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What Variation Identifies Payoffs in a Dynamic Game?
\maketitle
\begin{abstract}
Observed choice in a dynamic game mixes current profit with continuation value. A rival adds a second problem: the same comparison averages over the rival's equilibrium policy. Changing the primitive transition rewrites continuation technology; changing the rival's Markov policy, holding that law fixed, rewrites the mixture over rival-contingent payoffs. The two are not substitutes. For a rival-feature payoff of rank $K$, rank identification up to location requires $E_{\Phi}^{\star}=\lceil(MK-1)/(M-1)\rceil$ policy environments, and a second kernel when payoffs are saturated. Rank can still be restored by arbitrarily small policy differences. Independent private shocks force mixed rival actions to factor, so a payoff that depends jointly on $d$ rivals is visible only at order $\eta^{d}$ near a common interior baseline. Either rank fails or the smallest identified singular value is at most $\kappa\eta^{d_{\Phi}}$, independently of how many kernels are stacked. Oracle-GLS variance in that direction vanishes only if $n\eta^{2d_{\Phi}}$ diverges. An Anderson--Rubin set that carries first-stage error in the design matrix covers without a vanishing-risk condition. On U.S.\ airline entry, even among rank-identified directions, the most favorable rival-dependent contrast is several times wider than observed behavior.
\end{abstract}
\noindent\textit{Keywords:} dynamic games, identification, Markov-perfect equilibrium, weak instruments, Anderson--Rubin inference.
\noindent\textit{JEL classification:} C57, C73, L13.
\section{Introduction}
\label{sec:intro}
Imagine a firm deciding whether to enter a market. The observed entry probability records two objects at once: current flow profit, and the continuation value created by today's action. A subsidy that raises every action at a state by the same amount, and that is expected to persist in a way the transition law can price, does not change choice differences. Observed behavior therefore does not read off flow payoffs.
A rival creates a second observational problem. The comparison the researcher sees is an average over the rival's equilibrium policy. A payoff that is high when the rival is in and low when the rival is out can look identical to a payoff that never depends on the rival, provided the two objects share the same policy-weighted averages. The researcher who sees only those averages cannot tell them apart.
These are different information problems, and they are not substitutes. Changing the primitive transition $P(x'\mid x,a,b)$ rewrites how current actions move future states: it changes continuation technology. Changing the rival's Markov policy $q(b\mid x)$, holding $P$ fixed, rewrites the mixture over rival-contingent current payoffs. Policy mixing cannot cancel a shift that lives in a shared continuation technology. Extra kernels cannot manufacture a missing contrast across rival actions. The formal model below makes the two operators explicit. The economic distinction does not wait on that notation.
Even when enough distinct environments restore rank, identification need not be informative. Three rival policies such as $0.500$, $0.501$, and $0.502$ can be linearly independent while remaining arbitrarily close. Rank then answers a yes-or-no question: after one location normalization, can the stacked Hotz--Miller map be inverted? The smallest identified singular value answers a different question: how much sample does that inversion require? Identified is not the same as measured.
Two integers organize the rank question. For a finite public state space of cardinality $M\ge 2$, $A\ge 2$ own actions, $B\ge 2$ opponent action profiles, and an unrestricted flow payoff $u(x,a,b)$ at a known discount factor,
\begin{equation}
\label{eq:headline-sat}
R_{\min}=2,
\qquad
E_{\min}=E_{\star}(M,B):=\left\lceil\frac{MB-1}{M-1}\right\rceil.
\end{equation}
Applied specifications almost never leave rival-action payoffs unrestricted. Additive competition, a count of active rivals, or interactions up to a fixed degree constrain only the opponent-action coordinate. Coefficients $\theta(x,a)$ remain free across states and own actions, so
\[
u(x,a,b)=\phi(b)^\top\theta(x,a),
\]
with $\Phi$ of rank $K$ and $\mathbf{1}_B\in\operatorname{col}\Phi$. Economic structure replaces raw $B$ by $K$. The sharp policy count becomes
\[
E_{\Phi}^{\star}=\left\lceil\frac{MK-1}{M-1}\right\rceil.
\]
That number answers how many strategically distinct environments restore rank. When $K<B$, a generic strictly positive kernel already cuts the restricted dynamic-potential intersection down to location, so one transition regime can suffice. Two regimes and $E_{\Phi}^{\star}$ product policies always suffice. Those identifying policies may be taken arbitrarily close to a common interior baseline.
The main surprise is that this last fact is about rank, not about information. Private shocks that are independent across players force equilibrium opponent mixtures to factor opponent by opponent. A payoff component that depends on one rival's action, as in additive competition, is visible at first order in the policy gap. A component that depends on two rivals jointly is visible only when both marginals move, hence at second order. Three-way dependence is third order. Write $\eta$ for the radius of observed policies around a common interior product baseline, and $d_{\Phi}$ for the highest active interaction degree in $\operatorname{col}\Phi$. For every design built from any number of kernels and any number of policy environments inside that neighborhood, either rank identification fails or
\[
\sigma_{\min,+}\ \le\ \kappa\,\eta^{d_{\Phi}},
\]
with $\kappa$ depending only on $(M,A,\beta,\Phi)$ and the baseline. Extra regimes rewrite continuation technology; they cannot manufacture a missing policy contrast. The exponent is attained, so it is sharp. Feature dimension $K$ says how many environments restore rank. The degree $d_{\Phi}$ says how fast usable information disappears as those environments cluster. Additive rival effects have $d_{\Phi}=1$ for any number of rivals; saturation has $d_{\Phi}=J$. A restriction can be cheap on the first margin and expensive on the second.
Under a bounded measurement-covariance sequence, oracle minimum-distance variance in the weakest identified direction is of order $1/(n\sigma_{\min,+}^2)$. That variance vanishes only if $n\eta^{2d_{\Phi}}\to\infty$ for every rank-identifying design. Ten times less usable strategic variation, at degree one, requires roughly one hundred times as much independent information; at degree two the penalty is fourth-order. The radius $\eta$ can be read off opponent behavior before any payoff is estimated. Anderson--Rubin inversion of the Hotz--Miller moment, carrying first-stage error in the design matrix, covers at the truth without a vanishing-risk condition \citep{andersonrubin1949,stockwright2000,kleibergen2005,andrewsguggenberger2017,andrewsguggenberger2019}.
The airline section is a diagnostic of that prediction, not a structural estimate of Southwest's effect on American's payoffs. Southwest is active in $95.4$ percent of the $2{,}204$ route-quarters on $241$ routes in which both carriers appear. On three of five state grids the cross-stratum dispersion of rival policy does not exceed what sampling noise produces at the observed cell sizes. On the headline grid the saturated specification has rank $42$ of $48$: some rival-dependent directions are classified as identified, and some are not. The most favorably conditioned identified rival-dependent contrast still has a profiled interval of $52.7$ logit units against an observed behavioral range of $8.9$. Under the local $n^{-1/2}$ scaling that width implies an information-equivalent sample multiplier of about $35$ relative to that behavioral range. The multiplier is a design benchmark, not a claim about how many additional routes exist. Rank diagnostics separate identified from unidentified directions; they do not say how little information the identified directions contain.
\citet{hotzmiller1993}, \citet{rust1987}, and \citet{magnacthesmar2002} characterize what conditional choice probabilities reveal about flow utilities at a maintained discount factor. Dynamic-game estimators typically treat the discount as known and impose Markov-perfect play \citep{aguirregabiriamira2007,bajaribenkardlevin2007,pesendorferschmidt2008}. Exclusion restrictions and switching-cost structure identify components of payoffs or beliefs \citep{aguirregabiriamagesan2020,komarova2018}. Multi-environment inverse reinforcement learning and inverse-game theory give rank conditions under which transition variation, discount variation, or rival-strategy variation remove reward ambiguity, including with linear features and in Markov games \citep{ngharadarussell1999,skalse2023,aminsingh2016,cao2021,rolland2022,kleinebuening2024,linadamsbeling2019,futacchetti2021,freihautramponi2025,liao2025}. \citet{schlaginhaufenkamgarpour2024} replace a binary rank condition on transition laws with principal angles for whether a recovered reward transfers under regularized IRL. Their object is the geometry of transition subspaces. The geometry here is product equilibrium policy. Rank results ask when payoff ambiguity disappears. This paper asks which equilibrium-feasible variation removes it, and how quickly usable information disappears when those policies are close. Additional transition variation does not remove the product-policy rate. Once a player's payoff is known up to location, that player's best-response map is identified under a specified counterfactual primitive and a specified opponents' policy; absolute welfare is not \citep{aguirregabiriasuzuki2014,kalouptsidi2017,kalouptsidi2021}. In the weak-instrument literature, weak identification is a possibility to be guarded against. Given $\Phi$ and the observed dispersion of opponent policies, the weakly identified payoff directions here are known before estimation.
The paper proceeds as follows. Section~\ref{sec:model} records the game and a verified two-by-two toy. Section~\ref{sec:variation} states the two obstructions and the structure-and-variation theorem. Section~\ref{sec:strength} gives the attenuation bound. Section~\ref{sec:sampling} turns that bound into a sample-size floor. Section~\ref{sec:inference} records the associated local experiment and the confidence set. Section~\ref{sec:heterogeneity} treats unobserved types. Section~\ref{sec:empirical} is a design diagnostic on U.S.\ airline entry. Unknown patience, transfer implementation, the one-regime count, and lagged-action geometry are in the appendix.
\section{Why observed choices do not reveal flow payoffs}
\label{sec:model}
\subsection{Dynamic game and observables}
\label{sec:observables}
The economic object is an infinite-horizon discounted finite-state stochastic game with public states. A researcher who sees choice probabilities in that game sees a mixture of current profit and the value of future states. Identification statements in this paper concern one player's flow payoffs, taking opponents' Markov behavior as an environment. The remaining players may themselves be strategic; what matters for the identified player is the policy they play and the kernel that maps current actions into tomorrow's public state.
There are $N\ge 2$ players. The public state space $\mathcal X$ is finite, with cardinality $M\ge 2$. Player $i$'s action set has cardinality $A\ge 2$ and a designated reference action $a_0=0$. Write $B$ for the number of opponent action profiles. Flow payoffs of the identified player are $u\in\mathbb{R}^{MAB}$, with coordinates $u(x,a,b)$. The discount factor is $\beta\in(0,1)$. Saturation means that $u(x,a,b)$ is unrestricted across those coordinates. Many empirical specifications instead restrict only rival-action dependence,
\begin{equation}
\label{eq:feature}
u(x,a,b)=\phi(b)^\top\theta(x,a),
\end{equation}
where $\Phi\in\mathbb{R}^{B\times K}$ has rank $K$ and $\mathbf{1}_B\in\operatorname{col}\Phi$. The coefficients $\theta(x,a)\in\mathbb{R}^K$ remain unrestricted across states and own actions, so the unknown lives in a subspace $\mathcal U_\Phi\subset\mathbb{R}^{MAB}$ of dimension $MAK$. Only one global payoff location is normalized. The paper first records what variation must supply when $\Phi=I_B$, then shows how $\operatorname{col}\Phi$ replaces raw $B$ in the strategic count.
