“You will never meet the other player ever again”

Game-theoretic thought experiments often present the condition “you will never meet the other player ever again”, to abstract away questions of reputation or iterated games. For example, one might introduce the classical prisoner’s dilemma as follows:

“You have been chosen as one of the two players in a game show. Both players start with a pot of 400 euros. In the game, each player has the opportunity to press a button in secret. Pressing the button gives the presser an additional 100 euros, but reduces the other’s pot by 300 euros. After the game, you will never meet the other party again. Do you press the button?”

But this condition is insufficient in an important way! If I defect against Alice, Alice could draw the conclusion “oh, apparently people just defect, guess I will defect, too”. Even if Alice never plays again with me, she might later play with Bob, who even later does play with me, and through that my earlier defection can come back to bite me – more precisely, my earlier actions causally affect how my future opponents will play.

An essential property is the length of the causal cycle. It is possible to construct tournaments where, with $n$ players, the shortest such cycle is $n/2$ players. In practice, however, the world is a dense network where you’ll find cycles much shorter than four billion.

I’m a little surprised that I haven’t heard this argument before. To me that is an intuitively strong argument for why it’s worth acting prosocially, even if you wouldn’t run into the same person again.