Analyzing a decision problem: Engine belt replacement

This problem is true story which as communicated to me by Teddy Seidenfeld, who pointed out the nice features that make real problems so subtle.

Here is the problem, almost repeating the description by Teddy.

Suppose you have a car, and the car stops working. You go to a garage. Here is what the mechanic tells you. The engine timing belt broke while you were traveling at high speed. The mechanic sayes that he can't tell if there was resulting engine damage until completing most of a several hundred dollar replacement of the belt. So, you have to decide whether to gamble about $150 (out of a $230 total timing belt repair) to find out whether your car (with 98.5K miles on it) has sustained engine damage - in which case it wouldn't pay to replace the belt and the $150 would be lost! The car is crucial, and you estimate that replacing it with all the involved costs would take some $4,500. You also have extra info of a "subjective" kind to help with the decision: When the timing belt broke, you recognized what had happended and almost immediately shifted the car into neutral. So, the engine speed dropped to 0 rpm naturally, under its own weight, without the weight of the car forcing it to turn out of synch.
To solve this problem, consider the options: The first option has loss $4,500 in monetary utility; the second option has loss 230p + (1-p)4,650, where p is the probability that the engine is alright after all. The best action depends on p; the graph below contrasts the first option against the second, depending on p. The graph shows the expected loss of each action depending on p. It is almost always best to pay for the test; in fact it is best to test as long as p > 0.0339367.

Graph p vs. expected loss

This is a case in each we can be reasonably confident that decisions are robust, even if we are not very confident about the value of p. Say we think 0.1 < p < 0.4, which characterizes a set of probability distributions. Even then it is best to pay for the measurement.

Things change if you decide to buy a very cheap car. If the cost of the new car is $1000, then the graph of expected loss is as follows.

Graph p vs. modified expected loss

In this case, the interval 0.1 < p < 0.4 does not lead to a robust decision. It is best to test as long as p > 0.163043; at this point, you can either buy a new car or pay for the test. With the preferences and opinions you indicated, there is nothing that forces you to go either way.


Fabio Cozman, The Robotics Institute, Carnegie Mellon University
fgcozman@cs.cmu.edu
Last modified: Wed May 28 12:18:23 EDT 1997