A primitive transition regime is a row-stochastic array $P^r$ of shape $(A,B,M,M)$, indexed by $r=1,\ldots,R$. A strategic policy environment is a strictly interior opponent policy $q^e$, a map $x\mapsto q^e(\cdot\mid x)\in\mathrm{int}\Delta(B)$, indexed by $e=1,\ldots,E$. The identified player's own conditional choice probability in regime $r$ and policy environment $e$ is $p_i^{r,e}$. The two indexes are not interchangeable: $r$ changes the law of motion, $e$ changes the mixture over opponent actions.
\begin{assumption}[Known regular additive random utility]
\label{ass:rum}
Private shocks are additive, independent across players conditional on the public state, i.i.d.\ over time, independent of the controlled public transition, and drawn from a known state-independent distribution for which the surplus $\mathcal S(z)=\mathbb E[\max_{a'}(z_{a'}+\varepsilon_{a'})]$ is $C^2$ and translation-equivariant, and $p=\nabla\mathcal S(z)$ is a $C^1$ diffeomorphism from normalized deterministic value differences onto $\mathrm{int}\Delta(A)$.
\end{assumption}
Write $d(p)$ for the inverse from interior CCPs to normalized own-action value differences, and $g(p)$ for the surplus gap $\mathcal S(z(p))-z_0(p)$. Both are $C^1$ on the interior simplex. Logit is an example. Conditional independence of private shocks implies that, in a regular mixed-strategy Markov-perfect equilibrium, each opponent's mixed action is a function of the public state, so the joint opponent mixture factorizes as a product policy.
\begin{assumption}[Common primitives across measurements]
\label{ass:common}
Across primitive transition regimes $r$ and strategic policy environments $e$, the identified player's structural flow payoff $u$, discount factor $\beta$, and private-shock law are invariant. Only the primitive transition law $P^r$, equilibrium or opponent policies $q^e$, and known experimental transfers $(\tau_{ia}^{r,e})$ may vary across measurements.
\end{assumption}
Cross-environment stability of $(u,\beta)$ and the shock law is the economic restriction that makes pooling informative. Without it, each measurement would be a separate game. Arbitrary cross-market or calendar heterogeneity is not automatically identifying variation.
The experimenter, or the data, may include additive transfers $\tau_{ia}^{r,e}(x)$. Nonreference transfer differences enter action-value differences like a current payoff difference. The reference-action transfer enters continuation values. After Hotz--Miller inversion and subtraction of the observed nonreference transfer difference, the measurement in environment $(r,e)$ is affine in $u$:
\begin{equation}
\label{eq:z}
z^{r,e}
=
C^{r,e}(\beta)u
+
b^{r,e}(\beta).
\end{equation}
The matrix $C^{r,e}(\beta)$ is the payoff operator that maps flow payoffs into own-action value differences under $(P^r,q^e,\beta)$. The intercept $b^{r,e}(\beta)$ loads the surplus gap and the reference transfer through the continuation operator $S^{r,e}(\beta)$. Because $S^{r,e}(\beta)\mathbf{1}=0$, only the centered component of that continuation offset matters. Appendix~\ref{app:operators} records the stacked construction.
When $\beta$ is known, $b(\beta)$ is known from observed CCPs, the known shock law, and known transfers, and can be subtracted. Identification of $u$ is then a statement about $\mathrm{rank}\,C(\beta)$. When $\beta$ is unknown, $b(\beta)$ is a candidate-dependent affine correction, and observational equivalence is not a pure comparison of payoff column spaces.
\subsection{What one environment identifies}
\label{sec:one-env}
With only one primitive transition environment, a state-dependent transformation can change flow payoffs and continuation values together while leaving observed choice differences unchanged. Give each state an arbitrary potential $h(x)$ and shift current payoff by the current potential minus the discounted expected next-state potential. The Bellman value at $x$ then shifts by $h(x)$, so action-value differences do not move and choice probabilities are unchanged. The associated payoff perturbation is
\begin{equation}
\label{eq:G}
\bigl[G_{P,\beta}h\bigr](x,a,b)
=
h(x)-\beta\sum_{x'}P(x'\mid x,a,b)h(x').
\end{equation}
We call $\mathrm{Im}\,G_{P,\beta}$ the dynamic-potential gauge generated by $P$. One of its $M$ directions is a global location shift. For $M\ge 2$ at least $M-1$ non-location directions remain. Varying rival strategies inside the same $P$ does not remove the gauge, because every such observation inherits the same continuation technology.
\subsection{A small binary example}
\label{sec:example}
Take two states, two own actions, and two opponent actions, so the flow payoff is an array of length $8$. Then $E_{\Phi}^{\star}=E_{\star}(2,2)=3$ and the location-normalized target rank is $7$. A single mixed opponent policy produces two Hotz--Miller difference rows, and those rows never see $G_{P,\beta}h$. Any collection of policies inside one $P$ therefore leaves at least a two-dimensional class, one direction of which is the global constant.
Too few opponent policy mixtures expose only weighted averages of opponent-contingent payoffs. With two opponent policies the remaining averaging class is two-dimensional, distinct from the potential gauge. For binary policies that differ in every state it has the closed form
\begin{equation}
\label{eq:rho-example}
\rho_1=t\bigl(D(q^1)-D(q^2)\bigr)^{-1}\mathbf{1},
\qquad
\rho_0=c_1\mathbf{1}-D(q^1)\rho_1,
\end{equation}
with $D(q)=\mathrm{diag}(q_x)$ and $q_x=\Pr(b=1\mid x)$. The parameter $c_1$ is location; $t$ is the extra hidden direction. A third policy profile removes this averaging direction. A second strictly positive kernel then removes the remaining gauge direction. Two kernels, three policies, and one leftover constant are the binary case of Theorem~\ref{thm:structure}.
Figure~\ref{fig:toy} reports that architecture on the paper's operator, with $\beta=0.9$ and three interior product policies. One kernel yields a $6\times 8$ matrix of rank $6$ and nullity $2$. Two kernels yield a $12\times 8$ matrix of rank $7$ and nullity $1$: after location normalization the map is invertible. Holding those two kernels and the three-policy design fixed, and shrinking the policy radius, leaves rank equal to $7$ at every $\eta$ in the grid while $\sigma_{\min,+}$ falls in proportion to $\eta$. The numerical log-log slope is $1.000$, the degree-one rate of Theorem~\ref{thm:attenuation} for one binary rival. Rank stays solved; measurement does not.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{toy_rank_versus_strength.pdf}
\caption{A verified $M=A=B=2$ toy on the paper's Hotz--Miller operator. Panel A separates the rank question (can the location-normalized payoff be recovered) from the strength question (how small is the weakest identified singular value). Panel B: one primitive kernel leaves a two-dimensional kernel; a second kernel restores rank $7$. Panel C: the two-kernel, three-policy design remains rank $7$ as the policy radius shrinks, while $\sigma_{\min,+}$ tracks $\eta$.}
\label{fig:toy}
\end{figure}
The closed form \eqref{eq:rho-example} is only intuition for averaging. The general argument does not pass through it.
\section{How many environments restore rank?}
\label{sec:variation}
\subsection{Environmental obstruction}
\label{sec:env-obst}
The identification problem in one environment is that flow-payoff changes can be offset by state-dependent continuation-value changes. A subsidy that makes every action at a state more attractive by the same amount, and that is expected to persist in a way that the transition kernel can price, does not change observed choice differences. The analyst who sees only those differences cannot tell the subsidy apart from a change in continuation values. The previous subsection already gives this economic reason. The formal statement is that the dynamic-potential gauge has dimension $M$ and lies in the observational kernel of every mixed policy observed under that kernel.
\begin{proposition}[Environmental obstruction]
\label{prop:env}
Assume Assumptions~\ref{ass:rum} and~\ref{ass:common}, $\beta\in(0,1)$, and a saturated payoff $u\in\mathbb{R}^{MAB}$, with $M\ge 2$, $A\ge 2$, $B\ge 2$. Fix a row-stochastic kernel $P$. Then $\mathrm{Im}\,G_{P,\beta}$ lies in the observational kernel of every mixed opponent policy observed under $P$, $G_{P,\beta}$ is injective, and $\dim\mathrm{Im}\,G_{P,\beta}=M$. Consequently no collection of strategic policy environments inside a single primitive regime can reduce residual payoff ambiguity below dimension $M$ when the payoff is saturated. Identification of a saturated payoff up to only a global additive constant therefore requires $R\ge 2$.
\end{proposition}
Transfers, subsidies, or one-sided experiments that move $q$ while holding $P$ fixed are not a second primitive environment. Two kernels always share at least the constant-flow line; generic pairs share nothing more (Lemma~\ref{lem:E1}).
\subsection{Strategic obstruction}
\label{sec:strat-obst}
Even if continuation technology varies, observing too few rival-policy profiles only reveals averages of payoffs across rival actions. A payoff that is high when the rival enters and low when the rival stays out can look identical to a state-dependent payoff that does not depend on the rival at all, provided the two objects share the same policy-weighted averages. Some opponent-contingent payoff components therefore remain hidden. The formal object is an own-action-invariant payoff whose policy-weighted averages are state-constant. Current own-action differences vanish by construction. Under each observed policy the reference expected payoff is a state-constant, so every stochastic kernel annihilates the continuation term as well. Extra transitions cannot see that payoff, so the policy count is a separate requirement from a second kernel.
\begin{proposition}[Strategic-policy obstruction]
\label{prop:strat}
Let $C_E^\Phi(\beta)$ stack the feature-coordinate payoff operators over any finite collection of primitive regimes and $E$ interior policies. Then
\begin{equation}
\label{eq:avg-bound}
\dim\ker C_E^\Phi(\beta)
\ge
\max\bigl\{1,\,MK-(M-1)E\bigr\}.
\end{equation}
The bound does not depend on $R$, on heterogeneity of the kernels, on $A$, or on genericity of $P^r$. Identification up to a single global location therefore requires
\begin{equation}
\label{eq:Estar}
E\ge E_{\Phi}^{\star}=\left\lceil\frac{MK-1}{M-1}\right\rceil.
\end{equation}
When $K=B$, this is $E_{\star}(M,B)$. No amount of transition variation can compensate for observing too few distinct opponent policy profiles.
\end{proposition}
Propositions~\ref{prop:env} and~\ref{prop:strat} identify two universal lower-bound obstructions. They need not exhaust the observational kernel. Particular transition-policy geometries can create additional null directions. Appendix~\ref{app:lag} records one such family: lagged-action kernels leave a residual gauge of dimension at least $|\mathcal Z|$ no matter how many policies are stacked.
The two obstructions remain distinct. Fewer than $E_{\Phi}^{\star}$ policies leave a policy-averaging kernel no matter how many kernels are stacked. One transition regime leaves the full dynamic-potential gauge when the payoff is saturated. When $K<B$, that environmental obstruction is no longer universal. Table~\ref{tab:grid} records the saturated cells; Theorem~\ref{thm:structure} separates the restricted case.
\begin{table}[t]
\centering
\caption{Informational bottlenecks after rival-payoff restrictions. Averaging is indexed by $K$, not $B$. The cell $R=1$, $E\geE_{\Phi}^{\star}$ is a saturated impossibility; when $K<B$ one-regime identification is possible. The cell $R\ge 2$, $E\geE_{\Phi}^{\star}$ is attained at $R=2$.}
\label{tab:grid}
\small
\begin{tabular}{lcc}
\toprule
& $E<E_{\Phi}^{\star}$ & $E\geE_{\Phi}^{\star}$ \\
\midrule
$R=1$
& averaging remains
& saturated gauge remains; not a universal block if $K<B$ \\
$R\ge 2$
& averaging remains
& known-$\beta$ identification attainable \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Economic structure and variation}
\label{sec:sharp}
Definition~\ref{def:min} is understood in feature coordinates: $\mathfrak{I}(R,E)=1$ when some design attains $\mathrm{rank}\,C^\Phi=MAK-1$. The minima $R_{\min}$ and $E_{\min}$ remain useful marginal summaries. The object that records the tradeoff is the conditional frontier $\mathcal{E}_\Phi(R)$ in Section~\ref{sec:frontier}.
\begin{definition}[Minimal cardinalities of variation]
\label{def:min}
Let $\mathfrak{I}(R,E)=1$ when there exists a finite measurement design using at most $R$ distinct primitive transition regimes and at most $E$ distinct opponent-policy profiles such that, after payoff-location normalization, $\mathrm{rank}\,C^\Phi(\beta)=MAK-1$. Then
\[
R_{\min}=\min\{R:\exists E,\ \mathfrak{I}(R,E)=1\},
\qquad
E_{\min}=\min\{E:\exists R,\ \mathfrak{I}(R,E)=1\}.
