Solution of Wald's game using loadings and allowed strategies

TL;DR

This paper introduces a new approach using loadings and allowed strategies to resolve Wald's game, demonstrating its fairness and providing a natural solution, along with a novel concept of embedding games.

Contribution

It presents a new interpretation of Wald's game through loadings, proves the existence of a natural fair solution, and introduces the concept of embedding games into each other.

Findings

Wald's game admits a natural fair solution.

The concept of loadings clarifies strange phenomena in Wald's game.

An example of a game loadable in infinitely many ways is provided.

Abstract

We propose a new interpretation of the strange phenomena that some authors have observed about the Wald game. This interpretation is possible thanks to the new language of \emph{loadings} that Morrison and the author have introduced in a previous work. Using the theory of loadings and allowed strategies, we are also able to prove that Wald's game admits a \emph{natural} solution and, as one can expect, the game turns out to be fair for this solution. As a technical tool, we introduce the notion of \emph{embedding a game into another game} that could be of interest from a theoretical point of view. \emph{En passant} we find a very easy example of a game which is loadable in infinitely many different ways.