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Econ-256 · economics · computer science · Computational economics

Learning in games and convergence

LLM status Open unverified

Statement

Original atlas ID: M2. Formulation: editorial specification of a research family.

Definitions and assumptions

For a finite game $G$ with actions $A_i$ and bounded utilities $u_i\in[0,1]$, let $a^t=(a_i^t)_i$ be play at dates $t=1,2,\ldots$. Player $i$ uses an adapted randomized policy, depending only on a specified information filtration $\mathcal F_i^t$. Define empirical joint play $\mu_T=T^{-1}\sum_{t=1}^T\delta_{a^t}$. External regret and swap regret are respectively \[ R_i^{\rm ext}(T)=\max_{b\in A_i}\sum_{t=1}^T[u_i(b,a_{-i}^t)-u_i(a^t)],\qquad R_i^{\rm swap}(T)=\max_{\phi:A_i\to A_i}\sum_{t=1}^T[u_i(\phi(a_i^t),a_{-i}^t)-u_i(a^t)]. \] Regret statements concern the positive part of these quantities. A coarse-correlated equilibrium has no profitable fixed-action deviation under its joint law; a correlated equilibrium has no profitable deviation conditional on an action recommendation.

Formal research question

For a prespecified class $\mathcal L$ of heterogeneous learning policies, information structures, and permitted switching processes, identify the subclasses for which empirical play approaches the coarse-correlated or correlated-equilibrium set, and obtain valid finite-horizon rates. Separately ask whether the conditional mixed-action profile $p^t=(\Pr(a_i^t=\cdot\mid\mathcal F_i^t))_i$ approaches the Nash set under a clearly specified common history and conditional independence protocol. For example, seek $r_T\to0$ with \[ \sup_{L\in\mathcal L}\mathbb E_L\!\left[\operatorname{dist}(\mu_T,\mathcal E(G))\right]\le r_T, \] where $\mathcal E(G)$ is the declared equilibrium set and distance uses a fixed norm. Determine when such a uniform statement fails and whether measurable switching restrictions restore it.

Interpretation and scope

The distinction between empirical averages and last-iterate behavior is indispensable. Vanishing external regret implies that accumulation points of empirical play satisfy the coarse-correlated inequalities; vanishing swap regret supplies the correlated inequalities. Neither implication gives convergence of current play to Nash equilibrium. With finite actions, these statements follow by dividing the relevant deviation inequalities by $T$ and taking subsequential limits. Monitoring restrictions, changing payoffs, and switching between algorithms must enter $\mathcal L$; otherwise an arbitrary learner can cycle or keep choosing dominated actions. Empirical work should measure predictions at the same horizon and observation level as the mathematical result, and compare rules on held-out games.

Source audit and status

Fudenberg--Levine (1998) is a verified monograph. Hart--Mas-Colell (2000) already proves almost-sure convergence of empirical play to the correlated-equilibrium set for its regret-matching procedure. Young (1993) studies a specific adaptive convention process. Thus convergence to correlated equilibrium is not itself an unsolved general benchmark, and these references do not imply last-iterate Nash convergence. The open component is a restricted characterization and empirically disciplined analysis of heterogeneous learning and switching. The class $\mathcal L$, uniform rate, and prediction protocol are editorial specifications; an actual theorem must state their restrictions and exhibit its proof or counterexample.

References

  1. S1 Drew Fudenberg and David K. Levine (1998). The Theory of Learning in Games. MIT Press. https://mitpress.mit.edu/9780262061940/the-theory-of-learning-in-games/. Locator: publisher description; learning models. Establishes: Theory relating adaptive behavior and equilibrium. Does not establish: unconditional convergence of every adaptive process.