\]
\end{definition}
\begin{theorem}[Economic structure and variation]
\label{thm:structure}
Assume known $\beta\in(0,1)$, Assumptions~\ref{ass:rum} and~\ref{ass:common}, $M\ge 2$, $A\ge 2$, and a rival-feature payoff \eqref{eq:feature} with $\operatorname{rank}\Phi=K$ and $\mathbf{1}_B\in\operatorname{col}\Phi$.
\begin{enumerate}
\item[(A)] For any number of primitive regimes and any kernels,
\[
\dim\ker C_E^\Phi\ge\max\{1,MK-E(M-1)\}.
\]
Hence $E_{\min}=E_{\Phi}^{\star}$.
\item[(B)] There exist $R=2$ strictly positive kernels and $E=E_{\Phi}^{\star}$ strictly interior conditionally independent product-policy profiles, arbitrarily close to a common interior product baseline, such that $\operatorname{rank} C^\Phi=MAK-1$.
\item[(C)] For every fixed primitive kernel $P$,
\[
\dim\bigl(\mathcal U_\Phi\cap\operatorname{Im} G_{P,\beta}\bigr)
=
M-\operatorname{rank}\mathcal R_\Phi(P).
\]
If $K=B$, every kernel leaves the $M$-dimensional saturated gauge, so $R_{\min}=2$. If $K<B$, a generic strictly positive $P$ in the interior stochastic-kernel parameter space has $\operatorname{rank}\mathcal R_\Phi(P)=M-1$, and its restricted dynamic-potential intersection contains only location. The explicit one-regime construction then yields $R_{\min}=1$ when $K<B$. Thus
\[
E_{\min}=E_{\Phi}^{\star},
\qquad
R_{\min}=
\begin{cases}
2,&K=B,\\
1,&K<B.
\end{cases}
\]
The two-regime construction attains $(R,E)=(2,E_{\Phi}^{\star})$. An explicit one-regime construction attains $(R,E)=(1,K+E_{\Phi}^{\star})$.
\end{enumerate}
\end{theorem}
Ranks and cardinalities depend on $\operatorname{col}\Phi$ and on $K$, not on a particular basis. For $T\in\mathrm{GL}(K)$, $\Phi\mapsto\Phi T$ merely reparameterizes $\theta$.
Take $J$ binary opponents. Table~\ref{tab:menu} translates common rival-payoff restrictions into $K$ and $E_{\Phi}^{\star}$. Additive rival structure converts an exponential strategic-environment requirement into a linear one.
\begin{table}[t]
\centering
\caption{Rival-payoff restrictions and the sharp policy count $E_{\Phi}^{\star}$ for $J$ binary opponents ($B=2^J$). The last column is $E_{\Phi}^{\star}$ when $M\ge K$.}
\label{tab:menu}
\small
\begin{tabular}{lccc}
\toprule
restriction & $K$ & $E_{\Phi}^{\star}$ & $M\ge K$ \\
\midrule
no rival-payoff dependence & $1$ & $1$ & $1$ \\
linear active-rival count & $2$ & $3$ & $3$ \\
heterogeneous additive rivals & $J+1$ & $\lceil(M(J+1)-1)/(M-1)\rceil$ & $J+2$ \\
active-count indicators & $J+1$ & same & $J+2$ \\
interactions through degree $d$ & $\sum_{k=0}^{d}\binom{J}{k}$ & $\lceil(MK-1)/(M-1)\rceil$ & $K+1$ \\
saturated & $2^J$ & $\lceil(M2^J-1)/(M-1)\rceil$ & $2^J+1$ \\
\bottomrule
\end{tabular}
\end{table}
\begin{corollary}[Saturated benchmark]
\label{thm:sharp}
\label{cor:saturated}
If $K=B$, then $E_{\Phi}^{\star}=E_{\star}(M,B)$, the restricted gauge is the $M$-dimensional dynamic-potential class, and $R_{\min}=2$. Identification of the saturated payoff up to location is attained by $R=2$ strictly positive kernels and $E=E_{\star}$ interior product policies, which may be taken arbitrarily close to uniform.
\end{corollary}
The first regime can be thought of as a measurement in which continuation differences are shut down. The second reopens continuation and uses a cycle increment to recover the reference-payoff block. The $E_{\Phi}^{\star}$ policies invert the feature-moment averaging map. Appendix~\ref{app:restricted} records the constructions.
\begin{remark}[Open identifying neighborhood]
\label{rem:open}
At the two-regime construction, after one payoff-location normalization, $\mathrm{rank}\,C^\Phi(\beta)=MAK-1$. For every fixed known $\beta\in(0,1)$, an open set of strictly positive transition regimes and strictly interior product-policy profiles around that construction continues to identify the restricted payoff up to location.
\end{remark}
\label{sec:frontier}
Conditional on $R$ regimes, write $\mathcal{E}_\Phi(R)$ for the smallest $E$ such that some identifying design exists. Theorem~\ref{thm:structure} already gives $\mathcal{E}_\Phi(R)=E_{\Phi}^{\star}$ for every $R\ge 2$. One regime has only $EM(A-1)$ choice-difference rows, so identification up to location requires
\begin{equation}
\label{eq:E1LB}
\mathcal{E}_\Phi(1)\geE_{1}^{\mathrm{LB}}
:=
\max
\left\{
\left\lceil\frac{MAK-1}{M(A-1)}\right\rceil,\,
\left\lceil\frac{MK-1}{M-1}\right\rceil
\right\}.
\end{equation}
An explicit construction attains $K+E_{\Phi}^{\star}$. On a $47$-cell grid of model sizes and rival-feature families the two bounds meet, so those extra $K$ policies are an artifact of the construction (Proposition~\ref{prop:e1exact}). Whether $\mathcal{E}_\Phi(1)=E_{1}^{\mathrm{LB}}$ for every $(M,A,\Phi)$ outside that grid remains open. The two-regime count $\mathcal{E}_\Phi(R)=E_{\Phi}^{\star}$ for $R\ge 2$ holds for every such triple. Appendix~\ref{app:frontier} records the statements and the exact-arithmetic certificate.
\section{Why rank is not enough}
\label{sec:strength}
Theorem~\ref{thm:structure}(B) attains the sharp counts with policies that may be arbitrarily close to a common interior product baseline. Rank is restored at every positive spread. The smallest singular value is not. Private shocks that are independent across players force opponent mixtures to factor, and a $d$-way rival interaction is then exposed only when $d$ marginals move together. Extra transition regimes rewrite continuation technology. They cannot manufacture the missing policy contrast.
\subsection{Interaction filtration at an interior baseline}
\label{sec:filtration}
Write the opponent joint-action space as a product $B=\prod_{j=1}^{J}B_j$. The integer $J$ is the number of independently mixing opponent components. A conditionally independent product policy is the object that private-shock Markov-perfect play produces.
Fix an interior product baseline $\bar q=\bigotimes_{j=1}^{J}\bar q_j$ with $\bar q_j\in\operatorname{int}\Delta(B_j)$. Decompose each factor into its constant and $\bar q_j$-centered parts,
\[
\mathbb{R}^{B_j}=\operatorname{span}\{\mathbf{1}_{B_j}\}\oplus V_j,
\qquad
V_j=\Bigl\{v\in\mathbb{R}^{B_j}:\textstyle\sum_b \bar q_j(b)v(b)=0\Bigr\}.
\]
For $S\subseteq\{1,\ldots,J\}$ let $\mathcal{H}_S(\bar q)$ apply $V_j$ on $j\in S$ and the constant on $j\notin S$, and set
\begin{equation}
\label{eq:Hd}
\mathcal{H}_d(\bar q)=\bigoplus_{|S|=d}\mathcal{H}_S(\bar q),
\qquad
\mathbb{R}^{B}=\bigoplus_{d=0}^{J}\mathcal{H}_d(\bar q).
\end{equation}
The summands are orthogonal in the $\bar q$-weighted inner product. Let $F=\operatorname{col}\Phi$ and define the tail filtration
\[
F_{\ge d}=F\cap\Bigl(\bigoplus_{r=d}^{J}\mathcal{H}_r(\bar q)\Bigr),
\qquad
g_d=\dim F_{\ge d},\qquad g_{J+1}=0.
\]
The successive dimensions
\begin{equation}
\label{eq:md}
m_d^\Phi=g_d-g_{d+1}
\end{equation}
sum to $K$. Because $\mathbf{1}_B\in F$, one has $m_0^\Phi=1$. The highest active degree is
\begin{equation}
\label{eq:dPhi}
d_{\Phi}=\max\{d:g_d>0\}.
\end{equation}
Thus $K$ counts how much strategic variation is required, while $d_{\Phi}$ records how rapidly the weakest direction in $F$ deteriorates under local product mixing. Both integers, and the exponent $d_{\Phi}$, are properties of $\operatorname{col}\Phi$. The raw number $\sigma_{\min,+}(C_\Phi)$ is not: a nonorthogonal rescaling $\Phi\mapsto\Phi T$ reparameterizes $\theta$ and can change singular values without changing the identified payoff. At the uniform baseline $\bar q_j=B_j^{-1}\mathbf{1}$, \eqref{eq:Hd} is the usual orthogonal interaction decomposition and the mixing operator is
\[
R(\eta)=\bigotimes_{j=1}^{J}(\Pi_j+\eta H_j),
\qquad
\Pi_j=B_j^{-1}\mathbf{1}\mathbf{1}',\quad H_j=I_{B_j}-\Pi_j,
\]
which acts on $\mathcal{H}_d$ as $\eta^{d}$. Observed opponent policies cluster somewhere other than uniform, so the baseline is kept free throughout.
\begin{lemma}[Restricted mixing]
\label{lem:mix}
On $F$, the singular values of $R(\eta)|_F$ have orders $\eta^d$ with multiplicity $m_d^\Phi$ for $d=0,\ldots,J$.
\end{lemma}
The argument uses a basis $\{v_{d,\ell}\}$ adapted to $F_{\ge d}\supset F_{\ge d+1}$. Each complement vector $v_{d,\ell}\in F_{\ge d}\setminus F_{\ge d+1}$ has a nonzero degree-$d$ projection, and those projections are linearly independent, so
\[
R(\eta)v_{d,\ell}=\eta^d\bigl(P_d v_{d,\ell}+O(\eta)\bigr).
\]
Assembling columns yields $R(\eta)V=W(\eta)D_\eta$ with $W(\eta)\to W_0$ of full column rank and
\[
D_\eta=\operatorname{diag}(\eta^0 I_{m_0^\Phi},\eta^1 I_{m_1^\Phi},\ldots,\eta^J I_{m_J^\Phi}).
\]
Appendix~\ref{app:strength} records the details.
