Game Theory
Nash Equilibrium
A Nash equilibrium is a set of strategies, one for each player, such that no player has incentive to change his or her strategy Nash equilibrium are inherently stable A Nash equilibrium is a law that no one would want to break even in the absence of an effective police force - Pretend the police does not exist -The government passes a law
Best response
Given what all other players are doing, a strategy is a best response if and only if a player cannot gain more utility from switching to a different strategy A game is in a Nash equilibrium if and only if all players are playing best responses to what the other players are doing
Mixed Strategy Nash Equilibrium
If no Equilibrium exists in pure strategies, one must exist in mixed strategies A mixed strategy is a probability distribution over two or more pure strategies That is the players choose randomly among their options in equilibrium If mixtures are mutual best responses, the set of strategies is a mixed strategy Nash equilibrium
Strict Dominance and Mixed Strategies
If you find a mixture between two strategies strictly dominated another strategy, eliminate that last strategy immediately Strictly dominated strategies are irrational , whether pure strategies or mixed strategies dominate them
Extensive Form Games
Players take turns moving Items covered: Backward induction, sub game perfect equilibrium, credible threats, tying hands, commitment problems, forward induction
Strict Dominance
Strategy x strictly dominates strategy y for a player if strategy x generates a greater payoff than strategy y regardless of what other players do Rational players never play strictly dominated actions
Game Theory
Study of strategically interdependent behavior : What I do affect your outcome and what you do affect my outcome
Nash's Theorem
There must be at least one Nash Equilibrium for all finite games
Weak dominance
Weak dominance means that whatever other players are doing, player can get always higher or the same payoff for that action the player is playing A player will never play a strictly dominated action, however she may play a weakly dominated action
Simultaneous move games
games where players have to come out with their strategies independently without observing what the other guy does Items covered: Strict Dominance , iterated elimination of strictly dominated strategies, pure strategy Nash equilibrium, mixed strategy Nash equilibrium, weak dominance