Suppose that there are 3 consumers in the market who value a belt at $10. There are two firms inthemarket,firmsAandB,whocompetebysettingpricessimultaneously.Thereisnocostin making the belts. Eachfirmhasoneloyalcustomerwhowouldonlybuyfromthemandwouldneverbuyfromthe other firm. The remaining consumer does not care where he buys: He will buy wherever the best is cheaper (as long as the price is below $10); if the goods are equally-priced then she is equally likely to buy from either firm.

(i)Formally describe this as a game between the two firms (i.e. what are their strategies? Whar are payoffs for each combination of strategies?)(ii)Note that charging $10 guarantees a firm a profit of at least $10. Are there any dominated strategies? What are they? Explain.

(iii)There are no pure strategy equilibria of the game. Prove this as follows:

a.Show that it cannot be an equilibrium for them to both charge the same (high) price: for example, that it is not an equilibrium for them both to charge $9.

b.Show that it cannot be an equilibrium for them to both charge the same (low) price: for example, that it is not an equilibrium for them both to charge $1.

c.Show that it cannot be an equilibrium for Firm A to charge a high price and Firm B to charge a low price. For example, what happens if Firm A charges $10 and Firm B charges $6?