The Cardinality of Infinite Games

TL;DR

This paper rigorously analyzes infinite games, defining strong and weak types, and explores their properties, including the countability of moves and the implications for game structure and analysis.

Contribution

It introduces a formal distinction between strong and weak infinite games and establishes their countably infinite nature, providing foundational insights for game theory.

Findings

All infinite games have a countably infinite number of moves.

Strong and weak infinite games differ in move imputation rules.

The structure of infinite games underpins future theoretical analysis.

Abstract

The focus of this essay is a rigorous treatment of infinite games. An infinite game is defined as a play consisting of a fixed number of players whose sequence of moves is repeated, or iterated ad infinitum. Each sequence corresponds to a single iteration of the play, where there are an infinite amount of iterations. There are two distinct concepts within this broad definition which encompass all infinite games: the strong infinite game and the weak infinite game. Both differ in terms of imputations. The strong infinite game has a uniqueness qualification in that all moves must differ to the extent that no imputation (these occur at the end of any given iteration) may ever be the same. Conversely, there is no such qualification in a weak infinite game, any payout may equal another. Another property shared by strong and weak infinite games (apart from their fulfilling the criterion of an infinite game) is the fact that both consist of a countably infinite amount of moves. Therefore all infinite games have a countably infinite number of turns; the set of all infinite games is composed of each strong and weak infinite game. This result has a very important consequence: the ordinality of turns. That is, the moves of an infinite game have an order or structure which they adhere to. It is this structure which provides any future development or game theoretical analysis of these sorts of games with the necessary foundation.