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Econ-493 · economics · Economics

Equilibrium selection in games

LLM status Open unverified

Statement

Original atlas ID: M1. Formulation: editorial specification of a research family.

Definitions and assumptions

Fix a class $\mathcal G$ of finite normal-form games with players $N=\{1,\ldots,n\}$, finite action sets $A_i$, and utilities $u_i:A\to\mathbb R$, where $A=\prod_iA_i$. For game $G$, let $\mathcal N(G)\subseteq\prod_i\Delta(A_i)$ be its mixed-strategy Nash-equilibrium set: $\sigma\in\mathcal N(G)$ if every $\sigma_i$ maximizes expected $u_i$ against $\sigma_{-i}$. A context $c$ specifies the protocol, information, communication opportunities, and participant population. A selection rule is a probability kernel $q(\mathrm d\sigma\mid G,c)$ supported on $\mathcal N(G)$. This is a maintained behavioral hypothesis, not a consequence of equilibrium existence. Let $H_{G,c}(\mathrm dz\mid\sigma)$ be a specified observation kernel linking a selected equilibrium to measured session outcomes $z$.

Formal research question

Given population outcome laws $P_{G,c}$ and a prespecified restricted family $\mathcal Q$ of selection kernels, characterize \[ \begin{aligned} \mathcal Q(P)=\bigl\{q\in\mathcal Q:\;& P_{G,c}(B)=\int H_{G,c}(B\mid\sigma)q(\mathrm d\sigma\mid G,c)\\ &\text{for all observed }(G,c)\text{ and measurable }B\bigr\}. \end{aligned} \] Under what experimentally enforceable restrictions is this set a singleton, and which predictions for unobserved games or contexts are common to all its elements? When observations reject equilibrium support, enlarge the model explicitly to approximate equilibria or transient behavior and test that enlargement. For a declared distribution over held-out games, estimate predictive risk with uncertainty that includes selection-rule estimation.

Interpretation and scope

Identification of latent equilibrium selection is separate from prediction of observed action frequencies. Two selection kernels can be observationally equivalent when the observation kernel does not distinguish their induced outcomes. A unique selected equilibrium is therefore neither necessary nor sufficient for a well-identified empirical prediction. Context restrictions must say whether participant composition, payoff scaling, and information precision may change. Cross-game extrapolation requires shared parameters or invariance assumptions that can be tested on strategically related environments. A useful study randomizes these contextual variables and evaluates the resulting rule on games excluded from estimation; it also records finite-horizon departures from stationarity.

Source audit and status

Nash (1951) establishes existence in finite games, not empirical selection. Harsanyi--Selten (1988) supplies a theoretical selection construction. Carlsson--van Damme (1993) gives selection results for particular noisy-information perturbations; the conclusions cannot be transferred to arbitrary perturbations. All three author--year references are verified through the primary records below. The atlas question is a broad predictive agenda: many specified selection problems are solved, and these old sources do not certify that one precisely stated conjecture remains open in 2026. The kernel model and identification target above are editorial formalizations; no claim is made that they were stated in these sources.

References

  1. S1 John Nash (1951). Non-Cooperative Games. Annals of Mathematics 54(2), 286--295. https://www.jstor.org/stable/1969529. Locator: equilibrium definition and existence argument. Establishes: Existence of mixed-strategy equilibrium in finite games. Does not establish: a universal empirical equilibrium-selection rule.

  2. S2 John C. Harsanyi and Reinhard Selten (1988). A General Theory of Equilibrium Selection in Games. MIT Press. https://mitpress.mit.edu/9780262582384/a-general-theory-of-equilibrium-selection-in-games/. Locator: publisher description; selection construction. Establishes: A systematic theoretical selection proposal. Does not establish: that its selection is invariably observed.

  3. S3 Hans Carlsson and Eric van Damme (1993). Global Games and Equilibrium Selection. Econometrica 61(5), 989--1018. https://research.tilburguniversity.edu/en/publications/global-games-and-equilibrium-selection/. Locator: abstract and two-by-two coordination analysis. Establishes: Selection under specified noisy private-information perturbations. Does not establish: selection for every game or every information perturbation.

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-493-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-493

Identifying equilibrium-selection mixtures by repeated sessions

Problem ID: C72-489.

A finite, experimentally interpretable restriction

The general target is the identified set of selection kernels under a specified observation kernel, including common predictions in new games or contexts. We analyze a finite equilibrium dictionary and an experiment that repeatedly observes the same selected equilibrium. We derive a sharp fiber description, a necessary-and-sufficient criterion for identified predictions, an explicit repetition length guaranteeing identification, and simultaneous finite-sample uncertainty. We also show why a finite observation window cannot identify an unrestricted continuum mixture.

