Optimal strategy in the game Risk or Safety

TL;DR

This paper analyzes the two-player stochastic game 'Risk or Safety,' proving its unique solution and developing methods to determine optimal strategies, with practical solutions for larger game sizes using iterative techniques.

Contribution

It introduces an analytical approach to solve 'Risk or Safety' and proposes an iterative method for larger instances where direct solutions are infeasible.

Findings

Unique solution for the game established

Analytical method based on transition equations

Iterative method enables strategy determination for larger game sizes

Abstract

In the paper it is proven that the two-players turn-based stochastic game "Risk or Safety" has a unique solution. Both players need to play the same strategy if they want to maximize their winning chances. An analytical method based on the transition equations depending on the players decisions at each game situation is presented. However, the method requires the solution of many large linear equation systems, which makes the problem practically unsolvable for more than 6 coins. Hence, an iterative method is used to determine the optimal strategy for up to 20 coins. A look-up table gives the number of coins a player should toss at each possible game situation.