On defining ex ante payoffs in games with diffuse prior

Abstract

While the diffuse prior has been widely used in applied economic theory for its technical convenience and as a way of modeling complete lack of knowledge, it is not formally defined, nor are ex ante payoffs in games under this prior. In this paper, we provide a formal treatment of the diffuse prior which can validate its application in games. We consider stationary games, in which players’ signals are translation invariant in the true state and players’ payoffs are translation invariant in actions together with the state. We show that strategies which admit well-defined expected payoffs under the diffuse prior are essentially stationary, being almost translation invariant in signals. Our analysis builds on two formal definitions. We define the diffuse prior through a limit construction, using sequences of well-defined priors that become increasingly dispersed. A class of strategy profiles is admissible if for any strategy profile, each player’s ex ante payoff along these sequences converges to a limit that does not depend on the particular sequence. A secondary contribution of the paper is an extension of the concept of distributional strategies (Milgrom and Weber in Math Oper Res 10:619–632, 1985) to a class of multistage games.

1 Introduction

The diffuse, or uninformative, prior is often interpreted informally as a uniform distribution on the real line. This prior has two advantages for use in economics: first, as it represents complete ignorance, it is appropriate for modeling situations in which agents have no advance knowledge of the environment; and second, it makes updating beliefs through Bayes’ rule computationally simpler. This tractability comes from ex ante symmetry across all states. When signals are the sum of the state and a conditionally i.i.d. noise, posterior beliefs about the state given two signal values are simply translations of one another. However, although the diffuse prior is commonly used, it is not formally defined: any uniform distribution must have constant density and must integrate to one, but any positive constant density, integrated over the real line, yields infinity, and zero density integrates to zero. This lack of a formal representation means that ex ante expected payoffs are not defined when driven by a random variable drawn from a diffuse prior distribution. The existing literature has circumvented this issue by leaving expected payoffs undefined and instead focusing on payoffs conditional on signal realizations (for example, Friedman 1991; Klemperer 1999; Morris and Shin 2002, 2003; Myatt and Wallace 2014).

In this paper, we develop a method for formally defining expected payoffs under a diffuse prior, and thereby bringing them into the realm of traditional game theory, where expected payoffs are assumed to be well-defined for all strategy profiles of a game. This can be directly useful in applications for various reasons. One might be interested in ex ante (or behind the veil of ignorance) welfare evaluation. Or in some contexts it is natural to allow for an ex ante participation constraint for one of the players. For example, when a management consultant is hired to examine and improve a firm’s performance, the management consultant has to decide whether to accept the task before inspecting the firm’s internal structure and production. In other contexts an agent might need her strategy approved by a principal ex ante, before acquiring private information, which only leads to a well-defined problem if the principal can form ex ante payoff expectations. Our method validates the conventional but informal approach of handling a diffuse prior, and providing rigorous foundations for diffuse prior might encourage more applied theory research to feature analytically tractable games with diffuse prior.

We claim that the diffuse prior can be rigorously constructed as a limit of well-defined distributions, and that expected payoffs under a diffuse prior can be defined in certain cases. We define a class of games with stationary information and payoff structures, where signals and payoffs are translation invariant in the following way: the distribution of signals is translation invariant in the true state, and payoffs are translation invariant in all actions together with the true state. We then show that as long as a class of strategy profiles includes all stationary strategies, then admissibility requires that all strategies are nearly stationary in a precise sense.

We capture the main features of this uninformative prior by using a sequence of (proper) measures that diffuse in a formal sense (Definition 4). We say that a class of strategies is admissible (Definition 5) when ex ante expected payoffs, taken along any diffusing sequence of proper priors, have a well-defined limit that does not depend on the particular sequence.¹ Stationary strategies are unsurprisingly admissible, as given any stationary strategy profile, expected payoffs conditional on all signal realizations are the same. Our main result (Theorem 1) states roughly that in any class of admissible strategy profiles that includes all stationary strategies, every strategy is nearly stationary in a particular sense (Definition 3). Furthermore, every such strategy is payoff-equivalent to some stationary strategy. We extend this result to a class of multistage games in Sect. 4.

An interpretation of our results is as follows. We offer a limit construction of the diffuse prior which allows ex ante payoffs under this prior to be defined as limits of payoffs under proper priors. However, the existence of such ex ante payoffs places limits on the strategy profiles available. To the extent that formal equilibrium concepts or welfare analyses require well-defined ex ante payoffs, it is useful to identify an admissible class of strategy profiles within which ex ante payoffs are guaranteed to be well-defined, and hence we define admissibility as a property of strategy profiles. In stationary games it is natural to assume that strategy sets include stationary strategies, and hence we require an admissible class to include all stationary strategies; in other words, strategies must yield well-defined payoffs when played against stationary strategies. This notion of admissibility then implies that all strategies are close to stationary strategies.²

We demonstrate the applicability of our results in two contexts: (i) in the context of beauty contest games introduced in Morris and Shin (2002),³ and (ii) in the delegation framework of Ambrus et al. (2019).

A secondary contribution of our paper is an extension of the concept of distributional strategies (Milgrom and Weber 1985) to a class of multistage games. The key additional feature of distributional strategies for multistage games is that a player’s actions in later stages can depend on actions by other players in earlier stages, and thus the distributional strategy includes dimensions for past actions.

Before proceeding, we briefly comment on the existing literature in statistics and probability theory on the subject of noninformative priors. Analysts have long debated the best way to impose a prior belief when performing parameter estimation and have recognized various desirable features. Laplace (1951) argued for a uniform or flat prior, from a principle of insufficient reason: without further information, any two possible values of the parameter should be considered equally likely. Jeffreys (1946, (1961) proposed a particular rule for selecting a prior, later known as the Jeffreys prior, for a given data-generating process which would be invariant to reparameterizations.⁴ While the Jeffreys prior need not be uniform, in the case of data drawn from a unidimensional normal distribution with known variance and unknown mean (as in many economic applications, including those considered in this paper), the Jeffreys prior is indeed uniform. (Of course, over the real line, this results in an improper prior.) Hence our use of a diffuse prior is consistent both with economic applications and with both approaches mentioned above. We refer the reader to Kass and Wasserman (1996) for an extensive review of the literature on noninformative priors, and to Yang and Berger (1998) for a catalog of such priors.

The paper is structured as follows. Section 2 gives an overview of the main ideas and outlines the steps of the main proof. Section 3 gives the formal analysis for single-stage games, and Sect. 4 extends this analysis to multistage games. Section 5 shows how our results can be applied to a beauty contest game and to a companion paper on delegation. Section 6 discusses alternative approaches to handling nonstationary strategies, and Sect. 7 concludes. Appendix contains proofs not provided in the body of the paper.

2 Overview

To capture the diffuse prior as a limit object, we define sequences of proper measures to be diffusing if, roughly, the measures become increasingly uniform and spread out over the real line. Our definition allows for a large class of diffusing sequences, including sequences of uniform distributions on $[-n,n]$ or sequences of normal distributions centered at zero with variance n, with n going to infinity.

We will define payoffs under a diffuse prior in cases where the limit of payoffs taken along any diffusing sequence exists and is independent of the sequence. Before defining the class of games we consider, we demonstrate how some concrete functions from ${\mathbb {R}}$ to ${\mathbb {R}}$ stand up to this criterion. Clearly, a constant function, when integrated with respect to any probability measure, integrates to that constant, and so all diffusing sequences result in the same limit, and thus a constant function is admissible. In addition, the function $x\mapsto {mathbb{1}}_{[0,1]}(x)$, which takes the value 1 if $x\in [0,1]$ and 0 otherwise, is also admissible; it is not difficult to show that along any diffusing sequence, the expected value approaches 0. On the other hand, a function like $x\mapsto mathbb{1}_{[0,\infty )}$ is not admissible. One could obtain a limit of 0 by defining a diffusing sequence $({\mathbb {P}} _{n}^{1})_{n\in {\mathbb {N}}}$ with the densities $\frac{mathbb{1} _{[-n^{2},n]}}{n^{2}+n}$ and a limit of 1 by a different diffusing sequence $({\mathbb {P}}_{n}^{2})_{n\in {\mathbb {N}}}$ with densities $\frac{mathbb{1} _{[-n,n^{2}]}}{n^{2}+n}$. The key property of admissible functions here is that they are constant or “nearly” constant in some formal sense. As we are interested in games of asymmetric information, and not real-valued functions per se, the exercise is more subtle than the above examples suggest. Nonetheless, the above intuition plays a key role in the analysis that follows.

