TITLE: Representing a real number as the value of a countably infinite game
QUESTION [9 upvotes]: Is it true that for any real number $p$ between 0 and 1, there exist finite or infinite sequences $x_m$ and $y_n$ of positive real numbers, and a finite or infinite matrix of numbers $\varphi_{mn}$ each of which is either 0 or 1, such that 
1) $\sum x_m=1$
2) $\sum y_n=1$
3) $\forall n\;\sum x_m\varphi_{mn}=p$
4) $\forall m\;\sum y_n\varphi_{mn}=p$

REPLY [5 votes]: Generalization of my previous answer.  Is this right?  $p$ irrational...
For $0\lt p \lt 1/2$ let $x_1(p)=y_1(p)=p$, $\varphi_{1,1}(p) = 1$,
$\varphi_{1,j}(p)=\varphi_{j,1}(p)=0$ for $j>1$,
$x_n(p)=(1-p)x_{n-1}(p/(1-p))$ for $n>1$. And
$\varphi_{n,m}(p) = \varphi_{n-1,m-1}(p/(1-p))$.
For $1/2 \lt p \lt 1$ let $x_n(p)=x_n(1-p)$ and $\varphi_{n,m}(p) = 1-\varphi_{n,m}(1-p)$.