Weighted voting games are classic cooperative games which provide compact representation for coalition formation models in human societies and multiagent systems. As useful as weighted voting games are in modeling cooperation among players, they are, however, not immune from the vulnerability of manipulations (i.e., dishonest behaviors) by strategic players that may be present in the games. With the possibility of manipulations, it becomes difficult to establish or maintain trust, and, more importantly, it becomes difficult to assure fairness in such games. For these reasons, we conduct careful experimental investigations and analyses of the effects of manipulations in weighted voting games, including those of manipulation by splitting, merging, and annexation . These manipulations involve an agent or some agents misrepresenting their identities in anticipation of gaining more power or obtaining a higher portion of a coalition's profits at the expense of other agents in a game. We consider investigation of some criteria for the evaluation of game's robustness to manipulation. These criteria have been defined on the basis of theoretical and experimental analysis. For manipulation by splitting, we provide empirical evidence to show that the three prominent indices for measuring agents' power, Shapley-Shubik, Banzhaf, and Deegan-Packel, are all susceptible to manipulation when an agent splits into several false identities. We extend a previous result on manipulation by splitting in exact unanimity weighted voting games to the Deegan-Packel index, and present new results for excess unanimity weighted voting games. We partially resolve an important open problem concerning the bounds on the extent of power that a manipulator may gain when it splits into several false identities in non-unanimity weighted voting games. Specifically, we provide the first three non-trivial bounds for this problem using the Shapley-Shubik and Banzhaf indices. One of the bounds is also shown to be asymptotically tight. Furthermore, experiments on non-unanimity weighted voting games show that the three indices are highly susceptible to manipulation via annexation while they are less susceptible to manipulation via merging. Given that the problems of calculating the Shapley-Shubik and Banzhaf indices for weighted voting games are NP-complete, we show that, when the manipulators' coalitions sizes are restricted to a small constant, manipulators need to do only a polynomial amount of work to find a much improved power gain for both merging and annexation, and then present two enumeration-based pseudo-polynomial algorithms that manipulators can use. Finally, we argue and provide empirical evidence to show that despite finding the optimal beneficial merge is an NP-hard problem for both the Shapley-Shubik and Banzhaf indices, finding beneficial merge is relatively easy in practice. Also, while it appears that we may be powerless to stop manipulation by merging for a given game, we suggest a measure, termed quota ratio, that the game designer may be able to control. Thus, we deduce that a high quota ratio decreases the number of beneficial merges.

Theory of multicriteria games is a special field of game theory, when one or more players have at least two payoff functions and want to maximize simultaneously. The work introduces a number of new findings. It examined the concept of finding equilibria in pure strategies in noncooperative multicriteria game. It is possible to find all the equilibria in pure strategies by full search and solving two linear programs for each point. Furthermore, two linear programs are formulated for verifying that a selected point is the equilibrium of the game or not. In the noncooperative games is also introduced the concept that with knowledge of the equilibrium of bimatrix game determines preferences of the players. Although finding the equilibrium point of the bimatrix game is nonlinear problem, finding the preferences is linear problem. The latest findings in the noncooperative games is a generalization of the concept that solves multicriteria game by assigning weights to each criterion of each player. The work demonstrates that it may not be necessarily linear weights, but it can be more general function that describes the player's preference. The remaining part is devoted to knowledge in cooperative games. There is considered that the players know their preferences and are able to express them by weights. The game with known preferences is defined and solved with the use of bargaining theory. Then it is generalized to a case where players have more payoff functions, from which they can choose. Finally, the multicriteria case of voting game is defined. It is designed completely new concept, which selects the winning coalition in the voting game. This concept is then applied to the real situation after the elections to the Chamber of Deputies in 2013.

Ce travail structuré en deux parties, porte sur l'étude des interactions entre des agents. Nous nous intéressons à la représentation conceptuelle du cadre et de ses règles de fonctionnement, à la mesure du pouvoir ou capacité des agents à influencer l'aboutissement des interactions et à l'analyse des aboutissements qui peuvent être considérés comme meilleurs.Dans la première partie, nous considérons des agents qui visent un objectif commun dépendant des facteurs distincts. Par exemple, dans un jeu avec abstention (Felsenthal et Machover (1997)) ou plus généralement dans les (j,k)-jeux, de Freixas et Zwicker (2003), des votes de natures différentes peuvent compter favorablement au résultat. Comme modèle, nous développons les jeux multi-types dans lesquels chacun tient un rôle précis dans un groupe. Ce modèle est proche de celui de Bolger (1986) où les rôles ne sont pas comparables. Dans ce cadre, nous proposons des extensions des indices de Shapley-Shubik (1954) et Banzhaf (1965). En prenant en compte une structure de coalitions sur l'ensemble des agents, nous reprenons l'étude avec les indices d'Owen-Shapley (1977) et Owen-Banzhaf (1981).Dans la deuxième partie, nous utilisons les positions des joueurs dans l'espace multidimensionnel pour modéliser des liens entre eux. Les indices d'Owen (1971) et de Shapley (1977) s'appliquent à ce cadre. Nous montrons que le second généralise le premier puis, nous les généralisons. Le cœur (Plott (1967)), le Yolk (Miller (1980), McKelvey (1986)) et le Finagle (Wuffle et al. (1989)) sont trois concepts de solution spatiale. Le Yolk est une région hypersphérique dont le centre est souvent supposé unique (Scott et Grofman (1988), Tovey (1992)). Nous le généralisons et nous montrons que son unicité n'est vraie que dans un espace bidimensionnel. En admettant qu'on peut se tromper sur la localisation spatiale, nous proposons une généralisation du cœur similaire à celle proposée par Bräuninger (2007), des études comparatives avec le Yolk et le Finagle sont faites. / This work structured into two parts, focuses on the study of interactions among agents. We are interested in the conceptual framework and its operating rules, the measurement of power or ability of agents to influence the outcome of interactions and analysis of outcomes which can be considered to be the best.In the first part, we consider that there is a set of agents who have a common objective which depends on different factors. For example, in a game with abstention (Felsenthal et Machover (1997)) or more generally in the (j,k)-games of Freixas and Zwicker (2003), the votes of different natures can contribute positively to the result. We use the model of multi-types games in which each agent has a specific role in a group. This model is similar to that of Bolger (1986) in which the roles are not comparable. In this context, we extend the Shapley-Shubik (1954) and Banzhaf (1981) power indices. We reconsider the multi-types games with the Owen-Shapley (1977) and Owen-Banzhaf (1965) power indices through the lens of a coalition structures on the set of agents.In the second part, we use the player positions in a multidimensional space to model the links among them. The Owen (1971) and Shapley (1977) power indices are developed in this framework. We show that the second generalizes the first and we extend them. The core (Plott (1967)), the Yolk (Miller (1980), McKelvey (1986)) and the Finagle (Wuffle et al. (1989)) are three concepts of spatial solution. The Yolk is an hyperspherical region whose center is often assumed unique (Tovey (1992), Scott et Grofman (1988)). We generalize this concept and show that uniqueness is only true on the bidimensional space. We consider the situation in which the social planner has a partial knowledge on the spatial location of agents and propose a generalization of the core similar to one of Bräuninger (2007). Comparative studies with the olk and the Finagle are made.