In the baseline model, we analyze static n-player games with asymmetric information. At the beginning of the game, the state of the world $\theta \in {\mathbb {R}}$ is drawn according to a diffuse prior, and players receive private, conditionally i.i.d. signals $s_{i}$ about $\theta$. Players then simultaneously choose real-valued actions. We assume that the game has a stationary structure in terms of signals and payoffs: (i) for each player there is some distribution $F_i$ such that for all $\theta$, $s_{i}-\theta$ is drawn from this distribution and (ii) payoffs are invariant to a translation of all actions and $\theta$ by a constant. The model also accommodates uninformed players.

Strategies must specify (distributions over) actions given a player’s private signal. Since we start from strategy sets that are not restricted to be stationary, we need to provide a flexible and careful formal definition of strategies. Since the state space is uncountable, it is not practical to define strategies as products of signal-dependent distributions over actions. The key tension is that desirable topologies should be both rich enough so that payoff functions are continuous in strategies, but also coarse enough so that the strategy space is compact. We follow the approach of Milgrom and Weber (1985) in using distributional strategies, which are measures $\mu _{i}$ over the product space of signals and actions. A distributional strategy induces a conditional distribution on actions for any given signal, and by integrating over signals, it induces a conditional distribution on actions for any given $\theta$.

It is convenient to normalize distributional strategies by setting the marginal distribution over the signal dimension to be some fixed, arbitrary distribution G with positive density g on the real line.⁵ To describe behavior and payoffs conditional on arbitrary $\theta$, it is useful to exploit the stationary structure of the game and define “recentered” strategies $\mu _{i}^{x}$, which, roughly speaking, specify behavior relative to the private signal as if that signal had been shifted down by the constant x. A stationary strategy then, as we define it, is one which is translation invariant, in the sense that if the signal is shifted by a constant, then actions are shifted by the same constant; it follows that for stationary strategies, all of its recentered strategies are the same strategy. Under a mild assumption on the action sets, we show that the space of all distributional strategies, denoted ${\mathcal {M}}_{i}$, is compact.⁶ We say that a strategy $\mu _{i}$ has a limit strategy $\mu _{i}^{*}\in {\mathcal {M}}_{i}$ if $\lim _{\theta \rightarrow -\infty }\mu _{i}^{\theta }=\lim _{\theta \rightarrow +\infty }\mu _{i}^{\theta }=\mu _{i}^{*}$ (Definition 3). We show that whenever this property holds, the strategy $\mu _{i}^{*}$ must be stationary, and hence we call $\mu _{i}$ nearly stationary.

We assume that the game is irreducible in the sense that there are no redundant strategies—there are no two strategies which always yield the same payoffs. Without this assumption, there would be no hope of disciplining the set of admissible strategies, since a player could combine such redundant strategies in arbitrary ways without affecting payoffs.

Our main result, Theorem 1, has two components. First, it says that the class of nearly stationary strategies is admissible, so near stationarity is sufficient for admissibility. Second, it says that in any admissible class of strategies which is at least as large as the set of stationary strategies, all strategies are nearly stationary. Since these strategies are payoff-equivalent to stationary strategies, we argue that for a game with diffuse prior to have well-defined ex ante expected payoffs, essentially all strategies have to be stationary.

We begin the proof of the necessity part of Theorem 1 by establishing the existence of some distribution $\mu _i^*\in {\mathcal {M}}_i$ with the following property: for all $\eta >0$, $\mu _i^{\theta }$ is within $\eta$ of $\mu _i^*$ for an infinite measure set of $\theta$ (Lemma 7). We then call $\mu _i^*$ an attraction. We show that this is a weaker condition than near stationarity; a necessary condition for $\mu _i$ to be near $\mu _i^*$ is that $\mu _i^*$ is an attraction.

Next, we argue that there can be at most one such attraction for any strategy which is part of an admissible class. The proof of this claim is by contradiction and contains several steps. We suppose that $\mu _i ^{*}$ and ${\hat{\mu }}_i$ are two distinct attractions for player i’s strategy $\mu _{i}$. We argue that by the irreducibility assumption, there must exist some profile of stationary strategies of the rivals, $\mu _{-i}$, against which these distributions yield distinct expected payoffs. Given $\eta >0$, we can construct a sequence of measures $({\mathbb {P}}_{n}^{1})_{n\in {\mathbb {N}}}$ (resp. $({\mathbb {P}}_{n}^{2})_{n\in {\mathbb {N}}}$) that places increasing mass on $\theta$ such that $\mu _i ^{\theta }$ is within $\eta$ of $\mu _i ^{*}$ (resp. ${\hat{\mu }}_i$). By continuity and translation, the limit of expected payoffs can be close to either of two distinct values. This violates admissibility, giving the desired contradiction.

Given the unique attraction $\mu _i^*$, we show that $\mu _i^*$ is a limit strategy for $\mu _i$. If $\mu _i$ has any limit strategy, that strategy must be an attraction, so $\mu _i^*$ is the only candidate. We show that if $\mu _i^*$ is not a limit strategy for $\mu _i$, then there is a compact set of strategies that does not contain $\mu _i^*$ but contains $\mu _i^{\theta }$ for a measurable set of infinitely many $\theta$. This compact set itself contains an attraction, and this contradicts the uniqueness of the attraction $\mu _i^*$.

To complete the proof, we argue that $\mu _i^*$ is stationary, so that $\mu _i$ is nearly stationary.

3 Model

In this section, for expositional reasons, we consider single-stage games; we later extend the results to multistage games in Sect. 4. Before analyzing the diffuse prior, we specify the class of games we consider, which consists of an information structure and a payoff structure.⁷

3.1 Setup

Let $\varGamma$ denote the game. There are n players indexed by $i\in {\mathcal {I}}:=\{1,2,\dots ,N\}$. All players assign a diffuse prior (formalized in Sect. 3.3) to the state of the world, $\theta \in \varTheta :={\mathbb {R}}$. Players are categorized as either informed or uninformed. Informed players receive signals $s_{i}\in S_{i}:= {\mathbb {R}}$ which are conditionally independent given $\theta$ with distributions $s_{i}-\theta =:{\hat{s}}_{i}\sim F_{i}$ for some cumulative distribution function $F_{i}$ on ${\mathbb {R}}$ admitting a bounded, positive density $f_i$; uninformed players do not receive such a signal. We use ${\mathcal {I}}^{\textit{inf}}$ and ${\mathcal {I}}^{un}$ to denote the sets of informed and uninformed players, respectively. Let $S:=\times _{i\in {\mathcal {I}}^{\textit{inf}}}S_{i}$ (and likewise for s and ${\hat{s}}$) and let F be the joint CDF, $F({\hat{s}})= \times _{i\in {\mathcal {I}}^{\textit{inf}}}F_{i}({\hat{s}}_{i})$. We let $X_{i}$ denote a copy of $\mathbb { R}$, one for each player; each player simultaneously chooses an action $a_{i}$ from a set of available actions, denoted $A_{i}(s_{i})\subset X_{i}$ for $i\in {\mathcal {I}}^{\textit{inf}}$ and $A_i\subset X_i$ for $i\in {\mathcal {I}}^{\textit{un}}$. Given $\theta$ and a vector of actions $a=(a_{i},a_{-i})$ where $a_{-i}:=(a_{1},\dots ,a_{i-1},a_{i+1},\dots ,a_{N})$, players receive payoffs $u_{i}(a_{i},a_{-i},\theta )$. We assume that the payoff functions $u_{i}$ are continuous in all arguments.

