Non-Cooperative Game Theory To define a game, you need to know three things: –The set of players –The strategy sets of the players (i.e., the actions they can take) –The payoff functions

Game A Players Strategy sets for each player Payoffs for each player, for each possible outcome Payoff to RowPayoff to Column

Game B

What happened in these two games? –What were the strategies? –What were the outcomes? Why did we get these outcomes? Should we have expected these outcomes? In other words -- How do we solve these games?

Solving Games We are looking for the equilibrium. What is equilibrium? –Equilibrium is a strategy combination where no one player has an incentive to change her strategy given the strategies of the other players. Huh?

Game A

Nash Equilibrium (NE) Formally, a set of strategies forms a NE if, for every player i,  i (s i, s -i )   i (s i*, s -i ). Note that the equilibrium is defined in terms of strategies, not payoffs. Why is this a solution? Because it’s a rest point - no incentive for one player to change unilaterally.

How Do We Find NE? Elimination of Dominated Strategies. A player has a dominated strategy if there is one action/strategy which always provides a lower payoff than another strategy, no matter what other players do. If you cross off all dominated strategies, sometimes you are left with only NE.

Game A

Repeated elimination can find the NE

Elimination of dominated strategies only works if the strategies are strictly dominated –Always worse, not just equal to or worse

Sometimes there aren’t dominated strategies so you have to check for NE cell by cell

Sometimes there aren’t any NE

We can use the “Normal” or matrix form if: –There are only 2 (sometimes 3) players –There are a finite number of strategies –Actions approximately simultaneous If actions are sequential, must use another form, the “Extensive” form: –Still only really feasible for 2 or 3 players, although can accommodate “chance” –Still must have finite number of strategies

Extensive Form Games Use a game “tree” to depict the order in which players make decisions and the choices that they have at each decision point. Decision points are called “nodes”. Players’ strategies or choices branch off from each decision node. At the end of each branch on the game tree are the payoffs the players would receive if that branch were the path followed.

US vs. Saudi Arabia Oil “Game” Quota Tariff Nothing R R R N N N 90,80 100,60 75,50 100,60 40,80 50,100 US Saudi Arabia

Solving Extensive Form Games Nash Equilibrium has the same meaning in extensive form games as in normal form games. There is also another solution concept in extensive form games, the Subgame Perfect Equilibrium (SPE) strategy which has some advantages over Nash Equilibrium.

US vs. Saudi Arabia Oil “Game” Quota Tariff Nothing R R R N N N 90,80 100,60 75,50 100,60 40,80 50,100 US Saudi Arabia Subgame = part of larger game that can stand alone as a game itself.

Sub-Game Perfect Equilibrium A subgame can be defined for any node other than a terminal (payoff) node, and includes all of the subsequent “branches” of the tree that emanate from that node. For a strategy to be a Subgame Perfect Equilibrium (SPE) strategy, it can only contain actions that are optimal for their respective subgames.

Quota Tariff Nothing R R R N N N 90,80 100,60 75,50 100,60 40,80 50,100 US Saudi Arabia

Quota Tariff Nothing R R R N N N 90,80 100,60 75,50 100,60 40,80 50,100 US Saudi Arabia To find all of the Subgame Perfect Equilibria: For each subgame, determine the optimal strategy. Find the optimal strategy for the “pruned” tree.

Quota Tariff Nothing R R R N N N 90,80 100,60 75,50 100,60 40,80 50,100 US Saudi Arabia Compare Subgame Perfect Equilibria (SPE) to NE: NE can include incredible threats, along as unilateral changes are not optimal. Example:Quota; R if Quota or Tariff, N if Nothing

Another Example Enter Stay Out High P Low P High P Low P 2,2 -1,0 0,5 0,0 Entrant Incumbent Find optimal strategy for each subgame (prune the tree). Find Entrant’s optimal action.

Repeated Games In repeated games, strategies are much richer. In a one-shot Prisoner’s Dilemma game, players can either cooperate or defect. In a repeated game, players choose whether to cooperate or defect each period. Players can have strategies that are contingent on the other player's actions. –Cooperate if the other player cooperated last period. –Defect if the other player has ever defected. Note: In repeated games, must discount future payoffs. – (1/(1+r)) t =  t is the discount factor for period t.

Solving Repeated Games If the game has a finite horizon (that is, it ends after a specified number of rounds), you use backwards induction. –Start by finding the optimal strategy in the last period –Move to the next to the last period, and find the optimal strategy, recognizing the effects on the final round. If the game has an infinite horizon, you can't use backwards induction because there is no last period. To solve infinite horizon games, you check different strategies to see if they meet the requirements of equilibrium. –For each player, changing strategies unilaterally will not make the player better off.

Solving Repeated Prisoner’s Dilemma Games For finite horizon repeated PD, use backwards induction. –In the last period, always optimal to defect –If your action in the next-to-the-last period does not affect the optimal strategy in the last period, you do better by defecting in the next to the last period –And so on…. For finite horizon repeated PD, collusion is never optimal.

Solving Repeated Prisoner’s Dilemma Games For infinite horizon repeated PD, consider different strategies. “Grim Trigger” strategy: –Cooperate as long as other player cooperates, but once he defects, defect forever. –His defection “triggers” the punishment. –“Grim” because punishment lasts forever. To check if there is a symmetric equilibrium with trigger strategies: –Make sure that cooperating is better than defecting if other player has cooperated. –Make sure that “punishment” is a credible threat, that you will actually go through with it.

Prisoner’s Dilemma

When Are Trigger Strategies are NE? Assume other player also using a trigger strategy. If neither has defected, both cooperate this period. If you follow the trigger strategy, i.e. cooperate, you get  C this period (the payoff from cooperation) and you get  C each period in the future. –An infinite stream of payments of  C can be written as 1/(1-  )*  C. If you defect, you get  D this period (the increased payoff from unilateral defection) but in all future periods you get  P (the punishment payoff level) –Total earnings thus are  D +  /(1-  )*  P. Thus following the strategy is optimal if: 1/(1-  )*  C >  D +  /(1-  )*  P.

When Are Trigger Strategies are NE, con’t? The condition 1/(1-  )*  C >  D +  /(1-  )*  P can be rewritten as:  > (  D -  C ) / (  D -  P ) So the discount factor, , must be sufficiently large for collusion to be sustainable. How do we interpret this? –A high discount factor means that payoffs in the future are relatively important. –You are willing to forsake immediate, but transitory gains from defection for higher payoffs in the future.

When Are Trigger Strategies are NE, con’t? Is punishment a credible threat? Once again, assume other player also using a trigger strategy. If either has defected, both will punish this period. If you follow the trigger strategy, i.e. punish, you get  P this period and you get  P each period in the future. If you don’t punish, you will get a lower payoff, since defecting is a best response to other players playing defecting. Therefore the punishment is a credible threat.