PUMPING JACK FOR MONEY

What You Have What Jack Has Transaction

A and $0 B, C, and $99

A for B and $1

B and $1 A, C, and $98

B for C and $1

C and $2 A, B, and $97

C for A and $1

A and $3 B, C, and $96

A for B and $1

B and $4 A, C, and $95

B for C and $1

C and $5 A, B, and $94

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A and $99 B, C, and $0

10 CHAPTER 1: INTRODUCTION TO STRATEGIC REASONING

by more than a dollar), Jack will make the trade. Now you possess C and two dollars. The next step is to offer C in exchange for A and a dollar. Since Jack prefers A to C (say, by at least a dollar), he’ll make the trade. Now you have A and three dollars, whereas if you recall, you started with A and no money. Trading with Jack is a money pump! It gets even better: you can continue to execute this sequence of trades while accumulating three dollars in each round. Eventually, you’ll have taken all of Jack’s money. Such is the sad life of someone whose preferences are not transitive, so take this cautionary tale to heart and always have your preferences be transitive!

If a person’s preferences are complete and transitive, then there is a way in which to assign numbers to all of the feasible items—where the associated number is referred to as an item’s utility—so that a person’s preferences can be represented as choosing the item that yields the highest utility. To be more concrete, suppose there are four cell phone providers available to Grace: AT&T, Verizon, Sprint, and T-Mobile. Her preferences are as follows:

AT&T is better than Verizon. Verizon is better than Sprint.

Sprint and T-Mobile are equally appealing.

This set of preferences implies the following ordering of plans: AT&T is best, Verizon is second best, and Sprint and T-Mobile are tied for third best. The next step is to as- sign a utility to each of these choices so that choosing the plan with the highest utility is equivalent to choosing the most preferred plan. Such an assignment of utilities is shown in TABLE 1.3.

We can now describe Grace’s behavior by saying that she makes the choice which yields the highest utility. If all four plans are available in her area, we know by her pref-

erences that she’ll choose AT&T. If we say that she chooses the plan with the highest utility, it means that she chooses AT&T, because the utility of choos- ing AT&T is 10, which is higher than 6 from Verizon and 2 from either Sprint or T-Mobile. Now suppose that AT&T is unavailable in her area, so she can choose only between Verizon, Sprint, and T-Mobile. Her preferences rank Verizon higher than the other two, so that is what she will buy. Choosing Verizon is also what maximizes her utility—it delivers utility of 6—when she can choose only between Verizon, Sprint, and T-Mobile.

To ensure that choosing the option with the highest utility is equivalent to choosing the most preferred option, numbers need to be assigned so that the utility of option A is greater than the utility of option B if and only if A is pre- ferred to B and the utility of A is equal to that of B if and only if the individ- ual choosing is indifferent between A and B. Note that there is no unique way to do that. Rather than assigning 10, 6, 2, and 2 to AT&T, Verizon, Sprint, and T-Mobile, respectively, it would have worked just as well to have used 14, 12, 11, and 11 or 4, 3, 0, and 0. As long as the utility is higher for more preferred items, we’ll be fine.