Informally, a mixed strategy is a randomization over pure strategies. However, in games of incomplete information with uncountable type spaces (such as in our class of games), topological problems arise from interpreting strategies as mixed strategies (i.e., distributions over pure strategies) or behavioral strategies (i.e., products of history-contingent distributions over actions). Although the standard way to define mixed strategies in games with finite or countable signal spaces is as products, over signals, of signal-dependent distributions, this approach runs into problems when signal spaces are uncountable. As an alternative, Milgrom and Weber (1985) (hereafter “MW”) introduce the notion of distributional strategies, whereby strategies are defined as joint distributions over signals and actions. We adapt this concept to our needs in Definition 1.

Toward Definition 1, let G be an arbitrary distribution over ${\mathbb {R}}$ (the common signal space for informed players) with bounded density $g>0$, and let $\phi$ denote the measure induced by this distribution G. Let $\lambda$ denote the Lebesgue measure; note that $\phi$ and $\lambda$ are mutually absolutely continuous, that is, they have the same zero-measure sets.⁸ We use $\mu _i(\cdot |s_i)$ to denote (versions of) the conditional distribution over actions given signals. We are now ready to define strategies, which for informed players take the form of distributional strategies.

Definition 1

(Strategies) A strategy for a player $i\in {\mathcal {I}}^{\textit{inf}}$ is a probability measure $\mu _i$ on $S_i\times X_i$ such that:

• (Marginal Distribution over Signals) For all measurable $T\subseteq S_i$ , $\mu _i(T\times X_i)=\phi (T)$.

• (Proper Support) For any version of the conditional distribution $\mu _{i}(\cdot |s_i)$, for $s_i\in S_i$ except on a set of measure zero, the support of $\mu _{i}(\cdot |s_i)$ is a subset of $A_i(s_i)$.

A strategy for a player $i\in {\mathcal {I}}^{\textit{un}}$ is a probability measure $\mu _i$ on $X_i$ whose support is a subset of $A_i$.

A few comments on Definition 1 are in order. The first property is a normalization and represents a departure from MW. The key feature is that the distribution G (which defines $\phi$) has full support over the reals. The strategic content of a distributional strategy is in the conditional distributions $\mu _{i}(\cdot |s_{i})$ which describe how the player behaves given a particular signal $s_{i}$; this content is independent of the particular marginal distribution G. In contrast, MW use the actual distribution over types (here, signals) to play the role of G. We argue that this is not necessary, since the strategic content is contained in the conditional distributions, and the marginal distribution is merely a tool for “packaging” these into a joint distribution. In addition, it is not possible to use the actual type distribution in our setting since, prior to the $\theta$ realization, signals for informed players have ex ante diffuse distributions over ${\mathbb {R}}$. The same applies to our multistage version in Sect. 4.

The second property in Definition 1 ensures that with probability one, actions are chosen from $A_i(s_i)$. For uninformed players, we can dispense with the signal dimension and define strategies as measures over actions alone.

Under the above definition, a pure strategy for an informed player i is one such that for almost all $s_i$, $\mu _i(\cdot |s_i)$ places all mass on a single point (which can depend on $s_i$), where $\mu _i(\cdot |s_i)$ here denotes the regular conditional probability. That is, for almost every $s_i$, there exists $x\in X_i$ such that for all measurable $Y\subseteq X_i$, $\mu _i(Y|s_i)=1$ if $x\in Y$ and 0 otherwise. (A pure strategy for an uninformed player has the analogous property.)

Since $X_i$ and $S_i\times X_i$ are complete and separable, the spaces of probability measures $\varDelta (X_i)$ and $\varDelta (S_i\times X_i)$, under the topology of weak convergence, are metrized by the Prokhorov distance.⁹ We let ${\mathcal {M}} _i$ denote the space of all strategies for player i, where ${\mathcal {M}}_i\subset \varDelta (X_i)$ if player i is uninformed and ${\mathcal {M}}_i\subset \varDelta (S_i\times X_i)$ if player i is informed; on each ${\mathcal {M}}_i$, we use the same Prokhorov distance.

3.2 Stationarity

As the diffuse prior implies symmetry across states, we focus on games where this symmetry holds for all components of the game. We label these games stationary. Below we define stationarity of signals, payoffs and strategies.

We assume that the signal structure of the game is stationary so that conditional on any realization $\theta$, the values $s_i$ are drawn i.i.d. with $s_i-\theta \sim F_i$ for some distribution $F_i$ on ${\mathbb {R}}$ as described earlier. In addition, we assume that payoffs are stationary, in the sense that for any fixed strategy profile of the uninformed players, all players’ payoffs are invariant to a shift in the informed players’ actions and the state shifted by the same constant: for all $i\in {\mathcal {I}}$, and strategy profiles a, $u_i(a,\theta )=u_i(a-\theta ^{\textit{inf}},0)$, where we use $\theta ^{\textit{inf}}$ to denote the vector of length N whose ith component takes the value $\theta$ if $i\in {\mathcal {I}}^{\textit{inf}}$ and 0 otherwise.¹⁰

Compactness of the strategy space is useful throughout our analysis. We therefore assume that the action space available after each signal is a compact set; since we are focusing on stationary games, we also assume that action sets after different signals are translations of one another.

Assumption 1

For any player $i\in {\mathcal {I}}^{\textit{inf}}$, the action space available after signal 0, $A_i(0)$, is compact, and for all $s_i\ne 0$, $A_i(s_i)=A_i(0)+s_i$. For $i\in {\mathcal {I}}^{\textit{un}}$, the action space available $A_i$ is compact.

For example, suppose action sets are compact intervals, centered at the signal for informed players; specifically, suppose that for each informed player i there exists some constant $M_i>0$ such that for all $s_i$, $A_i(s_i)=[s_i-M_i,s_i+M_i]$, and for each uninformed player i there exists $M_i>0$ such that $A_i=[-M_i,M_i]$. Then Assumption 1 is satisfied. An implication of Assumption 1 with our notion of strategies is the following.

Lemma 1

Under Assumption 1, each ${\mathcal {M}}_i$ is compact.

For informed players, a strategy as defined above gives a distribution over signals and actions with an arbitrary fixed marginal distribution over the signal dimension G. To describe joint distributions over actions and signals (and ultimately, to describe payoffs) conditional on arbitrary $\theta$ values, it is useful to define a “recentering function.” Formally, for any $i\in {\mathcal {I}}^{\textit{inf}}$, strategy $\mu _i\in {\mathcal {M}}_i$, and $x \in {\mathbb {R}}$, we define a strategy $\mu _i^{x}$ by specifying its conditional distributions $\mu _i^x(Y|s_i)=\mu _i(Y-x|s_i-x)$ for all $Y\subseteq X_i$ and all $s_i\in S_i$ except on a set of measure zero under $\phi$, where $Y-x$ denotes the set Y shifted by $-x$. We then say that $\mu _i^x$ is a recentered strategy; note that in the case of $x=0$, we have $\mu _i^0=\mu _i$. For intuition, it is helpful for a moment to interpret strategies as inducing distributions over markups, that is, actions minus signals. Then the strategy $\mu _i^x$ gives the same distribution over markups after signal $s_i$ as strategy $\mu _i$ does after signal $s_i-x$. Example 1 gives more detail. For completeness, we define $\mu _i^x=\mu _i$ for uninformed players in this section.

The recentering function also allows us to readily define stationarity of strategies.

