overweighting of low probabilities reverses the pattern described above: It
enhances the value of long shots and amplifies the aversiveness of a small
chance of a severe loss. Consequently, people are often risk seeking in
dealing with improbable gains and risk averse in dealing with unlikely
losses. Thus, the characteristics of decision weights contribute to the
attractiveness of both lottery tickets and insurance policies.
The nonlinearity of decision weights inevitably leads to violations of
invariance, as illustrated in the following pair of problems:
Problem 5 ( N = 85): Consider the following two-stage game. In
the first stage, there is a 75% chance to end the game without
winning anything and a 25% chance to move into the second
stage. If you reach the second stage you have a choice between:A. a sure win of $30 (74%)
B. 80% chance to win $45 (26%)Your choice must be made before the game starts, i.e., before
the outcome of the first stage is known. Please indicate the
option you prefer.Problem 6 ( N = 81): Which of the following options do you prefer?C. 25% chance to win $30 (42%)
D. 20% chance to win $45 (58%)Because there is one chan ce i toce in four to move into the second
stage in Problem 5, prospect A offers a .25 probability of winning $30, and
prospect B offers .25 × .80 = .20 probability of winning $45. Problems 5
and 6 are therefore identical in terms of probabilities and outcomes.
However, the preferences are not the same in the two versions: A clear
majority favors the higher chance to win the smaller amount in Problem 5,
whereas the majority goes the other way in Problem 6. This violation of
invariance has been confirmed with both real and hypothetical monetary
payoffs (the present results are with real money), with human lives as
outcomes, and with a nonsequential representation of the chance process.
We attribute the failure of invariance to the interaction of two factors: the
framing of probabilities and the nonlinearity of decision weights. More