Game theory basics A Game describes situations of strategic interaction, where the payoff for one agent depends on its own actions as well as on the actions of other agents. Example. In an industry with two firms, each firm’s profit depends not only on its own price but also on the price charged by the other firm.

Game theory basics Ingredients of a simple game (with complete information): The players. The actions. feasible courses of actions for each player The rules. who moves and when. The payoffs. the utility each player gets for every possible combination of the players’ actions

Strategy sets for each player Game A Strategy sets for each player Players This form of representing games in a matrix is known as the “normal form”. Later we will consider different forms for representing games. Payoff to Row Payoff to Column

Dominant strategies A strategy s'i for player i is dominant if, regardless of the other players’ strategies, player i’s payoff from playing s'i is larger than his payoff from playing any other strategy A rational player will choose his dominant strategy (if there is one!) Dominant strategies in Game A (Red for player Row and Red for Player Column) give us a unique pair of strategies (Red, Red)

Game B: The Prisoners dilemma

Prisoner’s dilemma Game B (Prisoners dilemma): same logic as in Game A with different payoffs Players have a dominant strategy Conflict between individual goals and social goals

Game C: no Dominant strategies

Dominated strategies A strategy s'i for player i is dominated by another strategy s''i if for each feasible combination of the other players’ strategies, player i’s payoff from playing s'i is smaller than his payoff from playing s''i. A rational player will not choose a dominated strategy So, we can eliminate dominated strategies in a successive manner. Game C: successive (iterated) elimination of dominated strategies leaves a unique pair of strategies (B,R).

Game D: no dominant or dominated Strategies

Nash Equilibrium For games A, B, and C, we can use dominant strategies and iterated elimination of dominated strategies to find unique solutions (we could say something definite about how players will choose): these are the equilibrium of these Games! For Game D, we need the concept of Nash Equilibrium

Nash Equilibrium (NE) A pair of strategies is a Nash Equilibrium if no player can improve his payoff by unilaterally changing his strategy. A pair of strategies is a Nash Equilibrium, if each player’s strategy is a best response to the other player’s strategy Game D: Nash Eq. leads to a unique pair of strategies (B,R)

Game E: The battle of the sexes Game E has two Nash Equilibria (multiple equilibria) Existence of Nash Equilibrium is guaranteed: if the number of players is finite and the number of available strategies for every player is also finite, then there exists at least one equilibrium.