Golden games

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We consider extensive form win-lose games over a complete binary-tree of depth n where players act in an alternating manner. We study arguably the simplest random structure of payoffs over such games where 0/1 payoffs in the leafs 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 that is 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.