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\begin{center}
{\Large \bf Testing for Monotone Equilibrium Strategies\\ in Games of Incomplete Information}\footnote{This paper supersedes an earlier version that is entitled ``Testing for Monotone Equilibrium Strategies in First Price Auctions.'' We thank Phil Haile, Matt O'Keefe and Andres Santos for helpful comments on the early version of the paper. Yu-Chin Hsu gratefully acknowledges the research support from the
National Science and Technology Council of Taiwan (NSTC113-2628-H-001-001 and NSTC114-2410-H-001-080-MY3) and the Academia Sinica
Investigator Award of the Academia Sinica, Taiwan (AS-IA-110-H01). Chu-An Liu gratefully acknowledges the research support by the Academia Sinica Career Development Award
(AS-CDA-110-H02). Yu-Chin Hsu and Chu-An Liu gratefully acknowledge the research support by the Center for Research in Econometric Theory
and Applications, Taiwan (Grant No. 113L8601). Tong Li and Hidenori Takahashi gratefully acknowledge financial support from the Joint Usage/Research Center, Institute of Economic Research, Kyoto University. Hidenori Takahashi gratefully acknowledges financial support from KAKENHI (Grant No. 24K00252).}
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\textbf{\large Yu-Chin Hsu$^{\dag}$}{\large{} }
\vspace*{1mm}
Institute of Economics, Academia Sinica\\
Department of Finance, National Central University\\
Department of Economics, National Chengchi University and\\
CRETA, National Taiwan University
\vspace*{3mm} \textbf{\large Tong Li$^{*}$}
\vspace*{1mm} Department of Economics, Vanderbilt University
\vspace*{3mm} \textbf{\large Chu-An Liu$^{\ddag}$}
\vspace*{1mm} Institute of Economics, Academia Sinica
\vspace*{3mm}
\textbf{\large Hidenori Takahashi$^{\star}$}
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Kyoto Institute of Economic Research, Kyoto University
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This version: \today
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\noindent $^{\dag}$ {\fontsize{8.5pt}{9pt}\selectfont [email removed]. }, $^{*}$ {\fontsize{8.5pt}{9pt}\selectfont [email removed]}, $^{\ddag}$ {\fontsize{8.5pt}{9pt}\selectfont [email removed]},
\noindent $^{\star}$ {\fontsize{8.5pt}{9pt}\selectfont [email removed]}
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\begin{abstract}
This paper develops a unified framework for testing monotonicity of Bayesian Nash equilibrium strategies in unobserved types in games of incomplete information. We show that, under symmetric independent private types, monotonicity of differentiable equilibrium strategies is equivalent to monotonicity of a quasi-inverse strategy identified from observed actions. This allows the problem to be reformulated as testing a countable set of moment inequalities involving unconditional expectations. We propose a Cramér–von Mises–type statistic with bootstrap critical values. The method accommodates covariates and game heterogeneity. Monte Carlo simulations demonstrate finite-sample performance, and an application to procurement auctions illustrates cartel detection.
\medskip
\medskip
\noindent \textit{Keywords:} Monotone equilibrium strategies, Bayesian games, Moment inequalities, Rationalization,
Testing for monotonicity, Game heterogeneity
\noindent \textit{JEL codes: C12, C14, L70}
\end{abstract}
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\section{Introduction}
In games of incomplete information, or Bayesian games, players' strategies are functions mapping their private types such as signals, beliefs, valuations, or costs, to actions. A Bayesian Nash equilibrium (BNE) is a profile of such strategy functions where each player's strategy is a best response to others, given beliefs about their types. Monotonicity of equilibrium strategies, which means that a player's action is monotone with their type, is one of the most empirically informative predictions in games of incomplete information: higher valuations, lower costs, or stronger signals lead to more aggressive actions. This “sorting” property underlies how economists interpret actions as revealing private information in a wide range of environments, from auctions to contests, public good provision, and quantity competition with private information. It is also a cornerstone of the theory: monotonicity is typically the condition that delivers well-behaved equilibria (existence and uniqueness), intuitive comparative statics, and tractable mechanism design.
On the theoretical ground, \cite{Athey2001} establishes seminal results on the existence of a monotone pure strategy Bayesian Nash equilibrium under the Spence-Mirrlees single-crossing condition, which are further extended to accommodate multi-dimensional type and action spaces by \cite{McAdams2003}, and to allow for more general spaces by \cite{Reny2011}. From an econometric perspective, monotone equilibrium strategies are equally central because they justify the inversion logic that maps observed actions back into latent types, enabling identification and inference in structural models.
See, e.g., \cite{GPV2000}, \cite{LPV2002} and a large related literature in first price auctions, \cite{Kline2025} in allocation-transfer games, \cite{LiZhangZhao2025} and \cite{LiZhangZhao2026} for Bayesian games with continuous payoffs, and \cite{AryalZincenko2024} for Cournot competition with private information, to name only a few.
While there has been significant progress in identification and estimation of Bayesian games, there is remarkably little work on testing the monotonicity of Bayesian Nash equilibrium strategies, despite its role as a key equilibrium condition and as a cornerstone in identification and estimation. This gap is consequential because monotonicity is routinely imposed as an equilibrium restriction and as a prerequisite for inversion-based identification and estimation. If monotonicity fails in the data, the structural model and its implied counterfactuals may be invalid. Moreover, monotonicity has independent empirical content in contexts where departures from competitive behavior are of interest. For example, \citet{BajariYe2003} derive a set of sufficient and necessary conditions for the symmetric and competitive bidding equilibrium including monotonicity of the bidding equilibrium, thus rejection of monotonicity can constitute a test for collusion. However, they bypass testing for monotonicity, and only test for other conditions such as exchangeability, largely because tools for testing monotone equilibrium strategies have been absent.
Testing monotone equilibrium strategies in types is challenging because types are unobserved to the econometrician. To date, the only development is in the auction setting considered in \cite{LV2021}, who exploit the equivalence between monotonicity of Bayesian Nash equilibrium bidding strategies in private values and monotonicity of the quasi-inverse of the bidding strategy, which is identified from observed bids.\footnote{In a somewhat related yet different vein, noting that the nonparametric estimator of the quasi-inverse bidding strategy in \cite{GPV2000} does not impose the monotonicity condition, \cite{HLMPP2012} and \cite{MMSX2021} develop monotonicity constrained nonparametric estimators of the quasi-inverse strategies.} They reformulate the test as one for the concavity of the integrated quasi-inverse strategy to avoid nonparametric estimation of bid densities in the quasi-inverse formulation. They
then construct a statistic based on the difference between this function and its least concave majorant (LCM).
This paper aims to fill the gap in the literature by developing a new and easy-to-implement test for monotone equilibrium strategies in general Bayesian games. To this end, we first characterize the sufficient and necessary conditions for the
existence of a private type distribution that can rationalize the observed action distribution in general Bayesian games considered in \cite{Athey2001} within the symmetric independent private type paradigm. As such, we extend the rationalization logic in the empirical auction literature to games of incomplete information, which include not only auctions that have discontinuous payoffs, but also other games with continuous payoffs such as contests, public good provision, and Cournot competition with private information, among others.\footnote{The rationalization result in the empirical auction literature was first established in \cite{GPV2000} for the symmetric independent private value setting, and has been extended to the symmetric affiliated private value setting in \cite{LPV2002}, and the asymmetric affiliated private value setting in \cite{CPV2003}, as well as for risk-averse bidders in \cite{GPV2009}.}
Our rationalization result delivers an equivalence that forms the basis of our test: within the maintained framework, monotonicity of the equilibrium strategy in private types holds if and only if the quasi-inverse equilibrium strategy is monotone in actions. The quasi-inverse is a ratio of the derivatives of two reduced-form objects—the expected allocation rule and the expected payment rule—that are functions of observed actions and can therefore be identified from the distribution of actions given the known institutional mapping from actions to allocations and transfers. As a result, the testing problem becomes to test whether this identified quasi-inverse map is monotone in the action.
Although this insight is in a similar spirit to
\cite{LV2021}, our testing approach differs fundamentally. First, we move beyond first-price auctions and develop a single and unified framework that covers a broad class of Bayesian games with continuous actions, including contests, public good provision, and Cournot competition with private information, among others. Second, rather than relying on an LCM-based concavity test, we reformulate the monotonicity null into a countable collection of moment inequalities that involve only unconditional expectations and can be estimated by simple sample analogs, following the approach of \cite{HLS2019}, \cite{HsuShen2021} and \cite{HHLL2025}, who reformulate the null hypothesis of monotonicity in regressions, conditional treatment effects under regression discontinuity designs, and mean potential outcomes in continuous treatment effect models, respectively, to moment inequalities. Third, this moment-inequality formulation naturally extends to heterogeneous games, allowing us to control for game-specific covariates and to develop procedures that allow for both observed and unobserved game heterogeneity—features that are essential in empirical work and difficult to accommodate with LCM-type approaches.
Building on this moment-inequality representation, we construct a Cram\'{e}r–von Mises (CvM) type statistic based on the estimated inequalities.
We combine a nonparametric bootstrap and the generalized moment selection method of \cite{AndrewsShi2013} to construct the critical values. We show that our test controls size asymptotically and is consistent against any fixed alternative. We also provide a practical guide by specializing the general procedure to four widely used models: first-price auctions, Tullock contests, public good provision, and Cournot competition with private information. Monte Carlo evidence indicates that the test performs well in finite samples, including sample sizes typical in auction applications.
We apply our test to the asphalt paving procurement datasets studied in Aaltio et al. (2025), where cartel detection is a central concern. The empirical findings are consistent with the evidence reported in \cite{Aaltio_etal2025}: we do not reject monotonicity in the dataset where no collusion was found, while in datasets where collusion was detected, our results provide evidence against monotonicity. To our knowledge, this is the first empirical application to formally test monotonicity of equilibrium strategies in a Bayesian Nash equilibrium framework using real data.
An important advantage of our moment inequalities-based test for monotonicity is it naturally accommodates heterogeneous games. Controlling for game-specific covariates is essential in almost all empirical applications. Moreover, our approach can accommodate unobserved game heterogeneity, noting that it has been an important and yet challenging issue in the structural auction literature to develop methods in identification and inference with unobserved auction heterogeneity.\footnote{While it is important to control for unobserved game heterogeneity, little attention has been paid to controlling for unobserved heterogeneity in Bayesian games other than auctions. For identification and estimation of auction models with unobserved heterogeneity, see \cite{LiZheng2009}, \cite{Krasnokutskaya2011}, \cite{Roberts2013}, \cite{CHS2020}, and \cite{LSX2024}, among others, and \cite{HaileKitamura2019} for a survey.} Specifically, We develop fully nonparametric tests that can control for observed heterogeneity and also provide tests based on implications that remain valid under unobserved heterogeneity. Because fully nonparametric procedures can suffer from the curse of dimensionality, we additionally propose semiparametric tests based on action homogenization, extending the bid homogenization method proposed in \cite{HHS2003} beyond auctions to general Bayesian games. As such, our paper provides a monotonicity testing framework for incomplete-information games that can accommodate both observed and unobserved heterogeneity in a way suitable for applied work. .
Our work contributes to the econometrics of games of incomplete information. Our focus is on Bayesian games with continuous actions, which have witnessed vast development in auctions, and recent emerging literature in other games. See \cite{LarzenZhang2018}, \cite{ALS2023}, \cite{Kline2025}, \cite{LiZhangZhao2025}, and \cite{LiZhangZhao2026} for identification of general Bayesian games with different payoff/information structures; \cite{HeHuang2021} for estimation of a contest model, \cite{AryalGabrielli2020} for estimation of a competitive nonlinear pricing model, and \cite{AryalZincenko2024} for estimation of a Cournot competition model with private information.\footnote{Notably, there has been significant development in the econometrics of games of incomplete information with discrete actions, e.g., \cite{Seim2006}, \cite{Aradillas-Lopze2010}, \cite{Tang2010}, \cite{dePaulaTang2012}, \cite{WanXu2014}, \cite{AguirregabiriaMira2019}, among others.}
Our work also contributes to the econometric literature on testing shape restrictions, and monotonicity in particular.\footnote{See, e.g., \cite{CSS2018} for a review on the econometrics of shape restrictions.} The recent projection-based approach of \cite{FS2021}, for example, provides a general framework for testing shape restrictions defined by convex cones. As \cite{LV2021} emphasize, most existing tests focus on monotonicity in observed variables, whereas both their work and ours address monotonicity in latent variables.
While \cite{LV2021} rely on the LCM in the spirit of \cite{FS2021}, our test employs a transformation that leads to moment inequalities akin to \cite{HLS2019}, \cite{HsuShen2021} and \cite{HHLL2025}; the advantages of our test highlight a flexible route to testing shape restrictions in structural models and demonstrate the practical value of such methods in applications.
This paper is organized as follows. Section 2 characterizes differentiable symmetric equilibrium strategies within the symmetric independent private-type paradigm and establishes the rationalization result linking monotonicity in types to monotonicity of the quasi-inverse map in actions. It then develops our benchmark test for homogeneous games and derives its asymptotic properties. Section 3 illustrates implementation of our benchmark test in auctions, contests, public good provision, and Cournot competition with private information. Section 4 considers the extensions of our tests in Section 2 to allow for heterogeneous games, controlling for observed and unobserved game heterogeneity. Section 5 reports Monte Carlo evidence. Section 6 presents the asphalt paving procurement auction application. Section 7 concludes. Technical proofs are collected in the Appendix.
