ECO290E: Game Theory Lecture 4 Applications in Industrial Organization

Review of Lecture 2 Outcomes of games, i.e., Nash equilibria may not be Pareto efficient. (e.g., Prisoners’ Dilemma) There can be multiple equilibria. (e.g., Battle of the sexes) One equilibrium can be less efficient than (Pareto dominated by) the other equilibrium. (e.g., Coordination game)  Coordination failure

Review of Lecture 3 When players are rational and share the correct belief about the future play, NE will emerge. In some cases, however, NE can be reached only by rationality.  Dominant strategy (e.g., PD)  Focal Point (e.g., Class room experiment)  Iterated elimination of strictly dominated strategies (e.g., Spatial competition model)

Spatial Competition Model Players: Two ice cream shops Strategies: Shop location along a beach (any integer between 0 and 100) Payoffs: Profits=The number of customers Assumptions: Customers are located uniformly on the beach. Each customer goes to the nearest shop (and buys exactly one ice dream). If both shops choose the same location, each receives half of the customers.

Nash Equilibrium There is a unique NE in which both shops open at the middle. Why? Choosing separate locations never becomes a NE. Choosing the same locations other than the middle point also fails to be a NE. If both shops choose the middle, then no one has an incentive to change the location.

Solution by Iterated Elimination Step 1: A rational player never takes the edges, since 0 (100) is strictly dominated by 1 (99). Step 2: 1 and 99 are never chosen if the players know their rival is rational. Step 3: 2 and 98 are never chosen if the players know that their rival knows that you are rational. Step 50: 49 and 51 are never chosen if the players know that their rival knows that …  Both players choose 50 in the end!

Common Knowledge Each step requires a further assumption about what the players know about each other’s rationality. We need to assume not only that all the players are rational, but also that all the players know that all the players are rational, and that all the players know that all the players know that all the players are rational, and so on. For an arbitrary number of steps, we need to assume that it is common knowledge that the players are rational.

Weak Predictive Power The process often produces a very imprecise prediction about the play of the game. Nash equilibrium is a stronger solution concept than iterated elimination of strictly dominated strategies, in the sense that the players’ strategies in a Nash equilibrium always survive during the process, but the converse is not true.  If the elimination processes pick up a unique strategy profile (e.g., Spatial competition model), then that must be a NE.

Bertrand Model Players: Two firms Strategies: Prices they will charge Payoffs: Profits Assumptions: A linear demand function: P=a-bQ Common marginal cost, c. The firm with lower price must serve the entire market demand. If the firms choose the same price, then each firm sells the half of the market demand.

Bertrand-Nash Equilibrium There is a unique NE in which both firms charge the price equal to their (common) marginal cost. Why? Choosing different prices never becomes a NE. Choosing the same price other than the marginal cost also fails to be a NE. If both firms choose p=c, then no firm has an (strict) incentive to change the price.

Cournot Model Players: Two firms Strategies: Quantities they will charge Payoffs: Profits Assumptions: A linear demand function: P=a-bQ Common marginal cost, c. Firms cannot decide their prices to charge, but the unique market price is determined so as to clear the market.

Important Remarks Bertrand and Cournot models are different games, i.e., price competition vs. quality competition. The unique equilibrium concept (=NE) can explain different market outcomes depending on the models. That is, we don’t need different assumptions about firms’ behaviors.  Once a model is specified, then Nash equilibrium gives us the result of the game.