A reinforcement learning process in extensive form games

The CPR (“cumulative proportional reinforcement”) learning rule stipulates that an agent chooses a move with a probability proportional to the cumulative payoff she obtained in the past with that move. Previously considered for strategies in normal form games (Laslier, Topol and Walliser, Games and Econ. Behav., 2001), the CPR rule is here adapted for actions in perfect information extensive form games. The paper shows that the action-based CPR process converges with probability one to the (unique) subgame perfect equilibrium.Received: October 2004     

A dynamical method for the calculation of Nash-equilibria in n-person games

In this paper, we are concerned with the calculation of Nash-equilibria in non-cooperative n-person games. For this purpose, we construct a continuous mapping of the Cartesian product of the strategy sets of the players into itself such that the fixed points of this mapping are Nash-equilibria. This gives rise to an iteration method for the calculation of fixed points of this mapping which leads to Nash-equilibria, if it converges. As important special cases Bi-matrix games and evolution matrix games are considered.     

Dynamic stability in symmetric extensive form games

Dynamic stability under the replicator dynamic of evolutionary game theory is investigated for certain symmetric extensive form games whose subgame structure exhibits a high degree of decomposability. It is shown that a pervasive equilibrium strategy is locally asymptotically stable (l.a.s.) if and only if it is given by backwards induction applied to the l.a.s. pervasive equilibria of the subgames and their corresponding truncations. That is, this dynamic backwards induction procedure provides a rational basis on which to predict the evolutionary outcome of the replicator dynamic for these symmetric games.     

Convention evolution in organizations and markets

Conventions are essential for the coordination of multi-agent systems. However, in many systems conventions can not be legislated in advance and need to emerge during the system's activity. As designers of such systems we may wish to ensure that conventions will evolve rapidly. Given a classical model for convention evolution where agents tend to mimic agents they interact with, the designer can control the organizational structure of the system in order to speedup the evolution of conventions. This paper introduces a study of convention evolution in the context of basic organizational structures. Our study sheds light on a basic aspect of organizational design which has not been discussed in the literature, and which is crucial for efficient design of non-trivial multi-agent systems.     

Conditional Viability for Impulse Differential Games

We introduce in this paper the concept of “impulse evolutionary game”. Examples of evolutionary games are usual differential games, differentiable games with history (path-dependent differential games), mutational differential games, etc. Impulse evolutionary systems and games cover in particular “hybrid systems” as well as “qualitative systems”. The conditional viability kernel of a constrained set (with a target) is the set of initial states such that for all strategies (regarded as continuous feedbacks) played by the second player, there exists a strategy of the first player such that the associated run starting from this initial state satisfies the constraints until it hits the target. This paper characterizes the concept of conditional viability kernel for “qualitative games” and of conditional valuation function for “qualitative games” maximinimizing an intertemporal criterion. The theorems obtained so far about viability/capturability issues for evolutionary systems, conditional viability for differential games and about impulse and hybrid systems are used to provide characterizations of conditional viability under impulse evolutionary games.     

Pure strategy equilibria in symmetric two-player zero-sum games

We observe that a symmetric two-player zero-sum game has a pure strategy equilibrium if and only if it is not a generalized rock-paper-scissors matrix. Moreover, we show that every finite symmetric quasiconcave two-player zero-sum game has a pure equilibrium. Further sufficient conditions for existence are provided. Our findings extend to general two-player zero-sum games using the symmetrization of zero-sum games due to von Neumann. We point out that the class of symmetric two-player zero-sum games coincides with the class of relative payoff games associated with symmetric two-player games. This allows us to derive results on the existence of finite population evolutionary stable strategies.     

Highway games on weakly cyclic graphs

A highway problem is determined by a connected graph which provides all potential entry and exit vertices and all possible edges that can be constructed between vertices, a cost function on the edges of the graph and a set of players, each in need of constructing a connection between a specific entry and exit vertex. Mosquera (2007) introduce highway problems and the corresponding cooperative cost games called highway games to address the problem of fair allocation of the construction costs in case the underlying graph is a tree. In this paper, we study the concavity and the balancedness of highway games on weakly cyclic graphs. A graph G is called highway-game concave if for each highway problem in which G is the underlying graph the corresponding highway game is concave. We show that a graph is highway-game concave if and only if it is weakly triangular. Moreover, we prove that highway games on weakly cyclic graphs are balanced.     

On minimum sum representations for weighted voting games

A proposal in a weighted voting game is accepted if the sum of the (non-negative) weights of the ??yea?? voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. In Freixas and Molinero (Ann. Oper. Res. 166:243?C260, 2009) the authors have classified all weighted voting games without a unique minimum sum representation for up to 8 voters. Here we exhaustively classify all weighted voting games consisting of 9?voters which do not admit a unique minimum sum integer weight representation.     

A demand adjustment process

The aspiration approach to cooperative games, which has been studied by a number of authors, including Cross, Turbay, Albers, Selten and Bennett, presumes that players in a game bargain over their reservation prices, or aspirations. A number of aspiration-based solution concepts have been put forth, and aspiration solutions have been connected to non-cooperative bargaining models. Missing in this approach has been theory of how aspirations themselves arise. The present paper is an attempt to fill this gap. It describes a very general demand adjustment process, using the framework of set-valued dynamical systems developed by Maschler and Peleg. This demand adjustment process always converges; sufficient conditions are given in order that it converge to an aspiration, and that it converge in a finite number of steps.     