\subsection{A bound that no design can beat}
\label{sec:attenuation}
Call a measurement design \emph{$\eta$-clustered at $\bar q$} if every opponent-policy environment is a strictly interior conditionally independent product policy whose marginals satisfy
\begin{equation}
\label{eq:cluster}
\max_{e\le E}\ \max_{j\le J}\ \max_{x\in\mathcal{X}}\ \bigl\|q^{e}_j(\cdot\mid x)-\bar q_j\bigr\|_1\le\eta .
\end{equation}
We measure policy dispersion in $L^1$ distance. This equals twice the conventional total-variation distance and avoids an irrelevant factor of two in the rate statements. No restriction is placed on the number of environments, on the number of primitive transition regimes, or on the kernels themselves. Because stacking more measurement blocks mechanically inflates every singular value of the raw stack, $C^{\Phi}$ is normalized per environment, $C^{\Phi}=(RE)^{-1/2}\bigl[C^{r,e}\bigr]_{r,e}$. This is also the normalization under which a fixed total sample is split across environments, so that the factor cancels in the oracle variance of Section~\ref{sec:sampling}.
\begin{theorem}[Design-free strategic attenuation]
\label{thm:attenuation}
Assume Assumptions~\ref{ass:rum} and \ref{ass:common}, a known $\beta\in(0,1)$, and a rival-feature payoff \eqref{eq:feature} with $\operatorname{rank}\Phi=K$ and $\mathbf{1}_B\in\operatorname{col}\Phi$. Fix an interior product baseline $\bar q$ and let $d_{\Phi}\ge 1$ be given by \eqref{eq:dPhi}. There is a finite constant
\[
\kappa=\kappa(M,A,\beta,\Phi,\bar q)
\]
such that every $\eta$-clustered design at $\bar q$, using any number $R$ of strictly positive primitive transition regimes and any number $E$ of policy environments, satisfies the following dichotomy: either the restricted payoff is not identified modulo the global payoff location, or
\[
\sigma_{\min,+}\bigl(C^{\Phi}(\beta)\bigr)\ \le\ \kappa\,\eta^{d_{\Phi}},
\]
where $\sigma_{\min,+}$ is the smallest singular value after the global payoff location is quotiented out. Thus rank failure is already a stronger failure, while every rank-identifying design is subject to the same $\eta^{d_{\Phi}}$ strength bound. The constant does not depend on $R$, on $E$, on the transition kernels, or on the placement of the policies inside the neighborhood \eqref{eq:cluster}. One admissible choice is
\begin{equation}
\label{eq:kappa}
\kappa=\frac{2\beta}{1-\beta}\cdot\sqrt{M(A-1)}\cdot\frac{c_\Phi(\bar q)}{\delta_\Phi},
\end{equation}
where $c_\Phi(\bar q)$ is the attenuation constant of the top filtration layer and $\delta_\Phi$ is the distance from the normalized test direction to the payoff-location line, both defined in Appendix~\ref{app:attenuation}.
\end{theorem}
The proof isolates why transition variation cannot help. Choose a payoff direction $\psi$ that is invariant across the identified player's own actions and whose feature image $\Phi\psi$ lies in the top layer $F_{\ged_{\Phi}}$. Three facts then combine. First, own-action invariance makes every current-payoff difference vanish identically, so $\psi$ can reach the data only through continuation values. Second, the continuation term loads on $q^{e}(\cdot\mid x)'\Phi\psi$, and $\Phi\psi$ is $\bar q$-centered on at least $d_{\Phi}$ factors, so the product expansion $\bigotimes_j(\bar q_j+\eta h^{e}_j)$ annihilates every term with fewer than $d_{\Phi}$ moving marginals and leaves $O(\eta^{d_{\Phi}})$. Third, the continuation operator $\beta(P_a-P_0)(I-\beta P_0)^{-1}$ has norm at most $2\beta/(1-\beta)$ for every row-stochastic kernel. Hence $\|C^{\Phi}\psi\|\le\kappa\eta^{d_{\Phi}}\|\psi\|$ uniformly. If the design fails to identify the restricted payoff modulo location, that is already a stronger failure. If it identifies, $\sigma_{\min,+}$ inherits the bound. Appendix~\ref{app:attenuation} gives the details.
In the identifying constructions of Theorem~\ref{thm:structure}, rank is restored at every positive spread, but the smallest nonzero singular value collapses at the rate governed by the interaction filtration. Designs that retain additional null directions are weaker still. The cardinalities are therefore necessary for identification and still leave precision governed by $\eta^{d_{\Phi}}$; no redesign of the transition environment closes that gap among identifying designs. Adding regimes, adding policies, or separating the kernels sharply all leave $\eta^{d_{\Phi}}$ untouched.
The bound is uniform over transition geometries: it concerns the strategic margin alone, and the environmental spread $\alpha$ in Section~\ref{sec:strength-thm} is a property of the canonical construction, not of the problem. It is also uniform in $E$, so collecting more policy environments inside a fixed neighborhood cannot help. Finally it depends on the payoff restriction only through $d_{\Phi}$, not through $K$. The number of environments needed and the precision obtainable are governed by different features of $\operatorname{col}\Phi$.
\begin{corollary}[Attenuation is attained]
\label{cor:attain}
For the canonical construction of Appendix~\ref{app:strength} with environmental spread bounded away from zero, $R=2$, and $E=E_{\Phi}^{\star}$, $\sigma_{\min,+}\asymp\eta^{d_{\Phi}}$. The exponent in Theorem~\ref{thm:attenuation} is therefore sharp.
\end{corollary}
\subsection{Canonical restricted spectrum}
\label{sec:strength-thm}
The canonical two-regime construction refines Theorem~\ref{thm:attenuation} by resolving the whole spectrum, at the cost of specializing the transition geometry. Let $\eta$ be the strategic policy spread and $\alpha$ the environmental transition spread. After the global payoff constant is quotiented out, current-payoff differences contribute $M(A-1)m_d^\Phi$ directions of order $\eta^d$. The reference payoff contributes $M-1$ directions of order $\alpha$ and, for $d\ge 1$, $M m_d^\Phi$ directions of order $\alpha\eta^d$.
\begin{theorem}[Canonical restricted spectrum]
\label{thm:strength}
Under the filtration-preserving canonical two-regime, $E_{\Phi}^{\star}$-policy construction adapted to $F=\operatorname{col}\Phi$ and specified in Appendix~\ref{app:strength}, let the strategic policy spread be $\eta$ and the environmental transition spread be $\alpha$. After quotienting the global payoff constant,
\[
\sigma_{\min,+}\asymp\alpha\eta^{d_{\Phi}}.
\]
On the isotropic path $\alpha=\eta=\varepsilon$,
\[
\sigma_{\min,+}\asymp\varepsilon^{d_{\Phi}+1}.
\]
Current-payoff-difference blocks contribute $M(A-1)m_d^\Phi$ directions of order $\eta^d$. The reference-payoff block contributes $M-1$ directions of order $\alpha$ and $M m_d^\Phi$ directions of order $\alpha\eta^d$ for $d\ge 1$.
\end{theorem}
The stacked two-regime operator is not block diagonal. A bounded row operation with condition number independent of $(\alpha,\eta)$ subtracts the first regime from the second and permits the current-difference and reference blocks to be read separately; Appendix~\ref{app:strength-R} records that step. The extra factor $\alpha$ relative to Theorem~\ref{thm:attenuation} is the price of near-uniform kernels: a well-separated transition pair recovers the design-free rate $\eta^{d_{\Phi}}$ but not more.
Additive heterogeneous binary rivals have $F=\mathcal{H}_0\oplus F_1$, hence $d_{\Phi}=1$ and $\sigma_{\min,+}\asymp\varepsilon^2$ for any $J$. Interactions through degree $d$ have $d_{\Phi}=d$ and rate $\varepsilon^{d+1}$. Saturation has $d_{\Phi}=J$ and recovers $\varepsilon^{J+1}$. Table~\ref{tab:strength-menu} collects these cases.
\begin{table}[t]
\centering
\caption{Examples for $J$ binary opponent components. The integer $K$ governs the sharp policy count $E_{\Phi}^{\star}$; the highest active degree $d_{\Phi}$ governs local strength. The design-free column is the exponent of Theorem~\ref{thm:attenuation}, valid for every rank-identifying transition geometry; the canonical column adds the environmental spread of Theorem~\ref{thm:strength} on the isotropic path $\alpha=\eta=\varepsilon$.}
\label{tab:strength-menu}
\small
\begin{tabular}{lcccc}
\toprule
restriction & $K$ & $d_{\Phi}$ & design-free & canonical \\
\midrule
no rival-payoff dependence & $1$ & $0$ & $1$ & $\varepsilon$ \\
heterogeneous additive rivals & $J+1$ & $1$ & $\varepsilon$ & $\varepsilon^{2}$ \\
interactions through degree $d$ & $\sum_{k=0}^{d}\binom{J}{k}$ & $d$ & $\varepsilon^{d}$ & $\varepsilon^{d+1}$ \\
saturated & $2^{J}$ & $J$ & $\varepsilon^{J}$ & $\varepsilon^{J+1}$ \\
\bottomrule
\end{tabular}
\end{table}
\begin{corollary}[Saturated recovery]
\label{cor:spectrum}
If $K=B$, then $m_d^\Phi=\dim\mathcal{H}_d=c_d$ as in \eqref{eq:cd} below, $d_{\Phi}=J$, and Theorem~\ref{thm:strength} recovers $\sigma_{\min,+}\asymp\alpha\eta^{J}$.
\end{corollary}
The combinatorial counts used for saturation are
\begin{equation}
\label{eq:cd}
c_d
=
[z^d]
\prod_{j=1}^{J}
\bigl(1+(B_j-1)z\bigr).
\end{equation}
Figure~\ref{fig:strength} plots the product-versus-joint comparison for the saturated construction. Product slopes track $\varepsilon^{J+1}$ and the joint comparison tracks $\varepsilon^{2}$ inside this geometry; Theorem~\ref{thm:attenuation} is the statement that applies to every geometry among rank-identifying designs.
\begin{figure}[t]
\centering
\includegraphics[width=0.82\textwidth]{product_vs_joint_strength.pdf}
\caption{Why product equilibrium policy is costly. Local singular-value scaling in the canonical sharp product-policy construction and an unrestricted joint-policy comparison, under isotropic environmental and strategic spread $\varepsilon$. Product series correspond to the saturated case of Corollary~\ref{cor:spectrum} with $J=1,2,3,4$ independently mixing opponent components. The remaining series is the unrestricted joint-policy comparison for the same environmental perturbation. Product slopes track $\varepsilon^{J+1}$ and the joint comparison tracks $\varepsilon^{2}$ inside this geometry. Unrestricted joint policies are not feasible under Assumption~\ref{ass:rum}; they isolate the product restriction.}
\label{fig:strength}
\end{figure}
\begin{remark}[Joint-policy comparison]
\label{rem:joint}
\label{cor:joint}
Under Assumption~\ref{ass:rum}, equilibrium opponent policies are conditionally independent product policies. Unrestricted correlated joint-policy perturbations are not feasible equilibrium-policy designs under the maintained model. Their role is to isolate the effect of relaxing the product restriction. In the canonical sharp geometry, product feasibility generates the interaction hierarchy. Relaxing it removes that hierarchy, although other transition geometries can also improve local strength while retaining product policies.
