I shall begin by saying that I thought about this today in my Detection and Estimation Class while looking at the mse calculations for the MMSE estimator.

Let’s suppose we have a vector of discrete RVs player of elements playeri  with the following rule:

ijPr{playeri=playerj}=0

which means

ijplayeriplayerj

Suppose we are given a N×N square matrix R.
A square matrix has the operator Trace (tr) defined as the sum of the elements on its main diagonal:

tr(R)=Ni=1rii

Suppose that the value tr(R) is part of the set of values that playeri can assume.

Now suppose that the variable playerA assumes the value of tr(R). Indeed:

playerA=tr(R)

So playerA is Trace(R).

Let’s now suppose that also another random variable playerx, with xA wants to assume the role of Trace(R).

However,

Pr{playerx=tr(R)|playerA=tr(R)}=0

Therefore, the variable playerx cannot assume the value of Trace(R) and should proceed to choose another value, since playerA is already Trace(R).

Sorry about this stupid meme.

Meru.