\section{Games of Incomplete Information}
\label{sec: incomplete information}
In this section, we first characterize the sufficient and necessary conditions for the
existence of a private type distribution that can rationalize the observed action distribution in general Bayesian games considered in \cite{Athey2001} within the symmetric independent private type paradigm, in which all agents play a symmetric, monotone and differentiable strategy. The framework here covers a large class of Bayesian games, including the trading games with auction games as special cases in \cite{LarzenZhang2018}, the allocation-transfer games considered in \cite{Kline2025}, which include auctions, contests, and public good provision, and general Bayesian games with continuous payoffs in \cite{LiZhangZhao2025}, which include contests, public good provision, and Cournot competition with incomplete information.
\subsection{Strictly Monotone Bayesian Nash Equilibrium and Rationalization Result}
\label{sec: necessary and sufficient conditions}
We consider a model with symmetric independent private types. We impose the following conditions on the types.
\begin{assumption} \label{assu: value}
Assume that (i) Agent $i$, $i = 1, \ldots, N,$ draws his/her type independently from the other agents from a CDF $F(\cdot)$ with the PDF $f(\cdot)$.
(ii)
$f(\cdot)$ is strictly positive and bounded away from zero on its support, a compact
interval $\mathcal{V}=[\underline{v}, \overline{v}] \subseteq R$ , and is twice continuously differentiable on $(\underline{v}, \overline{v})$.
\end{assumption}
Each agent only observes his/her own type, but not the other agents' types, while the number of agents $N$, and $F(\cdot)$ ($f(\cdot)$) are common knowledge to all agents, who are assumed to be risk neutral. Let $D_{i}$ be the allocation indicating $i$ attaining the portion of the good or the probability of attaining the good, and let $C_{i}$ be any net payment made or cost incurred by $i$.
The outcome allocations and transfers for all agents are calculated as functions of all agents' actions and we introduce an individual allocation function $D_i \equiv D(B_i,B_{-i})$, which determines the allocation of the item for a given vector of actions $B_1, ...,B_N$. The eventual payment $C_{i}$ is also a function of all agents' actions and thus we introduce an individual payment function $C(B_i,B_{-i})$ such that $C_i \equiv C(B_i,B_{-i})$, which is the final payment made by agent $i$. The outside option is normalized to zero. If agent $i$ has type $V_{i}$, agent $i$'s utility is linear in type: $V_{i}D_{i} - C_{i}$.
Having observed their $V_{i}$'s, agents choose a monotone strategy mapping values into an action $b \in \mathcal{B}$, where $\mathcal{B}$ is the space of actions available to agents. We assume that $\mathcal{B} = [\underline{b}, \overline{b}]$ with $0 \leq \underline{b} < \overline{b} < \infty$. For a given strategy $s$, we write $s(v)$ for the action played by type $v$ under $s(\cdot)$.
We define $P(b)$ and $T(b)$ to be the expected outcome allocations and expected payment, respectively, for agent $i$ when agent $i$ plays action $b$ that is integrated over the distribution of rivals' strategies $B_{-i}$. Specifically,
\begin{align}
P(b)=E_{B_{-i}}[D_i(B_i,B_{-i})|B_i=b],~~T(b)=E_{B_{-i}}[C_i(B_i,B_{-i})|B_i = b].
\label{eq: P and T}
\end{align}
We make the following assumption on the smoothness of both $P(\cdot)$ and $T(\cdot)$.
\begin{assumption} \label{assu: PT}
Both $T(b)$ and $P(b)$ are twice continuously differentiable in $b$ on $(\underline{b}, \overline{b})$.
\end{assumption}
A similar assumption is made in \cite{LarzenZhang2018} for point identification of trading games and \cite{Kline2025} for point identification of allocation-transfer games, respectively.
Now define the interim expected payoff for an agent with type $v$ and action $b$ as $\Pi(b,v) \equiv vP(b)-T(b)$. we make the following assumptions:
\begin{assumption} \label{assu: SSM}
$\frac{\partial^2 \Pi(b,v)}{\partial b \partial v} > 0$ for all $v\in(\underline{v}, \overline{v})$ and $b\in(\underline{b}, \overline{b})$.
\end{assumption}
\begin{assumption} \label{assu: SC}
$\frac{\partial^2 \Pi(b,v)}{\partial b^2} < 0$ for all $v\in(\underline{v}, \overline{v})$ and $b\in(\underline{b}, \overline{b})$.
\end{assumption}
Assumption \ref{assu: SSM} is the strict supermodularity assumption, which is stronger than the single crossing condition in \cite{Athey2001} who establishes existence of monotone Bayesian Nash equilibrium in games of incomplete information with continuous action spaces. With the specification of $\Pi (b, v)$ as $vP(b)-T(b)$, Assumption \ref{assu: SSM} is equivalent to $P'(b) > 0$, which means that the expected outcome allocation is strictly increasing in actions. Assumption \ref{assu: SC} assumes the expected payoff to be strictly concave. These two assumptions are needed to get strictly monotone Bayesian Nash equilibrium (SMBNE) in our framework where we use the first order condition (FOC) approach in considering differentiable equilibrium strategies.
\begin{lemma}\label{lemma: SMBNE}
Under Assumptions \ref{assu: value}, \ref{assu: PT}, \ref{assu: SSM} and \ref{assu: SC}, $s'(v) > 0$ for all $v\in(\underline{v}, \overline{v})$.
\end{lemma}
For an SMBNE where the FOC is satisfied, each agent's strategy $s(v_i)$ needs to be a best response given that all rival players play according to $s(v)$ for all $v \in [\underline{v}, \overline{v}]$. This means that incentive compatibility holds:
\begin{align}
s(v_i) \in \arg\max_{b' \in \mathcal{B}} \; v_{i} P(b') - T(b').
\label{eq: IC 1}
\end{align}
Suppose that the SMBNE is an interior solution. The FOC-based lemma below applies on intervals where best responses are in the interior and $s(\cdot)$ is differentiable.
With agents behaving non-cooperatively, a common strategy $s_i = s(v_i)$ is an equilibrium strategy if, when adopted by all agents but one, say agent $1$, it is optimal for that agent to adopt the given monotone strategy as well. Hence, we can write any action as $s_1 = s(x)$ and view agent $1$ as choosing $x$. It then follows that $s(v)$ is an equilibrium strategy if agent $1$ can do no better than choosing $x = v_1$, and so its action is $s(v_1)$. To this end, the expected gain to agent $1$ can be expressed as
\[
\Pi(x,v_1) = v_1 \, P\!\big(s(x)\big) - T\!\big(s(x)\big).
\]
Thus, the first-order condition for an interior optimum for any agent with valuation $v$ is to choose $x = v$, such that
\[
\frac{\partial \Pi}{\partial x}(x,v)
= \big[ v \, P'\!\big(s(x)\big) - T'\!\big(s(x)\big) \big] \, s'(x) = 0
\qquad \text{at } x = v .
\]
Because $s'(x) > 0$, we get
\begin{align}
v = \frac{T'\!\big(s(v)\big)}{P'\!\big(s(v)\big)} \;\equiv\; \xi\!\big(b\big),\label{eq:implicit}
\end{align}
where $b = s(v)$.
(\ref{eq:implicit}) is established in Corollary 2 in \cite{LarzenZhang2018} for trading games, and in equation (26) in \cite{Kline2025} for allocation-transfer games in the independent valuation paradigm. We refer to $\xi(\cdot)$ in (\ref{eq:implicit}) as the quasi-inverse equilibrium strategy in all the Bayesian games we consider.
For a continuously differentiable CDF $G$ on $[\underline b,\bar b]$ with density $g>0$ on $(\underline b,\bar b)$,
let $P(\cdot;G)$ and $T(\cdot;G)$ denote the reduced-form expected allocation and transfer
when opponents' actions are i.i.d. with the CDF \ $G(\cdot)$, and assume $P(\cdot;G),T(\cdot;G)\in C^{2}(\underline b,\bar b)$
with $P'(b;G)>0$ for all $b\in(\underline b,\bar b)$. Define
\[
\xi(b;G):=\frac{T'(b;G)}{P'(b;G)}, \qquad b\in(\underline b,\bar b),
\]
and assume $\xi(\cdot;G)$ extends continuously to $[\underline b,\bar b]$ with $\underline\xi:=\xi(\underline b;G)$ and $\bar\xi:=\xi(\bar b;G)$.
\begin{thm}[Rationalizability and Quasi-Inverse Equilibrium Strategy]
\label{thm: LZ2018}
Fix a continuously differentiable CDF $G(\cdot)$ on $[\underline b,\bar b]$ with density $g>0$ on $(\underline b,\bar b)$. Then $G$ is rationalizable by a strictly monotone, symmetric, and differentiable equilibrium if and only if the following (C1) and (C2) hold.
\begin{itemize}
\item[] (C1) Actions $(B_1,\ldots,B_N)$ are i.i.d.\ with common CDF $G(\cdot)$.
\item[] (C2) $\xi(\cdot;G)$ is strictly increasing on $(\underline b,\bar b)$.
\end{itemize}
In addition, for any strictly monotone interior symmetric equilibrium rationalizing $G(\cdot)$, the type distribution $F(\cdot)$ is unique.
\end{thm}
Theorem \ref{thm: LZ2018} extends Theorem 1 in \cite{GPV2000} in first price auctions within the symmetric independent private value paradigm to general Bayesian games considered in \cite{Athey2001} within the symmetric independent private type paradigm and with differentiable equilibrium strategies. First, it provides necessary and sufficient conditions for the existence of a private type distribution that can rationalize the observed action distribution. These two conditions are independence of actions and also monotonicity of the quasi-inverse equilibrium strategy, with the latter being the basis for our tests for monotone equilibrium strategies. Second, it establishes nonparametric point identification of the private type distribution from the distribution of the observed actions in the model considered. Our result extends the point identification result in \cite{LiZhangZhao2026} in Bayesian games with continuous payoff functions to general Bayesian games. Relatedly, under somewhat different set of assumptions, \cite{Kline2025} establishes the point identification result in allocation-transfer games.
\subsection{Proposed Test}\label{sec: proposed test}
Based on Theorem \ref{thm: LZ2018}, we formulate the null hypothesis as
\begin{align}\label{eq: general H0}
H_0:~ \xi(b)=\frac{T'(b)}{P'(b)}~\text{is weakly increasing in $b$.}
\end{align}
In (\ref{eq: general H0}), we suppress the dependence of $\xi$, $T$ and $P$ functions on $G$ for notational simplicity.
Ideally, the null should be that $\xi(b)$ is strictly increasing, which usually results in a degenerate case for the asymptotic distribution of the test statistics. Therefore, we include the equality in the null hypothesis, as is usually done in the literature, e.g., \cite{LV2021}.\footnote{Another reason to consider weakly increasing in the null hypothesis is that any test of the null of strictly increasing is likely to have power equal to size against alternatives that are weakly (but not strictly) increasing. This is related to the problems in \cite{Romano2004}. We thank Andres Santos for pointing this out.}
We propose a test for the null hypothesis in (\ref{eq: general H0}) and our test is similar to those in \cite{HLS2019}, \cite{HsuShen2021} and \cite{HHLL2025}. We first adapt the transformation in Lemma 3.1 of \cite{HsuShen2021} to transform the null hypothesis in (\ref{eq: general H0}) to countably many inequalities without loss of information and we summarize this in the following lemma. Define $a=\overline{b}-\underline{b}$, which is the length of $[\underline{b},\overline{b}]$.
\begin{lemma}\label{lemma: general H0 transformation}
Let $h(b)$ be a known function such that $0 <h(b)P'(b) \leq M<\infty$.
Then $H_0$ in (\ref{eq: general H0}) is equivalent to
\begin{align}
&H_0': \nu(b_1,b_2, q)=M(b_2,q)W(b_1,q)-M(b_1,q)W(b_2,q)\leq 0~\text{for all $(b_1,b_2, q)\in\mathcal{L}$} \label{eq: general H0'},
\end{align}
where
\begin{align}
&M(b,q)= \int_{b}^{b+\frac{a}{q}} h(\tilde{b})T'(\tilde{b}) d\tilde{b},
~~~
W(b,q)=\int_{b}^{b+\frac{a}{q}} h(\tilde{b})P'(\tilde{b}) d\tilde{b},~\text{and}\label{eq: general M W}\\
&\mathcal{L}=\Big\{(b_1,b_2, q): \Big(\frac{b_1-\underline{b}}{a}, \frac{b_2-\underline{b}}{a}\Big)\cdot q\in(0,1,\ldots,q)^2, b_1>b_2~\text{for q=2,3,\ldots} \Big\}.\label{eq: L}
\end{align}
\end{lemma}
Lemma \ref{lemma: general H0 transformation} shows that the transformation of the null hypothesis $H_0$ to $H_0'$ in (\ref{eq: general H0'}) which includes a countable many moment inequalities is without loss of information. The $h(b)$ function can be any function that satisfies $0 <h(b)P'(b) \leq M<\infty$, therefore, in practice, $h(b)$ can be case specific and one can pick $h(b)$ in a way such that $\nu(b_1,b_2, q)$ can be estimated and weakly converges to a Gaussian process.
