A Centipede for Intransitive Preferrers* Wlodek Rabinowicz Department of Philosophy, Lund University Kungshuset, Lundagård, 222 22 Lund, Sweden e-mail:
[email protected]
Suppose that the agent's preferences over three alternatives, a, b, and c, are cyclical: a ? b ? c ? a. Suppose that these cyclical preferences are not reversible by small side rewards. That is, a ? b + ε ? c + 2ε ? a +3ε ? a, where ε is a small positive monetary value. Now, assume that the agent originally holds a and is confronted with a decision problem in which he can exchange his holding for b + ε. If he does, he can subsequently exchange b + ε for c + 2ε, and then exchange c + 2ε for a +3ε, whereupon the sequence of exchanges will stop. If he refuses any of the exchanges, no new exchange offer will be forthcoming. Clearly, if the agent is myopic, he will do badly in this decision problem: he will refuse the first exchange, which immediately stops the whole sequence. This means that the myopic agent will forgo a sure benefit of 3ε. His behaviour will be self-defeating. If the agent is resolute (cf. McClennen 1990 and 1997), each of the four possible outcomes is available to him. He can simply choose one of them and then go for it. Even though his pairwise preferences are cyclical, he may well have definite views as to which of these four outcomes are choiceworthy. It seems clear that outcome a will not be one of them. It is also quite possible that a +3ε will be be seen by him as choiceworthy. Therefore, it is quite possible that the resolute agent will make the three exchanges. Now, what *
I am indebted to Michael Almeida for getting me think on the problem discussed in this note. The work on this note was supported by a generous research grant from The Bank of Sweden Tercentenary Foundation.
about the sophisticated agent, i.e., an agent who solves his problem using backward induction? The decision tree for the problem that confronts the agent looks as follows: FIGURE 1. a + 3ε
a
b + εc + 2ε
Moves across stand for exchanges, downward moves stand for refusals to exchange. The bold lines mark the moves recommended by backward induction. Each such move goes down for the following reason: At the last choice node, the rational thing to do is to go down, since c + 2ε ? a +3ε. As the agent at the next-to-last choice node expects he would make the rational move at the last node, the rational thing for him to do at the next-to last node is to go down, since b + ε ? c + 2ε. As the agent at the initial node expects that he would at the next choice node trust his future rationality and act accordingly, i.e., move down, the rational thing for him to do at the initial node is again to go down, since a ? b + ε. Thus, the sophisticated chooser goes down in the first move and thereby forgoes a sure benefit: he opts for a even though he would get a +3ε if he made the three exchanges. In this sense, then, his behavior is self-defeating, just as in the case of the myopic chooser. Comparison with the Centipede Note that the situation, as described, closely reminds of the Centipede game:
FIGURE 2. X
Y
X 3, 3
1, 1 2, 3 4, 2 In this game between two players, X and Y, whose payoffs are given at the end of each branch (X’s payoff first, Y’s payoff second), backward induction prescribes terminating moves at each choice node, exactly as in the decision tree described in Figure 1. This means that the game terminates in the first move, even though the outcome of that move is dispreferred by each player to the outcome that would result if each of them moved across at each node. In other words, the decision problem described in Figure 1 is the Centipede for the Intransitive Preferrer. Would the outcome of sophisticated choice be any different if we changed the decision problem in such a way as to allow the exchange offers to be repeated after they have been rejected by the agent? The answer is No. Giving the sophisticated chooser another chance does not get him off the hook. To see this suppose that he will have three occasions to make the exchanges, just as in the original problem, but a refusal to accept the exchange at one of the two earlier stages will not be final: the same exchange will be available to him at the next stage. The decision tree for this modified problem has the following form:
FIGURE 3. a + 3ε
c + 2ε c + 2ε b+ε c + 2ε b+ε b+ε a
