Optimal Search

First, consider the low-search-cost treatment. Obviously any amount above 85 cents should be accepted, since the potential gain from searching again is at most 5 cents (if a very lucky draw of 90 is obtained) and the cost of the new draw is 5 cents. Thus there is only a very low chance (1/91) that the new draw will cover the cost. At the opposite extreme, a low draw, say 0, would be rejected since the cost of another draw is only 5 cents and the expected value of the next draw is 45 cents (the midpoint of the distribution from 0 to 90). Notice the use of an expected value here, which is justified by the assumption of risk neutrality. To summarize, the benefits of further search exceed the cost when the best current draw is very low, and the cost exceeds the expected benefit when the best current draw is near the top of the range. The optimal reservation prize level is found by locating the point at which the expected benefits of another search are equal to the search cost.

Suppose that the best current draw is 60. There is essentially a 2/3 chance that the next draw is at 60 or below, in which case the net gain is 0. This situation is represented in Figure 8.2, where the flat horizontal line represents the fact that each of the draws will be observed with the same probability. The area below this line represents probability. For example, two-thirds of the area below the dashed line is to the left of 60, and one third is to the right of 60. In other words, starting from 60, there is a 1/3 chance of obtaining better draw.

0 0.005 0.01 0.015

0 10 20 30 40 50 60 70 80 90

Draw in Pennies

Probability

Figure 8.2. A Uniform Distribution on the Interval [0, 90]

(With One Third of the Probability to the Right of 60)

Next consider the expected gains from search when the best current draw is 60 (with recall and a 5-cent search cost). A draw below 60 will not produce any gain, so the expected value of the gain will have a term that is the product:

(2/3)0 = 0. The region of gain is to the right of the vertical line at 60 in Figure 8.2, and the area below the dashed line is one third of the total area. Thus, there is essentially a 1/3 chance of an improvement, which on average will be half of the distance from 60 to 90, i.e. half of 30. The expected value of an improvement, therefore, is 15 cents. An improvement occurs with a probability about 1/3, so the expected improvement is approximately (1/3)(15) = 5 cents. Any lower current draw will produce an expected improvement from further search that is above 5 cents, and any higher current draw will produce a lower expected improvement.

It follows that 60 is the reservation draw level for which the cost of another draw equals the expected improvement. The same reasoning applied to the 20-cent search cost treatment implies that the optimal reservation draw is 30 cents (see question 2).

The predicted reservation values of 60 and 30 are graphed as horizontal lines in Figure 8.1. The power of these predictions is quite clear. There is a break between the plus signs indicating acceptance and the dots indicating rejection, with little overlap. This break is slightly below the 30 and 60 levels predicted for a risk-neutral person. Actually, adding risk aversion lowers the predicted cutoff.

The intuition behind risk aversion effects is clear. While a risk-neutral person is approximately indifferent between searching again at a cost of 5 cents and stopping with a draw of 60, a risk-averse person would prefer to stop with 60, since it is a sure thing. In contrast, drawing again entails the significant risk (two

Conversely, draws above the reservation draw level are accepted, indicating a preference for those sure amounts of money over a continuation of the search process.

To summarize, the value of continuing to search is greater than any number below the reservation value, and it is less than any number above the reservation value, so the value of continuing to search must be equal to the reservation value. This implication was tested by Schotter and Braunstein (1981), who elicited a price at which individuals would be willing to sell the option to search. In one of their treatments, the draws were from a uniform distribution on the interval from 0 to 200 with a search cost of 5 cents. The optimal reservation value for a risk neutral person is 155 (see question 3). This should be the value of being able to play the search game, assuming that there is no enjoyment derived from doing an additional search sequence after several have already been completed. The mean reported selling price was 157, and the average accepted draw was 170, which is about halfway between the theoretical reservation value of 155 and the upper bound of 200. These and other results lead to the conclusion that the observed behavior is roughly consistent with the predictions of optimal search theory in this experiment.

