During my summer study I have thought of the following problem. My knowledge about game theory is at the level of "Introduction to game theory" by Osborne.
Problem: Given a two-players repeated prisoner's dilemma with N repetitions, where N $\sim$ $F(\cdot)$ and $F$ is a common knowledge discrete probability distribution (for example Poisson($\lambda$)). The discount factor for each repetition is $\delta \geq 0$. What is the optimal strategy for both players?
On Wikipedia in the article about repeated games it is stated that:
"Repeated games may be broadly divided into two classes, depending on the horizon. [...] A game repeated a finite number of times may be regarded as having an infinite horizon if the players in the game do not know how many times the game will be repeated"
However, in the case where the number of repetitions has a known distribution, the players do have some knowledge about when the game will end. I have not found any papers on this kind of games.
- Does this game differ in any way from a finitely-repeated prisoner's dilemma?
- Are there any papers describing a repeated game with a number of repetitions following a given distribution?
I will expand here on Pete Caradonna comment.
As long as the $supp(F)=\mathbb{N}$ you can treat it as an infinitely repeated game with appropriately adjusted discount factor.
To be more precise, let $n \in \mathbb{N}$ denote the current round of play. Then, players will discount next period with a discount factor $\delta \times \mathbb{P}(N \geq n+1|N\geq n)$ rather than $\delta$, the payoff from interactions in round $n+2$ with a discount factor $\delta \times \mathbb{P}(N \geq n+2|N\geq n)$, and so on. If $N\sim Poisson(\lambda)$ simply compute required probabilities using Poisson distribution.
If you are interested in the lietarture on this topic, go to google scholar and search for "Uncertain-Horizon Repeated Game." There are several papers that treat that topic, though somewhat surprisingly they are all recent. But you can check literature review in those papers to find earlier work.