Specifically, let $\widehat{M}(b,q)$ and $\widehat{W}(b,q)$ denote the estimators for ${M}(b,q)$ and ${W}(b,q)$, respectively. Denote the estimator for $\nu(b_1,b_2, q)$ as
\begin{align}
&\widehat{\nu}(b_1,b_2, q)=\widehat{M}(b_2,q)\widehat{W}(b_1,q)-\widehat{M}(b_1,q)\widehat{W}(b_2,q). \label{eq: general nu estimator}
\end{align}
\begin{assumption}\label{assu: general DGP}
Assume that we observe actions
$\{B_{i,\ell}: i=1,\ldots, N,~\ell=1,\ldots, L \}$
that are i.i.d.\ random variables across $L$ games and the agents in each game.
\end{assumption}
Assumption \ref{assu: general DGP} imposes conditions on the data that econometricians observe. Let $S=NL$ denote the total number of observed actions.
\begin{assumption}\label{assu: general gaussian limit}
Assume that \begin{align*}
\sqrt{S}(\widehat{\nu}(\cdot,\cdot,\cdot)-{\nu}(\cdot,\cdot,\cdot))\Rightarrow
\Phi_\kappa(\cdot,\cdot,\cdot)
\end{align*}
in which $\Rightarrow$ denotes weak convergence and $\Phi_\kappa(\cdot,\cdot,\cdot)$ is a Gaussian process with covariance kernel being
$\kappa\big((b'_1,b'_2,q'),(b''_1,b''_2,q'')\big)$ for $(b'_1,b'_2,q'),(b''_1,b''_2,q'')\in \mathcal{L}$. In addition, $\kappa$ is not a zero function.
\end{assumption}
Assumption \ref{assu: general gaussian limit} requires that there exists a consistent estimator for $\nu(\cdot)$ which weakly converges to a Gaussian process with proper rescaling. In addition, that $\kappa$ is not a zero function implies that $\widehat{\nu}(\cdot)$ is not degenerate.
Define $\sigma^2_\nu(b_1,b_2,q)=\kappa\big((b_1,b_2,q),(b_1,b_2,q)\big)$ which is the asymptotic variance of
$\sqrt{S}(\widehat{\nu}(b_1,b_2,q)-{\nu}(b_1,b_2,q))$ and its estimator as $\widehat{\sigma}^2_
\nu(b_1,b_2,q)$
\begin{assumption} \label{assu: general consistent variance}
Assume that $\widehat{\sigma}^2_\nu(b_1,b_2,q)$ is uniformly consistent for $\sigma^2_\nu(b_1,b_2,q)$ in that $\sup_{(b_1,b_2,q)\in \mathcal{L}}|\widehat{\sigma}^2_\nu(b_1,b_2,q)-{\sigma}^2_\nu(b_1,b_2,q)|\stackrel{p}{\rightarrow}0$. In addition, ${\sigma}^2_\nu(\underline{b}, (\underline{b}+\overline{b})/2,2)>0$.
\end{assumption}
Assumption \ref{assu: general consistent variance} requires that there exists a uniformly consistent estimator for the asymptotic variance. We specifically assume that one of the moments is not degenerate. In practice, if the influence functions of the $\widehat{\nu}(\cdot)$ are known, then one can use the estimated influence functions to construct an estimator for the variance function. Or one might use the bootstrap estimator to estimate the variance function.
For some small $\epsilon>0$, define
\begin{align}
\widehat{\sigma}^2_{\nu,\epsilon}(b_1,b_2,q)=\max\{\widehat{\sigma}^2_
\nu(b_1,b_2,q),\epsilon\cdot
\widehat{\sigma}^2_\nu(\underline{b}, (\underline{b}+\overline{b})/2,2)\}.\label{eq: general estimated sigma epsilon}
\end{align}
For a weighted function $Q(b_1,b_2,q)>0$ and $\sum_{(b_1,b_2,q)\in\mathcal{L}}Q(b_1,b_2,q) <\infty $, define our test statistic as
\begin{align}
\widehat{T}_{S}=\sum_{(b_1,b_2,q)\in\mathcal{L}} \max\Big\{ \sqrt{S} \frac{\widehat{\nu}(b_1,b_2,q)}
{\widehat{\sigma}_{\nu,\epsilon}(b_1,b_2,q)},0\Big\}^2 Q(b_1,b_2,q). \label{eq: general test statistics}
\end{align}
We propose a nonparametric bootstrap combining with the generalized moment selection (GMS) method introduced by \cite{AndrewsSoares2010} and \cite{AndrewsShi2013} to construct the critical values for our test.\footnote{The GMS method is similar to the
recentering method in \cite{Hansen2005} and \cite{DonaldHsu2016}, as
well as the contact set approach in \cite{LSW2010}.} For the nonparametric bootstrap, we bootstrap games. That is, let $\{(B^*_{1\ell},\ldots, B^*_{N\ell})
:\ell \leq L \}$ be a
bootstrap sample in which $B^*_{i,\ell}=B_{i\ell^*}$ and $\{
\ell^*: \ell \leq L\}$ is an i.i.d.\ bootstrap sample
drawn from the empirical distribution of $\{\ell : \ell\leq L\}$.
Define the bootstrap estimator for $\nu(b_1,b_2, q)$ as
\begin{align}
&\widehat{\nu}^*(b_1,b_2, q)
=\widehat{M}^*(b_2,q)\widehat{W}^*(b_1,q)-\widehat{M}^*(b_1,q)\widehat{W}^*(b_2,q), \label{eq: general bootstrap nu estimator},
\end{align}
where $\widehat{M}^*(b,q)$ and $\widehat{W}^*(b,q)$
are bootstrap estimators for ${M}(b,q)$ and ${W}(b,q)$, respectively, based on the bootstrap sample.
The bootstrapped process is defined as $\Phi^*(\cdot,\cdot,\cdot)=\sqrt{S}(\widehat{\nu}^*(\cdot,\cdot,\cdot)-\widehat{\nu}(\cdot,\cdot,\cdot)).$
\begin{assumption}\label{assu: general bootstrap limit}
Assume that $\Phi^*(\cdot,\cdot,\cdot)\Rightarrow \Phi_\kappa(\cdot,\cdot,\cdot)$ conditional on the sample path with probability approaching 1 where $\Phi_\kappa(\cdot,\cdot,\cdot)$ is as in Assumption \ref{assu: general gaussian limit} and we denote it as $\Phi^*(\cdot,\cdot,\cdot)\stackrel{p}{\Rightarrow} \Phi_\kappa(\cdot,\cdot,\cdot)$.
\end{assumption}
Assumption \ref{assu: general bootstrap limit} requires that we have a bootstrap method that approximates the limiting process $\Phi_\kappa(\cdot,\cdot,\cdot)$ well.
As most papers in the moment inequality literature, we use
the GMS method to construct the critical value.
By doing this, one can construct a more powerful test without resorting
to the least favorable configuration. Let $\{\beta_{\ell},~\ell=1,2,\ldots\}$ and $\{\kappa_{\ell},~\ell=1,2,\ldots\}$ be sequences of positive numbers.
Define the GMS
function $\psi_S(b_1,b_2,q)$ as
\begin{align}
\psi_S(b_1,b_2,q)=-\beta_S \cdot 1\Big(\frac{\sqrt{S}\widehat{\nu}(b_1,b_2,q)}{\widehat{\sigma}_{\nu,\epsilon}(b_1,b_2,q)}<-\kappa_S\Big). \label{eq: general GMS-function}
\end{align}
We impose conditions on $\beta_S$ and $\kappa_S$.
\begin{assumption}\label{assu: general GMS}
Assume that (i) $\lim_{S\rightarrow\infty}\kappa_S=\infty$ and
$\lim_{S\rightarrow\infty}\kappa_S/\sqrt{S}=0$ and (ii)
$\beta_S$ is non-decreasing, $\lim_{S\rightarrow\infty}\beta_S=\infty$ and
$\lim_{S\rightarrow\infty}\beta_S/\kappa_S=0$.
\end{assumption}
For a significance level $\alpha<1/2$, define the bootstrapped critical value
$\hat{c}_\eta$ as
\begin{align*}
\hat{c}_\eta=\sup\Big\{c\big|P^*\Big(\sum_{(b_1,b_2,q)\in\mathcal{L}} \max\Big\{
\frac{{\Phi}^*(b_1,b_2,q)}
{\widehat{\sigma}_{\nu,\epsilon}(b_1,b_2,q)}+\psi_S(b_1,b_2,q),0\Big\}^2 Q(b_1,b_2,q)\leq c\Big)\leq 1-\alpha+\eta\Big\}+\eta,
\end{align*}
where $\eta>0$ is an arbitrarily small positive number. Note that $\hat{c}_\eta$ is the $(1-\alpha+\eta)$-th
quantile of the simulated null distribution plus $\eta$ which is
called an infinitesimal uniformity factor in \cite{AndrewsShi2013}.
Following \cite{HLS2019}, we set $\beta_{S}=0.85\cdot\ln(S)/\ln\ln(S)$, $\kappa_{S}=0.15\cdot\ln(S)$ and
$\eta=10^{-6}$.
The decision rule is: ``Reject $H_0$ ($H_0'$) when $\widehat{T}_{S}>\hat{c}_\eta$."
The following theorem shows the asymptotic size control and the power against fixed alternatives of our test.
\begin{thm} Suppose Assumptions \ref{assu: general DGP}, \ref{assu: general gaussian limit}, \ref{assu: general consistent variance}, \ref{assu: general bootstrap limit}, \ref{assu: general GMS} hold. Then,\\
(a) under $H_0$,
${\lim}_{S\rightarrow\infty}
P(\widehat{T}_{S}>\hat{c}_\eta)\leq \alpha$;\\
(b) under $H_1$,
${\lim}_{S\rightarrow\infty}
P(\widehat{T}_{S}>\hat{c}_\eta)= 1.$ \label{thm: general test size and power}
\end{thm}
Theorem \ref{thm: general test size and power} shows that our test has asymptotic size control under the null hypothesis and is consistent against fixed alternatives.
\begin{remark}\label{remark: hetero number}
In practice, the data may include games with varying numbers of agents. In the Appendix, we discuss how to conduct tests that allow for such heterogeneity. The key issue is that the quasi-inverse equilibrium strategy, the $\xi$ function, implicitly depends on the number of agents in a game. If this strategy varies with the number of agents, then the corresponding moment conditions must be treated separately for each group. Accordingly, when games involve different numbers of agents, we implement a joint test across these groups. For further details, see Appendix \ref{sec: hetero numbers}.
\end{remark}
\section{Examples} \label{sec: examples}
In this section, we consider four examples of Bayesian games. The first example is the first-price sealed-bid auction model within the symmetric independent private value paradigm, the second one is a Tullock contest model, the third one is a public good provision model, and the last one is the Cournot competition model with incomplete information.
\subsection{First-price Sealed-bid Auction Model}\label{subsec: auction}
As the first example, we consider a first-price sealed-bid auction model for homogeneous goods with a fixed number of bidders.
The bids submitted by the bidders are the observed actions in each auction. The econometricians observe $N$ bids $(B_{1\ell},\ldots, B_{N\ell})$ in the $\ell$-th auction for $\ell=1,\ldots, L$, but not bidders' valuations be $\{V_{i,\ell}:~i=1,\ldots, N,~\ell=1,\ldots, L \}$.
Thus, the allocation function of the auctioned product for player $i$ is determined by the indicator function $D(B_{i}, B_{-i})$ defined as
\begin{align*}
D(B_i, B_{-i}) =
\left\{
\begin{array}{ll}
1, & \text{if } B_i > B_j \ \ \forall j \neq i, \\
0, & \text{otherwise.}
\end{array}\right.
\end{align*}
Also, the payment function of the auctioned product for player $i$ is determined by the cost function defined as $C(B_{i}, B_{-i})=B_i\cdot D(B_{i}, B_{-i})$.
Integrating over the distribution of $B_{-i}$ gives the monotone equilibrium expected allocation for bidder $i$, i.e., $E_{B_{-i}}[D(B_i, B_{-i})]$.
Let $V_i$ be the value for agent $i$. Then for agent $i$, his/her expected equilibrium payoff $\pi(B_i, V_i)$ is
\begin{equation*}
\pi(B_i, V_i) = V_i\cdot E_{B_{-i}}[D(B_i, B_{-i})] - B_i\cdot E_{B_{-i}}[D(B_i, B_{-i})]
=V_i P(B_i)-T(B_i).
\end{equation*}
The following assumption is summarized from \cite{GPV2000} on the distribution of the valuations, which gives rise to the symmetric independent private value paradigm.
\begin{assumption}\label{assu: auction DGP}
Assume that (i) the unobserved valuations
$\{V_{i,\ell}: i=1,\ldots, N,~\ell=1,\ldots, L \}$ are i.i.d.\ with probability density function (PDF), $f(\cdot)$, and cumulative distribution function (CDF), $F(\cdot)$; (ii)
$f(\cdot)$ is strictly positive and bounded away from zero on a convex and compact support,
$[\underline{v}, \overline{v}] \subseteq R$ , and is twice continuously differentiable on $(\underline{v}, \overline{v})$.
