write:Samuelson’s ProblemThe great Paul Samuelson—a giant among the economists of thetwentieth century—famously asked a friend whether he would accept agamble on the toss of a coin

but Humansare by nature narrow framers.The ideal of logical consistency
as this example shows
is notachievable by our limited mind. Because we are susceptible to WY SIATIand averse to mental effort
we tend to make decisions as problems arise
even when we are specifically instructed to consider them jointly. We haveneither the inclination nor the mental resources to enforce consistency onour preferences
and our preferences are not magically set to be coherent
as they are in the rational-agent model.Samuelson’s ProblemThe great Paul Samuelson—a giant among the economists of thetwentieth century—famously asked a friend whether he would accept agamble on the toss of a coin in which he could lose $100 or win $200. Hisfriend responded
“I won’t bet because I would feel the $100 loss morethan the $200 gain. But I’ll take you on if you promise to let me make 100such bets.” Unless you are a decision theorist
you probably share theintuition of Samuelson’s friend
that playing a very favorable but riskygamble multiple times reduces the subjective risk. Samuelson found hisfriend’s answer interesting and went on to analyze it. He proved that undersome very specific conditions
a utility maximizer who rejects a singlegamble should also reject the offer of many.Remarkably
Samuelson did not seem to mind the fact that his proof
which is of course valid
led to a conclusion that violates common sense
ifnot rationality: the offer of a hundred gambles is so attractive that no saneperson would reject it. Matthew Rabin and Richard Thaler pointed out that“the aggregated gamble of one hundred 50–50 lose $100/gain $200 betshas an expected return of $5
0
with only a 1/2
300 chance of losing anymoney and merely a 1/62
000 chance of losing more than $1
000.” Theirpoint
of course
is that if utility theory can be consistent with such a foolishpreference under any circumstances
then something must be wrong with itas a model of rational choice. Samuelson had not seen Rabin’s proof ofthe absurd consequences of severe loss aversion for small bets
but hewould surely not have been surprised by it. His willingness even toconsider the possibility that it could be rational to reject the packagetestifies to the powerful hold of the rational model.Let us assume that a very simple value function describes thepreferences of Samuelson’s friend (call him Sam). To express his aversionto losses Sam first rewrites the bet
after multiplying each loss by a factor