Definition 2

(Stationary strategies) For $i\in {\mathcal {I}}^{\textit{inf}}$, a strategy $\mu _i$ is stationary if the distribution over i’s action and i’s private signal is together translation invariant: for all $x\in {\mathbb {R}}$, $\mu _i=\mu _i^x.$ For $i\in {\mathcal {I}}^{\textit{un}}$, all strategies are stationary.

A stationary strategy in the case of single-stage games can be described by a single distribution over $A_i(0)$, interpreted as a (possibly random) markup x so that the action chosen after receiving signal $s_{i}$ is $a_i=s_{i}+x$. As we prove later (see Lemma 5 in appendix), for an arbitrary strategy $\mu _{i}$, the map $\theta \mapsto \mu _{i}^{\theta }$ from ${\mathbb {R}}$ to ${\mathcal {M}}_i$ is uniformly continuous.

Example 1

Suppose the (nonstationary) strategy $\mu _i$ specifies

• if $s_i<1$, play $a_i \sim U[s_i,s_i+1]$ (i.e., a markup distributed U[0, 1], )

• if $s_i\ge 1$, play $a_i \sim U[s_i+2,s_i+3]$ (i.e., a markup distributed U[2, 3]. )

The recentered strategy $\mu _i^{1/2}$ (where $x=1/2$) prescribes the following behavior. If $s_i<3/2$ is observed, apply a random markup distributed U[0, 1] (as the original strategy $\mu _i$ specifies for a signal $s_i-x=s_i-1/2<1$), that is, play $a_i \sim U[s_i,s_i+1]$; if $s_i\ge 3/2$ is observed, apply a markup distributed U[2, 3] (as $\mu _i$ specifies for a signal $s_i-x=s_i-1/2\ge 1$), that is, play $a_i\sim U[s_i+2,s_i+3]$. Note that $\mu _i$ here is not stationary, but given the first bullet point, it would be stationary if for all $s_i$, $a_i \sim U[s_i,s_i+1]$.

As mentioned, the recentering function allows us to easily express ex interim expected payoffs (that is, conditional on $\theta$) for arbitrary $\theta$ realizations. Since we consider a stationary payoff and signal structure, we can calculate expected payoffs under strategies $\mu _i$ conditional on $\theta =x$ as the expected payoffs under strategies $\mu _i^{-x}$ conditional on $\theta =0$ (see Eq. (1)). Let $\mu ^{x}$ denote the strategy profile where each player $i\in {\mathcal {I}}$ plays $\mu _i^{x}$ (and in particular, each player $i\in {\mathcal {I}}^{\textit{un}}$ plays $\mu _i)$.

A profile of strategies induces ex interim expected payoffs for each player as follows:

$$\begin{aligned} u_i(\mu ,\theta )&:=\int _{{\hat{s}}\in {\mathbb {R}}^{|{\mathcal {I}}^{\textit{inf}}|}}\int _{a\in X} u_i(a,\theta )\,d((\times _{j\in {\mathcal {I}}^{\textit{inf}}} \mu _{j,X_j}(a_j|{\hat{s}} _j+\theta ))\nonumber \\&\quad \times (\times _{j\in {\mathcal {I}}^{\textit{un}}} \mu _{j}(a_j)))\mathrm{d}F({\hat{s}}) \nonumber \\&=\int _{{\hat{s}}\in {\mathbb {R}}^{|{\mathcal {I}}^{\textit{inf}}|}}\int _{a\in X} u_i(a-\theta ^{\textit{inf}},0)\,d((\times _{j\in {\mathcal {I}}^{\textit{inf}}} \mu _{j,X_j}(a_j|{\hat{s}} _j+\theta ))\nonumber \\&\quad \times (\times _{j\in {\mathcal {I}}^{\textit{un}}} \mu _{j}(a_j)))\mathrm{d}F({\hat{s}}) \nonumber \\&=\int _{{\hat{s}}\in {\mathbb {R}}^{|{\mathcal {I}}^{\textit{inf}}|}}\int _{a\in X} u_i(a,0)\,d((\times _{j\in {\mathcal {I}}^{\textit{inf}}} \mu _{j,X_j}(a_j+\theta |{\hat{s}} _j+\theta ))\nonumber \\&\quad \times (\times _{j\in {\mathcal {I}}^{\textit{un}}} \mu _{j}(a_j)))\mathrm{d}F({\hat{s}}) \nonumber \\&=\int _{{\hat{s}}\in {\mathbb {R}}^{|{\mathcal {I}}^{\textit{inf}}|}}\int _{a\in X} u_i(a,0)\,d((\times _{j\in {\mathcal {I}}^{\textit{inf}}} \mu _{j,X_j}^{-\theta }(a_j|{\hat{s}} _j))\nonumber \\&\quad \times (\times _{j\in {\mathcal {I}}^{\textit{un}}} \mu _{j}(a_j)))\mathrm{d}F({\hat{s}}) \nonumber \\&=u_i(\mu ^{-\theta },0) , \end{aligned}$$

(1)

where the second line uses stationarity of payoffs, the third line performs a change of variables for each $a_j$, $j\in {\mathcal {I}}^{\textit{inf}}$, the fourth line uses the definition of the recentering function, and the fifth line uses the definition from the first line. Note that if all players play stationary strategies, then it can be seen from (1) that they obtain the same interim payoffs for all $\theta$ realizations, as $\mu ^{-\theta }=\mu$ and hence $u_i(\mu ^{-\theta },0)=u_i(\mu ,0)$ in that case.

In general, ex post payoffs can vary widely due to the noise in the signals. To ensure that the ex interim payoffs (given a realization of $\theta$, but not the signal realization) are finite, we impose the following assumption, which is a joint condition on the payoff function, action sets and signal distribution. Here, for each noise realization ${\hat{s}}$, $A({\hat{s}}+\theta )$ denotes the set of action profiles a such that $a_j\in A_j({\hat{s}}_j+\theta )$ for all $j\in {\mathcal {I}}^{\textit{inf}}$ and $a_j\in A_j$ for all $j\in {\mathcal {I}}^{\textit{un}}$.

Assumption 2

(Bounded Interim Payoffs) Together, the signal distributions, action sets, and payoff functions are such that $\int _{{\hat{s}} \in {\mathbb {R}}^{|{\mathcal {I}}^{\textit{inf}}|}} \sup _{a \in A({\hat{s}}+\theta )} |u_i(a_i,a_{-i},\theta )| \mathrm{d}F({\hat{s}})$ is bounded over all $i\in {\mathcal {I}}$ and $\theta \in {\mathbb {R}}$.

Assumption 2 holds, for example, when payoff functions are finite polynomial, the signal distributions conditional on the state are normal (which includes the commonly analyzed case of normally distributed signals and quadratic loss functions), and action sets are intervals of bounded size containing the signal. This assumption ensures that expected interim payoffs are bounded.

Lemmas 5 and 6 in appendix establish useful continuity properties of the recentering function and interim payoffs.

Next we define a weakening of stationarity that we will later prove to be both necessary and sufficient for admissibility. Recall that our arbitrary specification G for the marginal distribution over signals allows us operate with distributional strategies, which are joint distributions over signals and actions. Hence, to describe the distance between strategies, we must use a suitable metric for distances between distributions, and thus we use the Prokhorov metric. Convergence of strategies in the Prokhorov metric means that the measure these strategies assign to well-behaved sets (of pairs of signals and actions) also converge. Toward defining near stationarity, we begin by defining a limit strategy (if it exists) as a strategy that a given strategy approaches in the limit when recentering at extreme states, with distance between strategies being measured by the Prokhorov metric. In other words, the original strategy behaves increasingly similar to the limit strategy when the state becomes very high or very low.

