Robert Samuel Simon
Games of Incomplete Information, Ergodic Theory, and the Measurability of Equilibria
Preprint series: Mathematica Gottingensis
MSC:
90D10 Noncooperative games
Abstract: 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
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.
Keywords: Bayesian Equilibria, Belief Spaces, Sequence Spaces, Non-measurable Sets