Behavioral Finance for Private Banking, 2nd Edition

CHAPTER 15Appendix: Mathematical Arguments

15.1 PROOF THAT EXPECTED UTILITY SATISFIES THE AXIOMS OF RATIONAL CHOICE

Transitivity:

Suppose so that lottery A is chosen over lottery B, so that lottery B is chosen over lottery C. Since and are real numbers these numbers can be ordered from high to low so that from it follows that . That is, lottery A is chosen over lottery C.

Independence Axiom:

To prove that expected utility satisfies the independence axiom, we need to be a bit more formal. A lottery is given by a collection of payoffs x₁,…,x_n and their probabilities p₁,…,p_n. Without loss of generality, we can assume that all lotteries have the same payoffs but might differ in the probabilities.¹ Thus, given the possible outcomes the lotteries are determined by the probabilities of these outcomes. To distinguish the lotteries we will then index the probabilities with the lottery, i.e., ...