Definition 3

(Limit strategies and near stationarity) For any $i\in {\mathcal {I}}$ and any strategy $\mu _i$, we say that a strategy $\mu _i^*$ is a limit strategy for $\mu _i$ if $\lim _{\theta \rightarrow -\infty } \mu _i^{\theta }=\lim _{\theta \rightarrow +\infty } \mu _i^{\theta }=\mu _i^*$, with limits taken with respect to the Prokhorov metric. If $\mu _i^*$ is stationary, then we classify $\mu _i$ as nearly stationary.

Note that stationarity implies near stationarity, as a stationary strategy is its own limit strategy. In particular, for $i\in {\mathcal {I}}^{\textit{un}}$ in static games, any $\mu _i$ is nearly stationary.

It is immediate from Definition 3 that a strategy can have at most one limit strategy. However, a strategy need not have any limit strategy. For a counterexample, consider a game with a single player receiving some signal $s_i$ where $s_i-\theta$ has distribution $F_i$ conditional on $\theta$, and suppose the action space given $s_i$ is $\{s_i,s_i+1\}$. Then the following strategy, call it $\mu _i$, has no limit strategy: assign action $a_i=s_i+1$ for all signals $s_i\ge 0$ and action $a_i=s_i$ for all signals $s_i<0$. As $\theta \rightarrow +\infty$, $\mu _i^{\theta }\rightarrow \kappa _0$, where we use $\kappa _x$ to denote the stationary strategy characterized by $a_i(s_i)=s_i+x$ with probability 1 for all $s_i\in {\mathbb {R}}$. Hence $\kappa _0$ is the only candidate for a limit strategy. But as $\theta \rightarrow -\infty$, $\mu _i^{\theta }\rightarrow \kappa _1\ne \kappa _0$, and thus $\mu _i$ has no limit strategy.

To see that nearly stationary does not imply stationary, modify the example above so that $\mu _i$ is characterized by $a_i=s_i+1$ for all $s_i\in [-K,K]$ for some $K>0$, and $a_i=s_{i}$ otherwise. We have $\lim _{\theta \rightarrow -\infty } \mu _i^{\theta }=\lim _{\theta \rightarrow +\infty } \mu _i^{\theta }=\kappa _0$, so $\mu _i$ is nearly stationary, but it is not stationary.

The next lemma provides a useful property equivalent to the one in Definition 3. It says that a strategy $\mu _i^*$ is a limit strategy for $\mu _i$ if and only if $\mu _i$ is close to $\mu _i^*$ for “most” $\theta$, that is for all $\theta$ except for some set of finite measure. Recall that $\lambda$ denotes the Lebesgue measure.

Lemma 2

For $i\in {\mathcal {I}}^{\textit{inf}}$, a strategy $\mu _i^*$ is a limit strategy for $\mu _i$ if and only if $\lambda \left( \{\theta :d_{P,i}(\mu _i^{\theta },\mu _i^*)\ge \eta \}\right) <\infty$ for all $\eta >0$, where $d_{P,i}$ is the Prokhorov metric.

The following lemma shows that the first part of Definition 3 implies the second.

Lemma 3

If $\mu _i^*$ is a limit strategy of $\mu _i$ in the sense of Definition 3, then $\mu _i^*$ is stationary and thus $\mu _i$ is nearly stationary.

Proof

If $\mu _i^*$ is not stationary, then there exists $\theta$ such that $\mu _i^{*,\theta }\ne \mu _i^*$. Now if $\mu _i^*$ is a limit strategy of $\mu _i$, then $\mu _i^{*,\theta }$ is a limit strategy of $\mu _i^{\theta }$. But also note that if $\mu _i^*$ is a limit strategy of $\mu _i$, then $\mu _i^*$ is a limit strategy of $\mu _i^{\theta '}$ for all $\theta '\in {\mathbb {R}}$, including $\theta '=\theta$, and thus $\mu _i^*$ is a limit strategy of $\mu _i^{\theta }$. Since a strategy can have at most one limit strategy, we have $\mu _i^{*,\theta }=\mu _i^*$, a contradiction. We conclude that $\mu _i^*$ is stationary. $\square$

For ease of exposition, from now on we require that the game involves no redundant strategies, i.e., strategies, which are indistinguishable from other strategies in their payoff implications.

Assumption 3

(Irreducibility) The game is irreducible in that there are no distinct strategies $\mu _i\ne \mu _i^{\prime }$ for any player i such that for all $\theta \in {\mathbb {R}}$ and all profiles $\mu _{-i}$ of stationary strategies for the rivals, $u_i(\mu _i,\mu _{-i},\theta )=u_i(\mu _i^{\prime },\mu _{-i},\theta )$.

3.3 The diffuse prior

The informal concept of diffuse prior has two key properties. The first property is that all real numbers are in the support of the prior. The second property is uniformity—all points are weighted equally. Hence, we define a sequence of proper measures to be diffusing if these properties hold in the limit—that is, sufficiently far into the sequence, the properties hold arbitrarily closely. The following definition formalizes this idea.

Definition 4

(Diffusing sequence) Consider a sequence $({\mathbb {P}}_n)_{n\in {\mathbb {N}}}$ of Borel probability measures on ${\mathbb {R}}$. We say that this sequence is diffusing if for any set W with $\lambda (W)\in (0,\infty )$ and any $\eta >0$, there exists $N\in {\mathbb {N}}$ such that for all $n\ge N$,

• ${\mathbb {P}}_n(W)>0$, and

• for all measurable $Y\subseteq W$, $\left| \frac{{\mathbb {P}}_n(Y)}{ {\mathbb {P}}_n(W)}-\frac{\lambda (Y)}{\lambda (W)}\right| <\eta$.

From the definition, we can establish the following property of diffusing sequences as a result.

Lemma 4

If $({\mathbb {P}}_n)$ is diffusing, then for any set $E\subset {\mathbb {R}}$ with $\lambda (E)<\infty$, $\lim _{n\rightarrow \infty } {\mathbb {P}}_n(E)=0$.

Proof

Consider any set $E\subset {\mathbb {R}}$ with $\lambda (E)<\infty$ and any diffusing sequence $({\mathbb {P}}_n)$. We establish the result for $\lambda (E)>0$, from which the result for $\lambda (E)=0$ immediately follows. Choose any arbitrarily small $\eta >0$. Choose $M>1$ sufficiently large that (i) $\frac{1}{M}<\eta$ and (ii) $\lambda (E\setminus [-M,M])<\eta \lambda (E)$. Let $W_M:=[-M,M]$ and $W_{M^2}:=[-M^2,M^2]$. By applying Definition 4 twice, first with the set E playing the role of W in the definition and again with $W_{M^2}$ playing the role of W, there exists K such that for all $n\ge K$,

$$\begin{aligned} {\mathbb {P}}_n(W_{M^2})&>0\nonumber \\ {\mathbb {P}}_n(E)&>0\nonumber \\ \left| \frac{{\mathbb {P}}_n(E\setminus W_M)}{{\mathbb {P}}_n(E)}-\frac{\lambda (E\setminus W_M)}{\lambda (E)}\right|&<\eta \end{aligned}$$

(2)

$$\begin{aligned} \left| \frac{{\mathbb {P}}_n(W_M)}{{\mathbb {P}}_n(W_{M^2})}-\frac{\lambda (W_M)}{\lambda (W_{M^2})}\right|&<\eta . \end{aligned}$$

(3)

In (2), recall that by construction $\frac{\lambda (E\setminus W_M)}{\lambda (E)}<\eta$ and thus rearranging (2) yields

$$\begin{aligned} {\mathbb {P}}_n(E\setminus W_M)&<2\eta {\mathbb {P}}_n(E)<2\eta . \end{aligned}$$

(4)

In (3), we have $\frac{\lambda (W_M)}{\lambda (W_{M^2})}=\frac{1}{M}<\eta$ and thus

$$\begin{aligned} {\mathbb {P}}_n(W_M)&<2\eta {\mathbb {P}}_n(W_{M^2})<2\eta . \end{aligned}$$

(5)

Adding (4) and (5) then yields

$$\begin{aligned} {\mathbb {P}}_n(E)\le {\mathbb {P}}_n(E\setminus W_M)+{\mathbb {P}}_n(W_M)<4\eta . \end{aligned}$$

Since $\eta$ is arbitrary, we have ${\mathbb {P}}_n(E)\rightarrow 0$. $\square$

For illustrative purposes, we highlight two specific diffusing sequences (see Fig. 1). As one would expect, flattening sequences of the uniform distribution or normal distribution are diffusing according to our definition.