Further Reading and Extensions

The search game discussed here is simple, but it illustrates the main intuition behind the determination of the reservation draw level in a sequential search problem. There are many interesting and realistic variations of this problem. The planning horizon may not be infinite, e.g. when you have a deadline for finding a new apartment before you have to move out of the current one. In this case, the reservation value will tend to fall as the deadline nears and desperation takes over. In the Cox and Oaxaca (1989) experiments with a finite horizon, subjects stopped at the predicted point about three-fourths of the time, and the deviations were in the direction of stopping too soon, which would be consistent with risk aversion.

In all situations considered thusfar, the person searching has been assumed to know the probability distribution from which draws were being made. This is a strong assumption, and therefore, it is interesting to consider cases in which people learn about the distribution of draws as they search. Suppose that you think the draws will be in the range from 0 to 100, and you see a draw of 180.

This would have been well above your reservation value if the distribution from which draws were made had an upper limit of 100, but now you realize that you do not know what the upper limit really is. In this case, a high draw may be rejected in order to find out whether even higher draws are possible (Cox and Oaxaca, 2000).

Although the aggregate data discussed here are roughly consistent with theoretical predictions, this leaves open the issue of how people learn to behave in this manner. Certainly they do not generally do any mathematical analysis. Hey (1981) has analyzed individual behavior in search of heuristics and adaptive patterns that may explain behavior at the individual level. Also, see Hey (1987, when he is “still searching”). He specified a number of rules of thumb, such as:

One-Bounce Rule:

Buy at least two draws, and stop if a draw received is less than the previous draw.

Modified One-Bounce Rule:

Buy at least two draws, and stop if a draw received is less than the sum of the previous draw and the search cost.

With a search cost of 5 and initial draws of 80 and 50, for example, the modified one-bounce rule would predict that the person would stop if the third draw turned out to be less than 55. The experiment involved treatments with and without recall, and with and without information about the distribution (a truncated normal distribution) from which draws were drawn. Behavior of some subjects in some rounds corresponded to one or more of these rules, but by far the most common pattern of behavior was to use a reservation draw level. In the fifth and final round, about three-fourths of the participants exhibited behavior that conformed to the optimal reservation-value rule alone.

Questions

1. The computerized experiments discussed in this chapter were set up so that draws were equally likely to be any amount on the interval from 0 to 90. When throwing 10-sided dice with numbers from 0 to 9, the throws will be 0, 1, 2, ... 99. In order to truncate this distribution at 90, is it is convenient to ignore the first throw if it is a 9, so that all of the 90 integers from 0 through 89 are equally likely, i.e. each has a probability of 1/90.

(This is the approach taken for the hand-run experiment instructions in the appendix to this chapter.) If the current draw is 60, then the chances that the next draw will be as good or better are: a 1/90 chance of a draw of 60, for a gain of 0, a 1/90 chance of a draw of 61 for a gain of 1 cent; a 1/90

the current draw (of either 59 or 60). Finally, weight each gain by the probability (1/90) of that draw, and add up all of the weighted gains to get the expected gain, which can then be compared with the search cost of 5 cents.

2. With a search cost of 20 cents per draw from a distribution that is uniform from 0 to 90, show that the person is approximately indifferent between searching again and stopping when the current best draw is 30.

3. With a search cost of 5 cents for draws from a uniform distribution on the interval from 0 to 200, show that the reservation value is about 155.

4. Evaluate the data in Table 8.1 in terms of the “One-Bounce Rule” and the

“Modified One-Bounce Rule.” Do these rules work well, and if not, what seems to be going wrong?

5. (Open-ended) Rules of thumb have more appeal in limited-information situations where rational behavior is less likely to be observed. Can you think of other rules of thumb that might be good when the distribution of draws is not known and the person searching has not had time to learn much about it?

Part III. Game Theory Experiments: Treasures