(Solved): 260 [CH. 8] SIMULTANEOUS-MOVE GAMES WITH MIXED STRATEGIES 7. Recall the game from Exercise 11 in Ch ...

260 [CH. 8] SIMULTANEOUS-MOVE GAMES WITH MIXED STRATEGIES 7. Recall the game from Exercise 11 in Chapter 4. Three contestants (A, B, and C) are participating in a game in which there is a prize worth $\$30$. Each can buy a ticket worth $\$15$ or $\$30$ or not buy a ticket at all. They make these choices simultaneously and independently. Then, knowing the ticketpurchase decisions, the game organizer awards the prize. If none of the contestants has bought a ticket, the prize is not awarded. If only one has bought a ticket, that player gets the prize. If two or more have bought tickets, then the one with the highest-cost ticket gets the prize if there is only one such contestant. If there are two or three contestants all having bought highestcost tickets, the prize is split equally among them. Find a symmetric equilibrium in mixed strategies for this game, where each contestant mixes over the ticket-purchase choices and the mixture probabilities are the same for all three contestants.