Take it or leave it

Here is the game. There are 10 coins on the table. One player, the proposer, says how to split them: "I keep 7, you get 3." The other player, the responder, says yes or no. Yes: the split happens. No: the coins go back in the box and nobody gets anything. One offer, one answer, no haggling. It is called the ultimatum game (say: ul-ti-MAY-tum), because an ultimatum is a take-it-or-leave-it offer.

You go first, as the proposer. This Robo is not a pure backward-inducer. It is built to act like a person: it has a secret minimum, some number from 1 to 5, and it says yes only when your offer is at least that. Below its minimum it says no, and you both get nothing. You get 10 rounds against the same Robo. Your job is to find its minimum without wasting too many rounds, and then never pay one coin more than you have to.

The star works like this. A player who knew the minimum m from the start could keep 10 − m coins every round, 10 × (10 − m) in total. You earn the star if you keep at least 80 percent of that. So you can afford to lose about two rounds' worth of coins finding out. Think about where to start probing: from the top, from the bottom, or from the middle. One of those is a bad idea.

Now switch chairs. Robo proposes and you respond. Robo will be stingy. Before each answer, notice what you want to do, and then do that. There is no wrong answer in this box. This is the part where you find out what kind of responder you are.

Look at your total next to the "yes to all" total. Backward induction says: when Robo offers 1 coin, you are choosing between 1 coin and 0 coins, and 1 is more than 0, so say yes. Say yes to everything. Most people do not. In experiments with real money, offers below about 20 to 30 percent of the pot get turned down a lot of the time, even though turning them down costs the responder real money. If you said no to a 1-coin offer, you were in good company. If you said yes to everything, you were following the theory. Both are honest, and the gap between them is the whole point of this chapter.

Solve the ultimatum game the way you solved the 21 Game: start at the end.

And it is wrong about people. When the ultimatum game is played for real money, proposers usually offer 4 or 5 coins out of 10, and responders usually refuse 1 or 2. Is the theory broken? Not quite. Backward induction was solving a game whose payoffs were "coins, and nothing else." The people in the experiment are playing a game with different payoffs. They care about coins, and they also care about being treated fairly, and about punishing someone who treats them unfairly, even when the punishment costs them. Put those things into the payoffs and the same backward induction gives a different answer: the responder refuses 1 coin because "nothing, and I taught a cheapskate a lesson" is worth more to them than 1 coin, and a proposer who knows that offers more.

Remember Chapter 1: a game is the rules plus what each player wants. The math was fine. The mistake was in the payoffs. The hard part of playing with people is not the backward induction. It is knowing what the other player actually wants.

Five pirates have found a chest of 100 gold coins. They are ranked by seniority: A is the captain, then B, C, D, and E. The rules for splitting the loot:

• The most senior pirate proposes a split.
• Everybody votes, including the proposer.
• If at least half the pirates vote yes, the split happens and the game is over.
• Otherwise the proposer is thrown overboard, and the next most senior pirate proposes.

Every pirate wants, in this order: to stay alive, to get as many coins as possible, and, when nothing else is at stake, to throw someone overboard. They are all perfect backward-inducers, and they all know it.

What should A propose? Guess before you compute. Most people guess that A has to give away a lot to avoid a swim. The truth is stranger. The trick is Chapter 5's trick: do not start with 5 pirates. Start with 2, and work backward. At each step, predict the proposal in the boxes, then check. A pirate votes yes only when the offer is strictly better than what they would get if the proposer went overboard. Equal is not enough. With equal coins, the pirate takes the option with a splash.

Once you have seen all four steps, switch to Play as pirate A. The other four pirates vote for real, each one comparing your offer to what they would get if you went overboard. If you want to be sure the pattern is not a coincidence, try 6 or 7 pirates.

In the ultimatum game there is one offer. Real bargaining goes back and forth, and while it goes on, something is usually lost: time, patience, or ice cream cake. Here is the simplest version.

An ice cream cake worth 100 is melting. In round 1 you propose a split and Robo accepts or rejects. If Robo rejects, the cake loses 20 percent of its value (it is now worth 80) and in round 2 Robo proposes and you respond. If you reject, it loses another 20 percent (worth 64) and in round 3 you propose one last time. If that fails too, the cake is a puddle and nobody gets anything.

This Robo is a pure backward-inducer. It accepts an offer exactly when the offer is at least what it could guarantee itself by saying no and playing on. Before you play, work out what that number is in round 1. It is the same reasoning as the pirates: start from round 3. The star is for a first offer that Robo accepts and that keeps at least 84 for you.