Two oligarchs, A and B had an argument about the money they "earned", K billion US dollars. In order to solve the problem, they went to a court. Instead of using a court in city L (which was a common practice), they opted for a court in city M. A justice system of city M is famous for its transparency: a judge collects "contributions" from both parties. The party which made the largest "contribution" wins the case, another looses, "contributions" are not returned. In case of a tie, the judge flips a coin.

1. Formulate this situation as a strategic game

2. Show that there is no pure strategy equilibrium

The Model n identical firms produce and sell homogeneous good with marginal production costs c. There is a unit mass of consumers. Each consumer has a valuation of the good 1/ > c now it is the same for all consumers! Fraction A of consumers are informed. These consumers know both the prices and buy at the lowest price. If the prices are equal they split evenly. Fraction 1 A of consumers are uninformed, they just buy the good at the closest store. We assume that they split evenly among the firms. What is the equilibrium price(s) in this model? What are the options left??? There is nothing better than to try mixed strategies! We look at a symmetric candidate equilibrium, in which all firms play a mixed strategy F (p) Note, that the supports of this distributions can include only prices between c and v. The profit of firm one if it charges p comes from two sources: ' Uninformed consumers: 1g)'(p c) ' Informed consumers. if it charges the lowest price. Because each other firmj chooses prices according to F(p) the probability that we have a lower price is 1 F(p). This must hold for all firms. Therefore. our expected profit from uninformed is A[1 F(p)]"'1(p C)- 0 The total expected profit is m(p):A[1F(p)r-1