A Brief Introduction to the Basics of Game Theory Paper by Matthew O. Jackson Presented by Wei Lu Feb 2018 

What’s the game, dawg? - Players - Actions - Timing - Simultaneous or sequential - Order of action - Repeated? - Payoffs/Utilities - Cost - Benefit 

Normal Form Game aka. Strategic Form Game 

Definitions, Yo! - Pure strategy: single action chosen - Mixed strategy: a randomization over actions - Dominant strategy: a strategy that produces the highest payoff of any strategy available for every possible action by the other players. - Strictly dominant strategy: > than every other alternative 

Nash Equilibrium A pure strategy Nash equilibrium is a profile of strategies such that each player’s strategy is a best response (results in the highest available payoff) against the equilibrium strategies of the other players. 

Confused much? Dominant Strategy of player i, for all a’ i and all a -i ∈ a -i Best Response of player i, for all a’ i to a profile of strategies for the other players, Nash Equilibrium is a profile of strategies a ∈ A. for each i, a i is a best reply to a −i 

Nash Equilibrium - Stable: no incentive for anyone to deviate from his/her current action - Multiple equilibria possible 

Nash Equilibrium 

Nash Equilibrium 

Nash Equilibrium - Stable: no incentive for anyone to deviate from his/her current action - Multiple equilibria possible - No equilibrium possible? 

Nash Equilibrium 

Mixed-strategy Nash Equilibrium A profile of mixed strategies (s 1 , . . . , s n ) for or all i and a′ i , where s i (a i ) is the probability that a i is chosen. 

Mixed-strategy Nash Equilibrium 

Extensive Form Game Games in extensive form include a complete description of - who moves in what order and - what they have observed when they move 

Extensive Form Game 

Finite Games of Perfect Information Let firm 1 choose first 

Non-credible Threats 

Backward Induction 

Subgame Perfect Equilibrium Subgame perfection requires that the stated strategies constitute a Nash equilibrium in every subgame. 

Minimax Theorem (von Neumann, 1928) In any finite, two-player, zero-sum game, in any Nash equilibrium each player receives a payoff that is equal to both his maxmin value and his minmax value. - The maxmin strategy for player i is arg max si min s−i u i (s 1 , s 2 ), and - The maxmin value for player i is max si min s-i u i (s 1 , s 2 ). - In a two-player game, the minmax strategy for player i against player −i is: arg min si max s−i u −i (s i , s −i ), and - player −i’s minmax value is min si max s−i u −i (s i , s −i ) 

What’s the point of all these?