(D1) Winning $50 with probability .5
(D2) Winning $30 with probability .7
(E1) Winning $50 with probability .8
(E2) Winning $30 with probability 1.0
Consider a jar with 50 green balls, 30 yellow balls, and 20 red balls.
Green
(50)
Yellow
(30)
Red (20)
D1
$50
$0
$0
D2
$30
$0
$30
Green
(50)
Yellow
(30)
Red (20)
E1
$50
$50
$0
E2
$30
$30
$30
Most people choose D1 and E2.
but the payoff for yellow favors E1 over E2, but does not favor
D1 or D2.
Going from Game D to Game E only makes option 1 more favorable..
In terms of utility:
If you prefer D1 to D2 then .5D(50) > .7D(30) or D(30) < .7D(30)
If you prefer E2 to E1 then .8D(50) < D(30) or D(30) > .8D(30)
Either one can be rational depending on your risk aversion, bot they can't both be rational for the same person.
The explanation is that there is a preference for certainty separate
from the utility of the payoff. D1 &E1, D2&E2, and D2&E1
can all be explained with different utility functions, but explaining the
most popular choice cannot.