31
Subjectivity and Correlation in Randomized Strategies
1
Introduction
Subjectivity and correlation, though formally related, are conceptually
distinct and independent issues. We start by discussing subjectivity.
A mixed strategy in a game involves the selection of a pure strategy by
means of a random device. It has usually been assumed that the random
device is a coin ﬂip, the spin of a roulette wheel, or something similar;
in brief, an ‘‘objective’’ device, one for which everybody agrees on
the numerical values of the probabilities involved. Rather oddly, in spite
of the long history of the theory of subjective probability, nobody
seems to have examined the consequences of basing mixed strategies
on ‘‘subjective’’ random devices, i.e. devices on the probabilities of whose
outcomes people may disagree (such as horse races, elections, etc.).
Even a fairly superFcial such examination yields some startling results, as
follows:
a. Two-person zero-sum games lose their ‘‘strictly competitive’’ charac-
ter. It becomes worthwhile to cooperate in such games, i.e. to enter into
binding agreements.
1
The concept of the ‘‘value’’ of a zero-sum game
loses some of its force, since
both
players can get more than the value (in
the utility sense).
b. In certain
n
-person games with
n
Z
3 new equilibrium points appear,
whose payo¤s strictly dominate the payo¤s of all other equilibrium
points.
2
Result (a) holds not just for certain selected 2-person 0-sum games, but
for practically
3
all such games. Moreover, it holds if there is any area
whatsoever of subjective disagreement between the players, i.e., any event
in the world (possibly entirely unconnected with the game under consid-
eration) for which players 1 and 2 have di¤erent subjective probabilities.
The phenomenon enunciated in Result (b) shows that not only the 2-
person 0-sum theory, but also the non-cooperative
n
-person theory is
modiFed in an essential fashion by the introduction of this new kind of
strategy. However, this phenomenon cannot occur
4
for 2-person games
This chapter originally appeared in
Journal of Mathematical Economics
1 (1974): 67–96.
Reprinted with permission.
1. Example 2.1 and sect. 6.
2. Example 2.3.
3. SpeciFcally, wherever there is one payo¤ greater than the value and another one less than
the value (for player 1, say).
4. Except in a very degenerate sense; see Example 2.2.