Game Theory Lecture Jan 18

In Bertrand’s model of oligopoly A)Each firm chooses its quantity as the best response to the quantity chosen by the other(s). B)Each firm chooses its price as the best response to the price chosen by the other(s). C)The firms set quantities sequentially. The second firm’s quantity is the best response to the first firm’s quantity. D)The first firm sets a quantity. The second firm follows by setting a price. E)The firms jointly set the price that maximizes industry profits.

Ex 42.2 (a joint project) Two players, choose effort levels x 1 and x 2 between 0 and 1. Cost of effort to player i is x i 2 Part a) Total output is 3x 1 x 2. They divide output equally. Payoff to Player 1 is:

Find Player 1’s best response Maximize

Player 1’s reaction function

The two reaction functions

Reaction Function Graph

Part b Total output is 4x 1 x 2 and is split between two players. Cost of effort x i is c(x i )=x i Payoff to player 1 is:

Find Player 1’s best response function Take derivative: What does it tell us?

Ex 58.2 (Cournot) Two firms, linear inverse demand function P=a-Q=a-q 1 -q 2. Firms have constant marginal costs, c 1 and c 2 Profit function for firm 1 is:

Reaction Functions

Graph reaction functions

Algebra

Ex 59.1 (Cournot 2) Linear inverse demand P=a-Q Quadratic Cost function: C(q i )=q i 2 What is profit function for firm 1?

Bertrand model Each player does best response to other’s price. Constant marginal cost Buyers will purchase only from seller with lowest price. If prices are equal, demands are split. What can be an equilibrium?

Ex 69.1 Bertrand with fixed costs

Mixed Strategies Matching pennies

Player 2’s best response If Player 1 plays heads with probability p>1/2, what is Player 2’s best response? What if p<1/2? What if p=1/2

Both players’ best responses

Hide and seek game