Possibilistic Beliefs and Higher-Level Rationality

We consider rationality and rationalizability for normal-form games of incomplete information in which the players have possibilistic beliefs about their opponents. In this setting, we prove that the strategies compatible with the players being level-k rational coincide with the strategies surviving a natural k-step iterated elimination procedure. We view the latter strategies as the (level-k) rationalizable ones in our possibilistic setting

Possibilistic Beliefs and Higher-Level Rationality

We consider rationality and rationalizability for normal-form games of incomplete information in which the players have possibilistic beliefs about their opponents. In this setting, we prove that the strategies compatible with the players being level-k rational coincide with the strategies surviving a natural k-step iterated elimination procedure. We view the latter strategies as the (level-k) rationalizable ones in our possibilistic setting

Possibilistic Boolean games: strategic reasoning under incomplete information

Boolean games offer a compact alternative to normal-form games, by encoding the goal of each agent as a propositional formula. In this paper, we show how this framework can be naturally extended to model situations in which agents are uncertain about other agents' goals. We first use uncertainty measures from possibility theory to semantically define (solution concepts to) Boolean games with incomplete information. Then we present a syntactic characterization of these semantics, which can readily be implemented, and we characterize the computational complexity

Games with incomplete information: a framework based on possibility theory

Les jeux probabilistes à information incomplète, appelés jeux Bayesiens, offrent un cadre adapté au traitement de jeux à utilités cardinales sous incertitude. Ce type d'approche ne peut pas être utilisé dans des jeux ordinaux, où l'utilité capture un ordre de préférence, ni dans des situations de décision sous incertitude qualitative. Dans la première partie de cette thèse, nous proposons un modèle de jeux à information incomplète basé sur la théorie de l'utilité qualitative possibiliste: les jeux possibiliste à information incomplète, nommés PI-games. Ces jeux constituent un cadre approprié pour la représentation des jeux ordinaux sous connaissance incomplète. Nous étendons les notions fondamentales de stratégie de sécurité et d'équilibres de Nash (pur et mixte). De plus, nous montrons que tout jeu possibiliste à information incomplète peut être transformé en un jeu à information complète sous la forme normale équivalent au jeu initial, dont les stratégies de sécurité, les équilibres de Nash purs et mixtes sont en bijection dans les deux jeux. Ce résultat de représentation est une contrepartie qualitative de celui de Harsanyi sur la représentation des jeux Bayésiens par des jeux sous forme normale à information complète. Cela est plus un résultat de représentation qu'un outil de résolution. Nous montrons que décider si un équilibre de Nash pur existe dans un PI-game est un problème NP-complet et proposons un codage de programmation linéaire mixte en nombres entiers (PLNE) du problème. Nous proposons également un algorithme en temps polynomial pour trouver une stratégie de sécurité dans un PI-game et montrons qu'un équilibre mixte possibiliste peut être également calculé en temps polynomial (en fonction de la taille du jeu). Pour confirmer la faisabilité de la formulation de programmation linéaire en nombres entiers mixtes et des algorithmes en temps polynomial, nous introduisons aussi un nouveau générateur pour les PI-games basé sur le générateur de jeux sous la forme normale: GAMUT. Représenter un PI-game sous forme normale standard nécessite une expression extensive des fonctions d'utilité et de la distribution des possibilités, à savoir sur les espaces produits des actions et des types. La deuxième partie de cette thèse propose une vue moins coûteuse des PI-games, à savoir la polymatrix PI-games basée sur min, qui permet de spécifier de manière concise les PI-games avec des interactions locales, en d'autre termes, lorsque les interactions entre les joueurs sont par paires et l'utilité d'un joueur dépend de son voisinage et non de tous les autres joueurs du PI-game. Ce cadre permet, par exemple, la représentation compacte des jeux de coordination sous incertitude où la satisfaction d'un joueur est élevée si et seulement si sa stratégie est cohérente avec celles de l'ensemble de ses voisins. Dans cette thèse, nous montrons que n'importe quel