  2. S2 Sergiu Hart and Andreu Mas-Colell (2000). A Simple Adaptive Procedure Leading to Correlated Equilibrium. Econometrica 68(5), 1127--1150. https://onlinelibrary.wiley.com/doi/10.1111/1468-0262.00153. Locator: abstract; regret-matching convergence result. Establishes: Almost-sure convergence of empirical play to the correlated-equilibrium set for the stated procedure. Does not establish: last-iterate convergence to Nash equilibrium.

  3. S3 H. Peyton Young (1993). The Evolution of Conventions. Econometrica 61(1), 57--84. https://www.econ2.jhu.edu/people/young/Publications.html. Locator: article; perturbed adaptive coordination process. Establishes: Stochastic selection of conventions under a particular learning process. Does not establish: a universal behavioral law.

Common conventions: Source mathematical conventions

Unless an entry states otherwise, agent and firm sets are finite. State and action spaces are measurable, strategies and outcome maps are measurable, probability laws are specified, and expectations exist for the targets under discussion. Infinite-horizon discount factors lie in $(0,1)$ and discounted objectives are finite. Norms, tolerances and complexity input sizes must be specified for computational comparisons. Constants or primitives that appear locally are inputs, not empirical facts asserted by this catalogue.

Identification. A statistical model $m\in\mathcal M$ implies an observable law $P_m$ and target $\theta(m)$. At law $P$, its identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Point identification holds when this set is a singleton, or equivalently when $P_{m_1}=P_{m_2}$ implies $\theta(m_1)=\theta(m_2)$ for all compatible models. Sharp bounds mean bounds attained, or approached when the boundary is not attained, by compatible models. Writing this set is a definition, not a solution: the substantive task is to establish informative restrictions and derive or estimate the set. An unrestricted-confounding model may give only trivial bounds.

Inference. Identification, estimability and valid uncertainty quantification are separate questions. Fix $\alpha\in(0,1)$. A confidence procedure $C_n$ over a declared law class $\mathcal P$ has asymptotic uniform coverage at level $1-\alpha$ when \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P}\Pr_P\{\theta(P)\in C_n\}\geq1-\alpha. \] For set-identified targets the coverage event must be defined separately, for example coverage of the full identified set. If an entry uses identification-indexed models instead of uniquely indexed laws, the same coverage convention is applied over those models. Pointwise limits do not establish uniform validity, and asymptotic validity does not guarantee finite-sample precision. Sample sizes, dependence structures and weak-identification sequences are part of each application.

Counterfactuals. $Y_i(a)$ is a potential outcome under a specified intervention, including its timing, implementation and versions. It is not $Y_i$ conditional on voluntarily choosing $a$. A full intervention vector permits interference; shorthand scalar treatment notation does not silently impose no interference. Assignment, selection, attrition, anticipation and measurement must be specified. A claim about an instrument's local effect is not automatically a population policy effect.

Economic equilibrium. A counterfactual model includes best responses, constraints, feasibility, market clearing, state transitions and expectations consistent with the stated information. When several equilibria exist, either a declared selector is an input or the target is set-valued. Comparisons across policy regimes must use the stated selection convention. Reduced-form relationships are not presumed invariant after an equilibrium-changing intervention.

Optimization and welfare. Optimization is over the stated feasible set. If attainment is not established, $\sup$ or $\inf$ and $\varepsilon$-optimal choices replace an asserted maximizer or minimizer. For example, with value $V=\sup_{a\in\mathcal A}W(a)<\infty$, an $\varepsilon$-optimal set is $\{a\in\mathcal A:W(a)\geq V-\varepsilon\}$ for $\varepsilon>0$. Benefits, costs, risk and health or environmental outcomes can be aggregated only using declared units and conversion weights. Interpersonal utility weights, the treatment of fiscal transfers and foreign profits, discounting, and the behavioral welfare criterion are explicit inputs. A comparative-static derivative additionally requires existence and appropriate differentiability of the selected counterfactual.

