Consider a game in which there is aprize worth $30. There are three contestants,

Question

Consider a game in which there is aprize worth $30. There are three contestants,

A, B, and C. 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 ticket-purchase decisions, the game organizer awards the

prize. If no one has bought a ticket, the prize is not awarded. Otherwise, the

prize is awarded to the buyer of the highest-cost ticket if there is only one

such player or is split equally between two or three if there are ties among

the highest-cost ticket buyers. Show this game in strategic form. Find all

pure-strategy Nash equilibria

Explanation / Answer

There are following 3 nash equilibria strategy:

Nash equilibrium in which all players choose not to buy a ticket.

If every contestant chooses not to buy a ticket, then everyone gets 0 payoff.

Then, one player would deviate to buying a $15 ticket, because then that player would win the prize, and a net payoff of 30-15=$15 which is positive.

Nash equilibrium in which all players choose to buy a ticket worth $15:

If every contestant chooses to buy a ticket worth $15,

then everyone gets payoff of 30/3-15=10-15=$-5. Then, one player would deviate to not buying a ticket, because then that player would get 0 which is larger than loss of $5.

Nash equilibrium in which any player chooses to buy a ticket worth $30:

Suppose in a strategy profile there is at least one contestant who chooses to buy a ticket worth $30.

A. If she is the only winner, then we have the following cases for the 3 respondents (in any order):

(15,15,30),

(0,15,30),

(0,0,30).

In the first two strategy profiles, the player who plays $15 gets a payoff of $-15, so deviates to 0 to get 0. In the last one, player who picks 30, gets a 0 payoff and deviates to 15 to get 15=30-15.

B. If she shares the prize with one other player, then we have either (0,30,30) or (15,30,30). In both strategy profiles any one of the players who picks 30, getting a payoff of -15, deviates to 15 to get 0 payoff.

C. If she shares the prize with both of the other players, then it is (30,30,30) in which case any one of the players deviates to 15 or 0.

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