A paradox of (vNM) rationality

181 views

Skip to first unread message

Mehmet Ismail

unread,

Jul 10, 2015, 7:37:41 AM7/10/15

to decision_t...@googlegroups.com

Dear everyone,

Attached I share my short note on the generality of rationality. I show that it may actually not be possible for a decision maker to satisfy von Neumann-Morgenstern utility axioms in both a game theoretic situation and the associated decision situation. I provide a condition that is necessary and sufficient for a decision maker to be rational in both situations.

Comments are more than welcome.

Best wishes,

Mehmet

A paradox of rationality a la von Neumann-Morgenstern.pdf

Mehmet Ismail

unread,

Nov 19, 2015, 11:08:51 AM11/19/15

to decision_theory_forum

Dear everyone,

Below is a revised version of my note with a simpler example:

Consider the following game played by Alice and Bob in which the numbers represent (non-utility) monetary payoffs:

L R

L 5, 10 5, 0

R 5, 0 5,10

Alice is inequality averse; she suffers a disutility from uneven expected payoff distributions but does not suffer any disutility from even ones -- no assumptions about Bob.

Having this in mind, consider the following decision problem in which the only player is Alice (player 2 is nature), and the monetary payoffs from the respective consequences are identical:

L R

L 5 5

R 5 5

The question is: Given her preferences, can Alice be rational (vN-M) in both situations simultaneously? It turns out she can't: If she'd like to be rational in the game, she has to give up her rationality in the decision problem, and vice versa.

Why is this? Suppose that she has vN-M utility functions U and u in both situations, respectively, and that u(L,L) = u(R,L) = 5 in the decision problem, and U(L,L) = U(R,L) = 4 in the game as she suffers a disutility of 1 in each case. Now consider the profile [(1/2,1/2),L], since she is inequality averse, her utility is maximized at this point because the expected payoffs are (5,5). But, by bilinearity we have u((1/2,1/2),L) = 5 and U((1/2,1/2),L) = 4 which is a contradiction to her preferences because 5-4 =1 which means her disutility is still 1 at this point.

In this note, I give a characterization of the situations in which she can actually be rational both in games and in decision problems with more general utility functions.

Download link: researchers-sbe.unimaas.nl/mehmetismail/wp-content/uploads/sites/32/2015/11/vnm_rationality.pdf