Consider the following game, which comes from James
Andreoni and Hal Varian at the University of Michigan.19 A neutral
referee runs the game. There are two players, Row and Column. The referee gives
two cards to each: 2 and 7 to Row and 4 and 8 to Column. This is common
knowledge. Then, playing simultaneously and independently, each player is asked
to hand over to the referee either his high card or his low card. The referee
hands out payoffs—which come from a central kitty, not from the players’
pockets—that are measured in dollars and depend on the
cards that he collects. If Row chooses his Low card, 2, then Row gets $2; if he
chooses his High card, 7, then Column gets $7. If Column chooses his Low card,
4, then Column gets $4; if he chooses his High card, 8, then Row gets $8.
(a) Show that the complete payoff table is as follows:
(b) What is the Nash equilibrium? Verify that this game
is a prisoners’ dilemma.
Now suppose the game has the following stages. The referee
hands out cards as before; who gets what cards is common knowledge. At stage I,
each player, out of his own pocket, can hand over a sum of money, which the
referee is to hold in an escrow account. This amount can be zero but cannot be
negative. When both have made their stage I choices, these are publicly
disclosed. Then at stage II, the two make their choices of cards, again
simultaneously and independently. The referee hands out payoffs from the
central kitty in the same way as in the single-stage game before. In addition,
he disposes of the escrow account as follows. If Column chooses his high card,
the referee hands over to Column the sum that Row put into the account; if
Column chooses his low card, Row’s sum reverts back to him. The disposition of the
sum that Column deposited depends similarly on Row’s card choice. All these
rules are common knowledge.
(c) Find the rollback (subgame-perfect) equilibrium of
this two-stage game. Does it resolve the prisoners’ dilemma? What is the role
of the escrow account?