Example 2

Both $({\mathbb {P}}^1_n)$ given by the density $\frac{1}{2n}mathbb{1}_{ [-n,n]}$ and $({\mathbb {P}}^2_n)$ given by N(0, n) are diffusing.

Proof

If $\lambda (W)\in (0,\infty )$, then for sufficiently large n, $W\cap [-n,n]\ne \emptyset$ and thus ${\mathbb {P}}^1_n(W)>0$. Moreover, for any measurable $Y\subseteq W$, $\left| \frac{{\mathbb {P}}^1_n(Y)}{{\mathbb {P}}^1_n(W)}-\frac{\lambda (Y)}{\lambda (W)} \right| =\left| \frac{\int _{ Y}\frac{1}{2n}mathbb{1}_{ [-n,n]}(y)\mathrm{d}\lambda }{\int _{W}\frac{1}{2n}mathbb{1}_{ [-n,n]}(w)\mathrm{d}\lambda }-\frac{\lambda (Y)}{\lambda (W)} \right| =\left| \frac{\int _{Y}mathbb{1}_{ [-n,n]}(y)\mathrm{d}\lambda }{\int _{ W}mathbb{1}_{ [-n,n]}(w)\mathrm{d}\lambda }-\frac{\lambda (Y)}{\lambda (W)} \right| \rightarrow 0$ by the monotone convergence theorem. Now ${\mathbb {P}}^2_n(W)>0$ for all $n\in {\mathbb {N}}$. Using a similar argument, for any $Y\subseteq W$, $\left| \frac{{\mathbb {P}}^2_n(Y)}{{\mathbb {P}}^2_n(W)}-\frac{\lambda (Y)}{\lambda (W)} \right| =\left| \frac{\int _{Y}\frac{1}{\sqrt{2\pi n}}e^{-y^2/(2n)}\mathrm{d}\lambda }{\int _{W}\frac{1}{\sqrt{2\pi n}}e^{-w^2/(2n)}\mathrm{d}\lambda }-\frac{\lambda (Y)}{\lambda (W)} \right| =\left| \frac{\int _{Y}e^{-y^2/(2n)}\mathrm{d}\lambda }{\int _{W}e^{-w^2/(2n)}\mathrm{d}\lambda }-\frac{\lambda (Y)}{\lambda (W)} \right| \rightarrow 0$. $\square$

Fig. 1

Examples of diffusing sequences

3.4 Admissibility

Recall the example from Sect. 2, where we asked which functions of $\theta$ are “admissible” in the sense that they yield consistent limits when integrated along certain kinds of sequences of distributions. We suggested that admissible functions are nearly constant in a particular way: there exists some constant such that for any $\epsilon >0$, the set of $\theta$ on which the function deviates from that constant by more than $\epsilon$ has finite measure. As we show in our main result, the spirit of this example extends to admissibility of strategies. Since players only observe noisy signals of $\theta$, their strategies are not directly functions of $\theta$ but functions of their private signals. Nonetheless, having well-defined expected payoffs will place restrictions on players’ strategies, similar to how admissible functions are nearly constant.

Definition 5

(Admissibility) A class ${\mathcal {M}}^0\subset {\mathcal {M}}$ of strategy profiles is said to be admissible if for any profile $(\mu _1,\mu _2,\ldots ,\mu _N)$ of strategies in ${\mathcal {M}}^0$, there exists a vector $u^*\in {\mathbb {R}}^N$ such that for any diffusing sequence $({\mathbb {P}} _n)$ and all $i\in {\mathcal {I}}$, $\lim _{n\rightarrow \infty }\int _{\theta \in {\mathbb {R}}} u_i(\mu _i,\mu _{-i},\theta )\, \mathrm{d}{\mathbb {P}}_n(\theta )=u_i^*$.

Let ${\mathcal {K}}_i$ denote the class of stationary strategies for player i. Note that for profiles $\mu =(\mu _i,\mu _{-i})$ of stationary strategies, the mapping $\theta \mapsto u_i(\mu , \theta )$ is constant, and hence the sequence of integrals in Definition 5 trivially converges to the same limit for every diffusing sequence. Thus the class of profiles of stationary strategies is admissible.

Given the definition of admissibility, we can define (ex ante) payoff equivalence between two strategies.

Definition 6

(Payoff equivalence) Let ${\mathcal {M}}^0$ be a class of admissible strategy profiles. Two strategies $\mu _i$ and $\mu _i^{\prime }$ are said to be payoff equivalent for player i if for all strategy profiles $\mu _{-i}$ of the remaining players such that $(\mu _i,\mu _{-i}),(\mu _i^{\prime },\mu _{-i})\in {\mathcal {M}}^0$, player i gets the same payoff under $\mu _i$ as under $\mu _i^{\prime }$: for all diffusing sequences $({\mathbb {P}}_n)$,

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{\theta \in {\mathbb {R}}} u_i(\mu _i,\mu _{-i},\theta )\, d {\mathbb {P}}_n(\theta )=\lim _{n\rightarrow \infty }\int _{\theta \in {\mathbb {R}}} u_i(\mu ^{\prime }_i,\mu _{-i},\theta )\, \mathrm{d}{\mathbb {P}}_n(\theta ). \end{aligned}$$

We now present our main result for single-stage games. For the necessity direction, our proof technique makes use of the assumption that the class of strategy profiles contains stationary strategies. The reason is that when we consider payoffs conditional on various $\theta$, stationarity of the rivals’ strategies (together with stationarity of signals and payoffs) means that an informed player i’s payoff is changing as if one is holding $\theta$ fixed and changing the actual strategy of player i. By irreducibility, this means that if i’s strategy is not nearly stationarity, then i’s payoff conditional on $\theta$ varies significantly with $\theta$, which we then show is inconsistent with admissibility.

Theorem 1

Suppose the game $\varGamma$ satisfies Assumptions 1–3. Then

• (Sufficiency) The class of profiles of nearly stationary strategies is admissible.

• (Necessity) If ${\mathcal {M}}^{0}=\times _{i\in {\mathcal {I}}} {\mathcal {M}} ^0_i$ is admissible and ${\mathcal {K}}_i\subseteq {\mathcal {M}}^{0}_i$ for all i, then every $\mu _i \in {\mathcal {M}}^{0}_i$ is nearly stationary and payoff equivalent to some stationary strategy.

4 Multistage games

We now extend the analysis to a class of multistage games. The game has stages indexed by $t=1,2,\dots , \tau$ for $\tau \in {\mathbb {N}}$. In each stage t, some nonempty subset of players choose actions $a_{i}^{t}\in {\mathbb {R}}$; we use ${\mathcal {I}}(t)$ to denote this subset. As in the static model, a subset of players ${\mathcal {I}}^{\textit{inf}}$ are informed and obtain a private signal $s_i$ with the same properties as in the static game, while the remaining players receive no such signal. Each player acts in exactly one stage, denoted T(i). We assume that informed players act in stage 1, while all uninformed players act in later stages. We assume that uninformed players have a finite action set. Players observe other players’ past actions perfectly. The public history at stage $t>1$ is a record of all players’ past actions through stage $t-1$, and it is denoted $h^{t}$. An informed player’s history is simply that player’s initial signal $s_{i}$. To economize on notation, we define the history observed by player $i \in {\mathcal {I}}(t)$ by $h^t_i:=h^t$ when $t>1$ (and thus $i\in {\mathcal {I}}^{\textit{un}}$) and $h^t_i:=s_i$ when $t=1$ (and thus $i\in {\mathcal {I}}^{\textit{inf}}$). The set of available actions for $i\in {\mathcal {I}}$ moving in stage t is $A_i^t(h_i^t)$.

