# Characterization of n-grade, m-variables polynomials $P$ over $\mathbb{R}$, with $P(x_1,\ldots,x_m)\in [0,1]$ if $\forall x_i \in [0,1]$ .

I'll write an introduction. This problem came to me while I was doing an experiment with the prisoners dilemma. I'm codifing the agents behavior in genes (I want to measure the correlation between the "stability/peace" of the system and many characteristics of the agents.

But there are three characteristics that are the most important in my experiment:

• Memory capacity (how many past games can remember the agents). Every "memory slot" represents a variable in the polinomial (which can be 0 or 1).
• Recognition of other individuals (another form of memory capacity). In case there are recognition of individuals, every game we associate a new variable to every memory slot to indicate if the current adversary was the adversary in the past game associated with the memory slot. Evidently, the values are again 0 or 1.
• "Cognitive sofistication": This is... the grade of the polynomial. Yes, this is not a good representation of cognitive sofistication, but it's better than nothing. In any case, I prefer polynomials over boolean operators because the "mutations" can be smoother, I can capture more complex behaviors and I can work with probabilistic values.

Then, the problem:

It's possible to find the characterization of n-grade, m-variables polynomials $P$ over $\mathbb{R}$, with $P(x_1,\ldots,x_m)\in [0,1]$ if $\forall x_i \in [0,1]$ ?

Why polynomials, why over reals? And, do you mean the 2 element set $\{0,1\}$ instead of the interval $[0,1]$? What should the value $P(x_1,..,x_m)$ indicate? Some kind of probablistic or what? – Berci Oct 13 '12 at 9:50
I mean $[0,1]$, not $\lbrace 0,1\rbrace$.Over reals because in many cases it's possible to have undefined values in many slots (example: in the first game, the slots of "older games" are undefined). And yes, the result should be the probability of betray the opponent. – castarco Oct 13 '12 at 12:54