\end{assumption}
Assume that the observed bids are generated from the valuations satisfying Assumption \ref{assu: auction DGP} and by a BNE bidding strategy, $s(V_{i,\ell})$. It then follows that
\begin{align}
B_{i,\ell}= s(V_{i,\ell})\equiv V_{i,\ell}- \frac{1}{F(V_{i,\ell})^{N-1}} \int_{\underline{v}}^{V_{i,\ell}}F(u)^{N-1} du,
\label{eq: BNE strategy}
\end{align}
and the observed bids
$\{B_{i,\ell}: i=1,\ldots, N,~\ell=1,\ldots, L \}$ are i.i.d.\ with PDF $g(\cdot)$ and CDF $G(\cdot)$ with a convex and compact support, $[\underline{b},\overline{b}]\subseteq R$.\footnote{Note that $g(\cdot)$ and $G(\cdot)$ depend on $N$, the number of bidders in the auction, because the equilibrium bidding strategy in (\ref{eq: BNE strategy}) is a function of $N$.}
The BNE requires that the bidding strategy $s(\cdot)$ is strictly increasing.\footnote{Moreover, \cite{GPV2000} show that $s(\cdot)$ is at least three times continuously differentiable on $(\underline{v}, \overline{v})$.}
It can be shown that $P(b)=G(b)^{N-1}$ and $T(b)=bG(b)^{N-1}$. As a result, $\xi(\cdot)$ in (\ref{eq:implicit}) becomes the quasi-inverse bidding strategy in \cite{GPV2000}:
\begin{align}
\xi(b) = b+\frac{1}{N-1}\frac{G(b)}{g(b)}, \label{eq: gpv}
\end{align}
where $G(\cdot)$ and $g(\cdot)$ are the CDF and PDF of bids, respectively.
\cite{GPV2000} show that the observed bid distribution $G(\cdot)$ is an equilibrium bid distribution if and only if the quasi-inverse strategy, $\xi(b)$, in (\ref{eq: gpv})
is strictly increasing, which is implied by Theorem \ref{thm: LZ2018}.
Therefore, we define the null hypothesis as
\begin{align}
&H_{0,au}: \xi(b)= b+ \frac{1}{N-1}\frac{G(b)}{g(b)}~\text{is weakly increasing in $b$}. \label{eq: auction H0}
\end{align}
We apply Lemma \ref{lemma: general H0 transformation} to transform the null hypothesis in (\ref{eq: auction H0}) by setting $h(b)=G(b)^{2-N}(N-1)^{-1}$, so we have
\begin{align*}
&M(b,q)=\int_{b}^{b+\frac{a}{q}} h(\tilde{b}) T'(\tilde{b}) d\tilde{b}=
\int_{b}^{b+\frac{a}{q}} \big(\tilde{b} g(\tilde{b})+(N-1)^{-1}G(\tilde{b})\big) d\tilde{b},\\
&W(b,q)=\int_{b}^{b+\frac{a}{q}} h(\tilde{b}) P'(\tilde{b}) d\tilde{b}=\int_{b}^{b+\frac{a}{q}} g(\tilde{b}) d\tilde{b}.
\end{align*}
Note that $M(b,q)$ and $W(b,q)$ are both identified and can be expressed as unconditional means. We summarize the results in the following lemma.
\begin{lemma}\label{lemma: auction H0 transformation}
The $H_{0,au}$ in (\ref{eq: auction H0}) is equivalent to
\begin{align}
&H_{0,au}': \nu(b_1,b_2, q)=M(b_2,q)W(b_1,q)-M(b_1,q)W(b_2,q)\leq 0~\text{for all $(b_1,b_2, q)\in\mathcal{L}$} \label{eq: auction H0'},
\end{align}
where
\begin{align}
&M(b,q)= E\Big[B_{i,\ell} 1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big)\Big]\notag\\
&~~~~~~~~~~~~~~~~+
\frac{1}{N-1} E\Big[ 1\Big(B_{i,\ell}\leq b+\frac{a}{q} \Big) \Big(b+\frac{a}{q} - B_{i,\ell}\Big)
- 1(B_{i,\ell}\leq b ) (b - B_{i,\ell}) \Big], \label{eq: auction M}\\
&W(b,q)=E\Big[1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big) \Big]\label{eq: auction W}.
\end{align}
\end{lemma}
Based on (\ref{eq: auction M}) and (\ref{eq: auction W}), we can apply the test in Section \ref{sec: proposed test}. In Appendix, we provide a uniform consistent estimator for the asymptotic variance for each moment and show the sufficient conditions hold for Theorem \ref{thm: general test size and power} in our case.
\begin{remark} \label{remark: procurement} {\bf (Procurement Auctions)}
The example above has considered high-bid first-price auctions to be in line with the literature where the identification results starting \cite{GPV2000} have been established for high-bid auctions. For low-bid first-price auctions such as procurement auctions considered in our empirical application, we can similarly show that $P(b)=(1-G(b))^{N-1}$ and $T(b)=b(1-G(b))^{N-1}$. Therefore, it is straightforward to see that
\begin{align*}
\xi(b)=b-\frac{1}{N-1}\frac{1-G(b)}{g(b)}.
\end{align*}
In this case, the $M(b,q)$ and $W(b,q)$ are expressed as
\begin{align}
&M(b,q)= E\Big[B_{i,\ell} 1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big)\Big]-\frac{1}{N-1}\frac{a}{q}\notag\\
&~~~~~~~~~~~~~~~~+
\frac{1}{N-1} E\Big[ 1\Big(B_{i,\ell}\leq b+\frac{a}{q} \Big) \Big(b+\frac{a}{q} - B_{i,\ell}\Big)
- 1(B_{i,\ell}\leq b ) (b - B_{i,\ell}) \Big], \label{eq: Pro auction M}\\
&W(b,q)=E\Big[1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big) \Big]\label{eq: Pro auction W},
\end{align}
respectively.
\end{remark}
\subsection{Tullock Contest Model}\label{subsec: contest model}
In the context of a Tullock contest such as in \cite{Tullock1980}, \cite{Ryvkin2010} and \cite{Ewerhart2014}, let the contestants' valuations be $\{ V_{i,l}: i=1,...,N \}$ which are unobservable to the econometrician. The efforts exerted by contestants are the observed actions.
To be specific, $N$ contestants choose effort levels $B_i \ge 0$ to compete for a prize. The winning probability, conditional on rivals' effort levels $B_{-i}$, is given by the symmetric contest–success function
\begin{equation}
D(B_i, B_{-i}) = \frac{B_i}{\sum_{s=1}^{N} B_s}.
\label{eq:csf}
\end{equation}
Also, the contestant $i$ has a linear effort cost $C(B_i,B_{-i}) = B_i$.
Integrating over the distribution of $B_{-i}$ gives the monotone equilibrium expected allocation $P(B_i)=E_{B_{-i}}[D(B_i, B_{-i})]$ and expected effort cost $T(B_i)=E_{B_{-i}}[C(B_i, B_{-i})]=B_i$.
The expected payoff is
\begin{equation*}
\pi(B_i, V_i) = V_i \, E_{B_{-i}}[D(B_i, B_{-i})] - E_{B_{-i}}[C(B_i, B_{-i})]= V_iP(B_i)-T(B_i).
\end{equation*}
Let the contestants' valuations be $\{ V_{i,l}: i=1,...,N, \ell=1,..,L \}$ which are unobservable to the econometrician. We impose the following assumption.
\begin{assumption}\label{assu: contest DGP}
Assume that (i) the unobserved valuations
$\{V_{i,\ell}: i=1,\ldots, N,~\ell=1,\ldots, L \}$ are i.i.d.\ with probability density function (PDF), $f(\cdot)$, and cumulative distribution function (CDF), $F(\cdot)$; (ii)
$f(\cdot)$ is strictly positive and bounded away from zero on a convex and compact support,
$[\underline{v}, \overline{v}] \subseteq R$ , and is twice continuously differentiable on $(\underline{v}, \overline{v})$.
\end{assumption}
\cite{Ewerhart2020} show the existence and uniqueness of a monotone equilibrium. The FOC for an interior optimum is
\[
\frac{\partial \Pi(B_{i}; V_{i})}{\partial B_{i}}
= V_{i,\ell}\,P'(B_{i}) - 1 = 0
\quad\Rightarrow\quad
V_{i}=\frac{1}{P'(B_{i}) }.
\]
In this case, we have $P(b)=E_{B_{-i}}[D(b,B_{-i})]$ and $T(b)=b$.
It is straightforward to see that
$P'(b)=b^{-1}E_{B_{-i}}[D(b,B_{-i})(1-D(b,B_{-i}))]$ which is strictly positive.
We then have
\begin{align*}
\xi(b)=\frac{T'(b)}{P'(b)}=
\frac{1}{b^{-1}E_{B_{-i,\ell}}[D(b,B_{-i})(1-D(b,B_{-i}))]}
\end{align*}
which is the same as in \cite{Ewerhart2020}.
Therefore, we define the null hypothesis as
\begin{align}
&H_{0,con}: \xi(b)= \frac{1}{b^{-1}E_{B_{-i}}[D(b,B_{-i})(1-D(b,B_{-i}))]}~\text{is weakly increasing in $b$}. \label{eq: contest H0}
\end{align}
We apply Lemma \ref{lemma: general H0 transformation} to transform the null hypothesis in (\ref{eq: contest H0}) by setting $h(b)=b g(b)$ with $g(b)$ being the density function of the actions under BNE,
so we have
\begin{align*}
&M(b,q)=\int_{b}^{b+\frac{a}{q}} h(\tilde{b}) T'(\tilde{b}) d\tilde{b}=
\int_{b}^{b+\frac{a}{q}} \tilde{b} g(\tilde{b}) d\tilde{b},\\
&W(b,q)=\int_{b}^{b+\frac{a}{q}} h(\tilde{b}) P'(\tilde{b}) d\tilde{b}=\int_{b}^{b+\frac{a}{q}}E_{B_{-i}}[D(\tilde{b},B_{-i})(1-D(\tilde{b},B_{-i}))] g(\tilde{b}) d\tilde{b}.
\end{align*}
Note that $M(b,q)$ and $W(b,q)$ are both identified and can be expressed as unconditional means. We summarize the results in the following lemma.
\begin{lemma}\label{lemma: contest H0 transformation}
The $H_{0,con}$ in (\ref{eq: contest H0}) is equivalent to
\begin{align}
&H_{0,con}': \nu(b_1,b_2, q)=M(b_2,q)W(b_1,q)-M(b_1,q)W(b_2,q)\leq 0~\text{for all $(b_1,b_2, q)\in\mathcal{L}$} \label{eq: contest H0'},
\end{align}
where
\begin{align}
&M(b,q)= E\Big[B_{i,\ell} \cdot 1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big)\Big]\label{eq: contest M}\\
&W(b,q)=E\Big[1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big) \frac{B_{i,\ell}}{\sum_{j=1}^N B_{j,\ell} }\Big(1-\frac{B_{i,\ell}}{\sum_{j=1}^N B_{j,\ell}}\Big)\Big]\label{eq: contest W}.
\end{align}
\end{lemma}
Based on (\ref{eq: contest M}) and (\ref{eq: contest W}), we are able to apply the test in Section \ref{sec: proposed test}.
\subsection{Public Good Provision Model}
\label{subsec: public}
Suppose there are $N$ contributors, and contributor $i$ voluntarily contributes $B_i$ to provide a public good.
Let $\Omega$ be the common benefit function of the total contributions. We assume that $\Omega$ is known, and is strictly increasing and concave. Therefore, we assume $D(B_i, B_{-i})=\Omega ( \sum_{s=1}^N B_s).$ Also, if contributor $i$ decides to contribute $B_i$, the net payment will be $B_i$ for contributor $i$, so we have $C(B_i, B_{-i})=B_i$. Contributor $i$'s private value of the public good $V_i$ affects the marginal utility of consuming the public good. Then given the actions $(B_1,\ldots, B_N)$, the utility for contributor $i$ is $V_i D(B_i, B_{-i})- C(B_i, B_{-i}).$
Define $P(b)=E_{B_{-i}}[D(b, B_{-i})]$ and $T(b) =E_{B_{-i}}[C(b, B_{-i})]=B_i$.
Following \cite{BagRoy2011} who extend \cite{Varian1994}'s model with complete information to allow for incomplete information , the expected payoff for contributor $i$ is
\begin{equation*}
\pi(B_i, V_i) = V_i \, E_{B_{-i}}[D(B_i, B_{-i})] - B_i= V_i P(B_i)-T(B_i),
\end{equation*}
and we have
\begin{align*}
\xi(b) = \frac{T'(b)}{P'(b)} =
\frac{1}{E_{B_{-i}}\big[ \Omega'\big( b+ \sum_{s\neq i} B_s \big) \big]}.
