The Cardinality of Infinite Games

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.

💡 Research Summary

The paper sets out to give a rigorous mathematical treatment of a class of games it calls “infinite games.” An infinite game is defined as a play involving a fixed, finite set of players whose moves are organized into a repeated pattern called an iteration; each iteration consists of a finite sequence of moves followed by a payoff (imputation) and the whole pattern is repeated endlessly. Within this broad definition the author distinguishes two sub‑classes: strong infinite games and weak infinite games.

In a strong infinite game the payoff at the end of each iteration must be unique – no two iterations may generate the same imputation. This uniqueness condition forces every iteration to occupy a distinct state in the game’s history. By contrast, a weak infinite game imposes no such restriction; the same payoff vector may appear in multiple iterations, allowing repetitions in the outcome sequence.

The central technical result is that both strong and weak infinite games contain a countably infinite number of moves (or turns). The proof proceeds by constructing a bijection between the set of iterations and the natural numbers ℕ. For strong games the uniqueness of payoffs guarantees that each iteration can be assigned a distinct natural number; for weak games the same assignment is possible because the underlying structure of moves – independent of the payoff – is still a finite sequence repeated ad infinitum. Consequently the union of the two families – the set of all infinite games – is also countably infinite.

From this cardinality result the author derives an “ordinality of turns” property: the turns of an infinite game are not merely an unstructured infinite set, but they inherit the well‑ordered structure of the natural numbers. This ordering provides a foundation for defining time‑dependent concepts such as strategy profiles, sub‑game perfect equilibria, and dynamic consistency in a setting where the horizon never ends. In strong infinite games the uniqueness of each iteration makes it possible to treat every turn as a distinct sub‑game, thereby allowing the classic sub‑game perfect equilibrium concept to be applied without modification. In weak infinite games, because payoffs may repeat, equilibrium concepts may need to be relaxed or reinterpreted, but the underlying turn order remains well‑defined.

The paper also situates its contribution relative to existing literature on infinite repeated games and dynamic games. Traditional infinite repeated games assume a single stage game that is played over and over, often introducing a discount factor or averaging payoffs over time to obtain equilibrium results. The present framework, by allowing each iteration to have its own (potentially distinct) payoff structure, generalizes this setting. It can model scenarios where external conditions, rules, or payoff functions change from one iteration to the next – a feature absent from the classic repeated‑game model.

Beyond the formal definitions, the author discusses the implications of countable cardinality and natural‑number ordering for future research. Because the set of turns is countable, standard mathematical tools from set theory, combinatorics, and analysis remain applicable. Researchers can therefore extend familiar solution concepts (Nash equilibrium, Pareto optimality, etc.) to infinite games by working with sequences indexed by ℕ. Moreover, the strong‑game subclass offers a fertile ground for studying long‑run cooperation, punishment strategies, and the emergence of complex dynamic patterns, since each iteration can be uniquely identified and conditioned upon.

In summary, the paper provides a clean axiomatic definition of infinite games, separates them into strong and weak variants, proves that every such game possesses a countably infinite sequence of turns, and shows that this endows the game with a natural‑number order. This order supplies the essential scaffolding for any subsequent game‑theoretic analysis, opening the door to a richer theory of games that extend indefinitely while retaining a mathematically tractable structure.