EXAM (MT)

• A Nash equilibrium is a strategy profile where all players are best responding to each other

• From the normal form, we can see the payoffs corresponding to best responses are underlined. (AB,XY) is a strategy profile, where both players are playing best responses, hence it is a Nash-equilibrium

• For a strategy mixing between A and B to be a best response, player 1 must be indifferent between a pure strategy A and pure strategy B

When decision nodes of the same player belong to the same information set, this player is unable to tell which node is the realized history

A strategy is a complete contingent plan of action. A strategy for a player specifies the action that the player takes at each of the player’s information sets

Contains all the possible strategies of that player

A strategy profile assigns a strategy to every player

A belief is a probability distribution over the strategies played by the other player

A mixed strategy for player i is a probability distribution over his/her strategy set Si

• It is optimal for a decision maker to randomize between two options only when he/she is indifferent. Because of indifference, any randomization has the same expected utility

• For a strategy mixing between A and B to be a best response, player 1 must be indifferent between a pure strategy A and pure strategy B

• The set of mixed best responses contains mixed strategies that achieve an expected payoff greater than or equal to the expected payoff of any other mixed strategy

• Mixed strategy best responses assign positive probability only to pure strategies that are best responses (for some belief)

• In a Nash-equilibrium each player chooses a best response to the strategies of the other players

• Players have correct beliefs about the strategies of the other players and choose optimally against these beliefs

• Given the belief, each player chooses a complete contingent plan of action that is a best response

• No player has an incentive to deviate unilaterally

• Pure or mixed strategy of player i that maximize player i’s expected payoff given the belief about the other players’ strategies

• A strategy of player i is dominated if there is a mixed strategy that always yields a higher payoff, irrespective of the pure strategies of the opposing players

• A dominated strategy is never a best response, regardless of a players’ belief

• A strategy profile is pareto efficient if we cannot find another strategy profile such that (i) No player is worse off, (ii) some players are better off

• If we maximize aggregated utility, the resulting strategy profile is necessarily efficient

• Small number of firms competing by choosing prices independently and simultaneously (homogeneous good)

• Consumers differ in their preferences over similar products, even if they have similar prices (phones, toothpaste, etc.)

• Modeled by having two prices in the demand function

• Firms choose quantities independently and simultaneously and compete over quantities

• Products differ in the perceived quality. If all products had the same price, consumers would purchase the one with the highest quality

• A mixed strategy is a best reply if and only if it randomizes over pure strategies that are best replies

• It is optimal for player 1 to randomize between L and R if and only if they yield the same expected payoff

• A strategy for a player is sequentially rational if at each of his/her information sets it is a best response to the strategies of the other players for the remainder of the game, regardless of whether the player believes that the information set will actually be reached

A subgame perfect equilibrium is a strategy profile that specifies a Nash-equilibrium in every subgame of the original game

• A subgame originates at node x if no successor of x shares an information set with a node that is not a successor of x. A subgame consists of a node and all its successors and inherits the players and the information sets of the original game