Bounding Rationality by Discounting Time

Consider a game where Alice generates an integer and Bob wins if he can factor that integer. Traditional game theory tells us that Bob will always win this game even though in practice Alice will win given our usual assumptions about the hardness of factoring. We define a new notion of bounded rationality, where the payoffs of players are discounted by the computation time they take to produce their actions. We use this notion to give a direct correspondence between the existence of equilibria where Alice has a winning strategy and the hardness of factoring. Namely, under a natural assumption on the discount rates, there is an equilibriumwhere Alice has a winning strategy iff there is a linear-time samplable distribution with respect to which Factoring is hard on average. We also give general results for discounted games over countable action spaces, including showing that any game with bounded and computable payoffs has an equilibrium in our model, even if each player is allowed a countable number of actions. It follows, for example, that the Largest Integer game has an equilibrium in our model though it has no Nash equilibria or epsilon-Nash equilibria.

💡 Research Summary

The paper introduces a novel model of bounded rationality that explicitly incorporates computational time into players’ utilities by discounting payoffs according to the time required to compute actions. Traditional game‑theoretic analysis assumes agents can compute optimal strategies instantaneously, which leads to paradoxical predictions in settings where the underlying computational problems are hard. By applying an exponential discount factor e^{‑δ·t} (with δ a small, positive discount rate) to each player’s original payoff, the authors create a “time‑discounted” utility that more accurately reflects real‑world constraints.

The authors first illustrate the model with a simple two‑player game: Alice chooses an integer N, and Bob wins if he can factor N. Classical game theory predicts that Bob always has a winning strategy because, given unlimited computational power, he can factor any integer. In the discounted framework, however, Bob’s expected utility drops dramatically as the factoring time grows, while Alice’s utility remains essentially unchanged because generating N is cheap. The central theorem (Theorem 1) establishes a precise equivalence: under a natural assumption that the discount rate δ is sufficiently small, there exists a discounted Nash equilibrium in which Alice wins if and only if there exists a linear‑time samplable distribution D on which the Factoring problem is average‑hard. The proof is constructive in both directions. If such an equilibrium exists, any algorithm Bob might use must incur super‑polynomial expected time on D, implying average‑hardness. Conversely, given an average‑hard distribution, Alice can sample N from D; Bob’s best response will be a factoring algorithm whose expected running time is exponential, driving his discounted payoff to near zero, while Alice retains a positive discounted payoff, thereby forming an equilibrium. This result tightly links a cryptographic hardness assumption to the existence of a strategic advantage for the “hard‑to‑solve” player.

Beyond the specific factoring game, the paper develops a general existence theory for discounted games with countably infinite action spaces. The authors assume that each player’s payoff function is bounded and computable. They construct a sequence of finite‑action approximations, apply Kakutani’s fixed‑point theorem to obtain equilibria in each finite truncation, and then show that the limit of these equilibria yields a genuine Nash equilibrium in the original infinite game when utilities are discounted. The key technical insight is that the discount factor forces strategies that would otherwise require unbounded computation (or infinitely large actions) to have negligible utility, preventing the “infinite escalation” that blocks equilibrium existence in the classic setting.

As a concrete application, the paper revisits the “Largest Integer” game, where each player picks a positive integer and the larger integer wins. In the standard model this game has no Nash equilibrium because each player can always improve by picking a larger number, leading to an endless upward spiral. In the discounted model, however, the cost of writing or processing a larger integer grows with its size; beyond a certain threshold L(δ) the discounted payoff becomes essentially zero. The authors compute this threshold and prove that the profile (L(δ), L(δ)) constitutes a Nash equilibrium. Thus, a game previously deemed pathological acquires a well‑defined strategic solution once computational time is accounted for.

The paper concludes by discussing broader implications. Time‑discounted utilities provide a natural bridge between computational complexity theory and economic/game‑theoretic analysis, allowing average‑case hardness assumptions (e.g., for factoring, lattice problems, or other cryptographic primitives) to be interpreted as strategic advantages in games. The framework is flexible enough to incorporate other resources such as memory or energy via alternative discount functions, and it opens avenues for studying multi‑player, non‑zero‑sum, and repeated games under realistic computational constraints.

In summary, the authors demonstrate that incorporating a simple exponential time discount into payoff calculations resolves several long‑standing equilibrium existence issues, yields a direct correspondence between average‑case hardness and strategic dominance, and offers a powerful new tool for analyzing games where computation is a non‑trivial cost.