\end{align*}
Therefore, we define the null hypothesis as
\begin{align}
&H_{0,pg}: \xi(b)= \frac{1}{E_{B_{-i}}\big[ \Omega'\big( b+ \sum_{s\neq i} B_s \big) \big]}~\text{is weakly increasing in $b$}. \label{eq: public H0}
\end{align}
We apply Lemma \ref{lemma: general H0 transformation} to transform the null hypothesis in (\ref{eq: public H0}) by setting $h(b)= g(b)$ with $g(b)$ being the density function of the actions under BNE,
so we have
\begin{align*}
&M(b,q)=\int_{b}^{b+\frac{a}{q}} h(\tilde{b}) T'(\tilde{b}) d\tilde{b}=
\int_{b}^{b+\frac{a}{q}} g(\tilde{b}) d\tilde{b},\\
&W(b,q)=\int_{b}^{b+\frac{a}{q}} h(\tilde{b}) P'(\tilde{b}) d\tilde{b}=\int_{b}^{b+\frac{a}{q}}E_{B_{-i}}\big[ \Omega'\big( \tilde{b}+ \sum_{s\neq i} B_s \big) \big] g(\tilde{b}) d\tilde{b}.
\end{align*}
Note that $M(b,q)$ and $W(b,q)$ are both identified and can be expressed as unconditional means. We summarize the results in the following lemma.
\begin{lemma}\label{lemma: public H0 transformation}
The $H_{0,pg}$ in (\ref{eq: public H0}) is equivalent to
\begin{align}
&H_{0,pg}': \nu(b_1,b_2, q)=M(b_2,q)W(b_1,q)-M(b_1,q)W(b_2,q)\leq 0~\text{for all $(b_1,b_2, q)\in\mathcal{L}$} \label{eq: public H0'},
\end{align}
where
\begin{align}
&M(b,q)= E\Big[1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big)\Big]\label{eq: public M}\\
&W(b,q)=E\Big[1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big) \Omega'(\sum_{j=1}^N B_{j,\ell} )\Big]\label{eq: public W}.
\end{align}
\end{lemma}
\subsection{Cournot Competition Model with Private Information}
For the last example, we consider testing in Cournot competition models.
In the spirit of \cite{Vives2002}, let each firm $i$ choose output $B_i$ and have a private cost–efficiency parameter $V_i$ and $1/V_i$ denotes the cost for each unit produced for agent $i$.
Let $\mathcal{I}(\cdot)$ be the inverse demand function which is known.\footnote{\cite{AryalZincenko2024} extend \cite{Vives2002} to allow for stochastic demand and a common technology shock.}$^,$\footnote{In some cases,
the inverse demand function is unknown, but can be identified and estimated consistently from a separate sample. In this case, if the number of observations in the separate sample is large compared to the sample size of the testing sample so that we can ignore the estimation effect of the demand function asymptotically, then we can treat the estimated inverse demand function as a known function when we conduct the test. } Also we assume that $\mathcal{I}(\cdot)$ is strictly decreasing and convex, so that the revenue $D(B_i,B_{-i})\equiv \mathcal{I}(B_i, B_{-i}) B_i$ is strictly concave conditional on $B_{-i}$.
Let $C(B_i,B_{-i})=B_i$ denote the units produced by agent $i$, so $B_i/V_i$ is the total cost for agent $i$. Therefore, the total profit for agent $i$ is $D(B_i,B_{-i})-C(B_i,B_{-i})/V_i$. Integrating over the distribution of $B_{-i}$, the expected payoff can be rewritten as
\begin{equation*}
\ddot{\pi}(B_i, V_i) = E_{B_{-i}}[D(B_i, B_{-i})]- \frac{B_i}{V_i}
\equiv P(B_i)-\frac{T(B_i)}{V_i},
\end{equation*}
where $P(b)=E_{B_{-i}}[D(b,B_{-i})]$ and $T(b)=b$.
Assume that the demand curvature is known to the researcher and small enough, e.g., $-2 E[\mathcal{I}'] > E[\mathcal{I}''] \, b$ for all relevant $b$, where $\mathcal{I}'$ and $\mathcal{I}''$ denote the first and second derivatives, respectively.
Note that since maximizing $\ddot{\pi}(b, V_i)$ over $b$ is equivalent to maximizing ${\pi}(b, V_i)\equiv V_i\cdot \ddot{\pi}(b, V_i) = V_iP(B_i)-T(B_i)$ over $b$, we can still have the same form as before.
Therefore, we have
\begin{align*}
\xi(b) = \frac{T'(b)}{P'(b)} =
\frac{1}{E_{B_{-i}}\big[ \mathcal{I}\big( b+ \sum_{s\neq i}^N B_s \big)+\mathcal{I}'\big( b+ \sum_{s\neq i}^N B_s \big) b \big]}.
\end{align*}
Thus, we define the null hypothesis as
\begin{align}
&H_{0,cc}: \xi(b)= \frac{1}{E_{B_{-i}}\big[ \mathcal{I}\big( b+ \sum_{s\neq i}^N B_s \big)+\mathcal{I}'\big( b+ \sum_{s\neq i}^N B_s \big) b \big]}~\text{is weakly increasing in $b$}. \label{eq: Cournot H0}
\end{align}
We apply Lemma \ref{lemma: general H0 transformation} to transform the null hypothesis in (\ref{eq: Cournot H0}) by setting $h(b)= g(b)$ with $g(b)$ being the density function of the actions under BNE,
so we have
\begin{align*}
&M(b,q)=\int_{b}^{b+\frac{a}{q}} h(\tilde{b}) T'(\tilde{b}) d\tilde{b}=
\int_{b}^{b+\frac{a}{q}} g(\tilde{b}) d\tilde{b},\\
&W(b,q)=\int_{b}^{b+\frac{a}{q}} h(\tilde{b}) P'(\tilde{b}) d\tilde{b}=\int_{b}^{b+\frac{a}{q}}E_{B_{-i}}\big[ \mathcal{I}\big( \tilde{b}+ \sum_{s\neq i} B_s \big)+\mathcal{I}'\big( \tilde{b}+ \sum_{s\neq i} B_s \big) b \big] g(\tilde{b}) d\tilde{b}.
\end{align*}
Note that $M(b,q)$ and $W(b,q)$ are both identified and can be expressed as unconditional means. We summarize the results in the following lemma.
\begin{lemma}\label{lemma: Cournot H0 transformation}
The $H_{0,cc}$ in (\ref{eq: Cournot H0}) is equivalent to
\begin{align}
&H_{0,cc}': \nu(b_1,b_2, q)=M(b_2,q)W(b_1,q)-M(b_1,q)W(b_2,q)\leq 0~\text{for all $(b_1,b_2, q)\in\mathcal{L}$} \label{eq: Cournot H0'},
\end{align}
where
\begin{align}
&M(b,q)= E\Big[1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big)\Big]\label{eq: Cournot M}\\
&W(b,q)=E\Big[1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big)\Big( \mathcal{I}\big(\sum_{j=1}^N B_{j,\ell} \big)+\mathcal{I}'\big(\sum_{j=1}^N B_{j,\ell} \big)B_{i,\ell}\Big)\Big]\label{eq: Cournot W}.
\end{align}
\end{lemma}
\section{Heterogeneous Games of Incomplete Information}\label{sec: Games of Incomplete Information}
In this section, we extend the test under homogeneous games of incomplete information in Section \ref{sec: incomplete information} to allow for heterogeneous games of incomplete information including observed game heterogeneity and/or unobserved game heterogeneity. For both cases, we consider both nonparametric testing and semiparametric testing via homogenization.
\subsection{Observed Game Heterogeneity: Nonparametric Test} \label{sec: observed hetero nonparametric}
We extend our test in Section \ref{sec: incomplete information} to control for the observed heterogeneity in the model.
Specifically, let $X$ with support $\mathcal{X}$ denote a vector of covariates that is observed for each game. For example, $X$ represents the characteristics of a game, such as the observed quality of a good.
Without loss of generality, we assume that $X$ is a scalar and $\mathcal{X}=[0,1]$.\footnote{It is straightforward to allow for $X$ being vectors of observed characteristics.}
We impose the following conditions on the distribution of types.
\begin{assumption} \label{assu: value hetero}
Assume that (i) Conditioning on $X=x$, agent $i$, $i = 1, \ldots, N,$ draws his/her type independently from the other agents from a conditional CDF $F(\cdot|x)$ with the conditional PDF $f(\cdot|x)$.
(ii)
$f(\cdot|x)$ is strictly positive and bounded away from zero on its support, a compact
interval $\mathcal{V}_x=[\underline{v}_x, \overline{v}_x] \subseteq R$ , and is twice continuously differentiable on $(\underline{v}_x, \overline{v}_x)$.
\end{assumption}
We define $P(b,x)$ and $T(b,x)$ to be the conditionally expected outcome allocations and conditionally expected payment, respectively, for agent $i$ when agent $i$ plays action $b$ that is integrated over the distribution of rivals' strategies $B_{-i}$ given $X=x$. Specifically,
\begin{align}
&P(b,x)=E_{B_{-i}|X=x}[D_i(B_i,B_{-i})|B_i=b,X=x],\nonumber\\
&T(b,x)=E_{B_{-i}|X=x}[C_i(B_i,B_{-i})|B_i = b,X=x].
\label{eq: P and T covaraite}
\end{align}
Suppose all the conditions in Section \ref{sec: necessary and sufficient conditions} holds after we condition $X=x$ for all $x\in\mathcal{X},$ then it is true that we can formulate the null hypothesis as
\begin{align}\label{eq: general observed hetero H0}
H_{0,x}:~ \xi(b,x)=\frac{T'(b,x)}{P'(b,x)}~\text{is weakly increasing in $b$ for all $x\in\mathcal{X}$.}
\end{align}
Adapting Lemma A.1 of \cite{HsuShen2021}, we have the following equivalence result.
\begin{lemma}\label{lemma: general observed hetero H0 transformation}
Let $h(b,x)$ be a known function such that $0 <h(b,x)P'(b,x) \leq M<\infty$.
Then $H_0$ in (\ref{eq: general observed hetero H0}) is equivalent to
\begin{align}
H_{0,x}': \nu(b_1,b_2, x, q)&=M(b_2,x,q)W(b_1,x,q)-M(b_1,x,q)W(b_2,x,q)\leq 0\notag \\
&~\text{for all $(b_1,b_2,x, q)\in\mathcal{L}_x$} \label{eq: general observed hetero H0'},
\end{align}
where
\begin{align}
&M(b,x,q)= \int_{x}^{x+\frac{1}{q}}\int_{b}^{b+\frac{a}{q}} h(\tilde{b},\tilde{x})T'(\tilde{b},\tilde{x}) d\tilde{b}d\tilde{x} \notag\\
&
W(b,x,q)=\int_{x}^{x+\frac{1}{q}}\int_{b}^{b+\frac{a}{q}} h(\tilde{b},\tilde{x})P'(\tilde{b},\tilde{x}) d\tilde{b}d\tilde{x},~\text{and}\label{eq: general observed hetero M W}\\
&\mathcal{L}_x=\Big\{(b_1,b_2, x, q):
\Big(\frac{b_1-\underline{b}}{a}, \frac{b_2-\underline{b}}{a}, x\Big)
q\in(0,1,\ldots,q)^3, b_1>b_2~\text{for q=2,3,\ldots} \Big\}.\label{eq: L cov}
\end{align}
\end{lemma}
Once we get Lemma \ref{lemma: general observed hetero H0 transformation}, the general test will be similar to that in Section \ref{sec: proposed test}. Therefore, we omit the details.
One interesting thing to note is that (\ref{eq: general observed hetero H0}) actually implies the following:
\begin{align*}
&\nu(b_1,b_2, q)=M(b_2,q)W(b_1,q)-M(b_1,q)W(b_2,q)\leq 0~\text{for all $(b_1,b_2, q)\in\mathcal{L}$},
\end{align*}
where
\begin{align*}
&M(b,q)= \int_0^1\int_{b}^{b+\frac{a}{q}} h(\tilde{b},\tilde{x})T'(\tilde{b},\tilde{x}) d\tilde{b}d\tilde{x},
~~~
W(b,q)=\int_0^1\int_{b}^{b+\frac{a}{q}} h(\tilde{b},\tilde{x})P'(\tilde{b},\tilde{x}) d\tilde{b}d\tilde{x},~\text{and}\\
&\mathcal{L}=\Big\{(b_1,b_2, q): \Big(\frac{b_1-\underline{b}}{a}, \frac{b_2-\underline{b}}{a}\Big)\cdot q\in(0,1,\ldots,q)^2, b_1>b_2~\text{for q=2,3,\ldots} \Big\}.
\end{align*}
This is equivalent to the null hypothesis $H_0'$ in Lemma \ref{lemma: general H0 transformation} and this is true by integrating out the distribution of $X$ in Lemma \ref{lemma: general observed hetero H0 transformation} instead of restricting $X$ in an interval. This will be the foundation for our test under unobserved heterogeneity.