\end{remark}
\section{How much data the weakest direction needs}
\label{sec:sampling}
Work in the location-normalized identified payoff coordinates, and write $C_\varepsilon$ for the filtration-preserving canonical restricted design along a path of spreads. With fixed-length panels, $n$ denotes the number of independent market or route clusters. Suppose
\[
\widehat z_{n,\varepsilon}
=
C_\varepsilon\theta_0
+
n^{-1/2}\xi_{n,\varepsilon},
\]
and define $\Omega_{n,\varepsilon}=\operatorname{Var}(\xi_{n,\varepsilon})$. Assume uniformly along the considered sequence
\[
0<cI\preceq\Omega_{n,\varepsilon}\preceq CI.
\]
Oracle GLS uses the true covariance,
\[
\widehat\theta
=
\arg\min_\theta
(\widehat z_{n,\varepsilon}-C_\varepsilon\theta)'\Omega_{n,\varepsilon}^{-1}(\widehat z_{n,\varepsilon}-C_\varepsilon\theta).
\]
The estimator is linear, so
\begin{equation}
\label{eq:var}
\operatorname{Var}(\widehat\theta)
=
\frac1n
\bigl(C_\varepsilon'\Omega_{n,\varepsilon}^{-1}C_\varepsilon\bigr)^{-1}.
\end{equation}
The identity does not use a normality assumption. Uniform eigenvalue bounds yield
\begin{equation}
\label{eq:var-smin}
\lambda_{\max}\operatorname{Var}(\widehat\theta)
\asymp
\frac{1}{n\sigma_{\min,+}(C_\varepsilon)^2}.
\end{equation}
Theorem~\ref{thm:strength} then gives
\[
\lambda_{\max}\operatorname{Var}(\widehat\theta)
\asymp
\frac{1}{n\alpha^2\eta^{2d_{\Phi}}}.
\]
On an isotropic triangular sequence $\alpha=\eta=\varepsilon_n\to 0$,
\begin{equation}
\label{eq:var-eps}
\lambda_{\max}\operatorname{Var}(\widehat\theta)
\asymp
\frac{1}{n\varepsilon_n^{2(d_{\Phi}+1)}}.
\end{equation}
Along that sequence, $n\varepsilon_n^{2(d_{\Phi}+1)}\to\infty$ is necessary and sufficient for the largest oracle variance eigenvalue to vanish. Holding precision fixed along the design-free bound of Theorem~\ref{thm:attenuation} gives the iso-information relation $n'\eta'^{2d_{\Phi}}=n\eta^{2d_{\Phi}}$, or $n'/n=(\eta/\eta')^{2d_{\Phi}}$. This is a local scaling benchmark, not a finite-sample extrapolation of Anderson--Rubin widths.
\begin{proposition}[Oracle sampling-noise amplification]
\label{prop:noise}
Under the measurement-noise sequence above, \eqref{eq:var}--\eqref{eq:var-eps} hold for the filtration-preserving canonical restricted design of Theorem~\ref{thm:strength}.
\end{proposition}
The variance identity is algebraic. The Gaussian experiment below is what a triangular-array CLT for $\widehat z_{n,\varepsilon}$ delivers, and it supplies a minimax lower bound of the same order.
\begin{proposition}[Gaussian two-point lower bound]
\label{prop:twopoint}
In the location-normalized coordinates of this section, consider the Gaussian experiment
\[
\widehat z_n
=
C\theta
+
n^{-1/2}\xi,
\qquad
\xi\sim N(0,\Omega),
\]
with $cI\preceq\Omega\preceq CI$ and $\sigma_{\min,+}(C)>0$. There exist a pair $\{\theta_0,\theta_1\}$ with $\|\theta_1-\theta_0\|\asymp(n\sigma_{\min,+}^2)^{-1/2}$ and a constant $c_*>0$ depending only on $c$ such that
\[
\inf_{\widehat\theta}
\sup_{\theta\in\{\theta_0,\theta_1\}}
E_\theta\|\widehat\theta-\theta\|^2
\ \ge\
\frac{c_*}{n\sigma_{\min,+}^2}.
\]
The infimum is over every estimator of $\theta$, including those that use $C$ and $\Omega$.
\end{proposition}
\subsection{A sample-size floor that no design can lower}
\label{sec:floor}
Equation~\eqref{eq:var-smin} is a statement about whatever design is in hand. Combining it with Theorem~\ref{thm:attenuation} turns it into a requirement on the data that holds before any design is chosen. Proposition~\ref{prop:twopoint} upgrades the same quantity from an oracle-GLS variance to a minimax risk.
\begin{corollary}[Strategic sample-size floor]
\label{cor:floor}
Let $\{(R_n,E_n,P_n,q_n)\}$ be any sequence of measurement designs that is $\eta_n$-clustered at a fixed interior product baseline $\bar q$ and identifies the restricted payoff modulo global location for every $n$, with total sample size $n$. Under the measurement-noise conditions of Proposition~\ref{prop:noise},
\[
\lambda_{\max}\operatorname{Var}(\widehat\theta)
\ \ge\
\frac{1}{\kappa^{2}}\cdot\frac{1}{n\,\eta_n^{2d_{\Phi}}} ,
\]
so the largest oracle-GLS variance eigenvalue vanishes only if
\begin{equation}
\label{eq:floor}
n\,\eta_n^{2d_{\Phi}}\longrightarrow\infty .
\end{equation}
In the Gaussian experiment of Proposition~\ref{prop:twopoint} the same rate is necessary for vanishing minimax risk. Neither requirement depends on the number of transition regimes, on the number of policy environments, or on the transition kernels. Equivalently, a degree-$d_{\Phi}$ rival interaction read from opponent policies dispersed by $\eta$ requires at least an effective-sample factor of order $\eta^{-2d_{\Phi}}$ before that oracle variance, or that Gaussian risk, can vanish.
\end{corollary}
If a design is rank deficient beyond location, consistent estimation of the full location-normalized payoff already fails, so the rank-identifying case is the relevant boundary for the rate calculation. The theorem gives a necessary lower bound; a particular design can be worse.
Condition \eqref{eq:floor} is the empirically operative form of the theory, because $\eta$ is measurable directly from observed opponent behavior. It also separates the two integers cleanly. The feature dimension $K$ says how many distinct policy environments must be found at all; the highest active degree $d_{\Phi}$ says how much data each of them must carry. A restriction can be generous on the first margin and punitive on the second.
In the airline panel of Section~\ref{sec:empirical} the raw cross-stratum rival-policy dispersion is $\widehat\eta=1.24$, but after netting binomial sampling noise the corrected radius is $0.68$, and the width ratio of the confidence set suggests, as a heuristic scale comparison, an effective dispersion near $0.17$; this number is not used for inference. At that heuristic benchmark a degree-one rival effect already requires an effective sample of order $35$ relative to a rival-independent contrast, and a degree-two effect of order $10^{3}$. The panel contains $2{,}204$ route-quarters on $241$ route clusters; in the clustered asymptotics, the independent sampling unit is the route cluster. The frontier between what this design can and cannot measure therefore falls between a payoff that ignores the rival and a payoff that lets the rival enter, and it falls there for reasons that no reweighting or richer transition model can remove without genuinely more dispersed rival-policy variation.
\section{Inference that does not require vanishing risk}
\label{sec:inference}
Theorem~\ref{thm:attenuation} says that some payoff directions are measured arbitrarily badly, and Corollary~\ref{cor:floor} says how badly. Those directions, together with the observed radius $\eta$, index a local-to-zero experiment for the Hotz--Miller moment. A confidence interval built from an asymptotic normal approximation to $\widehat\theta$ presumes that the design pins $\theta$ down well enough for that approximation to bite. The set below is Anderson--Rubin inversion of the same moment, with first-stage error in $\widehat C_\Phi$ inside $\Omega_n(\theta)$. Proposition~\ref{prop:ar} gives coverage without a bound on $\sigma_{\min,+}$. Corollary~\ref{cor:width} and Proposition~\ref{prop:local} give the two sampling regimes the attenuation bound distinguishes: vanishing risk, and a noncentral $\chi^2$ experiment indexed by $\tau=\lim\sqrt n\,\sigma_{\min,+}$.
\subsection{The moment function and the variance that is usually omitted}
\label{sec:moment}
Fix $\beta$ and a feature matrix $\Phi$, and stack \eqref{eq:z} over the measured environments. Write $C_\Phi=C(\beta)J_\Phi$ for the operator restricted to $\mathcal U_\Phi$, where $J_\Phi$ embeds $\theta$ into $\mathbb{R}^{MAB}$, and write
\begin{equation}
\label{eq:amoment}
a
=
d(p_i)-\Delta\tau-S(\beta)\bigl[g(p_i)+\tau_{i0}\bigr]
\end{equation}
for the part of the measurement that is linear in the payoff, so that $a=C_\Phi\theta$ holds exactly at the truth. The sample analogue replaces the conditional choice probabilities and the transition kernels by estimates. Both objects in \eqref{eq:amoment} move: $\widehat a$ inherits first-stage error through $d(\widehat p_i)$, $g(\widehat p_i)$, and $\widehat S(\beta)$, and $\widehat C_\Phi$ inherits it through the estimated opponent policy and the estimated kernels. Define
\begin{equation}
\label{eq:mtheta}
m_n(\theta)
=
\widehat a-\widehat C_\Phi\theta .
\end{equation}
The variance of \eqref{eq:mtheta} is not the variance of $\widehat a$. It also contains the design-matrix variance
\[
\operatorname{Var}(\widehat C_\Phi\theta),
\]
which is quadratic in $\theta$, and the covariance between $\widehat a$ and $\widehat C_\Phi\theta$, which is linear in $\theta$. Standard practice plugs in estimated first-stage objects and then proceeds as though the design matrix were known. These omitted terms can be negligible when the payoff is well determined, but can become first-order or dominant in weakly measured directions, because a weakly identified direction is one along which $\theta$ must be large to move the measurement at all. In the environment of Theorem~\ref{thm:attenuation}, dropping first-stage error in the design matrix therefore reports a well-measured payoff along directions the design barely sees.
\subsection{The confidence set}
\label{sec:arset}
Let $\Omega_n(\theta)$ denote the asymptotic variance of $\sqrt n\,m_n(\theta)$ and let $\widehat\Omega_n(\theta)$ be a consistent estimator. Define the Anderson--Rubin statistic and set
\begin{equation}
\label{eq:ar}
\mathrm{AR}_n(\theta)
=
n\,m_n(\theta)'\widehat\Omega_n(\theta)^{+}m_n(\theta),
\qquad
\mathcal C_n(1-\alpha)
=
\bigl\{\theta:\ \mathrm{AR}_n(\theta)\le c_{L,1-\alpha}\bigr\},
\end{equation}
with $L=\operatorname{rank}\Omega_n$ and $c_{L,1-\alpha}$ the $(1-\alpha)$ quantile of $\chi^2_L$.
\begin{proposition}[Coverage that does not depend on identification strength]
\label{prop:ar}
Let the sample consist of $n$ independent markets. Suppose the first-stage estimator $\widehat\psi$ of $\psi=(p_i,q,P)$ satisfies $\sqrt n(\widehat\psi-\psi)\Rightarrow N(0,V)$, that the map $\psi\mapsto(a,C_\Phi)$ is continuously differentiable at $\psi$ with Jacobian $D(\theta)$ for the composite $\psi\mapsto a-C_\Phi\theta$, and that $\widehat\Omega_n(\theta_0)\to_p\Omega(\theta_0)=D(\theta_0)VD(\theta_0)'$. Let $L=\operatorname{rank}\Omega(\theta_0)$, and suppose
\[
\Pr\bigl\{\operatorname{rank}\widehat\Omega_n(\theta_0)=L\bigr\}\to 1.
