Jacques Durieu

Control costs and potential functions for spatial games

AbstractVan Damme and Weibull (1998, 2002) model the noise in games as endogenously determined tremble probabilities, by assuming that with some effort players can control the probability of implementing the intended strategy. Following their methodology, we derive logit-like adjustment rules for games played on quasi-symmetric weighted graphs and explore the properties of the ensuing Markov chain.

Farsighted coalitional stability in TU-games

We study farsighted coalitional stability in the context of TUgames. Chwe (1994, p.318) notes that, in this context, it is difficult to prove nonemptiness of the largest consistent set. We show that every TU-game has a nonempty largest consistent set. Moreover, the proof of this result points out that each TU-game has a farsighted stable set. We go further by providing a characterization of the collection of farsighted stable sets in TU-games. We also show that the farsighted core of a TU-game is empty or is equal to the set of imputations of the game. Next, the relationships between the core and the largest consistent set are studied in superadditive TU-games and in clan games. In the last section, we explore the stability of the Shapley value. It is proved that the Shapley value of a superadditive TU-game is always a stable imputation: it is a core imputation or it constitutes a farsighted stable set. A necessary and sufficient condition for a superadditive TU-game to have the Shapley value in the largest consistent set is given.

Adaptive play with spatial sampling

Abstract We consider a 2×2 coordination game where each agent interacts with his neighbors on a ring. Ellison (1993, Econometrica 61, 1047–1071) shows that the discrete dynamical system generated by the myopic best-reply rule converges to a Nash equilibrium or to a two-period limit cycle. Following Young (1993, Econometrica 61, 57–84), we consider a best-reply process with a sampling procedure. Particularly, we introduce a spatial sampling procedure: each agent observes a sample of information in his neighborhood and plays a best reply to it. We show that if the size of the sample of information is not too large, the best-reply process converges almost surely to a Nash equilibrium. If in addition agents experiment with small probabilities, we show that, in most cases, the risk-dominant equilibrium prevails in the long run. Furthermore, it turns out that the convergence is faster than in Ellison.

Complexity and stochastic evolution of dyadic networks

A strategic model of network formation is developed which permits unreliable links and organizational costs. Finding a connected Nash network which guarantees a given payoff to each player proves to be an NP-hard problem. For the associated evolutionary game with asynchronous updating and logit updating rules, the stochastically stable networks are characterized.The organization of agents into networks has an important role in the communication of information within a spatial structure. One goal is to understand how such networks form and evolve over time. Our agents are endowed with some information which can be accessed by other agents forming links with them. Link formation is costly and communication not fully reliable. We model the process of network formation as a non-cooperative game, and we then focus on Nash networks. But, showing existence of a Nash network with particular properties and computing one are two different tasks. The aim of this paper is to show that computing a connected Nash network is a computationally demanding optimization problem. The question then arises what outcomes might be chosen by agents who would like to form a connected Nash network but fail to achieve their goal because of computational limitations. We propose a stochastic evolutionary model. By solving a companion global optimization problem, this model selects a subset of Nash networks referred to as the set of stochastically stable networks.

Adaptive learning and p-best response sets

A product set of strategies is a p-best response set if for each agent it contains all best responses to any distribution placing at least probability p on his opponents’ profiles belonging to the product set. A p-best response set is minimal if it does not properly contain another p-best response set. We study a perturbed joint fictitious play process with bounded memory and sample and a perturbed independent fictitious play process as in Young (Econometrica 61:57–84, 1993). We show that in n-person games only strategies contained in the unique minimal p-best response set can be selected in the long run by both types of processes provided that the rate of perturbations and p are sufficiently low. For each process, an explicit bound of p is given and we analyze how this critical value evolves when n increases. Our results are robust to the degree of incompleteness of sampling relative to memory.

Contagion and Dominating Sets

Abstract Each agent of a finite population interacts strategically with each of his neighbours on a graph. All agents have the same pair of available actions. In every period, each agent chooses a particular action if at least a proportion p of his neighbours has chosen this action in the previous period. Contagion is said to occur if one action spreads from a particular group of agents to the entire population. We develop techniques for analysing contagion in connected regular graphs. These techniques are based on the concept of dominating set from graph theory. We also characterize the class of regular graphs where different agents may choose different actions forever and the class of regular graphs where two-period limit cycles may occur. Finally, we apply our results to the case of tori.

Stochastic Evolutionary Game Theory

This chapter first gives an overview of some developments in the area of adaptive learning in games. Next, we present a general framework to model adaptive learning processes with persistent noise, and define the notion of stochastic stability. We provide several leading examples.

Local Interactions and p-Best Response Set

We study a local interaction model where agents play a finite n-person game following a perturbed best-response process with inertia. We consider the concept of minimal p-best response set to analyze distributions of actions in the long run. We distinguish between two assumptions made by agents about the matching rule. We show that only actions contained in the minimal p-best response set can be selected provided p is sufficiently small. We demonstrate that these predictions are sensitive to the assumptions about the matching rule.

Location Games with Externalities

We propose a two step game of coalition or city formation. In a first step, each player chooses the location in which he wants to be. The payoff function, determined in the second step by a game between the different locations reflects two effects: a public effect such that payoffs decrease with the number of non-empty locations; a private effect such that payoffs to the inhabitants of a PartIcular location decrease with the size of the population at that location. We analyse the consequences for the set of stable profiles of an increase in the relative weight of the public effect in the payoff function. We show that the number of stable profiles increases with the public effect but that the newly added profiles are not always more concentrated.