PI-game à 2 joueurs peut être transformé en un jeu polymatriciel équivalent basé sur le min. Ce résultat est la contrepartie qualitative du théorème de Howson et Rosenthal reliant les jeux Bayésiens aux jeux polymatriciels. De plus, dès qu'une simple condition de cohérence des connaissances des joueurs sur le monde est satisfaite, tout polymatrix PI-game peut être transformé en temps polynomial en un jeu polymatriciel, basé sur le min, à information complète équivalent. Nous montrons que l'existence d'un équilibre de Nash pur dans un polymatrix PI-game est un problème NP-complet mais pas plus difficile que de décider si un équilibre de Nash pur existe dans un PI-game. Enfin, nous montrons que cette dernière famille de jeux peut être résolue grâce à une formulation de programmation linéaire en nombres entiers mixtes. Nous introduisons un nouveau générateur pour les polymatrix PI-games basés sur le générateur de PI-game. Les expérimentations confirment la faisabilité de cette approche.Probabilistic games with incomplete information, called Bayesian games, offer a suitable framework for games where the utility degrees are additive in essence. This approach does not apply to ordinal games where the utility degrees capture no more than a ranking, nor to situations of decision under qualitative uncertainty. In the first part of this thesis, we propose a representation framework for ordinal games under possibilistic incomplete information (PI-games). These games constitute a suitable framework for the representation of ordinal games under incomplete knowledge. We extend the fundamental notions of secure strategy, pure Nash equilibrium, and mixed Nash equilibrium to this framework. Furthermore, we show that any possibilis- tic game with incomplete information can be transformed into an equivalent normal form game with complete information. The fundamental notions such Nash equilibria (pure and mixed) and secure strategies are in bijection in both frameworks. This representation result is a qualitative counterpart of Harsanyi results about the representation of Bayesian games by normal form games under complete information. It is more of a representation result than the premise of a solving tool. We show that deciding whether a pure Nash equilibrium exists in a PI-game is a difficult task (NP-hard) and propose a Mixed Integer Linear Programming (MILP) encoding of this problem. We also propose a polynomial-time algorithm to find a secure strategy in a PI-game and show that a possibilistic mixed equilibrium can be computed in polynomial time (w.r.t., the size of the game), which contrasts with probabilistic mixed equilibrium computation in cardinal game theory. To confirm the feasibility of the MILP formulation and the polynomial-time algorithms, we introduce a novel generator for PI-games based on the well-known standard normal form game generator: GAMUT. Representing a PI-game in standard normal form requires an extensive expression of the utility functions and the possibility distribution on the product spaces of actions and types. This is the concern of the second part of this thesis where we propose a less costly view of PI-games, namely min-based polymatrix PI-games, which allows to concisely specify PI-games with local interactions, i.e., the interactions between players are pairwise and the utility of a player depends on her neighbors and not on all other players in the PI-game. This framework allows, for instance, the compact representation of coordination games under uncertainty where the satisfaction of a player is high if and only if her strategy is coherent with all of her neighbors, the game being possibly only incompletely known to the players. We show that any 2- player PI-game can be transformed into an equivalent min-based polymatrix game. This result is the qualitative counterpart of Howson and Rosenthal's theorem linking Bayesian games to polymatrix games. Furthermore, as soon as a simple condition on the coherence of the players' knowledge about the world is satisfied, any polymatrix PI-game can be transformed in polynomial time into an equivalent min-based and complete information polymatrix game. We show that the existence of a pure Nash equilibrium in a polymatrix PI-game is an NP-complete problem but no harder than deciding the existence of a pure Nash equilibrium in a PI-game. Finally, we show that the latter family of games can be solved through a MILP formulation. We introduce a novel generator for min-based polymatrix PI-games based on the PI-game generator. Experiments confirm the feasibility of this approach

