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JEFFREY A. BARRETT and FRANK ARNTZENIUS
AN INFINITE DECISION PUZZLE ABSTRACT. We tell a story where an agent who chooses in such a way as tomake the greatest possible profit on each of an infinite series of transactions ends up worse off than an agent who chooses in such a way as to make the least possibleprofit on each transaction. That is, contrary to what one might suppose, it is not necessarily rational always to choose the option that yields the greatest possibleprofit on each transaction.
KEY WORDS: Decision theory, Dutch book, Puzzles One might expect to do better acting rationally than acting ir-rationally. It is possible, however, to tell a story where one is
guaranteed to do better than a seemingly rational agent by alwaysacting in what appears to be an entirely irrational way.1
Suppose one has an infinite stack of dollar bills with consecutiveserial numbers: 1, 2, 3, etc. An agent, who starts with no money, is then offered the following choice, where n is equal to the totalnumber of times that the choice has been offered so far:
1. Get one dollar bill off the top of the stack. 2. Get 2n+1 dollar bills off the top of the stack, but you must thenreturn the bill with the smallest serial number that you currently
have. The returned bill is immediately destroyed. Suppose that the agent does not know how many times he will beoffered this choice.
The first time the agent is offered the choice (n = 1) he acceptsoption 2 because the expected return of option 1 is $1 while the expected return of option 2 is $3. The second time (n =...