Money Pump Research Articles

Axiomatic analysis of non-transitivity of preference and of indifference

This paper analyzes the usual Bayesian (money pump) argument for transitivity of preference and of indifference: that violation of transitivity is incoherent, since it requires an appreciable payment of money, with no benefit inreturn. The analysis shows that this argument is invalid, because the payment may be negligible. The analysis also shows that not only does non-transitivity require interminable trading (which is incoherent, because it wastes valuable time) but so does transitive indifference. This paradox motivates three axiomatic theories. With suitable axioms that uncover hidden assumptions, the first theory confirms the validity of the previous intuitive analysis. A technical modification of the axioms then replaces the first theory with the more reasonable second theory. The latter proves that trading is terminated under two of the three kinds of violation of transitivity, but remains interminable under cyclic preference. The third theory complements the second theory by studying a single free choice with neither trading nor money payments. It proves that cyclic preference is indecisive, while the other two violations of transitivity are decisive. Therefore, the axiom of transitivity is not a necessary truth for a general prescriptive decision theory, because it can be violated coherently. If a decision maker insists upon doing this, however, then he is simply unable to solve his decision problem under uncertainty by means of Bayesian decision theory.