\]
The nonzero eigenvalues of $\Omega(\theta_0)$ are uniformly bounded in $[c,C]\subset(0,\infty)$. Then for every $\theta_0$ satisfying the model,
\[
\mathrm{AR}_n(\theta_0)\Rightarrow\chi^2_L,
\qquad
\Pr\bigl\{\theta_0\in\mathcal C_n(1-\alpha)\bigr\}\to 1-\alpha .
\]
The convergence is uniform over any class of designs and data-generating processes on which those eigenvalue bounds hold and the first-stage remainder is uniformly negligible. No condition is imposed on $\sigma_{\min,+}(C_\Phi)$, and none is available: by Theorem~\ref{thm:attenuation} that quantity can be made arbitrarily small by clustering opponent policies, which is a property of the design rather than of the sampling problem.
\end{proposition}
The proposition says only that the set covers, whatever the design. Shape is governed by the attenuation theorem.
\subsection{Projections, unbounded directions, and the width the theorem forces}
\label{sec:projection}
Applied work reports scalar summaries: a switching cost, an entry cost, the payoff consequence of one more active rival. Each is a linear functional $\lambda'\theta$. The projection interval is
\[
\mathrm{CI}_\lambda(1-\alpha)
=
\Bigl[\inf_{\theta\in\mathcal C_n}\lambda'\theta,\ \sup_{\theta\in\mathcal C_n}\lambda'\theta\Bigr],
\]
which covers $\lambda'\theta_0$ with asymptotic probability at least $1-\alpha$ because $\theta_0\in\mathcal C_n$ implies $\lambda'\theta_0\in\mathrm{CI}_\lambda$. The set $\mathcal C_n$ is defined with $\widehat\Omega_n(\theta)$ evaluated at the candidate, which is the object whose coverage Proposition~\ref{prop:ar} proves. Holding $\widehat\Omega_n$ at a preliminary value makes the set an ellipsoid and the interval \eqref{eq:proj} explicit; that ellipsoid is a computational approximation, not the reported set. Write $\widehat W=\widehat C_\Phi'\widehat\Omega_n(\widehat\theta)^{+}\widehat C_\Phi$ and let $\widehat\theta$ be the minimizer of $\mathrm{AR}_n$ at that frozen covariance. Then
\begin{equation}
\label{eq:proj}
\mathrm{CI}_\lambda^{\mathrm{ellip}}
=
\lambda'\widehat\theta
\pm
\sqrt{\tfrac1n\bigl(c_{L,1-\alpha}-\mathrm{AR}_n(\widehat\theta)\bigr)\,\lambda'\widehat W^{+}\lambda}
\quad\text{if }\lambda\in\operatorname{range}\widehat W,
\end{equation}
and $\mathrm{CI}_\lambda^{\mathrm{ellip}}=\mathbb{R}$ otherwise. An unbounded interval reports that the observed variation does not restrict that payoff contrast: $\lambda$ lies in the null space the design failed to remove.
The reported interval inverts $\mathrm{AR}_n(\theta)\le c_{L,1-\alpha}$ with $\widehat\Omega_n(\theta)$ recomputed at each candidate. For a linear contrast the inversion is the profile
\begin{equation}
\label{eq:profile-ci}
\mathrm{CI}_\lambda(1-\alpha)
=
\bigl\{t:\ \min_{\theta:\,\lambda'\theta=t}\mathrm{AR}_n(\theta)\le c_{L,1-\alpha}\bigr\},
\end{equation}
which is the same as $\bigl[\inf_{\theta\in\mathcal C_n}\lambda'\theta,\ \sup_{\theta\in\mathcal C_n}\lambda'\theta\bigr]$. Restricting the minimizer to the ray $\theta=\widehat\theta+s\lambda$ yields a subset of that interval; ray inversion is used only as a computational lower bound on width. Because $m_n$ is affine in $\theta$, one cluster-bootstrap sample of $(\widehat a^{*},\widehat C_\Phi^{*})$ delivers $\widehat\Omega_n(\theta)$ everywhere. Bounded directions of that set are governed by the same spectrum that Theorem~\ref{thm:attenuation} bounds.
\begin{corollary}[Width under a slack condition]
\label{cor:width}
Under the conditions of Proposition~\ref{prop:ar} and the eigenvalue bounds of Proposition~\ref{prop:noise}, suppose $\sqrt n\,\sigma_{\min,+}(\widehat C_\Phi)\to_p\infty$. Let $\widehat\lambda$ be a unit right singular vector of $\widehat C_\Phi$ associated with $\sigma_{\min,+}(\widehat C_\Phi)$. For every fixed $\delta\in(0,c_{L,1-\alpha})$, on the event $\{\mathrm{AR}_n(\widehat\theta)\le c_{L,1-\alpha}-\delta\}$,
\[
\bigl|\mathrm{CI}_{\widehat\lambda}(1-\alpha)\bigr|
\ \ge\
\frac{\sqrt{\delta}\,c_\Omega^{1/2}}{\sqrt n\,\sigma_{\min,+}(\widehat C_\Phi)}\,(1+o_p(1))
\ \ge\
\frac{\sqrt{\delta}\,c_\Omega^{1/2}}{\kappa\sqrt n\,\eta^{d_{\Phi}}}\,(1+o_p(1)),
\]
where $c_\Omega$ lower-bounds the eigenvalues of $\Omega_n$ on its range. On that event,
\[
\bigl|\mathrm{CI}_{\widehat\lambda}\bigr|
=
\Omega_p\bigl((\sqrt n\,\sigma_{\min,+})^{-1}\bigr).
\]
The slack event has limiting probability one when the design is just identified. When it is overidentified by $L-r$ degrees the event has limiting probability $P(\chi^2_{L-r}\le c_{L,1-\alpha}-\delta)>0$, and on the complementary event the set can be arbitrarily thin because the minimized statistic can sit arbitrarily close to the cutoff. The requirement $\sqrt n\,\sigma_{\min,+}\to_p\infty$ is \eqref{eq:floor} in the operator's units: it makes the local comparison with the frozen-covariance ellipsoid take place on a shrinking neighborhood of $\widehat\theta$. Along a triangular array with $\sigma_{\min,+}\to 0$ faster than $n^{-1/2}$, that comparison is not available.
\end{corollary}
\begin{proposition}[Local-to-zero experiment]
\label{prop:local}
Under the conditions of Proposition~\ref{prop:ar}, consider a triangular array of rank-identifying, $\eta_n$-clustered designs, and write $\sigma_n=\sigma_{\min,+}(C_{\Phi,n})$. Theorem~\ref{thm:attenuation} gives
\[
\limsup_{n\to\infty}\sqrt n\,\sigma_n
\ \le\
\kappa\limsup_{n\to\infty}\sqrt{n\eta_n^{2d_{\Phi}}},
\]
allowing either limsup to be infinite. Coverage of $\mathcal C_n$ at $\theta_0$ continues to hold along the array, uniformly over the class of Proposition~\ref{prop:ar}.
Suppose in addition that $\sqrt n\,\sigma_n\to\tau\in[0,\infty)$, that $\lambda_n$ is a unit right singular vector of $C_{\Phi,n}$ for $\sigma_n$ with left vector $u_n=C_{\Phi,n}\lambda_n/\sigma_n$ when $\sigma_n>0$, and that $(\lambda_n,u_n)\to(\lambda,u)$. Let $\theta_n(a)=\theta_0+a\lambda_n$, and suppose the eigenvalue bounds of Proposition~\ref{prop:ar} hold uniformly for $\theta$ in a fixed ball about $\theta_0$. Then for every fixed $a\in\mathbb{R}$,
\[
\mathrm{AR}_n\bigl(\theta_n(a)\bigr)
\ \Rightarrow\
\chi^2_L\bigl(\delta(a,\tau)\bigr),
\qquad
\delta(a,\tau)
=
a^2\tau^2\,u'\Omega(\theta_0+a\lambda)^{+}u,
\]
with $\delta(a,0)=0$. Consequently:
\begin{enumerate}
\item[(i)] If $\tau=0$, then $\mathrm{AR}_n(\theta_n(a))\Rightarrow\chi^2_L$ for every fixed $a$, and $\bigl|\mathrm{CI}_{\lambda_n}(1-\alpha)\bigr|\to_p\infty$.
\item[(ii)] If $\tau\in(0,\infty)$, the ray inversion through $\lambda_n$ has width $O_p(1)$ and $\Omega_p(1)$, so the profiled interval is $\Omega_p(1)$.
\item[(iii)] If $\sqrt n\,\sigma_{\min,+}(\widehat C_{\Phi,n})\to_p\infty$, Corollary~\ref{cor:width} applies and the interval shrinks.
\end{enumerate}
\end{proposition}
The local parameter $a$ is order one in payoff units because $\sigma_n$ is order $n^{-1/2}$. The generic weakly identified GMM experiment is that of \citet{andrewsguggenberger2017,andrewsguggenberger2019}. Product-policy geometry supplies the sequence $\tau$ and the contrast $\lambda_n$: Theorem~\ref{thm:attenuation} bounds $\tau$ by $\kappa\sqrt{\gamma}$ along $n\eta^{2d_{\Phi}}\to\gamma$.
\subsection{Estimating the variance without an analytic Jacobian}
\label{sec:bootstrap}
The Jacobian $D(\theta)$ in Proposition~\ref{prop:ar} is available in closed form, but it is unpleasant, and it changes with every specification of the first stage. A market-level cluster bootstrap of the entire pipeline delivers $\widehat\Omega_n(\theta)$ at every candidate. Resample markets, re-estimate choice probabilities and kernels, and rebuild $(\widehat a^{*(b)},\widehat C_\Phi^{*(b)})$. Write $m_n^{*(b)}(\theta)=\widehat a^{*(b)}-\widehat C_\Phi^{*(b)}\theta$ and
\begin{equation}
\label{eq:boot-omega}
\begin{aligned}
\xi_n^{*(b)}(\theta)
&=
\sqrt n\bigl(m_n^{*(b)}(\theta)-m_n(\theta)\bigr),\\
\widehat\Omega_n(\theta)
&=
\frac1{B_{\mathrm{boot}}-1}
\sum_{b=1}^{B_{\mathrm{boot}}}
\bigl(\xi_n^{*(b)}(\theta)-\bar\xi_n^*(\theta)\bigr)
\bigl(\xi_n^{*(b)}(\theta)-\bar\xi_n^*(\theta)\bigr)',
\end{aligned}
\end{equation}
The factor $\sqrt n$ puts $\widehat\Omega_n$ on the scale of $\operatorname{Var}(\sqrt n\,m_n)$ in \eqref{eq:ar}; omitting it would estimate $\operatorname{Var}(m_n)$ and inflate the statistic by $n$. Here $\bar\xi_n^*(\theta)$ is the average of the $B_{\mathrm{boot}}$ scores. Because $m_n$ is affine in $\theta$, one set of draws serves every $\theta$. The reported set inverts that statistic. The ellipsoid \eqref{eq:proj} is computed from the same draws at a single preliminary $\theta$ and is not the reported object. Proposition~\ref{prop:ar} is a $\chi^2_L$ limit for a consistent covariance, which requires $B_{\mathrm{boot}}\to\infty$ or an analytic Jacobian. With $B_{\mathrm{boot}}>L$ held fixed, Lemma~\ref{lem:hotelling} replaces the cutoff by the Hotelling value $c_{L,1-\alpha}=L(B_{\mathrm{boot}}-1)(B_{\mathrm{boot}}-L)^{-1}F_{L,B_{\mathrm{boot}}-L,1-\alpha}$ in the Gaussian limit experiment for the bootstrap scores. As $B_{\mathrm{boot}}\to\infty$ the two cutoffs coincide.
