Oligopoly & Game Theory Lecture 27 Dr. Jennifer P. Wissink ©2018 John M. Abowd and Jennifer P. Wissink, all rights reserved. May 7, 2018

Game Theory: Setup List of players: all the players are specified in advance. List of actions: all the actions each player can take are spelled out. Rules of play: who moves and when is spelled out. Information structure: who knows what and when is spelled out. Strategies: the set of actions players can use. Payoffs: the amount each player gets for every possible combination of the players’ strategies. Solution or equilibrium concept: a way you reason that players select strategies to play, and then consequently how you predict the outcome of the game.

Dominant Strategy Equilibrium A Dominant Strategy for player “i” is a strategy such that player’s i’s payoff from playing that strategy is at least as large as the payoff player i would get from playing any other strategy, no matter what player i’s rivals choose as their strategies. A Dominant Strategy Equilibrium of a game occurs when each player of the game has and plays his/her dominant strategy. Lots of interesting games have dominant strategy equilibriums. And for lots of interesting and fun games DON’T.

The Prisoners’ Dilemma Game & Dominant Strategy Equilibrium Roger and Chris have been accused of a major crime (which they committed). They also have outstanding warrants based on minor crimes, too. They are held in isolated cells and offered the choice to either Lie or Confess. The payoff matrix shows the number of years of prison Roger (the row player) and Chris (the column player) will receive depending upon who confesses and who lies as (Roger’s prison time, Chris’ prison time). The game is played one-shot, simultaneously and non-cooperatively with full information.

The Prisoners’ Dilemma Is Very Distressing..., Or Is It? In the Prisoners’ Dilemma game, the “superior” outcome is when both prisoners lie – but that requires cooperation. When the game is only played once, simultaneously and non-cooperatively, (confess, confess) is the dominant strategy equilibrium and (-5, -5) is the dominant strategy equilibrium outcome. Could Roger & Chris sustain the (lie, lie) outcome of (-1, -1) somehow? change the payoffs in the matrix play the game repeatedly Do all games have at least one dominant strategy equilibrium? NO! Then what?

Nash Equilibrium Named after John Nash - a Nobel Prize winner in Economics. Did you read or see A Beautiful Mind? A Nash Non-cooperative Strategy (Best Response) for player “i” is a strategy such that player’s i’s payoff from playing that strategy is at least as large as the payoff player i would get from playing any other strategy, given the strategies the others are playing. A Nash Non-cooperative Equilibrium is a set of (Nash) strategies for all players, such that, when played simultaneously, they have the property that no player can improve his payoff by playing a different strategy, given the strategies the others are playing.

Hotelling's Location Game (Nash Equilibrium in Location)

The Price Game Chris Low High Low 20, 20 60, 0 Roger High 0, 60 100, 100 i>clicker question: Are there any dominant strategy equilibria? Yes No. i>clicker question: Are there any Nash equilibria? Yes No. i>clicker question: Which Nash equilibrium do you think is more likely? Low, Low High, High.

Maximin Equilibrium Chris Low High Low 20, 20 60, 0 Roger High 0, 60 More proactive than reactive. Pessimistic? The “min” part of maximin: for each of his options the player determines his worst outcome(s). The “max” part of maximin: the player then looks over all his worst case scenarios for each strategy and picks the best of the worst. The equilibrium part of maximin: When each player has a maximin strategy and when played against each other they are a Nash equilibrium. So all maximin equilibria are Nash equilibria. Not all Nash equilibria are maximin equilibria. Chris Low High Low 20, 20 60, 0 Roger High 0, 60 100, 100

Consider the following game between Roger and Chris Consider the following game between Roger and Chris. It is played one shot, simultaneously and non-cooperatively. CHRIS Left Right ROGER Top 10, 2 22, 2 Middle 20, 20 10, 20 Bottom 10, 5 5, 20 i>clicker question: Are there any dominant strategy equilibria? Yes No i>clicker question: Are there any Nash equilibria? Yes No i>clicker question: Are there any maximin equilibria? Yes No

Another Game To Try Player 1 is the row player and can select numbers. 10, 20 30, 15 10, 15 16, 12 2 20, 10 30, 40 20, 25 8, 24 3 20, 20 12, 20 15, 10 25, 29 4 10, 10 15, 24 11, 27 Player 1 is the row player and can select numbers. Player 2 is the column player and can select letters. The payoffs are (Player 1, Player 2) Game is one-shot, simultaneous, non-cooperative, full information. Are there any… Dominant strategy equilibriums? Nash? Maximin?