We conclude this subsection by summarizing the $M(b,x,q)$ and $W(b,x,q)$ in the forms of unconditional means in each example in Section \ref{sec: examples}.
\begin{enumerate}
\item {\bf First-price Sealed-bid Auction Model.} For example, in the context of timber auctions, a natural auction-level covariate is tract volume, which scales the economic stakes of the sale and can mechanically affect bids even in the absence of changes in strategic incentives.
\begin{align*}
&M(b,x,q)= E\Big[B_{i,\ell} 1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big) 1(x\leq X_\ell \leq x+\frac{1}{q})\Big]\notag\\
&~~~~~~+
\frac{1}{N-1} E\Big[ \Big(1\Big(B_{i,\ell}\leq b+\frac{a}{q} \Big) \Big(b+\frac{a}{q} - B_{i,\ell}\Big)
- 1(B_{i,\ell}\leq b ) (b - B_{i,\ell})\Big) 1(x\leq X_\ell \leq x+\frac{1}{q})\Big], \\
&W(b,x,q)=E\Big[1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big) 1(x\leq X_\ell \leq x+\frac{1}{q})\Big].
\end{align*}
For procurement auctions as in Remark \ref{remark: procurement}, a natural auction-level covariate is engineer's estimate, as in the empirical application in this paper. We have
\begin{align*}
&M(b,x,q)= E\Big[B_{i,\ell} 1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big) 1(x\leq X_\ell \leq x+\frac{1}{q})\Big]- \frac{1}{N-1}\frac{a}{q}
E\Big[1(x\leq X_\ell \leq x+\frac{1}{q})\Big]\notag\\
&~~~~~~+
\frac{1}{N-1} E\Big[ \Big(1\Big(B_{i,\ell}\leq b+\frac{a}{q} \Big) \Big(b+\frac{a}{q} - B_{i,\ell}\Big)
- 1(B_{i,\ell}\leq b ) (b - B_{i,\ell})\Big) 1(x\leq X_\ell \leq x+\frac{1}{q})\Big], \\
&W(b,x,q)=E\Big[1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big) 1(x\leq X_\ell \leq x+\frac{1}{q})\Big].
\end{align*}
\item {\bf Tullock Contest Model.} In a Tullock-type contests, prize shifters are one of the key covariates because they scale the marginal benefit of effort. The optimal effort responds mechanically to the prize, i.e., higher stakes raise equilibrium effort and change the distribution of observed actions.
\begin{align*}
&M(b,x,q)= E\Big[B_{i,\ell} \cdot 1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big)1(x\leq X_\ell \leq x+\frac{1}{q})\Big],\\
&W(b,x,q)=E\Big[1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big) \frac{B_{i,\ell}}{\sum_{j=1}^N B_{j,\ell} }\Big(1-\frac{B_{i,\ell}}{\sum_{j=1}^N B_{j,\ell}}\Big)1(x\leq X_\ell \leq x+\frac{1}{q})\Big].
\end{align*}
\item {\bf Public Good Provision Model.}
Let $\Omega(\cdot,x)$ be a known common benefit function of the total contributions conditional on $X=x$. A salient covariate in models of public good provision is the endowment. In experimental settings, the endowment is a core game-level primitive because it shapes both the maximum amount that can be allocated to the public good and the monetary payoffs associated with any given contribution profile.
\begin{align*}
&M(b,x,q)= E\Big[1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big)1(x\leq X_\ell \leq x+\frac{1}{q})\Big],\\
&W(b,x,q)=E\Big[1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big) \Omega'\Big(\sum_{j=1}^N B_{j,\ell}, X_\ell\Big)1(x\leq X_\ell \leq x+\frac{1}{q})\Big].
\end{align*}
\item {\bf Cournot Competition Model.} Let $\mathcal{I}(\cdot,x)$ be the inverse demand function conditional on $X=x$ which is known. Market level covariates include demand shifters such as population, income, and seasonality. It is important to control demand shifters in demand estimation because these variables shift the level (and often the slope) of market demand, and therefore move equilibrium prices and quantities even when firms’ costs and conduct are unchanged. Ignoring these covariates results in lack of comparability across markets or time, and can lead the econometrician to misattribute demand-driven variation in outputs and prices to differences in marginal costs, productivity, or the degree of market power.
\begin{align*}
M(b,x,q)=& E\Big[1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big)1(x\leq X_\ell \leq x+\frac{1}{q})\Big],\\
W(b,x,q)=&E\Big[1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big)\Big( \mathcal{I}\big(\sum_{j=1}^N B_{j,\ell},X_\ell \big)\Big)1(x\leq X_\ell \leq x+\frac{1}{q})\Big]\\
&+E\Big[1\Big( b \leq B_{i,\ell}\leq b+\frac{a}{q} \Big)\Big( \mathcal{I}'\big(\sum_{j=1}^N B_{j,\ell},X_\ell \big)B_{i,\ell}\Big)1(x\leq X_\ell \leq x+\frac{1}{q})\Big].
\end{align*}
\end{enumerate}
\subsection{Observed Game Heterogeneity: Semiparametric Test} \label{sec: observed hetero semiparametric}
In Section \ref{sec: observed hetero nonparametric}, we consider a nonparametric test for games of incomplete information covariates. While it is general, it may suffer from the curse of dimensionality as is usually the case with nonparametric estimation/testing, when the number of (continuous) covariates is large. In this section, we follow \cite{HHS2003}, \cite{MMSX2021} and \cite{CHS2020} and impose a semiparametric model to homogenize the actions. Specifically, let $\Gamma(x)$ be a strictly positive function of covariates. The rest of the framework follows Section \ref{sec: observed hetero nonparametric}.
\begin{assumption} \label{assu: homogenize}
Assume that (i) $\{(V_{1,\ell}=\Gamma(X_{\ell})u_{1,\ell}, V_{2,\ell}=\Gamma(X_{\ell})u_{2,\ell},\ldots, V_{N,\ell}=\Gamma(X_{\ell})u_{N,\ell}, X_{\ell}): \ell=1,\ldots, L\}$ are i.i.d.\ random vectors;
(ii) $\{u_{i,\ell}: i=1,\ldots, N,~\ell=1,\ldots, L \}$ are i.i.d.\ random variables with PDF $f_u(\cdot)$ and CDF $F_u(\cdot)$ and
are independent of
$X_{\ell}$.
(iii)
$f_u(\cdot)$ is strictly positive and bounded away from zero on its support, a compact
interval $[\underline{u}, \overline{u}] \subseteq R$ , and is twice continuously differentiable on $(\underline{u}, \overline{u})$.
\end{assumption}
\begin{assumption}\label{assu: D C homogeneous}
Assume that $D_{i,\ell}(B_{1,\ell},\ldots, B_{N,\ell})$ is a homogeneous function
of order zero and $T_{i,\ell}(B_{1,\ell},\ldots, B_{N,\ell})$ is a homogeneous function
of order one in that for any $r>0$
\begin{align*}
&D_{i,\ell}(rB_{1,\ell},\ldots, rB_{N,\ell})=D_{i,\ell}(B_{1,\ell},\ldots, B_{N,\ell}),\\
&T_{i,\ell}(rB_{1,\ell},\ldots, rB_{N,\ell})= r T_{i,\ell}(B_{1,\ell},\ldots, B_{N,\ell}).
\end{align*}
\end{assumption}
Let $S_v(v|x)$ denote the BNE strategy.
Let $B^u_{i}$ denote the actions when agent $i$'s value is $u_{i}$.
We claim that $S_v(v|x) =\Gamma(x)s_u(\Gamma^{-1}(x) v )$ for all $x\in\mathcal{X}$.
That is, if Assumption \ref{assu: D C homogeneous} holds, we want to show that ``$b^u\in s(u)$" $\Leftrightarrow$ ``$b=\Gamma(x)b^u\in s_v(v|x)= s_v(\Gamma(x)u|x)$".
Let $P^u_{i,\ell}(b^u)$ and $T^u_{i,\ell}(b^u)$ be the expected allocation and transfer that agent $i$ will attain if agent $i$ plays $b^u$.
To show ``$\Rightarrow$" direction,
note that if $b^u\in s(u)$, we have
\begin{align*}
&~u_{i,\ell}P^u_{i,\ell}(b^u)-T^u_{i,\ell}(b^u) \geq
u_{i,\ell}P^u_{i,\ell}(b^{u\prime})-T^u_{i,\ell}(b^{u\prime})~~\text{for all $b^{u\prime}$}\\
\Leftrightarrow &~
\Gamma(x)u_{i,\ell}P^u_{i,\ell}(b^u)-\Gamma(x)T^u_{i,\ell}(b^u) \geq
\Gamma(x)u_{i,\ell}P^u_{i,\ell}(b^{u\prime})-\Gamma(x)T^u_{i,\ell}(b^{u\prime})~~\text{for all $b^{u\prime}$}\\
\Leftrightarrow &~
V_{i,\ell}P^u_{i,\ell}(\Gamma(x)b^u)-T^u_{i,\ell}(\Gamma(x)b^u) \geq
V_{i,\ell}P^u_{i,\ell}(\Gamma(x)b^{u\prime})-T^u_{i,\ell}(\Gamma(x)b^{u\prime})~~\text{for all $b^{u\prime}$}
\\
\Leftrightarrow &~
V_{i,\ell}P_{i,\ell}(b|x)-T_{i,\ell}(b|x) \geq
V_{i,\ell}P_{i,\ell}(b'|x)-T_{i,\ell}(b'|x)
~~\text{for all $b^{\prime}$},
\end{align*}
where the first $\Leftrightarrow$ holds by rescaling everything by $\Gamma(x)$, the second one holds by the fact that $V_{i,\ell}=\Gamma(x)u_{i,\ell}$ and $P^u_{i,\ell}(\cdot)$ and $T^u_{i,\ell}(\cdot)$ are homogeneous functions of degree zero and one, respectively. The last $\Leftrightarrow$ holds because for agent $\ell$, if other players have $\Gamma(x)b^u\in s_v(v|x)$ being their best response functions, then $P_{i,\ell}(b|x)=P^u_{i,\ell}(\Gamma(x)b^{u\prime})$ and $T_{i,\ell}(b|x)=T^u_{i,\ell}(\Gamma(x)b^{u\prime})$. It follows that $b=\Gamma(x)b^u$ is agent $i$'s best responseheter too. That is, $b=\Gamma(x)b^u\in s_v(v|x)= s_v(\Gamma(x)u|x)$. The same arguments apply to the ``$\Leftarrow$" direction of the claim.
This result implies that
\begin{align}
&~\xi(b,x) \;\equiv\; \frac{T'(b,x)}{P'(b,x)} ~\text{is strictly increasing in $b$ for all $x\in\mathcal{X}$.} \notag\\
\Leftrightarrow
&~\xi(b^u) \;\equiv\; \frac{T^{u\prime }(b^u)}{P^{u\prime }(b^u)} ~\text{is strictly increasing in $b^u$.} \label{eq: semi equivalence}
\end{align}
(\ref{eq: semi equivalence}) further implies that
we can re-formulate the null hypothesis $H_{0,x}$ in (\ref{eq: general observed hetero H0}) as
\begin{align}\label{eq: null semi test}
H_{0,\text{semi}}:~\xi(b^u) \;\equiv\; \frac{T^{u\prime }(b^u)}{P^{u\prime }(b^u)} ~\text{is weakly increasing in $b^u$.}
\end{align}
Therefore, then $H_{0,\text{semi}}$ in (\ref{eq: null semi test}) is equivalent to
\begin{align*}
&H_{0,\text{semi}}': \nu(b^u_1,b^u_2, q)=M(b^u_2,q)W(b^u_1,q)-M(b^u_1,q)W(b^u_2,q)\leq 0~\text{for all $(b^u_1,b^u_2, q)\in\mathcal{L}$},
\end{align*}
where
\begin{align*}
&M(b^u,q)= \int_{b^u}^{b^u+\frac{a}{q}} h(\tilde{b})T'(\tilde{b}) d\tilde{b},
~~~
W(b^u,q)=\int_{b^u}^{b^u+\frac{a}{q}} h(\tilde{b})P'(\tilde{b}) d\tilde{b},~\text{and}\\
&\mathcal{L}=\Big\{(b^u_1,b^u_2, q): \Big(\frac{b^u_1-\underline{b}^u}{a}, \frac{b^u_2-\underline{b}^u}{a}\Big)\cdot q\in(0,1,\ldots,q)^2, b^u_1>b^u_2~\text{for q=2,3,\ldots} \Big\}.
\end{align*}
We note that first, Assumption \ref{assu: D C homogeneous} is crucial for the homogenization method to work. Therefore, among the four examples discussed in Section \ref{sec: examples}, the homogenization method can be applied to only the first-price sealed-bid auction model and Tullock contest model. Second, the intuition of the equivalence result is that when the actions are monetary activities such as the auction games, then if we use different currencies, the resulting bids should be the same in the sense that the resulting bids should have the relation as above.
If $\Gamma(x)$ is known, then we obtain
the rescaled actions, $B_{i,\ell}^u=\Gamma^{-1}(X_\ell)B_{i,\ell}$ and apply the test with the rescaled actions in Section \ref{sec: proposed test} to test $H_{0,\text{semi}}$ in (\ref{eq: null semi test}). If $\Gamma(x)$ follows a parametric specification $\Gamma(x,\theta)$ so that $\theta$ is identified and a consistent estimator $\widehat{\theta}$ is available, then we can apply a similar test in Section \ref{sec: proposed test} after accounting for the estimation effect of $\widehat{\theta}$ in the test. In the Appendix, we present a test in auction games and show its validity under regularity conditions.