Again, bold lines specify the moves recommended by backward induction. The sophisticated chooser persists in refusing to exchange. This means that we can just as well go back to the simple Centipede described in Figure 1. As is well-known, the method of backward induction is quite controversial. The assumptions needed for its defence, as a general method of solving extensive-form games and dynamic decision problems, are very strong, in fact, unacceptably strong, as many would say. One of these assumptions is that the agent’s trust in his own future rationality and the future rationality of his protagonists must be robust: He must expect to keep it under all counterfactual circumstances, and given all possible evidence about past behavior. That is, he must expect to keep it even at those choice nodes he is certain will never be reached because reaching them would require a sequence of irrational moves. (Cf Binmore 1987, Reny 1988 and 1989, Bicchieri 1989, Pettit and Sugden 1989. For some responses to this objection see Sobel 1993 and Aumann 1995.) However, in Rabinowicz (1998), I have shown that the defence of backward induction can be based on much less controversial assumptions, if we limit ourselves to what I have called terminating games and decision problems (cf. also Aumann 1998 and Broome &Rabinowicz 1999 for further discussion). A game or a decision problem is terminating, if, at each of its choice nodes, backward induction prescribes a terminating move. Unlike the decision problem from Figure 3 but just like the standard Centipede, the Centipede for the Intransitive Preferrer belongs to this
special class. In a terminating game or a decision problem the expectation of preserved trust in future rationality need not extend to all counterfactual circumstances. As I have shown, it is enough if this trust is expected to be preserved in the actual development, in which, we may assume, no irrational moves are expected to be observed. In the actual development, the agent does not expect to confront any evidence about past behavior that would give him reasons to revise his original trust in future rationality.
Comparison with the Money Pump The Centipede for the Intransitive Preferrer reminds somewhat of the wellknown Money Pump. In the latter, each exchange costs the agent a small amount ε but it gives him an option he prefers. After a series of exchanges, he is back to where he started but with less money in his pocket. Thus, suppose that his preferences are as before: a ? b ? c ? a, or, to put them in a different order, c ? a ? b ? c. These preferences are not reversed by small payments. Thus: c - 3ε ? a - 2ε ? b - ε ? c ? c - 3ε. The Money Pump against the agent looks as follows: FIGURE 4. c - 3ε
c
b - ε a - 2ε
While this pump works for the myopic choosers, the sophisticated chooser does not allow himself to be pumped. Backward induction prescribes making the first exchange but then requires the agent to stop (the bold lines mark the backward-induction moves). (Cf. McClennen 1990 and Rabinowicz 1995. For the intuition that underlies this argument, see Schick
1986.) Note that, unlike the Centipede, the Money Pump is not of terminating type. Thus, backward induction gets the sophisticated agent with cyclical preferences off the hook in this kind of a money pump: he expects what is kept in store for him and reacts accordingly. The Centipede from Figure 1 is thus more troublesome for such an agent. However, as I have shown in Rabinowicz (1999), one can set up a more complicated money pump in which the sophisticated intransitive preferrer would be pumped (cf. also Sobel, forthcoming). This more complicated pump closely reminds of the problem set up in Figure 3: an exchange offer refused at an earlier stage is being repeated at the next stage. Refusing an exchange does not automatically turn off the pump. This time, backward induction does get the agent into trouble: FIGURE 5. c + 3ε
a + 2ε a + 2ε b+ε a + 2ε b+ε b+ε c
Again, bold lines specify the moves recommended by backward induction. Conclusion: The sophisticated intransitive preferrer can be pumped. Sophisticated choice is not enough to get one of the hook, contrary to what might have been expected. But such an entrapment of a sophisticated chooser requires a rather complicated set-up. In particular, it requires a decision problem of a non-terminating kind. For such a problem, the backward-induction solution is quite controversial, and it it is not obvious that an agent who shows foresight, i.e., who solves his dynamic decision
problem with an eye to what his future moves are going to be, will necessarily reason in accordance with backward induction. The terminating Centipede from figure 1 may therefore be a more convincing illustration of the problems facing such an agent.
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