On pages 146-150 of my edition of Douglas Adams’ “Dirk Gently’s Holistic Detective Agency,” Dirk Gently has a phone conversation with one of his clients, whose cat he has not been able to find in the past seven years. They are debating the bill that the holistic detective had sent, and we only hear his side of the conversation. I will give you the key snippets: “Sadly, no sign as yet of young Roderick, I’m afraid, but the search is intensifying as it moves into what I am confident are its closing stages.” … “I grant you, Mrs. Sauskind, that nineteen years is, shall we say, a distinguished age for a cat to reach, yet can we allow ourselves to believe that a cat such as Roderick has not reached it?” And after some very entertaining bits about Dirk’s “quantum mechanical view” of the world and the psychological cost his client’s skepticism puts on him (also itemized on the bill) we get to Dirk’s decision-theoretic view: “I do appreciate, Mrs. Sauskind, that the cost of the investigation has somewhat strained from its original estimate, but I am sure that you will in your turn appreciate that a job that takes seven years to do must clearly be more difficult than one that can be pulled off in an afternoon and must therefore be charged at a higher rate. I have continually to revise my estimate of how difficult the task is in the light of how difficult it has so far proved to be.”

The pleasure one gets from reading this derives from the strong suspicion that Dirk’s implicitly stated model of the underlying problem is probably not appropriate, certainly different from his client’s model, and probably also not Dirk’s true model.

Dirk’s implicit model could be something like this. The cat is equally likely in any one of a large number of, say k, places. [Let us ignore the complication that arises from the possibility that the cat could, in reality, also move while the search is going on.] The cost of searching differs from place to place. Dirk seems to work under the hypothesis that Mrs. Sauskind attaches a very high value to finding her cat, so that the search should go on at any cost. Assuming the detective searches optimally (given his model), he goes to the place with the lowest search cost first, then the second lowest, and so on. Given this optimal behavior, the revised expected total cost of searching for the cat increases with every additional fruitless search. Moreover, the daily additional search cost goes up every time. Under this model, Dirk is then indeed right in saying that he has to “continually revise [his] estimate of how difficult the task is in the light of how difficult it has so far proved to be”.

Now, Mrs. Sauskind’s model of the problem seems to be different in at least one crucial aspect. She appears to have the view that, while there may be the same places that the cat could be in, there is also a non-negligible possibility that the cat is already deceased (or simply unfindable). This means she attaches a total probability to the cat being at any of the k places that is less than one. Assume, therefore, that Mrs. Sauskind’s model is such that the probability of the cat being at place i is with, importantly Then is the ex-ante probability that the cat is dead. Finally, it seems evident from the dialogue that Mrs. Sauskind attaches positive value to the cat being found, but that she is also quite cost-sensitive. That is, at any moment of time (as long as the cat is not yet found), she would weigh the future chance of finding the cat against the expected (additional) costs of searching for the cat.

Given her model, as time goes by without any sign of the cat, Mrs. Sauskind attaches a higher and higher probability of the cat being already deceased. Let us label the places in the order in which the detective searches them. Then, if the cat was not found in the first place, by Bayes’ law, the probability of the cat being dead increases to after two unsuccessful searches it increases to and so on. All this without even the concern that the cat is getting older and older, while the search is going on over, apparently, a number of years.

During all this time, Mrs. Sauskind’s expected total costs are continually rising, and, given the detective’s statements, it seems also her expected future costs are constantly rising. It would now depend on the exact utility function over the cat being found and over how much money she has left to determine Mrs. Sauskind’s optimal search policy, or optimal stopping time, as the literature on these problems likes to call it. But it seems clear from the conversation that Mrs. Sauskind would have preferred to have stopped the search some time ago.

So, the holistic detective and his client really have a fundamental disagreement about the true model of the world. It is therefore not surprising that they cannot agree on the reasonableness of the bill.