\subsection{Monte Carlo evidence}
\label{sec:mc}
We simulate a dynamic entry game with $J$ binary rivals and logit shocks, draw a panel of markets, estimate the first stage from counts, and invert \eqref{eq:ar} with $\widehat\Omega_n(\theta)$ evaluated at the candidate. Table~\ref{tab:mc} records the geometry and the conventional interval. Table~\ref{tab:mc-n} is the size check for Proposition~\ref{prop:ar}. Within each layer of the interaction filtration the reported contrast is the most favorably conditioned singular direction, so the widths are the most favorable local-conditioning statement each design supports. This selection is conservative for the local conditioning comparison. We do not claim that the selected direction globally minimizes the candidate-dependent profiled projection width. Joint coverage inverts $\mathrm{AR}_n(\theta_0)$ with $\widehat\Omega_n(\theta_0)$. The AR column of Table~\ref{tab:mc} inverts the profiled statistic \eqref{eq:profile-ci} at the true contrast. The width column inverts only along the identified ray through $\widehat\theta$; that interval is contained in the profiled projection, so the reported widths are a lower bound on the width of \eqref{eq:profile-ci}.
At $8{,}000$ markets the joint AR frequencies are $0.950$, $0.975$, and $0.950$ in the three designs (Table~\ref{tab:mc-n}). The two $J=1$ designs share a payoff and a baseline policy and differ only in the policy radius, by a factor of twelve in $\eta$ and ten in $\sigma_{\min,+}$; coverage is no worse in the weaker design. Projection intervals in Table~\ref{tab:mc} cover at $1.000$ everywhere, which is the expected conservatism of a projection. At $500$ markets, where $L$ is $36$ or $75$, the minimized statistic still carries first-stage bias and joint coverage is $0.845$, $0.895$, and $0.770$ in the same designs. Raising the number of bootstrap draws from $300$ to $600$ at the $J=2$ design leaves coverage at $0.73$ and $0.72$, so the covariance estimate is not the problem.
The conventional interval does not share the large-$n$ size. It is below nominal in every design at $500$ markets, at $0.905$ and $0.900$ for the degree-zero contrast in the two $J=1$ designs and as low as $0.780$ in the $J=2$ design. At the weakly identified contrasts the Wald interval is not obviously deficient, because those intervals are so wide that any procedure covers. Undercoverage appears at the well-measured contrasts, where the interval is short enough for the omitted design-matrix variance to matter, and it is worst in the design with the highest interaction degree. At $8{,}000$ markets Wald coverage is still $0.750$ and $0.775$ for the two rival-dependent $J=2$ contrasts, and $0.900$ for the well-measured $J=1$ contrast.
The widths in Table~\ref{tab:mc} reproduce the filtration. At $\eta=0.05$ the degree-one contrast is measured worse than the degree-zero contrast by a factor of $25.0$, against the $\eta^{-1}=20$ that Corollary~\ref{cor:width} predicts. At $\eta=0.6$ the same ratio is $2.00$ against a predicted $1.67$. In the two-rival design the degree-one and degree-two ratios are $16.6$ and $3756$ at $\eta=0.093$, where $\eta^{-1}=10.8$ and $\eta^{-2}=116$; the bound is an inequality, and both layers sit above it, with the gap widening in the degree exactly as the exponent requires.
The fifth column of Table~\ref{tab:mc} checks Theorem~\ref{thm:attenuation} across designs. The implied constant $\sigma_{\min,+}/\eta^{d_{\Phi}}$ is computed for designs that differ in their transition kernels and in their policy radius by more than an order of magnitude. If the attenuation exponent were an artifact of a particular construction, that constant would move with $\eta$. Between the two $J=1$ designs it moves by a factor of $1.2$ while $\eta$ moves by a factor of twelve.
Table~\ref{tab:mc-gamma} varies $\eta$ at a fixed panel of $2{,}000$ markets, so $\gamma=n\eta^{2}$ indexes the local-to-zero sequence of Proposition~\ref{prop:local} for $J=1$ and $d_{\Phi}=1$. Joint AR coverage stays between $0.888$ and $0.950$ as $\gamma$ runs from $0.8$ to $320$. The degree-zero ray width is stable. The degree-one ray width falls from $8.77$ to $0.30$, and the ratio to the degree-zero width tracks $\eta^{-1}$.
\begin{table}[t]
\centering
\caption{Monte Carlo geometry and conventional intervals. Panels of $500$ markets over $14$ periods, logit shocks, $\beta=0.9$, nominal $95$ percent, cluster bootstrap of the entire pipeline. $L$ is the number of moments and $p$ the payoff dimension. The AR column inverts the profiled statistic \eqref{eq:profile-ci} at the true contrast. Widths invert only along the identified ray through $\widehat\theta$ and are therefore a lower bound on the profiled projection. Wald is the conventional interval that treats the design matrix as known. Degree $d$ indexes the layer of the interaction filtration, and the reported contrast is the most favorably conditioned singular direction of that layer. Width ratios are relative to the degree-zero contrast of the same design. Joint coverage of the vector $\theta_0$ is Table~\ref{tab:mc-n}, because at $500$ markets the first-stage inversion still biases the minimized statistic.}
\label{tab:mc}
\small
\begin{tabular}{lrrrrrrrr}
\toprule
design & $L/p$ & $\eta$ & $\sigma_{\min,+}$ & $\sigma_{\min,+}/\eta^{d_{\Phi}}$ & $d$ & AR & Wald & width ratio \\
\midrule
$J=1$ separated & $36/16$ & $0.600$ & $3.6\times10^{-2}$ & $0.060$ & 0 & $1.000$ & $0.905$ & $1.00$ \\
& & & & & 1 & $1.000$ & $0.935$ & $2.00$ \\
\addlinespace
$J=1$ clustered & $36/16$ & $0.050$ & $3.7\times10^{-3}$ & $0.075$ & 0 & $1.000$ & $0.900$ & $1.00$ \\
& & & & & 1 & $1.000$ & $0.945$ & $25.0$ \\
\addlinespace
$J=2$ clustered & $75/24$ & $0.093$ & $1.3\times10^{-5}$ & $0.0015$ & 0 & $1.000$ & $0.840$ & $1.00$ \\
& & & & & 1 & $1.000$ & $0.780$ & $16.6$ \\
& & & & & 2 & $1.000$ & $0.840$ & $3756$ \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[t]
\centering
\caption{Size of the joint Anderson--Rubin set and of the conventional interval, designs held fixed. Proposition~\ref{prop:ar} is an $n\to\infty$ statement. At $8{,}000$ markets the AR joint frequencies are $0.950$, $0.975$, and $0.950$. At $500$ markets they are not, because $L$ is $36$ or $75$ and the Hotz--Miller inversion is nonlinear. Wald coverage in weakly measured directions does not rise with $n$. At $500$ and $2{,}000$ markets, $60$ replications; at $8{,}000$, $80$ for $J=1$ and $40$ for $J=2$. A dash means the design has no contrast of that degree.}
\label{tab:mc-n}
\small
\begin{tabular}{lrrrrr}
\toprule
design & markets & AR joint & Wald $d=0$ & Wald $d=1$ & Wald $d=2$ \\
\midrule
$J=1$ separated & $500$ & $0.850$ & $0.950$ & $0.967$ & --- \\
& $2{,}000$ & $0.967$ & $0.917$ & $0.950$ & --- \\
& $8{,}000$ & $0.950$ & $0.900$ & $0.888$ & --- \\
\addlinespace
$J=1$ clustered & $500$ & $0.933$ & $0.950$ & $0.900$ & --- \\
& $2{,}000$ & $0.917$ & $0.983$ & $0.867$ & --- \\
& $8{,}000$ & $0.975$ & $0.950$ & $0.912$ & --- \\
\addlinespace
$J=2$ clustered & $500$ & $0.733$ & $0.783$ & $0.783$ & $0.883$ \\
& $2{,}000$ & $0.900$ & $0.800$ & $0.700$ & $0.933$ \\
& $8{,}000$ & $0.950$ & $0.900$ & $0.750$ & $0.775$ \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[t]
\centering
\caption{Local-to-zero sequence, $J=1$, $d_{\Phi}=1$, $2{,}000$ markets, $80$ replications, nominal $95$ percent. The design and payoff are those of the $J=1$ rows in Table~\ref{tab:mc}; only the policy radius changes. $\gamma=n\eta^{2}$. Widths are ray inversions, as in Table~\ref{tab:mc}. Joint AR coverage does not fall with $\gamma$. The degree-one width does.}
\label{tab:mc-gamma}
\small
\begin{tabular}{rrrrrrr}
\toprule
$\eta$ & $\gamma$ & $\sqrt n\,\sigma_{\min,+}$ & AR joint & $|\mathrm{CI}|_{d=0}$ & $|\mathrm{CI}|_{d=1}$ & ratio \\
\midrule
$0.02$ & $0.8$ & $0.050$ & $0.950$ & $0.110$ & $8.77$ & $79.8$ \\
$0.05$ & $5$ & $0.155$ & $0.900$ & $0.115$ & $2.69$ & $23.3$ \\
$0.10$ & $20$ & $0.277$ & $0.912$ & $0.111$ & $1.21$ & $10.8$ \\
$0.20$ & $80$ & $0.387$ & $0.888$ & $0.121$ & $0.575$ & $4.74$ \\
$0.40$ & $320$ & $0.724$ & $0.938$ & $0.107$ & $0.296$ & $2.77$ \\
\bottomrule
\end{tabular}
\end{table}
\section{Unobserved heterogeneity}
\label{sec:heterogeneity}
Every result so far treats the identified player's payoff as common across the measurements that are pooled, which is Assumption~\ref{ass:common}. Applied work on dynamic games rarely believes that literally, and the standard remedy is a finite mixture over latent market types. When the mixture is resolved the bound survives intact and binds through the worst-served type. When it is not resolved, the apparent strategic variation that the design relies on can be an artifact of the mixture.
\subsection{Resolved types}
\label{sec:types-resolved}
Let markets carry a latent type $\omega\in\{1,\ldots,T\}$ with weights $\pi_\omega>0$, and let the payoff $u_\omega$, the kernels $P^r_\omega$, and the opponent policies $q^e_\omega$ all be type-specific. Suppose the mixture is resolved, in the sense that type-specific choice probabilities and kernels are identified from the panel by an auxiliary argument \citep{kasaharashimotsu2009,hushum2012}. Then \eqref{eq:z} holds type by type, the unknown is the concatenation $(u_1,\ldots,u_T)$, and the stacked operator is block diagonal because no measurement of one type restricts the payoff of another.
\begin{proposition}[Heterogeneity does not relax the attenuation bound]
\label{prop:het}
Suppose each type's design is $\eta_\omega$-clustered at an interior product baseline $\bar q_\omega$, and let $d_{\Phi,\omega}$ be the highest active degree of the interaction filtration of $F=\operatorname{col}\Phi$ at $\bar q_\omega$. Then the stacked restricted operator satisfies
\[
\sigma_{\min,+}\bigl(C^\Phi_{1:T}\bigr)
=
\min_{\omega\le T}\sigma_{\min,+}\bigl(C^\Phi_\omega\bigr)
\ \le\
\min_{\omega\le T}
\kappa_\omega\,\eta_\omega^{d_{\Phi,\omega}},
\]
with $\kappa_\omega=\kappa(M,A,\beta,\Phi,\bar q_\omega)$. Moreover, for every type $\omega$, vanishing oracle-GLS variance and vanishing Gaussian minimax risk require
\[
n\,\pi_\omega\,\eta_\omega^{2d_{\Phi,\omega}}\longrightarrow\infty.
