Coin-Flipping Games Immune Against Linear-Sized Coalitions

Perfect information coin-flipping and leader-election games arise naturally in the study of fault tolerant distributed computing and have been considered in many different scenarios. This paper answers a question of Ben-Or and Linial by proving that for every $c < 1$ there are such games on n players in which no coalition of $cn$ players can influence the outcome with probability greater than some universal constant times c. (Note that this paper actually proves this statement only for all $c < \frac{1}{3}$, but since its universal constant is bigger than 3 the above is trivial for $c \geqslant \frac{1}{3}$.) This paper shows that a random protocol of a certain length has this property and gives an explicit construction as well.