**MSC:**- 90D10 Noncooperative games

game of incomplete information

played on a sequence space $\{ 0,1\} ^{\bf Z}$ such that

the players' locally finite

beliefs are

conditional probabilities of the canonical Bernoulli

distribution on $\{ 0,1\}^{\bf Z}$, each player has only two moves, the

payoff matrix is determined by the $0$-coordinate and

all three players know that part of the payoff matrix pertaining to

their own payoffs. For this example there

are many equilibria (assuming the

axiom of choice)

but none that involve measurable selections of behavior by the players.

By measurable we mean with respect to the completion of the canonical

probability measure, e.g. all subsets of outer measure zero are measurable.

This example demonstrates that the existence of equilibria is as

much a philosophical issue as a mathematical one.

We consider

the double-shift Bernoulli probability space $B^{{\bf Z}^2}$,

where $T_1$ is the shift in the first coordinate,

$T_2$ is the shift in the second coordinate, and $x^{i,j}$ is the

$i,j$-coordinate of $x\in B^{{\bf Z}^2}$.

Let $C$ be a compact and convex set with

We present an example of a one-stage three player

game of incomplete information

played on a sequence space $\{ 0,1\} ^{\bf Z}$ such that

the players' locally finite

beliefs are

conditional probabilities of the canonical Bernoulli

distribution on $\{ 0,1\}^{\bf Z}$, each player has only two moves, the

payoff matrix is determined by the $0$-coordinate and

all three players know that part of the payoff matrix pertaining to

their own payoffs. For this example there

are many equilibria (assuming the

axiom of choice)

but none that involve measurable selections of behavior by the players.

By measurable we mean with respect to the completion of the canonical

probability measure, e.g. all subsets of outer measure zero are measurable.

This example demonstrates that the existence of equilibria is as

much a philosophical issue as a mathematical one.

We consider

the double-shift Bernoulli probability space $B^{{\bf Z}^2}$,

where $T_1$ is the shift in the first coordinate,

$T_2$ is the shift in the second coordinate, and $x^{i,j}$ is the

$i,j$-coordinate of $x\in B^{{\bf Z}^2}$.

Let $C$ be a compact and convex set with

compact subsets $(A_b\ | \ b\in B)$ indexed by the

set $B$ such

that $\cap_{b\in B} A_b =\emptyset$.

We conjecture that measurable functions

$f:B^{{\bf Z}^2}\rightarrow C$ can not keep

${1\over 4}(f(x)+f(T_1(x))+ f(T_2(x))+ f(T_1\circ T_2 (x)))$

in $A_{x^{0,0}}$ for all $x\in B^{{\bf Z}^2}$ and that the inability of

measurable functions to satisfy this property (in expectation) is

bounded below by a positive constant dependent on

the sets $(A_b\ | \ b\in B)$.

We give an example of a one-stage zero-sum game played on $B^{{\bf Z}^2}$

that would not have a value (but would have equilibria!) if

this conjecture were valid.