\subsection{Unobserved Game Heterogeneity} \label{sec: unobserved hetero nonparametric}
We extend our tests in Section \ref{sec: observed hetero nonparametric} to allow for unobserved heterogeneity in the model to the nonparametric case. For semiparametric case, please see Appendix \ref{sec: semi unobserved hetero} for details.
Suppose that for each game, in addition to an observed vector of covariates, $X$, there is an unobserved vector of $Z$ denoting the unobserved characteristics which is observable or is common knowledge among players, but not directly observable to econometricians. Without loss of generality, we assume both $X$ and $Z$ to be scalars with $\mathcal{X}=[0,1]$ and $\mathcal{Z}=[0,1]$.\footnote{It is straightforward to allow for $X$ and $Z$ being vectors of observed and unobserved characteristics, respectively.}
The model is the same as in Section \ref{sec: observed hetero nonparametric} except that we have an additional unobserved covariate $Z$.
For a given strategy $s$ and given $X=x$ and $Z=z$, we write $s(v,x,z)$ for the action played by type $v$.
Following Section \ref{sec: observed hetero nonparametric}, we would have
\begin{align}
\xi(b,x,z) \;\equiv\; \frac{T'(b,x,z)}{P'(b,x,z)}~\text{is strictly increasing in $b$.} \label{eq:implicit unobserved hetero}
\end{align}
However, we cannot formulate the null hypothesis as in \ref{eq: general observed hetero H0}, because $Z$ is unobservable to econometricians.
Instead, (\ref{eq:implicit unobserved hetero}) implies the following testable implications.
\begin{lemma}\label{lemma: general unobserved hetero H0 transformation}
Let $h(b,x,z)$ be a known function such that $0 <h(b,x,z)P'(b,x,z) \leq M<\infty$.
Then (\ref{eq:implicit unobserved hetero}) implies that
\begin{align}
H_{0,x,z}': \nu(b_1,b_2, x, q)&=M(b_2,x,q)W(b_1,x,q)-M(b_1,x,q)W(b_2,x,q)\leq 0\notag \\
&~\text{for all $(b_1,b_2,x, q)\in\mathcal{L}_x$} \label{eq: general unobserved hetero H0'},
\end{align}
where
\begin{align}
&M(b,x,q)= \int_{0}^1\int_{x}^{x+\frac{1}{q}}\int_{b}^{b+\frac{a}{q}} h(\tilde{b},\tilde{x},z)T'(\tilde{b},\tilde{x},z) d\tilde{b}d\tilde{x} dz \notag\\
&
W(b,x,q)=\int_{0}^1\int_{x}^{x+\frac{1}{q}}\int_{b}^{b+\frac{a}{q}} h(\tilde{b},\tilde{x},z)P'(\tilde{b},\tilde{x},z) d\tilde{b}d\tilde{x}dz,~\text{and}\label{eq: general unobserved hetero M W}\\
&\mathcal{L}_x=\Big\{(b_1,b_2, x, q):
\Big(\frac{b_1-\underline{b}}{a}, \frac{b_2-\underline{b}}{a}, x\Big)
q\in(0,1,\ldots,q)^3, b_1>b_2~\text{for q=2,3,\ldots} \Big\}.
\end{align}
\end{lemma}
The basic idea of Lemma \ref{lemma: general unobserved hetero H0 transformation} is to integrate out $Z$, so $Z$ will not show up in the expressions anymore. In this case, we may use the same tests as in Section \ref{sec: observed hetero nonparametric}.
The $M(b,x,q)$ and $W(b,x,q)$ for four examples are identical to those in Section \ref{sec: observed hetero nonparametric} and we omit the details.
Note that the testable implications in these cases are the same as those in Section \ref{sec: observed hetero nonparametric}, so we can apply the same test in Section \ref{sec: observed hetero nonparametric} to test the models with unobserved heterogeneity and we omit the details. The extension to settings without observed covariates
$X$ follows by a similar argument, so we omit the details.
\section{Simulation Studies}\label{sec: simulation}
We conduct Monte-Carlo simulations to examine the finite sample performance of the proposed tests.
To implement our test in practice, one has to choose several tuning parameters in advance. We first suggest the choices of these tuning parameters and then present the related Monte Carlo simulation results.
\begin{enumerate}
\item Instrumental functions: We opt for using a set of indicator functions of countable hypercubes. Define
\begin{align*}
\mathcal{L}=\{(b_1,b_2, x, q): (b_1,b_2,x )q\in\{0,1,\ldots,q-1\}^{2+d_x}, b_1>b_2~\text{for}~q=2,3,\ldots,q_1 \},
\end{align*}
where $q_1$ is a natural number and is chosen such that the expected sample size of the smallest cube is around 20. We also consider three alternative choices of $q_1$, each resulting in the expected sample size of the smallest cube $n_c=15$, $25$, and $30$.\footnote{To be specific, for a specific $n_c$, we set $q_1=\lfloor (S/n_c)^{1/(1+d_x)}\rceil$, where $\lfloor c \rceil$ gives the integer nearest to $c$ and $d_x$ is the dimension of (continuous) covariates. In addition, for heterogeneous numbers of bidders, we have $q_{1,t}$ a function of $t$ and $q_{1,t}=\lfloor (S_t/n_c)^{1/(1+d_x)}\rceil$.
} Our simulations show that the results are robust to various expected sample sizes.
\item {\bf $Q(b_1,b_2, x, q)$:} The distribution $Q(b_1,b_2, x, q)$ assigns weight $\propto q^{-2}$ to each $q$, and for each $q$, $Q(\ell)$ assigns an equal weight to each instrumental function with the last element of $\ell$ equal to $q^{-1}$. Recall that for each $q$, there are $(q(q+1)/2)\cdot q^{d_x}$ instrumental functions with the last element of $(b_1,b_2, x, q))$ equal to $q$.
\item {\bf $\kappa_{S}$, $\beta_{S}$, $\epsilon$, $\eta$:} We set $\kappa_{S}=0.15\cdot\ln(S)$, $\beta_{S}=0.85\cdot\ln(S)/\ln\ln(S)$, $\epsilon=10^{-6}$, and $\eta=10^{-6}$ as suggested by \cite{HLS2019}. These choices are used in all the simulations that we report below and seem to perform well. For the case with heterogeneous number of bidders, we set $\kappa_{S_t}=0.15\cdot\ln(S_t)$, $\beta_{S_t}=0.85\cdot\ln(S_t)/\ln\ln(S_t)$.
\item All our Monte Carlo results are based on 1000 simulations. In each simulation, the critical value is approximated by 1000 bootstrap replications. The nominal size of the test is set at 10\%.
\end{enumerate}
We first consider the case without covariates. To study the size and power properties, the CDF of the observed bids is specified as
\begin{align}
G(b)=\left( \frac{b}{k-(k-1)b}\right)^{1/5},\label{eq:CDF}
\end{align}
and the associated quantile function of the true bid distribution is
\begin{align}
Q(\tau)=\frac{k \tau^{5}}{1+(k-1)\tau^5}.\label{eq:quantile}
\end{align}
We set $k=0.5$ for the size analysis and set $k=5$, $10$, $15$, and $20$ for the power analysis. Note that $k=5$ corresponds to the example that violates the monotonicity condition provided in GPV. For all values of $k$, we set $N=2$ and $L=100$, 250, or 500. Figure \ref{fig:bidder_value} presents the quasi-inverse equilibrium strategy, $\xi(b)=b+{G(b)/(g(b)(N-1))}$
for different values of $k$. From this figure, we can observe that the monotonicity condition holds for $k=0.5$, but is violated for $k\geq5$. The figure also shows that it will be easier to detect the violation as $k$ increases.
\begin{figure}[t!]
\centering
\includegraphics[width=12cm,height=9cm]{fig_bidder_value}
\caption{Quasi-inverse equilibrium strategy, $\xi(b)$}\label{fig:bidder_value}
\end{figure}
Table \ref{table:DGP1} shows the rejection probabilities of our tests. The results show that the proposed test controls size well for $k=0.5$. Note that when $k=0.5$, the null hypothesis holds with strict inequality. Therefore, the size will converge to zero since every moment would hold with strict
inequality. It follows that the test statistics will converge to zero and the critical value is bounded away from zero. For the power analysis, the results show that the power increases with sample size and with $k$ for $k\geq 5$, but the rejection probabilities are a bit less than the nominal size $0.1$ for $k=5$. Table \ref{table:DGP1} also shows that the choices of $q_1$ do not affect the test performance much.
\begin{table}[H]
\begin{small}
\begin{center}
\caption{Rejection probabilities for the case without covariates}\label{table:DGP1}
\begin{tabular}{cccccc}
\toprule
k&L&$n_c$=15&$n_c$=20&$n_c$=25&$n_c$=30\\
\midrule
0.5&100&0.001&0.004&0.001&0.002\\
0.5&250&0.003&0.003&0.001&0.003\\
0.5&500&0.001&0.002&0.001&0.001\\
\midrule
5&100&0.080&0.084&0.084&0.067\\
5&250&0.080&0.070&0.073&0.077\\
5&500&0.084&0.084&0.098&0.106\\
\midrule
10&100&0.311&0.310&0.297&0.297\\
10&250&0.504&0.519&0.470&0.464\\
10&500&0.737&0.754&0.741&0.741\\
\midrule
20&100&0.710&0.690&0.700&0.682\\
20&250&0.953&0.939&0.958&0.951\\
20&500&1.000&1.000&1.000&1.000\\
\bottomrule
\end{tabular}
\end{center}
\end{small}
\end{table}
We next consider the case with covariates. Let $X$ to be a uniform distribution on [0, 1]. To consider the test with covariates, the quantile function of the true bid distribution is specified as
\begin{align*}
Q(x,\tau)=\frac{k(x)\tau^5}{1+(k(x)-1)\tau^5}.
\end{align*}
We consider the following cases for the size and power analyses:
\begin{align*}
\text{Case 1: }& k(x)=0.5+2x~~\text{for the size case},\\
\text{Case 2: }& k(x)=5+5x~~\text{for the power case},\\
\text{Case 3: }& k(x)=10+5x~~\text{for the power case},\\
\text{Case 4: }& k(x)=15+5x~~\text{for the power case},\\
\text{Case 5: }& k(x)=20+5x~~\text{for the power case}.
\end{align*}
For all cases, we set $N=2$ and $L=100$, 250, 500, or 1000. Table \ref{table:DGP2} shows the rejection probabilities of our tests for the case with covariates, and the results are consistent with our theoretical findings. The proposed test controls size well in Case 1, and the rejection probabilities increase with the sample size and are greater than the nominal size $0.1$ in Cases 2-5. Like the results in Table \ref{table:DGP1}, we do not find significant differences for different choices $q_1$.
\begin{table}[H]
\begin{small}
\begin{center}
\caption{Rejection probabilities for the case with covariates}\label{table:DGP2}
\begin{tabular}{cccccc}
\toprule
Case&L&$n_c$=15&$n_c$=20&$n_c$=25&$n_c$=30\\
\midrule
1&100&0.013&0.019&0.014&0.021\\
1&250&0.002&0.004&0.004&0.007\\
1&500&0.003&0.000&0.002&0.001\\
1&1000&0.000&0.000&0.000&0.000\\
\midrule
2&100&0.119&0.128&0.137&0.136\\
2&250&0.123&0.143&0.144&0.127\\
2&500&0.175&0.172&0.155&0.165\\
2&1000&0.256&0.256&0.240&0.248\\
\midrule
3&100&0.234&0.237&0.225&0.231\\
3&250&0.412&0.413&0.391&0.351\\
3&500&0.648&0.643&0.622&0.620\\
3&1000&0.910&0.896&0.910&0.885\\
\midrule
4&100&0.364&0.313&0.288&0.325\\
4&250&0.663&0.638&0.596&0.606\\
4&500&0.927&0.920&0.910&0.902\\
4&1000&0.998&0.997&0.997&0.999\\
\midrule
5&100&0.463&0.412&0.409&0.419\\
5&250&0.797&0.794&0.760&0.760\\
5&500&0.978&0.985&0.972&0.985\\
5&1000&1.000&1.000&1.000&1.000\\
\bottomrule
\end{tabular}
\end{center}
\end{small}
\end{table}
\section{Empirical Application}\label{sec:data}
As discussed in \citet{BajariYe2003}, monotonicity of equilibrium bidding strategies is one of the sufficient and necessary conditions for the symmetric and competitive bidding equilibrium. While they try to test for efficient collusion by testing these conditions, they do not test for monotonicity due to the lack of monotonicity tests.\footnote{\citet{AryalGabrielli2013} propose a stochastic dominance test to test for collusion. To implement the test, they first use the tests of \citet{BajariYe2003} to identify potential colluding bidders.} Clearly, rejection of monotonicity can be indicative of potential collusive behavior. To demonstrate the effectiveness of our monotonicity test, we use the asphalt paving auction datasets studied in \cite{Aaltio_etal2025}, who analyze bidding behavior in state-level road-paving procurements in Sweden, Finland and California.