For each observed game and context, suppose a known list of $K$ candidate equilibria is specified. The list may be the entire equilibrium set when finite, or a maintained restriction. A session selects index $k$ with probability $q_k$, then produces an observation in a finite alphabet $\mathcal Z$ according to the known column vector $h_k$. Write $H=(h_1,\ldots,h_K)$ and $q\in\Delta_K$. The population observation law is $p=Hq$. These are maintained sampling and support assumptions, not implications of Nash existence. Historical selection proposals and noisy-information selection theorems motivate restrictions but do not identify this $q$ without observation assumptions [hs,cvd].

Sharp fibers and identified predictions

Theorem: For the unrestricted finite-mixture family $q\in\Delta_K$, the sharp identified set is the polytope \[ Q(p)=\{q\geq0:Hq=p\}. \] Normalization is automatic because the columns of $H$ and $p$ sum to one. If $Q(p)$ is nonempty, let $S=\{k:\text{some }q\in Q(p)\text{ has }q_k>0\}$. For a given vector $\ell\in{\mathbb{R}}^K$, the target $\ell^\top q$ is identified at $p$ if and only if \[ \ell_S\in\operatorname{rowspan}(H_S). \] More generally, its sharp lower and upper bounds are the minimum and maximum of $\ell^\top q$ on $Q(p)$.
Proof: Every generating mixture must satisfy $Hq=p$, and every nonnegative solution is a valid selection distribution producing $p$. This proves sharpness. The set is a closed subset of the compact simplex, so the extrema exist.

If $\ell_S=b^\top H_S$ for some $b$, then every feasible $q$ vanishes outside $S$, and $\ell^\top q=b^\top p$ is constant. Conversely, if $\ell_S$ is not in the row span, elementary finite-dimensional orthogonality gives $d\in\ker H_S$ with $\ell_S^\top d\ne0$. For each $k\in S$, choose a feasible mixture positive at $k$ and average these finitely many mixtures. The resulting $q^\circ$ is strictly positive on $S$. Since columns of $H_S$ sum to one, $H_Sd=0$ implies $\sum_{k\in S}d_k=0$. Hence for sufficiently small $\epsilon>0$, both $q^\circ\pm\epsilon d$, extended by zero off $S$, are feasible mixtures with different target values. This contradicts identification and proves necessity. ∎

For a held-out outcome with a maintained common-mixture rule and known prediction vector $\ell$, Theorem Theorem applies directly. The common-mixture restriction across contexts must be stated. Without it, observations in one context put no restrictions on another context's freely varying selector.

Corollary (Observationally equivalent mixtures can predict identically): Failure to identify $q$ does not preclude identifying $\ell^\top q$. If $H$ has dependent columns, any $\ell$ in its row space is still identified at every compatible population law. Conversely, an interior compatible mixture admits unidentified predictions for every $\ell$ outside that row space.
Proof: Apply Theorem Theorem; in the interior case $S=\{1,\ldots,K\}$. The first assertion follows from the same row-space representation without an interior assumption. ∎

Repeated observations of one latent equilibrium

Suppose each session retains its selected equilibrium and yields $L$ conditionally independent draws from $h_k$. Independence is conditional on the equilibrium; unconditional observations within a session are generally dependent. Let $H^{(L)}$ have columns $h_k^{\otimes L}$.
Theorem (A finite repetition design): If $h_1,\ldots,h_K$ are pairwise distinct, then $H^{(L)}$ has full column rank for every $L\geq K-1$. Thus $K-1$ repeated observations per session identify all mixture weights. If two columns are equal, no number of such repetitions distinguishes the weights within that equal-column class.
Proof: There exists a vector $f\in{\mathbb{R}}^{|\mathcal Z|}$ such that the scalars $\theta_k=f^\top h_k$ are all distinct: each equality $f^\top(h_k-h_j)=0$ is a proper hyperplane, and a finite union of proper hyperplanes cannot fill the vector space. For completeness, take $f=(1,t,\ldots,t^{d-1})$ for $d=|\mathcal Z|$. Each pairwise difference becomes a nonzero polynomial of degree at most $d-1$. A finite set of their roots can be avoided by choosing an integer $t$.

From a session of length $L$, for each $r=0,\ldots,K-1$ observe the statistic $\prod_{s=1}^r f(Z_s)$, with empty product one. Its expectation conditional on $k$ is $\theta_k^r$. Thus a linear relation $\sum_k d_k h_k^{\otimes L}=0$ implies $\sum_k d_k\theta_k^r=0$ for all $r=0,\ldots,K-1$. The Vandermonde matrix is nonsingular. One direct proof uses interpolation polynomials $P_j(t)=\prod_{k\ne j}(t-\theta_k)/(\theta_j-\theta_k)$: taking the corresponding linear combination of the moment equations gives $d_j=0$. Hence all $d_k=0$, proving full column rank. Additional observations cannot reduce rank because marginalizing them recovers the length-$K-1$ law. Equal columns remain equal under every tensor power, proving the last assertion. ∎

The theorem is about a shared latent equilibrium within a session. If the equilibrium is redrawn independently at every date, the session law becomes $(Hq)^{\otimes L}$ and repetition adds no population identification beyond $Hq$. This difference is experimentally substantive.

