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Econ-568 · economics · Economics

Learning-based equilibrium selection

LLM status Open unverified

Statement

Original subject group: Game theory and strategic interaction

Audit classification: Published research agenda. Fudenberg–Levine pp.153–154 motivates equilibrium prediction through learning. The finite-horizon selection model is editorial.

Definitions and precise research target

Model and research target. Consider a preregistered class of finite coordination games $G$ with a known set $E_G$ of Nash equilibria. Groups of human players repeatedly interact for $T$ periods under randomly assigned feedback and initial-history treatment $Z$. Let $\widehat\sigma_T$ be the product of empirical action distributions, and fix disjoint neighborhoods $U_e$ around isolated equilibria $e\in E_G$. Define the selection probabilities \[ q_e(G,Z,P,T)=\Pr_P(\widehat\sigma_T\in U_e\mid do(Z)), \] where $P$ is the sampling population; preserve a residual category for paths lying outside every neighborhood. The target is to estimate these probabilities and identify a portable model relating them to learning-rule heterogeneity, payoff risk, information, and initial experience. Separate a treatment's effect on equilibrium selection from its effect on the speed of approaching any equilibrium by measuring several prespecified horizons. Elicit individual beliefs or learning parameters with measurement error explicitly modeled, and test the selection rule on new games and populations. Random assignment identifies treatment effects within sampled groups; transporting results requires stated population-overlap assumptions.

Definitions and maintained assumptions (editorial unless a source definition is named). For each coordination game select a finite set $E_G$ of isolated Nash equilibria; a game with a continuum of equilibria requires a different target. Use the $\ell^1$ metric on product strategy space and radii $r_e>0$ making $U_e=\{\sigma:\|\sigma-e\|_1<r_e\}$ pairwise disjoint. The event probabilities refer to independently sampled groups in population $P$ assigned treatment $Z$; $do(Z)$ is this randomized group assignment. Define the residual probability $q_0=1-\sum_e q_e$. It includes nonconvergence, other equilibria and points on the neighborhood boundaries. Specify empirical profile construction under fixed roles and the number of periods $T$. A portable model is a prespecified map from game features, treatment and participant measurements to the entire vector $(q_0,(q_e)_e)$, evaluated by a common proper scoring rule on unseen games. Randomized treatment identifies sampled-population selection probabilities, not structural learning heterogeneity without a measurement model.

Variants and assumption changes

  1. Two pure equilibria (editorial). Fix two-by-two coordination games and vary payoff risk and initial play independently. Estimate changes in basin-selection frequencies and time to enter either neighborhood.
  2. Heterogeneous learners (editorial). Admit a finite mixture of specified learning rules and imperfect belief reports. Determine which treatments identify mixture weights and which selection predictions remain partially identified when distinct rules generate the same history law.

References and source audit

  1. pp.153–154 and conclusion pp.165–166. Supports the stated research area; exact displayed specification is editorial. https://economics.mit.edu/sites/default/files/2022-09/Whither%20Game%20Theory%20Towards%20a%20Theory%20of%20Learning%20in%20Games.pdf
  1. Drew Fudenberg; David K. Levine. 2016. Whither Game Theory? Towards a Theory of Learning in Games. Introduction, p.153; conclusion, pp.165–166. https://economics.mit.edu/sites/default/files/2022-09/Whither%20Game%20Theory%20Towards%20a%20Theory%20of%20Learning%20in%20Games.pdf. DOI: https://doi.org/10.1257/jep.30.4.151.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-568-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-568

Learning-based equilibrium selection: Selection fibers and a two-horizon separation of speed

Target and statistical object

Fix finite coordination games, independent experimental groups, randomized feedback regimes, and disjoint open neighborhoods $U_e$ of selected isolated equilibria. The target uses the empirical mixed profile, not the last action: \[ q_e(G,z,T)=\Pr\{\widehat x_T\in U_e\mid\operatorname{do}(z)\}, \qquad q_0=1-\sum_e q_e. \] The learning and equilibrium-selection agenda in [fl, pp. 153--154,165--166] motivates the problem; the finite-mixture and timing results below are explicit restrictions, not claims that the source identified a universally portable learning rule.

