Next this result is used to get a theorem about the existence of Nash equilibria in bimatrix games with a possibility of payoffs equal to -∞. The first of these results is a discrete counterpart...

Equilibrium transitions in finite populations of players

J. Miękisz (2006)

Banach Center Publications

We discuss stochastic dynamics of finite populations of individuals playing symmetric games. We review recent results concerning the dependence of the long-run behavior of such systems on the number of players and the noise level. In the case of two-player games with two symmetric Nash equilibria, when the number of players increases, the population undergoes multiple transitions between its equilibria.

Estimating the supply chain efficiency loss when the seller has to estimate the buyer’s willingness to pay

Xavier Brusset (2014)

RAIRO - Operations Research - Recherche Opérationnelle

We study the pricing problem between two firms when the manufacturer’s willingness to pay (wtp) for the supplier’s good is not known by the latter. We demonstrate that it is in the interest of the manufacturer to hide this information from the supplier. The precision of the information available to the supplier modifies the rent distribution. The risk of opportunistic behaviour entails a loss of efficiency in the supply chain. The model is extended to the case of a supplier submitting offers to...

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