A pure strategy specifies for each player’s observed history $h_{i}^{t}$ an action in $A_{i}^{t}(h_{i}^{t})$. Given a realization of $\theta$ and sequence of action profiles taken in each stage, players receive payoffs $u_{i}(a_{i},a_{-i},\theta )$, where $a_{-i}$ is defined as before. In Definition 7, we extend the definition of distributional strategies from Sect. 3.

In order to extend Theorem 1, we must first adapt some other concepts from Sect. 3. As in Sect. 3, payoffs are assumed stationary in that they are invariant to shifts in $\theta$ and all informed players’ actions by the same constant. We adapt Assumption 1 to the following:

Assumption 4

For all $i\in {\mathcal {I}}(1)$, $A_i^1(h_i^1)$ is compact and satisfies $A_i^1(s_i)=A_i^1(0)+s_i$. For all $t>1$ and all $i\in {\mathcal {I}}(t)$, $A_i^t(h_i^t)$ is finite and independent of $h_i^t$.

Given Assumption 4, we use $A_i^t$ to denote the action sets available to uninformed players.

Given a realization of $\theta$ and sequence of action profiles taken in each stage, players receive payoffs $u_{i}(a_{i},a_{-i},\theta )$ for the game, where $a_{-i}$ is defined as before. As before, we assume that the payoff functions $u_{i}$ are continuous in all arguments, and we assume they are stationary in the same sense as in Sect. 3.

As in the static game, informed players’ only information is their private signal, so their strategies are defined exactly as in the static game. For uninformed players, their information is the past actions of other players. For $t>1$, let $H^t:=\left( \times _{i\in {\mathcal {I}}^{\textit{inf}}}X_i\right) \times \left( \times _{i\in {\mathcal {I}}^{\textit{un}}:T(i)<t}A_i^{T(i)}\right)$ denote the set of public histories at stage t. We formulate the strategies for players acting in stage $t>1$ as joint distributions over $H^t$ and their stage t actions, fixing a marginal distribution over $H^t$. Any version of the conditional distribution specifies behavior after a given history; under regularity conditions discussed below, any two versions of the conditional distribution are outcome-equivalent.

Definition 7

(Strategies—multistage version) A strategy for player $i\in {\mathcal {I}}^{\textit{inf}}$ is a probability measure $\mu _i$ on $S_i\times X_i$ satisfying the properties of Definition 1. A strategy for player $i\in {\mathcal {I}}^{\textit{un}}$ acting in stage t is a probability measure $\mu _i$ on $H^t\times A_i^t$ with the following property: the random variables induced by the marginal distributions over the dimensions for actions of players acting earlier are mutually independent and have distribution N(0, 1) for informed players and (discrete) uniform distribution for uninformed players.

For an uninformed player i, the distributional strategy $\mu _{i}$ includes dimensions for all the other players’ past actions. This results in some arbitrary detail which serves a technical purpose only. The strategically meaningful information from $\mu _{i}$ is in the distributions on the $A_i^t$ dimension conditional on each history. The marginal distributions for the past actions are not strategically meaningful, so for concreteness we normalize those to the normal distribution for past actions by informed players and discrete uniform distributions over finite sets for past actions by uninformed players; we wish to emphasize that $\mu _{i}$ does not contain any conjecture about other players’ behavior. Part (iii) of Assumption 5 ensures that all actions by informed players occur with probability zero (such as when all informed players play stationary strategies, due to the signal noise), and hence an uninformed player i’s behavior is well-defined, as all versions of the uninformed players’ strategies are outcome-equivalent.

Hence, there is an equivalence between distributional strategies under our formulation and behavioral strategies, given the regularity condition mentioned above. Specifically, for every distributional strategy, there exists an outcome-equivalent behavioral strategy, and conversely, for every behavioral strategy, there exists an outcome-equivalent distributional strategy. A distributional strategy specifies behavior at each decision node via the conditional distributions over that player’s actions, conditioning on the history; these conditional distributions and hence the player’s behavior are uniquely determined except on a set of measure zero. Given a behavioral strategy, one can construct a distributional strategy by specifying the marginal distributions on earlier actions as in Definition 7, and by defining the conditional distributions as those under the behavioral strategy.

Each player has a set of available strategies denoted ${\mathcal {M}}_i$; as in Sect. 3, it is metrized by the Prokhorov distance. In multistage games, some technical issues not present in static games arise. For instance, small changes in one player’s strategy can lead to discontinuous changes in the induced distribution over outcomes. To avoid such issues, we assume that ${\mathcal {M}}:=\times _{i\in {\mathcal {I}}} {\mathcal {M}}_i$ satisfies some regularity conditions (Assumption 5). Part (iii) ensures that each zero-measure set of actions occurs with zero probability, as described above. Part (v) of the definition adapts irreducibility from Assumption 3. As in the single-stage case, given any $\theta$ realization, a profile of distributional strategies $(\mu _i,\mu _{-i})$ induces a distribution over payoffs $u_i(a_i,a_{-i},\theta )$. We again abuse notation and let $u_i(\mu _i,\mu _{-i},\theta )$ denote ex interim payoffs conditional on $\theta$.

Assumption 5

The class ${\mathcal {M}}=\times _{i\in {\mathcal {I}}} {\mathcal {M}}_i$ of profiles of available strategies satisfies the following conditions: (i) ${\mathcal {M}}_i$ closed for all $i\in {\mathcal {I}}$, (ii) $\mu _i\in {\mathcal {M}}_i$ implies $\mu _i^{\theta }\in {\mathcal {M}}_i$ for all $\theta \in {\mathbb {R}}$ and $i\in {\mathcal {I}}$, (iii) $\mu _i({S_i}\times Y)=0$ for all $i\in {\mathcal {I}}$ and $Y\subset X_i$ such that $\lambda (Y)=0$, (iv) for all $i\in {\mathcal {I}}$, $\mu \mapsto u_i(\mu ,0)$ is continuous on ${\mathcal {M}}$, and (v) for all $i\in {\mathcal {I}}$ and strategies $\mu _i\ne \mu _i^{\prime }$ in ${\mathcal {M}}_i$, there exist $\theta _0\in {\mathbb {R}}$ and a profile $\mu _{-i}\in {\mathcal {M}}_{-i}$ of stationary strategies for the other players such that $u_i(\mu _i,\mu _{-i},\theta _0)\ne u_i(\mu _i^{\prime },\mu _{-i},\theta _0)$.

Given Definition 7 and Assumption 4, Lemma 1 can be applied to the space of measures satisfying Definition 7, and hence each ${\mathcal {M}}_i$, as a closed subspace by Assumption 5, is compact. By continuity of interim payoffs in strategies, interim payoffs are bounded.

Recentered strategies: The recentering function for informed players is exactly as in the static model. For uninformed players, let $h_i^{t,x}$ denote the history $h_i^t$ modified by adding x to all informed players’ (past) actions. We define the recentered strategy $\mu _i^x$ as the strategy which satisfies, for all $h_i^t\in H^t$ except on a set of measure zero and all $Y\subseteq A_i^{t}$, $\mu _i^x(Y|h_i^t)=\mu _i(Y|h_i^{t,-x})$. As in the single-stage case, for $x=0$, we have $\mu _i^0=\mu _i$. For strategy profiles, we define $\mu ^{x}$ as the profile in which each player plays $\mu _i^x$. Under the generalized recentering function defined above, we can now adapt Definition 2 to multistage games.

