# Is there a dutch book argument for the "independence of irrelevant alternatives" axiom?

There is a dutch book argument to show that nontransitive preferences are in a sense "unreasonable", which justifies why we pose the axiom of transitivity in the definition of "rational preferences", when it comes to no-uncertainty utility.

When it comes to utility functions over lotteries, Von Neumann and Morgenstern additionally pose the Independence axiom w.r.t. lotteries.

Are there Dutch-book arguments to justify the independence axiom?

• The Independence Axiom in vNM theory is different from Independence of Irrelevant Alternatives. Which one do you mean? Apr 14, 2023 at 16:15

Yes, and it's actually quite simple. Start with alternatives $${A, B}$$, where you prefer A to B. Let's say introducing C causes you to prefer B to A instead.

I start by offering $${A, B}$$, and you take $$A$$. Then I offer $$C$$ as well, and let you trade $$B$$ for $$A$$ as long as you give up some arbitrarily worthless object, like a one-in-a-million chance of losing a penny. All that matters is it's not enough to change your decision between $$B$$ and $$A$$. Let's call this $$-dx$$. Now your options are $${A, (B-dx), C}$$. You choose $$B-dx$$. Now notice that you could originally have chosen $$B$$, but instead took $$B-dx$$. This is a Dutch book.

I would think not. First, one can readily imagine cases where the Dutch book would hold, but which the independence axiom could not. Consider the presidential election of 1992. It would satisfy the necessary conditions for Bayesian reasoning and hence support the Dutch Book argument by default, but would clearly fail the independence of irrelevant alternatives criterion.

If someone chose to gamble on the election, then it is difficult to imagine why bookies could not place coherent odds. On the other hand, if memory serves me correctly, the majority polled preferred Bush over Clinton, but the plurality went for Clinton.

Part of that discussion, however, has to do with the particular form of von Neumann's particular axiom. He is adding zero, essentially, to a binary preference. There exist lotteries, such as those built around voting or other binary choice events, where the form of von Neumann's axiom is simply irrelevant to the problem.

If you take von Neumann's form, it implies that a third lottery has no impact on a binary choice of another lottery by implying that sufficient funds exist to purchase that lottery as well. In an election, you get only one outcome, you cannot "afford" two Presidents. This only holds if you can develop a "portfolio" of lotteries in any convex combination.

If you imposed additional restrictions on the type of lottery, then you may be able to get there from the Dutch book axioms, but that has more to do with the fact that you can always add zero to both sides of an equation in ordinary math. You should be able to walk backward from the utility function into preferences and arrive at von Neumann's axiom.

Of course, von Neumann's axioms do not assume the Dutch book argument and so you are dangerously mixing and matching. From the Dutch book argument, you can arrive at Kolmogorov's axioms. It doesn't mean you should, other than as a mental exercise. Indeed, Savage subsumes Kolmogorov into his justification of personalistic probability.

It does beg the somewhat interesting question as to whether the independence axiom is really an axiom and under what circumstances it is an axiom.