Sources. Local labels [S1], [S2], and so forth restart in every question. Locators identify the relevant abstract, model, section or result at the level actually checked; a bibliographic or abstract-level verification is not represented as inspection of an entire proof. Working papers are distinguished from later publications and changing versions. Source summaries are paraphrased. All URL checks and editorial formulations refer to the audit date stated above.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-256-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-256

Learning through predictable algorithm switches

Problem ID: C73-252. Status: Unresolved. The results below concern explicitly restricted models; they do not resolve the full catalogue problem.

Question and scope

Can populations using different learning algorithms, and changing them over time, converge to equilibrium? This requires distinguishing empirical distributions from current play. We prove a finite-time result for full-feedback learning restarted at predictable switch times, and give a last-iterate obstruction. Regret matching and correlated equilibrium originate in [HM]; the reset analysis below is supplied for an explicitly restricted class.

Consider a finite normal-form game with $N$ players, finite action sets $A_i$ of size $m_i\ge2$, and payoffs in $[0,1]$. At round $t$, players choose distributions using the public past and then sample independently conditional on that past. Full feedback reveals $g_{i,t}(a)=u_i(a,a_{-i,t})$ for every $a\in A_i$. Player $i$ partitions time into blocks, with switches determined before drawing that round's actions. Let $K_i(T)$ be the number of completed switches by round $T$. We assume a deterministic upper bound $K_i(T)\le K_i$ when stating probability bounds.

An explicit reset-tolerant learner

Within each block run exponential weights in epochs of scheduled lengths $1,2,4,\ldots$, resetting weights at each epoch. At an epoch of scheduled length $L$, use uniform initial weights and learning rate $\eta=\sqrt{8\log(m_i)/L}$. A block-ending switch may truncate its final epoch.
Lemma (An epoch and a block): Against every full-feedback payoff sequence, including adaptive sequences, the learner's expected-action payoff satisfies, on a block of actual length $\ell\ge1$, \[ \max_{a\in A_i}\sum_{t\text{ in block}}g_{i,t}(a) -\sum_{t\text{ in block}}\langle p_{i,t},g_{i,t}\rangle \le C_0\sqrt{\ell\log m_i},\qquad C_0=4. \] Here the inequality is pathwise for the realized vectors $g_{i,t}$; it does not average over the opponents.
Proof: For an epoch of actual length $l\le L$, let $W$ be the sum of exponential weights. The inequality $\log\sum_a p_a e^{\eta g_a}\le\eta\langle p,g\rangle+\eta^2/8$ for $g_a\in[0,1]$ follows by the bounded-variable exponential bound: the second derivative of the log moment-generating function is a variance at most $1/4$, and integrate twice from zero. Comparing the terminal weight of any fixed action to the total weight gives regret at most $\log m_i/\eta+\eta l/8\le\sqrt{L\log m_i/2}$. The sum of square roots of the scheduled lengths through the final epoch is at most $\sqrt{2L}/(\sqrt2-1)$. Since entering that epoch requires at least $L-1$ prior rounds, $L\le\ell$. Therefore the total is at most $(\sqrt2-1)^{-1}\sqrt{\ell\log m_i}<4\sqrt{\ell\log m_i}$. The maximum of a sum is no larger than the sum of the blockwise maxima, which completes the argument. ∎

Let $\widehat\mu_T=T^{-1}\sum_{t\le T}\delta_{a_t}$ be the empirical joint-action law. A law $\mu$ is an $\epsilon$-coarse correlated equilibrium if \[ \sum_a\mu(a)[u_i(b,a_{-i})-u_i(a)]\le\epsilon \quad\text{for all }i,b\in A_i. \]

