Game theory

A fiendish cyberspace wizard has locked you and Serendip into a diabolical game with the following rules:

On each turn of the game, you and Serendip must choose, without knowing the other's choice, between cooperating with each other and trying to take advantage of each other.

Following every turn of the game, you and Serendip will each receive a certain number of gold coins, the number depending for each of you on the choices made by both of you.

Should you both decide to cooperate, you will each receive 3 gold coins.

If one of you has decided to cooperate but the other has opted for competition, the successful competitor will receive 5 gold coins and the unsuccessful cooperator none.

Finally, if you both decide to outdo one another, each of you will receive a single gold coin.

The game will continue until the wizard tires of it, and includes this additional stipulation:

Your chances of surviving to play the next turn are closely related to the average number of coins you have received on previous turns. If this average on a given turn drops below a critical number (chosen in an unknown way by the wizard), some unknown butInconceivably Foul Fatewill befall you. The same is true for Serendip, of course, but since neither of you knows the critical number, neither of you has any choice but to try on each and every turn to maximize your own income.

The game is afoot. What should you do? Try it out. Can you find a successful strategy? A better one? A best one? Try keeping a record of not only your own score with each strategy but Serendip's too.

Suppose you play this game indefinitely and each time it is played represents a time period, say 1 year. Given Serendip's strategy, under what conditions (value ofr) is it worthwhile for you to cheat (compete) just once?

Suppose you play this game indefinitely. Given Serendip's strategy, under what conditions (value ofr) is it worthwhile for you to cheat (compete) forever?

Suppose that there is a 30% chance that Serendip will refuse to play the game in any given time period. In Serendip's own words, "...some Inconceivably Foul Fate will befall you." If this occurs, the repeated game ends. How does this knowledge change your answers to (d) and (e) if r = 0.1? Should you cheat forever, or even just once? What is the basis for your answer (be specific)?

How do answers change if there is a 50% chance that the game will end instead? Cheat forever? Cheat just once? What is the basis for your answer (be specific)?