# Bachet's game with lottery moves

## TL;DR

This paper studies a probabilistic variant of Bachet's game, where moves are determined by lotteries, and shows that as the number of objects grows large, the probability of the first player winning approaches 50% under certain conditions.

## Contribution

It introduces a lottery-based variant of Bachet's game and proves that the first player's winning probability converges to 1/2 as the pile size increases.

## Key findings

- First player's winning probability tends to 1/2 as n increases.
- Convergence holds under nondegeneracy conditions on lotteries.
- The game's outcome becomes fair in the limit for large n.

## Abstract

Bachet's game is a variant of the game of Nim. There are $n$ objects in one pile. Two players take turns to remove any positive number of objects not exceeding some fixed number $m$. The player who takes the last object loses. We consider a variant of Bachet's game in which each move is a lottery over set $\{1,2,\ldots, m\}$. The outcome of a lottery is the number of objects that player takes from the pile. We show that under some nondegenericity assumptions on the set of available lotteries the probability that the first player wins in subgame perfect Nash equilibrium converges to $1/2$ as $n$ tends to infinity.