Exercise 2. Backward Induction
There are five pirates with names 1,2,3,4,5. They have just seized a hundred gold coins, and now it’s time to share the loot. The bargaining rules are: Whoever has the lowest number as a name must propose an division of the one hundred coins to the remaining pirates. If the majority accepts the proposal, then the coins are allocated and the game ends. If the majority does not accept, then the proposer gets thrown overboard and the game is repeated with one less pirate. What should the first pirate propose?

We will assume that in the case of a tie, the pirate gets thrown overboard. Getting thrown overboard yields negative utility, and when indifferent pirates will reject an offer. We will proceed by backward induction. We start at the last subgame of the game.

Last subgame: There are only P4 and P5 left, as P1, P2, and P3 have been thrown overboard. P4 is the one to make an offer. Whatever he offers, however, he will be thrown to overboard. This is because P5 can always refuse, and in the case of a tie P4 gets thrown and P5 gets all the coins. So refuse is a dominant strategy for P5.

Second to last subgame: We substitute the last subgame with its equilibrium payoff. P4 knows that if P3’s offer gets refused, he will be thrown. P3, at the same time, wants to maximize his payoffs and offer P4 the strict minimum for him to accept. Hence, P3 offers 1 coin to P4, 0 to P5 and keeps 99. With this offer, P3 and P4 vote in favour and P5 against.
$$O f f_{P_3}=(\cdot, \cdot 99,1,0)$$

Third to last subgame: There are four pirates left. P2 will anticipate P3’s offer. Hence, he understands that he will need to lure two pirates to accept his offer. The cheapest ones are P4 and P5. He therefore proposes 1 gold coin to 5,2 gold coins to 4 and keeps 97 to himself, leaving 0 for P3. His proposal is accepted and this ends the game.
$$O f f_{P 2}=(\cdot, 97,0,2,1)$$

First subgame: P1 foresees P2’s proposal. He understands that he needs to get two more pirates to accept his proposal. He offers 1 gold coin to P3, 2 coins to P5 and keeps 97 to himself. The pirates will split the money this way and the game will end. The proposal by P1, which eventually will be the accepted one is:
$$O f f_{P 1}=(97,0,1,0,2)$$

The Single-Person Decision Problem

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Exercise 5. Brazil or the U.S.?
A long-lived government faces a short-run representative government employee. The government must choose whether to honor pensions $(H)$ or not $(N)$. At the beginning of the period, times are either “good” or “bad.” The probability times are “bad” is 90\%. In good times, pensions are always honored. In bad times they are honored or not depending on the government decision. The employee is informed and observes (after the fact, at the end of the period) whether or not times are good or bad. The choice of the employee is to guess whether or not her pension will be honored $(H)$ or $(N)$. The payoff of the employee is the sum of two parts: 1 if the pension is honored, 0 if it is not; and 1 for guessing right, 0 for guessing wrong. So guessing right when the pension is honored gives 2 , and so forth.
a. Find the extensive and normal forms of the stage-game.
It is always a good idea to start with the extensive form, which will help us in building the normal forms.

To have the normal form representation, we first write the two games and then mix them according to their probability of occurrence. The game in good time occurs with 0.1 probability, while the bad state of the world happens 0.9 probability. So players face a game of expected outcomes.

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