Noncooperative Games and Applications

A basic problem in optimization theory is to find the maximum value of a function φ over a set X: max x∈X φ(x). (1) One can think of X as the set of possible choices available to an individual, while φ(x) is his corresponding payoff. Game theory, on the other hand, is concerned with the more complex situation where two or more individuals, or " players " are present [13]. Each player can choose among his set of available options and seeks to maximize his own payoff. For simplicity, consider the case of two players. Player 1 chooses a strategy x 1 ∈ X 1 and seeks to maximize φ 1 (x 1 , x 2). Player 2 chooses a strategy x 2 ∈ X 2 and seeks to maximize φ 2 (x 1 , x 2). (2) Here the catch is that the payoff of each player also depends on the actions of the other player. This is indeed what happens in most economics models, where the profit of one agent also depends on the decisions of the other agents. In contrast with (1), it is clear that the problem (2) does not admit an " optimal " solution. Indeed, in general it will not be possible to find a couple (¯ x 1 , ¯ x 2) ∈ X 1 × X 2 which at the same time maximizes the payoff of the first player and of the second player, so that φ 2 (x 1 , x 2). For this reason, various alternative concepts of solutions have been proposed in the literature. These can be relevant in different situations, depending on the information available to the players and their ability to cooperate. For example, if the players have no means to talk to each other and do not cooperate, then an appropriate concept of solution is the Nash equilibrium [12], defined as a fixed point of the best reply map. In other words, (x * 1 , x * 2) is a Nash equilibrium if (i) the value x * 1 ∈ X 1 is the best choice for the first player, in reply to the strategy x * 2 adopted by the second player. Namely