Every atom in universe exists in equilibrium of different physical forces. Everyone in society lives in equilibrium of their work and family, of their enthusiasm and reality. Equilibrium is one of the main notions in Game Theory — the domain studies behaviors of entities according to their own interests. The conflicting interests, the lack of coordination and regulation may not lead a game to a pure equilibrium; even if pure equilibria exist, the result of local optimization of rational players in general does not have any type of global property. Our contributions in equilibria are twofold: we study the existence of a pure Nash equilibrium and analyze the inefficiency of equilibria in games. To prove the existence of equilibria, we use extensively the potential argument in which we figure out potential functions according to different dynamics in games. Moreover, for games which does not necessarily admit a pure equilibrium, we present an useful technique in settling the complexity of deciding whether the games possess an equilibrium. To quantify the loss caused by selfish behaviors regarding to a social objective function of a game, we study the two well-known measures: the price of anarchy which is defined as the worst-case ratio between the social objective value of an equilibrium and the optimum
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