State, Anarchy and Collective Decisions by Alex Talbot Coram (auth.)

By Alex Talbot Coram (auth.)

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Think of starting with f …mÃà † and taking away a contribution. f …mÃà À 1† is now equal to f …mÃà † À c. At m ˆ mà the curves intersect again at df =dm ˆ c. For m > mà ; f …m† À c < f …m À 1†: We can ignore points around mÃà for this argument. 6 Continuous pay-off function that violates vi (sdi , sdÀi ) > vi (sci , sdÀi ). It is easy to see that one Nash equilibrium for this game is for mà of the population to contribute to providing the good. Any additional individual gets a smaller pay-off than the no contribute pay-off, and any member of mà gets less if she withdraws.

This means that the value of  cannot be too small. As noted, the usual assumption in the Hobbesian game is that domination is the best outcome and being dominated is the worst. For example, let c ˆ 2; a ˆ 1 and b ˆ À3 and the probability that the other player is tit-for-tat be 12. For tit-for-tat to be the best ®rst move for player one, the probability that the game will be played again for every round must be given by 1 2 ! Àb…1 À †=‰a À …b ‡ c†…1 À †Š . This gives  ! 8. 8 It gets the label `backward' because the game unravels from the end point.

Let the good be order and the pay-offs vi …sci ; scÀi † ˆ a; vi …sdi ; scÀi † ˆ c, and vi …sdi ; sdÀi † ˆ 0. vi …sc1 ; sdÀi † ˆ b, and a; c > 0 and b < 0. 3). Let the probability that the opponent is a tit-for-tat player be p; p P ‰0; 1Š F F F. 3 Pay-offs with tit-for-tat and defecting players p‰a=…1 À †Š ‡ …1 À p†b ! pc; which requires p ! ii) has some interesting features. Observe that it requires a ! c…1 À † for p 1. By de®nition c > a. This means that the value of  cannot be too small. As noted, the usual assumption in the Hobbesian game is that domination is the best outcome and being dominated is the worst.

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