Infinitely Divisible Limit Processes for the Ewens Sampling Formula

The Ewens sampling formula in population genetics can be viewed as a probability measure on the group of permutations of a finite set of integers. Functional limit theory for processes defined through partial sums of dependent variables with respect to the Ewens sampling formula is developed. Using techniques from probabilistic number theory, it is shown that, under very general conditions, a partial sum process weakly converges in a function space if and only if the corresponding process defined through sums of independent random variables weakly converges. As a consequence of this result, necessary and sufficient conditions for weak convergence to a stable process are established. A counterexample showing that these conditions are not necessary for the one-dimensional convergence is presented. Very few results on the necessity part are known in the literature.     

Totally balanced combinatorial optimization games

Combinatorial optimization games deal with cooperative games for which the value of every subset of players is obtained by solving a combinatorial optimization problem on the resources collectively owned by this subset. A solution of the game is in the core if no subset of players is able to gain advantage by breaking away from this collective decision of all players. The game is totally balanced if and only if the core is non-empty for every induced subgame of it.?We study the total balancedness of several combinatorial optimization games in this paper. For a class of the partition game [5], we have a complete characterization for the total balancedness. For the packing and covering games [3], we completely clarify the relationship between the related primal/dual linear programs for the corresponding games to be totally balanced. Our work opens up the question of fully characterizing the combinatorial structures of totally balanced packing and covering games, for which we present some interesting examples: the totally balanced matching, vertex cover, and minimum coloring games. Received: November 5, 1998 / Accepted: September 8, 1999?Published online February 23, 2000     

Invertibility modulo dead-ending no-$$\mathcal {}$$-universes

In combinatorial game theory, under normal play convention, all games are invertible, whereas only the empty game is invertible in misère play. For this reason, several restricted universes of games were studied, in which more games are invertible. Here, we study combinatorial games under misère play, in particular universes where no player would like to pass their turn. In these universes, we prove that having one extra condition makes all games become invertible. We then focus our attention on a specific quotient, called \({\mathcal {Q}_{\mathbb {Z}}}\), and show that all sums of universes whose quotient is \({\mathcal {Q}_{\mathbb {Z}}}\) also have \({\mathcal {Q}_{\mathbb {Z}}}\) as their quotient.     

“Evolutionary” selection dynamics in games: Convergence and limit properties

This paper discusses convergence properties and limiting behavior in a class of dynamical systems of which the replicator dynamics of (biological) evolutionary game theory are a special case. It is known that such dynamics need not be well-behaved for arbitrary games. However, it is easy to show that dominance solvable games are convergent for any dynamics in the class and, what is somewhat more difficult to establish, weak dominance solvable games are as well, provided they are small in a sense to be made precise in the text. The paper goes on to compare dynamical solutions with standard solution concepts from noncooperative game theory.This paper is a revision of Chapter 1 of my Ph.D. thesis. It owes much to the guidance of Andreu Mas-Colell, Eric Maskin, Vijay Krishna, and Dilip Abreu. I wish also to express my thanks for the comments of an anonymous referee. Naturally, all remaining shortcomings are my responsibility.     

A limit theorem for equilibria under ambiguous belief correspondences

Previous literature shows that, in many different models, limits of equilibria of perturbed games are equilibria of the unperturbed game when the sequence of perturbed games converges to the unperturbed one in an appropriate sense. The question of whether such a limit property extends to the equilibrium notions in ambiguous games is not yet as clear as it seems; in fact, previous literature shows that the extension fails in simple examples.     

Perfect foresight dynamics in games with linear incentives and time symmetry

This paper investigates absorption and global accessibility under perfect foresight dynamics in games with linear incentives. An action distribution in the society is absorbing if there is no equilibrium path escaping from the distribution, and globally accessible if, from every initial distribution, there exists an equilibrium path which converges to the distribution. Using time symmetry of the dynamics, we show that every absorbing strict Nash equilibrium, if it exists, is globally accessible under zero rate of time preference. With the additional assumption of supermodularity, we prove that there generically exists an absorbing strict Nash equilibrium. Relations with a global game and a reaction-diffusion model also become clear. The definition of absorption used in this paper is slightly different from the original one in Matsui and Matsuyama (1995). This difference is neglected in Sect. 1, and will be discussed in Sect. 2.2.     

Self‐organization and selection in the emergence of vocabulary

Human language may have started from a consistent set of mappings between meanings and signals. These mappings, referred to as the early vocabulary, are considered to be the results of conventions established among the agents of a population. In this study, we report simulation models for investigating how such conventions can be reached. We propose that convention is essentially the product of self‐organization of the population through interactions among the agents and that cultural selection is another mechanism that speeds up the establishment of convention. Whereas earlier studies emphasize either one or the other of these two mechanisms, our focus is to integrate them into one hybrid model. The combination of these two complementary mechanisms, i.e., self‐organization and cultural selection, provides a plausible explanation for cultural evolution, which progresses with high transmission rate. Furthermore, we observe that as the vocabulary tends to convergence there is a uniform tendency to exhibit a sharp phase transition. © 2002 Wiley Periodicals, Inc.     

Genetic algorithms with noisy fitness

The convergence properties of genetic algorithms with noisy fitness information are studied here. In the proposed scheme, hypothesis testing methods are used to compare sample fitness values. The “best” individual of each generation is kept and a greater-than-zero mutation rate is used so that every individual will be generated with positive probability in each generation. The convergence criterion is different from the frequently-used uniform population criterion; instead, the sequence of the “best” individual in each generation is considered, and the algorithm is regarded as convergent if the sequence of the “best” individuals converges with probability one to a point with optimal average fitness.     

Communication, complexity, and evolutionary stability

In games with costless preplay communication, some strategies are more complex than others in the sense that they induce a finer partition of the set of states of the world. This paper shows that if the concept of evolutionary stability, which is argued to be a natural solution concept for communication games, is modified to take lexicographic complexity preferences into account, then for a class of games of common interest only communication strategies that induce payoff-dominant Nash outcomes of the underlying game are stable. Received April 1998/Final version September 1998