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) $\forall m\;\sum y\_n\varphi\_{mn}=p$



Representing a real number as the value of a countably infinite game
**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 $\varphi\_{n,m}(p) = 1-\varphi\_{n,m}(1-p)$.