A lower boundary for unrestricted equilibrium mixtures

Proposition: For every finite $L$, there is a finite game and two different selection distributions on its Nash equilibria that induce identical length-$L$ session laws under conditional independent play but different length-$(L+1)$ laws.
Proof: Use a two-player game where player 1 has two actions, player 2 has one, and every payoff is zero. Every Bernoulli mixed action $\theta\in[0,1]$ of player 1 is a Nash equilibrium. Select $L+2$ distinct numbers $\theta_1,\ldots,\theta_{L+2}$ in $(0,1)$. The $(L+1)\times(L+2)$ matrix with entries $\theta_k^r$, $r=0,\ldots,L$, has a nonzero null vector $d$. Since its first row is all ones, $\sum_kd_k=0$, so $d$ has both positive and negative components. Let $s=\sum_{d_k>0}d_k=-\sum_{d_k<0}d_k>0$, and form two probability distributions with masses $d_k^+/s$ and $d_k^-/s$ on these equilibria.

They have identical moments through degree $L$. The probability of every binary sequence of length $L$ is a polynomial $\theta^r(1-\theta)^{L-r}$ of degree at most $L$, so the induced session laws coincide. Their degree-$(L+1)$ moments differ: otherwise the square Vandermonde matrix for degrees $0,\ldots,L+1$ would annihilate $d$, contradicting nonsingularity. In particular the probability of $L+1$ successive ones differs. This proves the assertion with a valid game and a fully specified observation kernel. ∎

This establishes a precise finite-window limitation, rather than claiming that no selection rule can be learned. A finite dictionary plus persistent sessions supplies exactly the additional restriction used in Theorem Theorem (A finite repetition design).

Uncertainty that includes estimating the selector

Observe $B$ independent sessions under a fixed known finite observation matrix $H$ (possibly $H^{(L)}$). Let $d$ be its number of rows and $\widehat p$ the empirical observation law. For $0<\alpha<1$ set \[ r_B=\sqrt{\frac{\log(2d/\alpha)}{2B}},\qquad \widehat Q=\{q\in\Delta_K:\|Hq-\widehat p\|_\infty\leq r_B\}. \]
Theorem: With probability at least $1-\alpha$, $\widehat Q$ contains the entire population fiber $Q(p)$. Consequently the minimum and maximum of any fixed $\ell^\top q$ over $\widehat Q$ cover its entire sharp identified interval. If $H$ has full column rank and $L_HH=I_K$ is a chosen left inverse, then \[ \operatorname{diam}_{\infty}(\widehat Q) \leq2\|L_H\|_{\infty\to\infty}r_B \] whenever $\widehat Q$ is nonempty.
Proof: For each outcome cell the $B$ session indicators are independent Bernoulli variables. Hoeffding's inequality and a union bound give $\Pr(\|\widehat p-p\|_\infty>r_B)\leq2d e^{-2Br_B^2}=\alpha$. On the complementary event every $q$ with $Hq=p$ belongs to $\widehat Q$, proving coverage and the interval assertion. For any $q,q'\in\widehat Q$, the triangle inequality gives $\|H(q-q')\|_\infty\leq2r_B$. Multiplication by the left inverse proves the diameter bound. ∎

Poorly separated observation columns can make the inverse norm large even when population identification holds. This exposes an experimental precision issue: choosing sufficient repetition length does not itself guarantee useful sample size. An empty estimated set is a model-fit warning; the theorem bounds its probability under the maintained exact finite-mixture model. Held-out predictions using $\ell$ inherit the mixture confidence region rather than treating fitted weights as known.

Source check and unresolved scope

The Tilburg primary record for Carlsson--van Damme (1993) was inspected on 9 October 2026 and verified title, authors, journal, volume and pages; its linked full-text request failed. It was therefore a bibliographic check, not a check of that paper's perturbation theorem. Harsanyi--Selten is retained as a catalogue reference to a theoretical construction, not a newly verified empirical law. The results above have complete finite-dimensional proofs and do not invoke those theories as identification assumptions.

The unrestricted kernel family across games, equilibrium dictionaries that are unknown or continuous, departures from equilibrium support, and causal effects of randomized context changes remain outside the proved class. The finite model supplies a concrete experimental restriction and sharp prediction tests, rather than a universal behavioral selector.

Completion Estimate

60%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt characterizes sharp finite-dictionary selection fibers, identified predictions, persistent-session repetition designs, and simultaneous uncertainty, while proving a continuum-mixture limitation. General cross-game selection kernels, unknown equilibrium dictionaries, context invariance, and tested enlargements for transient or approximate equilibrium behavior remain outside the proved model.

References

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