Assume $n$ players have between $0$ and $k-1$ chips each. The total number of states for the game is then $N = k^n$, since there are $k$ options per player.
Now assume the number of chips is constant and predefined, say $s$, and players can only exchange chips between themselves. The number of possible states should drop drastically, reaching $N=1$ in the special cases where $s=0$ or $s=(k-1)n$.
What then is the general formula for the number of states given $k, n$ and $s$? is it still exponential?
You are asking for the coefficient of $x^s$ in the polynomial $$(1+x+\dots x^{k-1})^n =\Bigl(\frac{1-x^k }{1-x}\Bigr)^n= \Bigl(\sum_{j=0}^n {n \choose j} (-x^k)^{j}\Bigr)\cdot \sum_{\ell=0}^\infty {\ell+n-1 \choose n-1} x^\ell \,,$$ which is the convolution of the coefficients of the two power series in the product, evaluated at $s$:
$$\sum_{j \ge 0} {n \choose j} (-1)^{j} {s-kj+n-1 \choose n-1} \,.$$