Exactly what a battery identifies

Consider $K$ fully specified learning types. A type specifies initial conditions, updating, action randomization, and the conditional measurement law of every reported belief. Types have shared weights $w\in\Delta_K$ across a fixed battery of games and randomized regimes. Observed histories are finite, including finitely coded belief reports. For each battery arm $d$ and observable history $h$, let $A_{(d,h),k}$ be its probability under type $k$. Let $p_{d,h}$ be the population history probabilities identified by random assignment with positive arm probabilities. Thus $p=Aw$. For a new, fully specified game and horizon let $C_{e,k}$ be the type's probability of the actual empirical-profile classification, including $e=0$.
Theorem: Under the stated finite-mixture model the sharp joint identified set for new-game classification probabilities is \[ \{Cw:w\ge0,\ \mathbf1^{\mathsf T}w=1,\ Aw=p\}. \tag{1} \] Let $J$ contain those types receiving positive mass in at least one feasible weight vector. A scalar prediction $c^{\mathsf T}w$ is identified if and only if $c_J$ belongs to the row space of the matrix obtained by stacking $A_J$ and $\mathbf1_J^{\mathsf T}$.
Proof: The law of total probability gives both linear maps. Every feasible $w$ defines a population mixture with exactly the observed history laws and the displayed new-game prediction, proving both necessity and sharpness. Take, for each $j\in J$, one feasible vector with positive $j$th coordinate and average them. The resulting $w^*$ is positive on every coordinate of $J$. Every vector $d$ on $J$ satisfying $A_Jd=0$ and $\mathbf1_J^{\mathsf T}d=0$ can therefore be added and subtracted with a sufficiently small coefficient while preserving feasibility. A prediction is constant on the feasible set precisely when it annihilates this nullspace: necessity follows from these two perturbations and sufficiency from subtracting any two feasible vectors. The nullspace's orthogonal complement is the stated row space. ∎
Each coordinate bound follows from a finite linear program. Their Cartesian product need not equal (1); a common weight vector must generate every coordinate. The active-face qualification matters when data force some types to have zero mass. Full column rank is sufficient for identifying weights, but is unnecessarily strong for identifying a particular held-out prediction. Belief reports help only by adding genuinely different columns after applying their measurement kernel. An invertible-looking matrix of unobserved true beliefs is not enough.

A separation using the prescribed empirical profile

Here is a concrete two-equilibrium model in which observing two horizons separates eventual choice from speed. Both players have actions $0,1$, and the two selected equilibria are $00$ and $11$. Until commitment they play the off-diagonal profile $01$. A fraction $w_e$ of groups is destined to equilibrium $e$, where $w_{00}+w_{11}\le1$. Such a group plays $L$ off-diagonal rounds, then $e$ forever, with \[ \Pr(L=l\mid e)=\rho_e(1-\rho_e)^l, \qquad l=0,1,\ldots,\qquad 0<\rho_e\le1. \] The remaining groups play $01$ forever. Use radius $1/2$ in the sum of the two players' marginal $\ell^1$ distances. These two neighborhoods are disjoint. Commitment types and hazards may depend on the randomized regime, but the comparison below holds within one regime.
Proposition: For each equilibrium $e$ and integer $T\ge1$, \[ q_e(T)=w_e\{1-(1-\rho_e)^{\lceil T/4\rceil}\}. \tag{2} \] If $q_e(4)>0$, then the two horizons $4,8$ identify \[ \rho_e=2-\frac{q_e(8)}{q_e(4)},\qquad w_e=\frac{q_e(4)}{\rho_e}. \tag{3} \]
Proof: If a group destined to $e$ has committed by $T$, its empirical profile is at distance $2L/T$ from $e$: exactly one player's marginal changes from the off-diagonal action to its equilibrium action. The open-neighborhood condition is $L<T/4$. If commitment has not occurred, the distance is two, and the same inequality fails. A group destined to the other equilibrium cannot enter this neighborhood because one of its players always uses the wrong pure action. Hence summing the geometric probabilities for $l=0,\ldots,\lceil T/4\rceil-1$ proves (2). At $T=4,8$ the two probabilities are $w_e\rho_e$ and $w_e(2\rho_e-\rho_e^2)$; division yields (3). ∎
If $q_e(4)=0$, positive hazards imply $w_e=0$, and the unused hazard is unidentified. Allowing zero hazards makes even the latent destined mass unidentified. Observed values violating $1\le q_e(8)/q_e(4)<2$ reject this positive-hazard submodel rather than supplying an alternative interpretation of its parameters. The result is deliberately about empirical neighborhoods: using the last action would give a different clock and different formulas.

The finite-horizon obstruction

Proposition: Without restrictions on the continuation of learning, complete history laws through any finite $T_0$ do not identify eventual equilibrium selection, even when every group eventually converges to a pure equilibrium.
Proof: Take any common deterministic action and report history through $T_0$. In the first continuation every group plays $00$ forever from $T_0+1$; in the second every group plays $11$ forever. Generate identical reported beliefs during the observed period under both models. All observed laws, and hence every classification probability at observed horizons, coincide. Their empirical profiles converge respectively to $00$ and $11$, so their eventual selection distributions differ maximally. Mixtures of these continuations produce every intermediate eventual probability. ∎

Remaining empirical work

Equations (1)--(3) provide identified sets and a testable speed/selection separation for declared rules. They do not identify an unknown population's learning technology, establish stable weights across populations, or predict arbitrary new games. A portable model must commit to new-game history kernels before evaluation. Log scores require positive predicted probability on the evaluation support; any clipping or smoothing belongs in the prespecified predictor. Randomization identifies finite-horizon outcomes within the sampled population, while portability and measurement validity remain additional empirical restrictions.

Completion Estimate

65%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt characterizes sharp finite-mixture selection predictions and active-face identification, separates speed from eventual choice in a timing model, and proves a finite-history obstruction. Full portable equilibrium selection still requires validated learning and belief-report kernels, stable population weights, and independently scored predictions for unseen games, treatments, and horizons.

References

  • [fl] D. Fudenberg and D. K. Levine (2016), Whither Game Theory? Towards a Theory of Learning in Games, Journal of Economic Perspectives 30(4),151--170. https://doi.org/10.1257/jep.30.4.151.

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