Learning in Games
Learning in games is the study of how boundedly rational agents adjust their strategies over repeated play of a game, and whether — and under what conditions — those adjustments converge to equilibrium behavior. It occupies the intersection of game theory, economics, and machine learning, and it provides the natural home for a genuine theory of expectation formation that macroeconomics has failed to develop.
The field emerged from a simple question: if players do not know the equilibrium of the game they are playing, can they learn it through experience? John Nash's existence proof for equilibrium was non-constructive — it showed that equilibria exist but said nothing about how agents might discover them. The learning literature steps into this gap, asking not what rational agents would do if they already knew the game, but what actual agents do when they must learn it from repeated interaction.
Classical Models: Fictitious Play and Its Discontents
The earliest learning model, fictitious play, was introduced by Brown and Robinson in the early 1950s. Each player maintains a belief about the opponent's mixed strategy, updating it as if all past observations were drawn from a fixed distribution. In each period, the player best-responds to this empirical frequency. The model is elegant but flawed: it assumes players believe the world is stationary even while their own behavior is changing it. Fictitious play converges to Nash equilibrium in zero-sum games and some coordination games, but it can cycle indefinitely in games like matching pennies or Shapley's example. The lesson is immediate: learning is not guaranteed to find equilibrium, and the conditions under which it does are narrower than economists hoped.
Evolutionary Dynamics and Population Learning
An alternative framing treats learning not as individual belief updating but as population-level selection. The replicator dynamics of evolutionary game theory model a large population of agents, each playing a fixed strategy, with growth rates proportional to relative payoffs. Strategies that outperform the population average grow; underperformers shrink. The replicator dynamics converge to Nash equilibrium in many games, but they can also converge to non-equilibrium cycles or chaotic attractors. More importantly, the evolutionary framing reveals a deep connection: learning in games is formally analogous to natural selection. Both are processes by which a population of strategies is filtered by environmental feedback, and both can get stuck in local optima or fail to discover superior equilibria.
Modern Learning: Regret and No-Regret Algorithms
The most influential contemporary framework is no-regret learning. An agent's regret is the difference between the payoff it actually received and the payoff it would have received had it played some fixed strategy throughout. A learning algorithm is no-regret if, over time, its average regret approaches zero. Regret matching, the simplest no-regret algorithm, adjusts probabilities in proportion to accumulated regrets: if a strategy has been underplayed relative to its performance, its probability increases.
No-regret learning is powerful because it makes minimal assumptions about the environment. The agent does not need to know the game structure, the opponent's strategy, or even whether the game is stationary. It need only observe its own payoffs. This makes no-regret learning the natural bridge between bounded rationality and strategic interaction: the agent is not solving the game; it is adapting to feedback.
The Parallel Crisis in Economics and AI
The learning literature in game theory has made more progress than macroeconomics, but the two fields have developed in parallel without cross-pollination. Macroeconomics remains trapped between rational expectations — which assumes agents know the model — and adaptive expectations — which assumes backward-looking rules of thumb. Game-theoretic learning offers a third path: agents who are forward-looking but boundedly rational, who form beliefs about opponents' behavior and update them strategically.
The same structure appears in multi-agent reinforcement learning. When multiple AI agents interact — in competitive markets, in game-playing systems, in social media recommendation engines — they are playing a game they do not fully understand, updating their strategies based on reward signals. The convergence properties of these systems depend on the learning rules, the information structure, and the presence of regime shifts that invalidate past experience. The macroeconomist who ignores game-theoretic learning is not just missing a theoretical tool. They are ignoring the closest thing we have to an empirically grounded theory of how agents actually form expectations in strategic environments.
The rational expectations revolution assumed that agents know the equilibrium because it could not model how they might learn it. Game-theoretic learning provides that model. The tragedy is that macroeconomics, having built its edifice on the rational expectations assumption, now treats the learning literature as a specialty topic in game theory rather than the foundation its own theory desperately needs. This is not intellectual division of labor. It is intellectual Balkanization — and the cost is paid in the unrealism of models that guide real policy.