He who has the largest real number in $[0,1]$ wins

Question

Let's play a game:

Let $X,Y \sim U (0,1)$ be random variables uniformly distributed over $[0,1]$.

The game is as follows:

1. I obtain a realization of $X$. You obtain a realization of $Y$.
2. If dissatisfied with the realization we obtained, each of us can keep realizing $X$ or $Y$, respectively, until we get a real number we like. I don't know how many realizations you got, and you don't know how many realizations I got. If we get a new realization, the past realizations are forgotten. Only the latest one matters.
3. Once we decide to play, we compare the latest realizations we got. Whoever has the largest real number wins $\$1$.

What is the optimal strategy? More precisely, what is the threshold $\gamma \in [0,1]$ such that one stops obtaining new realizations once one obtains one that exceeds it?

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Hints:

• Think of this as a simultaneous game that is played over infinitely many rounds, a bit like rock-paper-scissors.
• At each round of the game, the loser does not pay the winner $\$1$. Rather, some third party pays the winner $\$1$
• One can think of player $i$'s strategy as a function $\gamma_i : \mathbb{N} \to [0,1]$, where $\gamma_i (k)$ is the threshold used by player $i$ at round $k$.
• Each player can adjust his threshold based on the other player's past thresholds, i.e., $\gamma_i (k)$ can be a function of $\gamma_j (0), \gamma_j (1), \dots, \gamma_j (k-1)$, where $j \neq i$.

Answer 1

Here is a variant of your game:

Each player secretly writes down a number less than $1.0$. The player who wrote down the higher number wins.

I hope it's clear that this is a very silly game! But it is isomorphic to the one that you describe.