Golden games

We consider extensive form 2-player win-lose games, with alternating moves, of perfect and complete information. The games are played over a complete binary-tree of depth n, where 0/1 payoffs in the leaves are drawn according to an i.i.d. Bernoulli distribution with probability p. Whenever p differs from the golden ratio, asymptotically as n→∞, the winner of the game is determined. In the case where p equals the golden ratio, we call such a random game a golden game. In golden games the winner is the player that acts first with probability equal to the golden ratio. We suggest the notion of fragility as a measure for “fairness” of a game's rules. Fragility counts how many leaves' payoffs should be flipped in order to convert the identity of the winning player. Our main result provides a recursive formula for asymptotic fragility of golden games. Surprisingly, golden games are extremely fragile. For instance, with probability ≈0.77 a losing player could flip a single payoff (out of 2ⁿ) and become a winner. With probability ≈0.999 a losing player could flip 3 payoffs and become the winner.