Golden games

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 $n\rightarrow \infty$, the winner of the game is determined. In the case where $p$ equals the golden ratio, we call such a random game a \emph{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 \emph{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 $\approx 0.77$ a losing player could flip a single payoff (out of $2^n$) and become a winner. With probability $\approx 0.999$ a losing player could flip 3 payoffs and become the winner.