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I do understand the gist of what infinitely repeated games are in that T=$\infty$ ; that is the stage game is played each period for an infinite number of periods.

In his book "Strategy", Watson makes the following claim:

Although such a game may not seem realistic at first (people do not live forever), infinitely repeated games are useful for modeling some real-world situations.

With reference to the part in bold, could you elaborate?

Thanks.

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In most situations it is not clear when a game ends, it is just clear that at every point in time there is a probability that this is the last period of them game.

Just try to say what is this $T$ for the life of a human? 150 years? Even if you are 150 years old there is a (probably small) chance that you will survive another year. And the next year? There is also another chance to live another year, and so on.

As long as there is no fixed endpoint to the game, infinitely repeated games are, in my opinion, more realistic as long as there is some chance of ending the game included (i.e., as a discount rate).

Infinitely repeated games are also a way to resolve some problems backward induction (for example in the Chain-store paradox). You really have to know that there is a very last situation to use backward induction.

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