Noncooperative Games

Figure 1 Strategic form game

Figure 2 Extensive form games and imperfect information

Figure 3 Extensive form game with incomplete information

To solve noncooperative games, we first need to specify the relevant players, the set of actions available to them, and the payoffs they would receive from all possible outcomes. In other words, we have to decide whose actions are relevant to explain a given event, what actions these actors can take to influence the outcome, and how they rank the possible outcomes. Figure 1 (p. 450) gives an example of a strategic form game with 2 players (Row, Column) who can use two actions (cooperate, defect).

The numbers in the cells denote the payoff each actor would get from this combination of actions. By convention, the payoffs of the row player are given first. If both actors choose to cooperate, Row would get a payoff of 3 in this example. The higher the payoff, the more utility an actor derives from an outcome. Thus, Row would prefer the situation in which he or she defects but Column cooperates with the situation of mutual cooperation because the former yields a higher payoff (4 > 3). For extensive form games, we also have [Page 452]to decide on the order in which actors have to move and their level of information at each stage. This information is commonly represented in a game tree.

Figure 2a (p. 450) provides an example of an extensive form game tree in which Row moves before Column and Column knows about Row's move. By convention, payoffs are listed in the order of moves. Uncertainty about previous moves by other players can be incorporated in the extensive form as well. Figure 2b (p. 450) provides an example of an extensive form game with imperfect information. The dashed line between Column's decision nodes depicts Column's uncertainty with regard to Row's previous move. Uncertainty about the payoffs (incomplete information) can be modeled by introducing a move by a neutral instance (Nature) that determines which game or subgame is being played. In Figure 3 (p. 451), Row plays the game of Figure 2a with probability p (upper branch) and against a column player who always prefers to cooperate with probability 1 - p (lower branch).

To solve a game we need to specify a principle which guides the choices of the players in this strategic setting. Most noncooperative games use the Nash equilibrium concept (and refinements thereof): An outcome is stable (in equilibrium) if no actor has an incentive to deviate from it unilaterally. In Figure 1 (p. 450), mutual defection is the Nash equilibrium. Both players would receive a lower payoff were they to change their actions individually. Because binding agreements are, by definition, not feasible, mutual cooperation cannot be reached by a coordinated move. Backward induction is a widely used solution concept for extensive form games.

...