\subsection{Data}
The Nordic data are drawn from contracts procured by the Swedish Road Administration and the Finnish Transport Infrastructure Agency, covering 1993-2009 in Sweden and 1994-2019 in Finland. Both datasets contain all submitted bids in each tender, the identity of the winning bidder, paving area (m$^2$), and the region where the pavement project took place.
Both Nordic markets experienced bid-rigging cartels during the late 1990s and early 2000s. To focus on bidding behavior during legally established cartel operation while avoiding contamination around detection and enforcement, we restrict attention to the court-documented conviction windows and exclude the dawn-raid years. Specifically, we use tenders from 1997–2000 in Sweden and 1995–2001 in Finland.\footnote{Both markets featured complementary bidding consistent with designated-winner coordination.} The California asphalt paving market is widely considered as competitive and there is no evidence of collusion during this period, and used in \cite{Aaltio_etal2025} as a control market. We use California asphalt paving contracts procured by the California Department of Transportation from 1999 to 2008.
For the monotonicity test, we work with normalized bids to make outcomes comparable across contracts of different scales. In Sweden and Finland, we normalize bids by paving area (price per $m^2$). In the test that control for observed auction heterogeneity, we control for paving area. The California dataset contains the similar information as the Nordic datasets, including all submitted bids, winner identity, and project region, except that the former contains the information on engineer's cost estimate but not paving area, while the latter contains the information on paving area but not engineer's cost estimate. In California, we normalize bids by engineer's cost estimate, which is also controlled for in the test that controls for observed auction heterogeneity.
\subsection{Test results}
We first conduct the monotonicity test for the California dataset. Table \ref{table:Cal} reports the test statistics and associated p-values for auctions with 2 to 10 bidders and heterogeneous numbers of bidders. The tuning parameters are chosen as in Section \ref{sec: simulation}, and the test with heterogeneous numbers of bidders is discussed in Appendix \ref{sec: hetero numbers}. The left panel of Table \ref{table:Cal} reports the test results based on the nonparametric test without controlling for the covariates, and the right panel of Table \ref{table:Cal} reports the test results based on the nonparametric test with covariates discussed in Section \ref{sec: observed hetero nonparametric}. The results show that there is no case with strong evidence against the monotonicity null hypothesis, which is consistent with the literature where the California market is usually modeled as competitive and no cartel investigation was conducted during 1999 to 2008.
We next apply the monotonicity tests to the Sweden and Finland datasets. Tables \ref{table:Swe} and \ref{table:Fin} report the test statistics and associated p-values for auctions with 3 to 8 bidders and heterogeneous numbers of bidders for Sweden and Finland, respectively. For Sweden, the p-values for the auction with 5 bidders and the cases associated with 5 bidders are less than 0.1, which provide strong evidence against the monotonicity. For Finland, the p-values for some cases in nonparametric tests with covariates are around 0.3, which provide weak evidence against the monotonicity. Interestingly, \cite{Aaltio_etal2025} find that the cartel had a larger impact on the distribution of bids in Finland than in Sweden, and they interpret this as that the Swedish cartel was more successful in mimicking competitive behavior, or it had more limited influence in scope. The strong evidence against monotonicity we find for the Swedish market shows the power of our test in detecting non-competitive behavior which other methods may not be able to detect.
\begin{table}[h]
\begin{small}
\begin{center}
\caption{Test statistics and p-values for California} \label{table:Cal}
\begin{tabular}{ccccccccc}
\toprule
&\multicolumn{4}{c}{without covariates} &\multicolumn{4}{c}{with covariates}\\
\cmidrule(lr){2-5}\cmidrule(lr){6-9}
&\multicolumn{2}{c}{$n_c$=15} &\multicolumn{2}{c}{$n_c$=20}
&\multicolumn{2}{c}{$n_c$=15} &\multicolumn{2}{c}{$n_c$=20} \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5}\cmidrule(lr){6-7}\cmidrule(lr){8-9}
$N_t$&stat&p-value&stat&p-value&stat&p-value&stat&p-value\\
\midrule
2&0.0318&0.3268&0.0233&0.3878&0.0001&0.6566&0.0001&0.6564\\
(2,3)&0.0414&0.6710&0.0322&0.7170&0.0001&0.9968&0.0001&0.9846\\
(2,3,4)&0.1306&0.7114&0.1210&0.7270&0.0122&0.8146&0.0111&0.7870\\
3&0.0096&0.6790&0.0089&0.6856&0.0000&1.0000&0.0000&1.0000\\
(3,4)&0.0987&0.6844&0.0977&0.6794&0.0121&0.7360&0.0110&0.7140\\
(3,4,5)&0.1165&0.8178&0.1153&0.8114&0.0128&0.8934&0.0115&0.8772\\
4&0.0892&0.5566&0.0888&0.5432&0.0121&0.5784&0.0110&0.5312\\
(4,5)&0.1069&0.7298&0.1064&0.7310&0.0128&0.7786&0.0115&0.7484\\
(4,5,6)&0.1512&0.8396&0.1506&0.8346&0.0172&0.8930&0.0153&0.8834\\
5&0.0178&0.6782&0.0176&0.6676&0.0007&0.9360&0.0005&0.9416\\
(5,6)&0.0620&0.7662&0.0619&0.7728&0.0051&0.9074&0.0043&0.9216\\
(5,6,7)&0.0626&0.8700&0.0624&0.8604&0.0051&0.9892&0.0043&0.9878\\
6&0.0442&0.6314&0.0442&0.6254&0.0044&0.7738&0.0038&0.7734\\
(6,7)&0.0449&0.7794&0.0448&0.7912&0.0044&0.9524&0.0038&0.9518\\
(6,7,8)&0.0489&0.8718&0.0470&0.8662&0.0044&0.9826&0.0038&0.9840\\
7&0.0006&0.8814&0.0006&0.8270&0.0000&1.0000&0.0000&1.0000\\
(7,8)&0.0047&0.7964&0.0028&0.8548&0.0000&1.0000&0.0000&1.0000\\
(7,8,9)&0.0562&0.7548&0.0508&0.7644&0.0000&1.0000&0.0000&1.0000\\
8&0.0041&0.5774&0.0022&0.6120&0.0000&1.0000&0.0000&1.0000\\
(8,9)&0.0555&0.5884&0.0502&0.6024&0.0000&1.0000&0.0000&1.0000\\
(8,9,10)&0.0586&0.7542&0.0530&0.7512&0.0000&1.0000&0.0000&1.0000\\
\bottomrule
\end{tabular}
\end{center}
Note: `Stat' denotes the test statistic, and the p-values of the tests are calculated based on 5000 bootstrap replications. The number of bidders $N_t$ and the number of auctions $L_t$ are: $\{(N_t,L_t):(2,204), (3,308), (4,276), (5,212), (6,163),(7,117),(8,58),(9,41),(10,33)\}$.
\end{small}
\end{table}
\begin{table}[h!]
\vspace{-1.5em}
\begin{small}
\begin{center}
\caption{Test statistics and p-values for Sweden} \label{table:Swe}
\begin{tabular}{ccccccccc}
\toprule
&\multicolumn{4}{c}{without covariates} &\multicolumn{4}{c}{with covariates}\\
\cmidrule(lr){2-5}\cmidrule(lr){6-9}
&\multicolumn{2}{c}{$n_c$=15} &\multicolumn{2}{c}{$n_c$=20}
&\multicolumn{2}{c}{$n_c$=15} &\multicolumn{2}{c}{$n_c$=20} \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5}\cmidrule(lr){6-7}\cmidrule(lr){8-9}
$N_t$&stat&p-value&stat&p-value&stat&p-value&stat&p-value\\
\midrule
3&0.1381&0.0744&0.0000&1.0000&0.0000&1.0000&0.0000&1.0000\\
(3,4)&0.2589&0.2034&0.1050&0.2434&0.0000&1.0000&0.0000&1.0000\\
(3,4,5)&0.9441&0.0508&0.7738&0.0636&0.0823&0.0698&0.0630&0.0766\\
4&0.1208&0.2340&0.1050&0.2316&0.0000&1.0000&0.0000&1.0000\\
(4,5)&0.8060&0.0616&0.7738&0.0636&0.0823&0.0688&0.0630&0.0874\\
(4,5,6)&0.8907&0.0530&0.8360&0.0484&0.0823&0.0590&0.0630&0.0768\\
5&0.6852&0.0438&0.6688&0.0378&0.0823&0.0836&0.0630&0.0866\\
(5,6)&0.7699&0.0346&0.7311&0.0352&0.0823&0.0838&0.0630&0.0786\\
(5,6,7)&1.4276&0.0722&1.3355&0.0796&0.1719&0.1198&0.0630&0.0842\\
6&0.0847&0.1866&0.0622&0.2058&0.0000&1.0000&0.0000&1.0000\\
(6,7)&0.7424&0.0982&0.6667&0.0786&0.0896&0.1372&0.0000&1.0000\\
\bottomrule
\end{tabular}
\end{center}
Note: `Stat' denotes the test statistic, and the p-values of the tests are calculated based on 5000 bootstrap replications. The number of bidders $N_t$ and the number of auctions $L_t$ are: $\{(N_t,L_t):(3,15), (4,58), (5,67), (6,28),(7,17)\}$.
\end{small}
\vspace{-1em}
\begin{small}
\begin{center}
\caption{Test statistics and p-values for Finland} \label{table:Fin}
\begin{tabular}{ccccccccc}
\toprule
&\multicolumn{4}{c}{without covariates} &\multicolumn{4}{c}{with covariates}\\
\cmidrule(lr){2-5}\cmidrule(lr){6-9}
&\multicolumn{2}{c}{$n_c$=15} &\multicolumn{2}{c}{$n_c$=20}
&\multicolumn{2}{c}{$n_c$=15} &\multicolumn{2}{c}{$n_c$=20} \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5}\cmidrule(lr){6-7}\cmidrule(lr){8-9}
$N_t$&stat&p-value&stat&p-value&stat&p-value&stat&p-value\\
\midrule
3&0.0000&1.0000&0.0000&1.0000&0.0000&1.0000&0.0000&1.0000\\
(3,4)&0.0000&1.0000&0.0000&1.0000&0.0000&1.0000&0.0000&1.0000\\
(3,4,5)&0.0474&0.7502&0.0293&0.7620&0.0374&0.3484&0.0000&1.0000\\
4&0.0000&1.0000&0.0000&1.0000&0.0000&1.0000&0.0000&1.0000\\
(4,5)&0.0474&0.7260&0.0293&0.7446&0.0374&0.3568&0.0000&1.0000\\
(4,5,6)&0.1595&0.6678&0.1217&0.6842&0.0374&0.3480&0.0000&1.0000\\
5&0.0474&0.6766&0.0293&0.7068&0.0374&0.3318&0.0000&1.0000\\
(5,6)&0.1595&0.6414&0.1217&0.6506&0.0374&0.3208&0.0000&1.0000\\
(5,6,7)&0.1597&0.8132&0.1217&0.8188&0.0374&0.6020&0.0000&1.0000\\
6&0.1121&0.4546&0.0924&0.4774&0.0000&1.0000&0.0000&1.0000\\
(6,7)&0.1124&0.7352&0.0924&0.7092&0.0000&1.0000&0.0000&1.0000\\
(6,7,8)&0.1124&0.8434&0.0924&0.8402&0.0000&1.0000&0.0000&1.0000\\
7&0.0002&0.9390&0.0000&1.0000&0.0000&1.0000&0.0000&1.0000\\
(7,8)&0.0002&0.9898&0.0000&1.0000&0.0000&1.0000&0.0000&1.0000\\
\bottomrule
\end{tabular}
\end{center}
Note: `Stat' denotes the test statistic, and the p-values of the tests are calculated based on 5000 bootstrap replications. The number of bidders $N_t$ and the number of auctions $L_t$ are: $\{(N_t,L_t):(3,15), (4,14), (5,41), (6,29),(7,15),(8,12)\}$.
\end{small}
\end{table}
\clearpage
\section{Conclusion} \label{sec: conclusion}
This paper develops a unified framework for testing whether Bayesian Nash equilibrium strategies are monotone in unobserved types in games of incomplete information. We first establish a rationalization result showing that, within the symmetric independent
private type paradigm, monotonicity of the differentiable equilibrium strategy in private types is equivalent to monotonicity of the quasi-inverse of the equilibrium strategy in actions; the quasi-inverse equilibrium strategy is a function of expected allocation
and expected payment rules, and can be identified from observed actions. As a result, our test reduces to testing monotonicity of the quasi-inverse equilibrium strategy, which can be reformulated as testing a countable collection of moment inequalities
that involve only unconditional expectations. The proposed testing procedure is easy to implement, and can accommodate game specific covariates and applies to both observed and unobserved game heterogeneity. The tests have good finite sample performance as demonstrated by the Monte-Carlo experiments. An application to asphalt paving auctions studied in \cite{Aaltio_etal2025} illustrates how the test can be used for cartel detection, and demonstrates the importance of testing for monotonicity and potential applications of our tests. For instance, our tests provide a key tool as a first step in detecting non-competitive behavior.
\vspace{5em}
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\baselineskip=12.75pt
\bibliographystyle{chicago}
\bibliography{test_auction.bib}
\end{small}}
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