Repeated games with decreasing marginal returns

Question

The standard analysis of repeated games assumes that the payoff of a player from a repeated game is a sum (or arithmetic mean, or discounted sum) of the payoffs in the basic games.

But what if the players have decreasing marginal returns?

For example, suppose the basic game is Matching Pennies. In each basic game, a player can either win 1 or lose 1. However, the average utility of a player from winning e.g. 10 times, is not 1 - it may be less than 1 if the player has decreasing marginal returns.

Are there references that deal with such repeated games?

Answer 1

It seems to me that this will not make a difference if you only consider pure strategies. Consider that player $i$'s winnings is $x$ and from this he gets utility $U_i(x)$. As long as $U_i()$ is increasing in $x$ player $i$ will prefer outcomes with larger winnings to lower ones, which is exactly what he would do if he had constant marginal returns.

Another way of forming this argument is that as long as $U_i()$ is increasing another utility function representing the same preferences over outcomes would be $\hat{U}_i(x) = x$. (You can perform a monotone transformation that transfers $U_i()$ to this.)