\]
\end{proposition}
Block diagonality gives the equality. Theorem~\ref{thm:attenuation} applies within each identified type-specific block with its own baseline-adapted degree $d_{\Phi,\omega}$ and constant $\kappa_\omega$. Taking the minimum across blocks gives the displayed bound. The effective sample size for type $\omega$ is $n\pi_\omega$, yielding the type-specific rate condition. Allowing heterogeneity multiplies the number of payoff parameters by $T$ and divides the effective sample by $\pi_\omega$, while the binding attenuation is set by whichever type sees the least rival-policy dispersion at its own filtration degree. A design can therefore look adequate on the pooled data and be hopeless within the type that matters. The mixture redistributes the variation that is already there.
\subsection{Unresolved types}
\label{sec:types-unresolved}
If types are not resolved, the analyst observes the mixed choice probability $\bar p=\sum_\omega\pi_\omega p_{i,\omega}$ and the mixed kernel. A mixture of type-specific equilibria is not, in general, an equilibrium of the pooled primitives: the Hotz--Miller map and the resolvent are both nonlinear, so the operator built from pooled objects need not be the operator of any homogeneous type. Assumption~\ref{ass:common} then fails, and the mixed objects do not supply identifying variation for a common payoff. The statistic of Section~\ref{sec:arset} can reject the pooled model when the design is overidentified, but an empty set does not isolate heterogeneity from other misspecification, and when $L=\operatorname{rank}C_\Phi$ the minimized statistic is zero by construction. We do not record a theorem for this case. The operational content is in Section~\ref{sec:composition} and in the airline diagnostic: composition across strata contaminates measured $\eta$, and the common-payoff assumption is maintained rather than tested.
\subsection{Composition is not policy variation}
\label{sec:composition}
Where $\eta$ comes from is part of the design. Applied work usually takes policy environments from strata: calendar windows, geographic cells, size classes. If markets in different strata differ in type composition, the estimated opponent policies differ across strata even when every type plays exactly the same policy everywhere. That difference is a composition effect. It contributes to a measured $\eta$, and it identifies nothing, because Assumption~\ref{ass:common} fails across the strata being pooled: the payoff attached to the measurement is a different mixture in each.
The comparison is testable. Under the null that all strata share one policy at each state, the estimated cross-stratum dispersion still has a nondegenerate distribution driven by cell sizes alone. Comparing the measured $\eta$ with that null distribution asks whether the design has any strategic variation to work with before asking what the variation identifies. Section~\ref{sec:empirical} runs the comparison on the airline panel.
Three further results sit in the appendix because they are not needed to read the attenuation bound. Unknown patience is identified at the same cardinalities by an arbitrarily small centered continuation-offset randomization (Appendix~\ref{app:unknown}). Those offsets, and the clustered product policies themselves, are locally implementable by transfers at a regular interior equilibrium (Appendix~\ref{app:transfer}). Lagged-action transition families preserve a residual gauge of dimension at least $|\mathcal Z|$ no matter how many policies are stacked (Appendix~\ref{app:lag}).
\section{A design diagnostic: U.S.\ airline entry}
\label{sec:empirical}
The section is a diagnostic of the interaction filtration in an observed, rank-deficient design, not a structural payoff estimate and not a direct application of the full-rank strength corollary. The observed design contains too little clean strategic variation to measure rival-dependent payoff components precisely. With one binary rival, $d_{\Phi}=1$. Rank classifies some rival-dependent directions as identified; it does not say those directions are measured. Calendar windows and distance terciles also differ in fuel prices and route composition, so Assumption~\ref{ass:common} does not hold exactly across the strata being pooled. Measured $\eta$ is therefore an upper bound on usable strategic dispersion, and the intervals below understate how weakly rival-dependent payoffs are measured.
The source is the T-100 Domestic Segment file for 2016--2019 \citep{bts_t100}. The identified player is American, the rival is Southwest, and the universe is undirected city-pairs among the forty largest 2014 cities. An action is at least twelve performed departures in the city-pair-quarter. After the lag and lead, $2{,}204$ route-quarters remain on $241$ routes where both carriers appear. Southwest is active in $95.4$ percent of those cells. The public state is $x=(s,a_{i,-1},a_{j,-1})$ with $s$ a 2014 demand tercile, so $M=12$ and $B=2$. The discount $\beta=0.95$ is maintained. With one binary rival the filtration has two layers, $d_{\Phi}=1$. In rank-restoring designs with one binary rival, the theory predicts first-order sensitivity of rival-dependent directions to rival-policy dispersion.
Markets are partitioned into six strata by calendar window and route-distance tercile. Within each stratum the rival CCP and the demand transition are estimated from counts. Nothing is perturbed. The six stacked measurements give $L=72$ moments.
\begin{table}[t]
\centering
\caption{Is there rival-policy variation beyond sampling noise? Measured $\widehat\eta$ against a null that redraws rival actions from the pooled policy at the observed cell sizes. Four hundred draws. $p$-values use $\hat p=(1+\#\{T_b\ge T_{\mathrm{obs}}\})/(B+1)$. The corrected radius nets binomial variance out of the cross-stratum spread.}
\label{tab:eta-placebo}
\small
\begin{tabular}{lrrrrrc}
\toprule
grid & strata & median cell & $\widehat\eta$ & corrected & $95\%$ of null & $p$ \\
\midrule
headline, $M=12$ & 6 & $8$ & $1.236$ & $0.684$ & $0.796$ & $<0.003$ \\
four strata, $M=12$ & 4 & $10$ & $0.894$ & $0.454$ & $0.908$ & $0.080$ \\
coarse demand, $M=8$ & 4 & $18.5$ & $0.924$ & $0.296$ & $0.938$ & $0.087$ \\
distance only, $M=12$ & 3 & $16$ & $1.139$ & $0.960$ & $0.775$ & $<0.003$ \\
sixty-state grid, $M=60$ & 6 & $0$ & $0.875$ & $0.503$ & $1.035$ & $0.342$ \\
\bottomrule
\end{tabular}
\end{table}
Raw cross-stratum dispersion is $\widehat\eta=1.24$. The median cell has eight observations, so most of that number is binomial noise. On the headline grid the observed value exceeds the $95$th percentile of the null; on three of five grids it does not (Table~\ref{tab:eta-placebo}). Calendar and distance strata also differ in route composition, and a composition effect contributes to measured $\eta$ without identifying anyone's payoff (Section~\ref{sec:composition}). The noise-corrected radius $0.68$ is therefore an upper bound on the strategic dispersion the design supplies. The width ratio in the confidence set below suggests, as a heuristic scale comparison, an effective dispersion near $0.17$; this number is not used for inference.
\begin{table}[t]
\centering
\caption{Headline grid, $\beta=0.95$, nominal $95$ percent. Rank is not full: $K=2$ has rank $42$ of $48$. Widths invert the profiled statistic \eqref{eq:profile-ci} in logit units for the most favorably conditioned singular direction of each interaction degree. Observed behavior spans $8.9$ such units. A dash means the specification has no contrast of that degree. The ellipsoid computed from a preliminary $\widehat\Omega_n$ is recorded in the appendix and is not this table.}
\label{tab:airline-ar}
\small
\begin{tabular}{lrrrrrrr}
\toprule
specification & $L$ & rank & $\sigma_{\min,+}$ & $\min\mathrm{AR}$ & cutoff & $|\mathrm{CI}|_{d=0}$ & $|\mathrm{CI}|_{d=1}$ \\
\midrule
$K=1$ & 72 & 20 & $2.2\times10^{-3}$ & $87.5$ & $106.0$ & $0.29$ & --- \\
$K=2$ & 72 & 42 & $8.8\times10^{-5}$ & $11.1$ & $106.0$ & $9.0$ & $52.7$ \\
\bottomrule
\end{tabular}
\end{table}
Table~\ref{tab:airline-ar} is the filtration in the panel. A payoff that ignores the rival is not rejected on this grid ($\mathrm{AR}=87.5$ against $106.0$), nor on the other four. Holding $\widehat\Omega_n$ at a preliminary least-squares $\theta$ produces a minimized statistic of $120.9$ and a rejection; that ellipsoid is not the set Proposition~\ref{prop:ar} covers, and the rejection does not survive the candidate-dependent inversion the proposition requires. Under saturation the reported set is never empty and never informative about the rival: the most favorably conditioned rival-dependent interval is $52.7$ units against a behavioral range of $8.9$, while the most favorably conditioned rival-independent interval in the same specification is $9.0$, already the width of observed behavior. Rank is $42$ of $48$. The sixty-state grid, whose median cell is empty, still reports rank $192$ of $240$. Rank on the stacked operator classifies some rival-dependent payoff directions as identified. The confidence set shows that even the most favorably conditioned such direction can remain extremely weakly measured: it records $\eta^{d_{\Phi}}$ at the layer Theorem~\ref{thm:attenuation} names, which rank does not.
Appendix~\ref{app:emp-robust} records the other grids, the two other discounts, and the rank-ceiling exercise in which the identifying policies were constructed rather than observed.
Figure~\ref{fig:airline} collects the design facts. On three of five grids, raw cross-stratum dispersion of rival policy does not exceed the sampling-noise benchmark (panel A). On every grid the most favorably conditioned rival-dependent profiled interval is wider than the range of observed behavior (panel B). Under the local $n^{-1/2}$ benchmark, matching the headline degree-one width of $52.7$ logit units to the $8.9$-unit behavioral range requires an information-equivalent sample multiplier $m=(52.7/8.9)^{2}\approx 35$. Matching it to the rival-independent width of $9.0$ gives $m\approx 34$. These numbers are not the exact number of additional routes required, and finite-sample Anderson--Rubin nonlinearity would make a literal $n$-extrapolation misleading. They are a local information-equivalent benchmark. Panel C records the theoretical iso-information curves: at degree one, ten times less usable variation requires roughly one hundred times as much independent information; at degree two the penalty is fourth-order.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{airline_information_frontier.pdf}
\caption{What the observed airline design measures. Panel A: rival-policy radius against a cell-size sampling null. Panel B: most favorably conditioned profiled widths by interaction degree, with the range of observed own-choice logits marked. Panel C: theoretical iso-information curves $n'/n=(\eta/\eta')^{2d}$ and the local sample multiplier implied by the headline degree-one interval. The application is a design diagnostic, not a structural estimate of Southwest's payoff effect.}
\label{fig:airline}
\end{figure}
\section{Discussion}
\label{sec:disc}
Feature dimension $K$ says how many distinct opponent-policy environments restore rank. The highest active degree $d_{\Phi}$ says how much data each of those environments must carry. Extra transition regimes do not close the second gap. A researcher who can read $\eta$ off opponent behavior can diagnose which interaction layers are intrinsically hard to measure before estimating payoffs; exact rank still depends on the full transition-policy geometry. Corollary~\ref{cor:width} and Proposition~\ref{prop:local} are the intervals that match that diagnosis in the two sampling regimes.
Once a player's pair $(u,\beta)$ is known up to a global additive constant, that player's action-value differences and best-response map are identified under a specified counterfactual primitive and a specified opponents' policy. Absolute lifetime welfare is not, because a constant added to flow payoffs shifts values by $c/(1-\beta)$ without changing behavior. A single player's identified payoff does not identify the equilibrium correspondence.
\clearpage