Palgrave Handbook of Econometrics: Applied Econometrics

John DiNardo 111

First gambling situation The second pair of gambles

[A.] You win $500,000 if you draw a

card numbered 1–11 (11% chance).

If you draw a number from 12–100,

you get the status quo (89% chance).

[B.] You win $2,500,000 if you draw

a card numbered 2–11 (10% chance.)

Draw a number from 12–100 or 1 and

you get the status quo (90% chance).

[C.] You win $500,000 for certain.

[D.] You win $2,500,000 if you draw

a card numbered 1–10 (10% chance),

$500,000 if you draw a card from

11–99 (89% chance), and the status

quo if you draw the card numbered

100.

As Allais found (and has been found repeatedly in surveys posing such gambles),
for most peopleBA(Bis preferred toA) andCDand, as Savage reports, the
same was true for him (Savage, 1972, pp. 101–4)!
As most economists will recognize, this is a “paradox” since, fromCD:

U(500, 000)>0.1U(2, 500, 000)+0.89U(500, 000)+0.01U( 0 ),

and fromBA:

0.1U(2, 500, 000)+0.9U( 0 )>0.11U(500, 000)+0.89U( 0 ),

and it is obvious that both inequalities can’t be true.^21 There are two ways to
handle this “paradox.”

1. One possibility (the one that appeals to me) is that – even after continued reflec-
   tion – my original preferences are just fine. For me, the fact that at the stated
   sums of money, etc., the comparison is inconsistent with Expected Utility The-
   ory is merely too bad for the theory, however plausible it sounds. Indeed, as
   is well-known, it is possible to axiomatize preferences so that Allais paradox
   behavior is consistent with “rational” behavior (Chew, 1983).


2. A second possibility is to conclude that something is “wrong” with your
   “preferences.” That was Savage’s conclusion; his solution was to “correct
   himself.”

Indeed, as befits a Bayesian, Savage analyzed the situation by rewriting the
problem in an equivalent, but different way:

Ticket number

1 2–11 12–100

First pair GambleA 55 0

GambleB 025 0

Second pair GambleC 55 5

GambleD 025 5