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

Optimization Models Using Fuzzy Sets and Possibility Theory

Optimization is of central concern to a number of disciplines. Operations Research and Decision Theory are often considered to be identical with optimization. But also in other areas such as engineering design, regional policy, logistics and many others, the search for optimal solutions is one of the prime goals. The methods and models which have been used over the last decades in these areas have primarily been "hard" or "crisp", i.e. the solutions were considered to be either feasible or unfeasible, either above a certain aspiration level or below. This dichotomous structure of methods very often forced the modeler to approximate real problem situations of the more-or-less type by yes-or-no-type models, the solutions of which might turn out not to be the solutions to the real problems. This is particularly true if the problem under consideration includes vaguely defined relationships, human evaluations, uncertainty due to inconsistent or incomplete evidence, if natural language has to be modeled or if state variables can only be described approximately. Until recently, everything which was not known with certainty, i.e. which was not known to be either true or false or which was not known to either happen with certainty or to be impossible to occur, was modeled by means of probabilities. This holds in particular for uncertainties concerning the occurrence of events. probability theory was used irrespective of whether its axioms (such as, for instance, the law of large numbers) were satisfied or not, or whether the "events" could really be described unequivocally and crisply. In the meantime one has become aware of the fact that uncertainties concerning the occurrence as well as concerning the description of events ought to be modeled in a much more differentiated way. New concepts and theories have been developed to do this: the theory of evidence, possibility theory, the theory of fuzzy sets have been advanced to a stage of remarkable maturity and have already been applied successfully in numerous cases and in many areas. Unluckily, the progress in these areas has been so fast in the last years that it has not been documented in a way which makes these results easily accessible and understandable for newcomers to these areas: text-books have not been able to keep up with the speed of new developments; edited volumes have been published which are very useful for specialists in these areas, but which are of very little use to nonspecialists because they assume too much of a background in fuzzy set theory. To a certain degree the same is true for the existing professional journals in the area of fuzzy set theory. Altogether this volume is a very important and appreciable contribution to the literature on fuzzy set theory

Operational Decision Making under Uncertainty: Inferential, Sequential, and Adversarial Approaches

Modern security threats are characterized by a stochastic, dynamic, partially observable, and ambiguous operational environment. This dissertation addresses such complex security threats using operations research techniques for decision making under uncertainty in operations planning, analysis, and assessment. First, this research develops a new method for robust queue inference with partially observable, stochastic arrival and departure times, motivated by cybersecurity and terrorism applications. In the dynamic setting, this work develops a new variant of Markov decision processes and an algorithm for robust information collection in dynamic, partially observable and ambiguous environments, with an application to a cybersecurity detection problem. In the adversarial setting, this work presents a new application of counterfactual regret minimization and robust optimization to a multi-domain cyber and air defense problem in a partially observable environment

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

B

Algorithms for Scheduling Problems

This edited book presents new results in the area of algorithm development for different types of scheduling problems. In eleven chapters, algorithms for single machine problems, flow-shop and job-shop scheduling problems (including their hybrid (flexible) variants), the resource-constrained project scheduling problem, scheduling problems in complex manufacturing systems and supply chains, and workflow scheduling problems are given. The chapters address such subjects as insertion heuristics for energy-efficient scheduling, the re-scheduling of train traffic in real time, control algorithms for short-term scheduling in manufacturing systems, bi-objective optimization of tortilla production, scheduling problems with uncertain (interval) processing times, workflow scheduling for digital signal processor (DSP) clusters, and many more