Philippe Solal

Rooted-tree solutions for tree games

In this paper, we study cooperative games with limited cooperation possibilities, represented by a tree on the set of agents. Agents in the game can cooperate if they are connected in the tree. We introduce natural extensions of the average (rooted)-tree solution (see [Herings, P., van der Laan, G., Talman, D., 2008. The average tree solution for cycle free games. Games and Economic Behavior 62, 77-92]): the marginalist tree solutions and the random tree solutions. We provide an axiomatic characterization of each of these sets of solutions. By the way, we obtain a new characterization of the average tree solution.

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.

THE RIVER SHARING PROBLEM: A SURVEY

The river sharing problem deals with the fair distribution of welfare resulting from the optimal allocation of water among a set of riparian agents. Ambec and Sprumont [Sharing a river, J. Econ. Theor. 107, 453–462] address this problem by modeling it as a cooperative TU-game on the set of riparian agents. Solutions to that problem are reviewed in this article. These solutions are obtained via an axiomatic study on the class of river TU-games or via a market mechanism.

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.

Axioms of Invariance for TU-games

We introduce new axioms for the class of all TU-games with a fixed but arbitrary player set. These axioms require either invariance of an allocation rule or invariance of the payoff assigned by an allocation rule to a specified player in two related TU-games. Combinations of these new axioms are used to characterize the Shapley value, the Equal Division rule, and the Equal Surplus Division rule. The classical axioms of Efficiency, Anonymity, Equal treatment of equals, Additivity and Linearity are not used.

Weighted component fairness for forest games

We study the set of allocation rules generated by component efficiency and weighted component fairness, a generalization of component fairness introduced by Herings et al. (2008). Firstly, if the underlying TU-game is superadditive, this set coincides with the core of a graph-restricted game associated with the forest game. Secondly, among this set, only the random tree solutions (Beal et al., 2010) induce Harsanyi payoff vectors for the associated graph-restricted game. We then obtain a new characterization of the random tree solutions in terms of component efficiency and weighted component fairness.

Preserving or removing special players: What keeps your payoff unchanged in TU-games?

If a player is removed from a game, what keeps the payoff of the remaining players unchanged? Is it the removal of a special player or its presence among the remaining players? This article answers this question in a complement study to Kamijo and Kongo (2012). We introduce axioms of invariance from player deletion in presence of a special player. In particular, if the special player is a nullifying player (resp. dummifying player), then the equal division value (resp. equal surplus division value) is characterized by the associated axiom of invariance plus efficiency and balanced cycle contributions. There is no type of special player from such a combination of axioms that characterizes the Shapley value.

Axiomatic characterizations under players nullification

Many axiomatic characterizations of values for cooperative games invoke axioms which evaluate the consequences of removing an arbitrary player. Balanced contributions (Myerson, 1980) and balanced cycle contributions (Kamijo and Kongo, 2010) are two well-known examples of such axioms. We revisit these characterizations by nullifying a player instead of deleting her/him from a game. The nullification (Beal et al., 2014a) of a player is obtained by transforming a game into a new one in which this player is a null player, i.e. the worth of the coalitions containing this player is now identical to that of the same coalition without this player. The degree with which our results are close to the original results in the literature is connected to the fact that the targeted value satisfies the null player out axiom (Derks and Haller, 1999).