Golden games

TL;DR

This paper analyzes binary-tree win-lose games with random payoffs, revealing that 'golden games' where payoff probability equals the golden ratio are highly fragile, with small payoff changes often flipping the winner.

Contribution

It introduces the concept of fragility in golden games and derives a recursive formula for their asymptotic fragility, highlighting their extreme sensitivity to payoff alterations.

Findings

Golden games are highly fragile, with small payoff flips often changing the winner.

Asymptotic fragility can be computed recursively for golden games.

Probability of flipping a few payoffs to change the winner approaches 1 as game depth increases.

Abstract

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.