Consider a game with the following payoff matrix:

```
3,5   0,0   0,0
0,0   5,3   0,0
0,0   0,0   0,0

```

Suppose the game is played infinitely many times, and both players have discount factor $\delta$. 

The players want to create the outcome $4,4$ using only pure strategies. Intuitively, they should alternate between the (3,5) and the (5,3). This will give the row player:

$$(1-\delta)\sum\_{t=0}^\infty \delta^{2t}\*3 + \delta^{2t+1}\*5 = \frac{3+5\delta}{1+\delta} $$

and the column player:
$$(1-\delta)\sum\_{t=0}^\infty \delta^{2t}\*5 + \delta^{2t+1}\*3 = \frac{5+3\delta}{1+\delta} $$

The payoff vector goes to $(4,4)$ when $\delta \to 1$, but otherwise it is not exactly $(4,4)$.

I am looking for references about the following questions: in what conditions is it possible to attain exactly a desired payoff vector with only pure strategies (for an arbitrary game and arbitrary number of players)? And how this payoff vector can be constructed?

I looked at some papers about folk theorems for discounted infinitely-repeated games. The problem is that they usually assume that $\delta\to 1$, which is not always true in practice.