Epistemic Implementation and The Arbitrary-Belief Auction

In settings of incomplete information we put forward an epistemic framework for designing mechanisms that successfully leverage the players' arbitrary higher-order beliefs, even when such beliefs are totally wrong, and even when the players are rational in a very weak sense. Following Aumann (1995), we consider a player i rational if he uses a pure strategy s_i such that no alternative pure strategy s_i' performs better than s_i in every world i considers possible, and consider him order-k rational if he is rational and believes that all other players are order-(k-1) rational. We then introduce an iterative deletion procedure of dominated strategies and use it to precisely characterize the strategies consistent with the players being order-k rational. We exemplify the power of our framework in single-good auctions by introducing and achieving a new class of revenue benchmarks, defined over the players' arbitrary beliefs, that can be much higher than classical ones, and are unattainable by traditional mechanisms. Namely, we exhibit a mechanism that, for every k greater than or equal to 0 and epsilon>0 and whenever the players are order-(k+1) rational, guarantees revenue greater than or equal to G^k-epsilon, where G^k is the second highest belief about belief about ... (k times) about the highest valuation of some player, even when such a player's identity is not precisely known. Importantly, our mechanism is possibilistic interim individually rational. Essentially this means that, based on his beliefs, a player's utility is non-negative not in expectation, but in each world he believes possible. We finally show that our benchmark G^k is so demanding that it separates the revenue achievable with order-k rational players from that achievable with order-(k+1) rational ones. That is, no possibilistic interim individually rational mechanism can guarantee revenue greater than or equal to G^k-c, for any constant c>0, when the players are only order-k rational

Contextual and Possibilistic Reasoning for Coalition Formation

In multiagent systems, agents often have to rely on other agents to reach their goals, for example when they lack a needed resource or do not have the capability to perform a required action. Agents therefore need to cooperate. Then, some of the questions raised are: Which agent(s) to cooperate with? What are the potential coalitions in which agents can achieve their goals? As the number of possibilities is potentially quite large, how to automate the process? And then, how to select the most appropriate coalition, taking into account the uncertainty in the agents' abilities to carry out certain tasks? In this article, we address the question of how to find and evaluate coalitions among agents in multiagent systems using MCS tools, while taking into consideration the uncertainty around the agents' actions. Our methodology is the following: We first compute the solution space for the formation of coalitions using a contextual reasoning approach. Second, we model agents as contexts in Multi-Context Systems (MCS), and dependence relations among agents seeking to achieve their goals, as bridge rules. Third, we systematically compute all potential coalitions using algorithms for MCS equilibria, and given a set of functional and non-functional requirements, we propose ways to select the best solutions. Finally, in order to handle the uncertainty in the agents' actions, we extend our approach with features of possibilistic reasoning. We illustrate our approach with an example from robotics

Egalitarian Collective Decision Making under Qualitative Possibilistic Uncertainty: Principles and Characterization

International audienceThis paper raises the question of collective decision making under possibilistic uncertainty; We study four egalitarian decision rules and show that in the context of a possibilistic representation of uncertainty, the use of an egalitarian collective utility function allows to get rid of the Timing Effect. Making a step further, we prove that if both the agents’ preferences and the collective ranking of the decisions satisfy Dubois and Prade’s axioms (1995), and particularly risk aversion, and Pareto Unanimity, then the egalitarian collective aggregation is compulsory. This result can be seen as an ordinal counterpart of Harsanyi’s theorem (1955)

Multi-Unit Auction Revenue with Possibilistic Beliefs

The revenue of traditional auction mechanisms is benchmarked solely against the players' own valuations, despite the fact that they may also have valuable beliefs about each other's valuations. Not much is known about generating revenue in auctions of multiple identical copies of a same good. (In particular the celebrated Vickrey mechanism has no revenue guarantees.) For such auctions, we (1) put forward an attractive revenue benchmark, based on the players' possibilistic about each other, and (2) construct a mechanism that achieves such benchmark, assuming that the players are two-level rational (where the rationality is in the sense of Aumann)