Consider the game depicted below. Player 1 decides between going L or R in stage 1 and 3 of the game. Player 2 decides between going l and r in stage 2 of the game.
a. List the possible pure strategies for each player in this game and illustrate the payoffs from each pair of strategies in a matrix.
b. Is there a dominant strategy for either player?
c. Identify the subgame perfect equilibrium strategies and outcome.
d. Identify the Nash Equilibria that are not subgame perfect.
e. For each Nash Equilibrium that is not subgame perfect, explain which parts of the Nash Equilibrium strategies are non-credible.
f. Suppose you have developed a drug that can be administered without the victim being aware of it. The effect of the drug is that the victim suddenly becomes gullible and believes anything he is told. You only have 1 dose of the drug and decide to auction it off to the two players right before they play each other in the game you have analyzed so far. Each player is asked to submit a sealed bid, and the highest bidder will be sold the drug at a price equal to the highest bid. In case of a tie in bids, a coin is flipped to determine who wins and pays the price that was bid. Suppose in this part that payoffs are in terms of dollars and that bids can be made in one cent increments. Suppose further that players do not consider bidding above the maximum they are willing to pay. Given that the players know each other's payoffs in the above game, what is the equilibrium price that you will be able to sell the drug for? (Hint: There are two possible answers.)
g. In part (f), we said "Suppose further that players do not consider bidding above the maximum they are willing to pay." Can you think of a Nash equilibrium to the auction that would end in a price of $8 if we had not made that statement in (f)?
What will be an ideal response?
b. There is no dominant strategy for either player.
c. In stage 3, player 1 should go L -- getting 5 rather than 0. On the left node in stage 2, player 2 therefore should move r -- getting 10 rather than 5; and on the right node in stage 2, player 2 should move r -- getting 5 rather than zero. Finally, this implies player 1 should go R in stage 1, getting 10 rather than 5. The subgame perfect equilibrium strategies are therefore (R, L) for player 1 and (r, r) for player 2, leading to the outcome of player 1 moving R followed by player 2 moving r -- with payoff of 10 for player 1 and payoff of 5 for player 2.
d. There are 3 Nash equilibria that are not subgame perfect:
Equilibrium 1: {(L, R), (l, l)}
Equilibrium 2: {(L, R), (l, r)}
Equilibrium 3: {(L, L), (r, l)}
e. In Equilibrium 1 and 2, it is not credible for player 1 to plan to go R in stage 3 -- because it would mean he would choose 0 over 5. And it is only because player 2 takes the threat seriously that he best responds by going l in stage 2. Player 1 therefore uses the threat of going R in the last stage to get his most preferred outcome of 20.
In Equilibrium 3, player 2 non-credibly threatens to go l on the right node in stage 2. This is not credible because player 2 would be picking payoff 0 over 5. But if player 1 believes the threat, it leads him to best respond by going L in stage 1 in order to avoid a 0 payoff from being R -- and this gets player 2 to his preferred outcome with payoff of 10.
f. If player 1 uses the rug on player 2, he'll be able to threaten R in stage 3 and thus get a payoff of 20 rather than the subgame perfect payoff of 10. If player 2 uses the drug on player 1, he'll be able to threaten l on the right node of stage 2, thereby getting a payoff of 10 rather than the subgame perfect payoff of 5. The drug is therefore worth 10 to player 1 and 5 to player 2. The equilibrium then has player 1 bidding $5.01 and player 2 bidding $5, or it has player 1 bidding $5 and player 2 bidding $4.99. The equilibrium price is thus either $5 or $5.01. (Actually, the drug should be worth even more to player 1 -- because he ends up with only 5 if player 2 gets it while he ends up with 20 if he gets it. The equilibrium is still the same, though.)
g. Yes: Player 1 bids $8, player 2 bids $7.99.
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