Theorem (Sublinear switching preserves empirical convergence): With probability at least $1-\alpha$, $\widehat\mu_T$ is an $\epsilon_T$-coarse correlated equilibrium, where \[ \epsilon_T=\max_i\left\{ 4\sqrt{\frac{(K_i+1)\log m_i}{T}} +\sqrt{\frac{\log(N/\alpha)}{2T}}\right\}. \] In particular, if $K_i(T)=o(T)$ deterministically, the coarse-correlated-equilibrium deviation inequalities vanish in probability.
Proof: Sum the block lemma and apply Cauchy--Schwarz to the block lengths to bound pseudo-regret by $4\sqrt{(K_i+1)T\log m_i}$. The difference between pseudo-payoff and realized payoff is \[ M_{i,T}=\sum_{t\le T}\left(\langle p_{i,t},g_{i,t}\rangle-u_i(a_{i,t},a_{-i,t})\right). \] Conditional on the past and the opponents' current actions, each increment has mean zero and range of length at most one. Conditional independence of current draws is used precisely here. The same exponential bound as in the lemma, followed by iteration and Markov's inequality, gives ${\mathbb{P}}(M_{i,T}\ge z)\le\exp(-2z^2/T)$. A union bound over players proves the formula after division by $T$. Each fixed-action deviation of $\widehat\mu_T$ is exactly the corresponding realized average regret. ∎

This implies distance to the equilibrium set in any fixed norm tends to zero in probability. To justify that implication without a quantitative polyhedral constant, let $r(\mu)$ be the maximum of zero and all displayed deviation expressions. On the compact probability simplex, it is continuous and its zero set is the coarse correlated equilibrium set $E$. If measures with $r\to0$ stayed a positive distance from $E$, a convergent subsequence would have a limit in $E$, a contradiction. The theorem gives a rate for the economically meaningful deviation incentives; it does not claim a game-uniform distance constant.

The same deterministic summation works for swap-regret blocks: if each block of length $\ell$ guarantees realized swap regret at most $B_i\sqrt\ell$ for every map $\phi:A_i\to A_i$, then total swap regret is at most $B_i\sqrt{(K_i+1)T}$. Consequently the empirical law satisfies every correlated-equilibrium deviation inequality with that divided by $T$. This is a conditional extension, not a construction or a claim that external-regret learners automatically control swap regret.

Two limitations that cannot be removed by terminology

Proposition (Empirical success need not imply convergence of current play): There is a two-player zero-sum game and a deterministic play path with bounded external and swap regret at every horizon, whose empirical distribution converges to a correlated equilibrium, but whose current mixed-action profile never approaches the unique Nash equilibrium.
Proof: Use matching pennies, with row payoff one on matches and zero otherwise, and column payoff its complement. Cycle through the four action pairs once each and repeat. In a full cycle, each player's actual payoff is two. Every fixed action also earns two. Conditional on either own action, the opponent uses each action once, so every swap map earns two as well. A partial cycle contains at most three rounds, and every payoff difference per round has absolute value at most one. Hence all cumulative regrets are at most three. The empirical law tends to the uniform law on the four pairs, a correlated equilibrium. At each date the current strategies are point masses. The unique Nash strategies are both uniform, so each current marginal is at $\ell^1$ distance one from its Nash marginal. Uniqueness follows because an opponent not mixing equally induces a strict pure best response, which cannot be part of equilibrium in matching pennies. ∎
The constructed path has low regret along itself; it is not asserted to be a universally no-regret algorithm. It suffices to show that realized regret inequalities alone cannot imply last-iterate convergence.

Switching restrictions also matter. In a one-player game with payoffs one and zero, resetting a uniform-initialized learner before every round yields expected regret $T/2$. Each uninterrupted exponential-weights instance has the block guarantee, but linear resetting prevents learning. Additional assumptions on policy retention, dwell times, feedback, and opponents are therefore substantive. Arbitrary heterogeneous switching and convergence to Nash equilibrium remain outside the result.

Completion Estimate

45%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt proves finite-time coarse-correlated incentive bounds for predictable full-feedback algorithm resets and shows why low regret does not ensure current-play Nash convergence. General heterogeneous learning classes, feedback restrictions, correlated-equilibrium constructions and uniform distance or last-iterate rates remain uncharacterized.

References

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