Definition 8

A strategy $\mu _i$ is stationary if for all $x\in {\mathbb {R}}$, $\mu _i=\mu _i^x.$

We can import several statements from Sect. 3 with no change in notation, and therefore, there is no need to reproduce them here. In particular, we maintain the following definitions: (i) the definition of limit strategies and nearly stationary strategies (Definition 3), (ii) the definition of a diffusing sequence (Definition 4), (iii) the definition of admissible strategies (Definition 5), and (iv) the definition of payoff equivalence (Definition 6). We define ${\mathcal {K}}_i$ as the class of stationary strategies in ${\mathcal {M}}_i$.

We now extend Theorem 1 to the class of multistage games presented in this section.

Theorem 2

Suppose Assumptions 4 and 5 are satisfied.

• (Sufficiency) The class of profiles of nearly stationary strategies is admissible.

• (Necessity) If ${\mathcal {M}}^{0}=\times _{i\in {\mathcal {I}}} {\mathcal {M}} ^0_i$ is admissible and ${\mathcal {K}}_i\subseteq {\mathcal {M}}^{0}_i$ for all i, then every $\mu _i \in {\mathcal {M}}^{0}_i$ is nearly stationary and payoff equivalent to some stationary strategy.

5 Applications

In this section, we apply our single-stage model to a beauty contest game and our multistage model to a delegation (sender–receiver) game.

5.1 Beauty contests

An application of the single-stage version of our model is an adaptation of the beauty contest model of Morris and Shin (2002). We assume that there is a finite number $N\ge 2$ of agents $i\in \{1,2,\dots ,N\}$ and an underlying state of the world $\theta$ drawn from a diffuse prior distribution. Each agent has a bias parameter $b_{i}$ and receives a private signal $s_{i}=\theta +\epsilon _{i}$, where $\epsilon _{i}\sim N(0,\sigma _{\epsilon }^{2})$ and where the $\epsilon _{i}$ and $\theta$ are mutually independent. Each agent chooses an action $a_{i}\in [s_{i}-M,s_{i}+M]$ for some large constant $M>0$ and receives a payoff

$$\begin{aligned} u(a_{i},a_{-i},\theta )=-\lambda (a_{i}-\theta -b_{i})^{2}-(1-\lambda )(a_{i}-{\bar{a}})^{2}, \end{aligned}$$

where $\lambda \in (0,1)$ is a constant and ${\bar{a}}:=\frac{\sum _{j}a_{j}}{N }$ is the average of all players’ actions, including player i. In other words, players want to choose actions that balance their idiosyncratic preferences with a desire to match other agents’ actions.

The game described here has stationary payoffs and satisfies Assumptions 1–3, and thus Theorem 1 applies. Hence, in order to guarantee well-defined ex ante expected payoffs, the analyst must restrict strategy sets to nearly stationary strategies. In the game we have described, stationary strategies are mixtures over constant markups: each player chooses a (possibly random) markup $k_{i}\in [-M,M]$ and, given any $s_{i}$, plays the action $a_{i}(s_{i})=s_{i}+k_{i}$.

If we restrict players to stationary strategies, there is a unique Bayesian Nash equilibrium of the game with diffuse prior, in which each player plays a pure strategy characterized by $a_i(s_i)=s_i+k_i$, where

$$\begin{aligned} k_i=\lambda b_i+(1-\lambda )\frac{\sum _j b_j}{N}. \end{aligned}$$

(6)

Each player’s signal $s_i$ is that player’s posterior mean belief about the state $\theta$ and about all other agent’s signals $s_j$ for $j\ne i$, and thus the mean action of each other player is $s_i+k_j$. Each player’s payoff is a concave function and thus has a unique maximizer given the strategies of the other players. It follows that any equilibrium is in pure strategies. Equation (6) says that each player’s equilibrium markup is a convex combination of all players’ biases, and the weight assigned to one’s own bias is an increasing function of $\lambda$. In the extreme case $\lambda =1$, each player cares only about his idiosyncratic preferences, and optimally sets $k_i=b_i$. For the other extreme, $\lambda =0$, the game is a pure coordination game and multiple equilibria exist, but in the limit as $\lambda \rightarrow 0$, $k_i\rightarrow \frac{\sum _j b_j}{N}$; with agents coordinating on the average bias of all agents.

5.2 Delegation

As a direct application of our model, we can also consider sender–receiver games. To illustrate, consider games with two stages: one stage in which n senders simultaneously choose actions after observing private signals, and a second stage in which the receiver observes the senders’ actions and chooses an action. Depending on the application in mind, the receiver’s action could be interpreted as a policy choice (as in many games of cheap talk) or as a sender’s identity (as in games of delegation with multiple experts).

In the case of delegation, the receiver’s available action choices may depend on the actions chosen by the senders. For example, in a companion paper by Ambrus et al. (2019), the senders are two experts who propose action choices, and the receiver must choose one of them. All players have quadratic loss functions; senders have biases $b_i$ relative to the receiver. The game unfolds as follows:

• 1. A state of the world $\theta \in {\mathbb {R}}$ is realized, drawn from a diffuse prior distribution.

• 2. Senders (experts) $i=1,2$ receive conditionally i.i.d. signals $s_i\sim N(\theta ,\sigma ^2)$.

• 3. Senders choose markups $k_i\in [-M,M]$ where M is some large constant.

• 4. The receiver (principal or decision maker, labeled player 3) observes $a_{i}=s_{i}+k_{i}$ for $i\in \{1,2\}$ and chooses the expert whose action will be implemented. Specifically, the receiver chooses an action j from $\{1,2\}$.

• 5. Payoffs are realized: receiver gets $-(s_j+k_j-\theta )^2$, sender j gets $-(s_j+k_j-\theta -b_j)^2$, sender $i\ne j$ gets $-(s_j+k_j- \theta -b_i)^2$.

A stationary strategy for the receiver must have the property that $C(a_{1},a_{2})=C(a_{1}+x,a_{2}+x)$ for all x. This includes mixtures of the following fundamental strategies: (i) always choose the higher offer, (ii) always choose the lower offer, (iii) always choose the offer from sender 1, and (iv) always choose the offer from sender 2. Ambrus et al. (2019) show that these are essentially the only possible best responses of the principal to a pair of stationary strategies of the experts.

Suppose the receiver is restricted to the four pure strategies above, and denote this strategy space by ${\mathcal {R}}$. We show that this strategy space alone is enough to establish irreducibility for the senders. Specifically, we show that any class ${\mathcal {M}}={\mathcal {M}}_1\times {\mathcal {M}}_2\times {\mathcal {R}}$ of strategy profiles that satisfies parts (i)–(iv) of Assumption 5 must also satisfy part (v), and hence (as Assumption 4 is clearly satisfied) Theorem 2 applies: all admissible strategies are nearly stationary, and in particular, are nearly “constant markup strategies” — that is, each sender’s strategy is nearly a stationary strategy in which there is a single “markup” distribution $H_i$ over $[-M,M]$ and actions are simply $s_i$ plus the draw from $H_i$.

Proposition 1

In the stationary game of Ambrus et al. (2019), suppose that strategy spaces ${\mathcal {M}}_1$ and ${\mathcal {M}}_2$ each contain all the constant markup strategies and are such that the class of strategy profiles ${\mathcal {M}}={\mathcal {M}}_1\times {\mathcal {M}}_2\times {\mathcal {R}}$ satisfies parts (i)–(iv) of Assumption 5. Then part (v) is also satisfied, and Theorem 2 applies to the game.

6 Alternative approaches to nonstationary strategies

In this section, we consider alternative approaches to handling ex ante payoffs under the diffuse prior.¹¹

First, in some cases it is possible to make comparisons across strategy profiles — for example, when an individual player considers a deviation — without requiring that ex ante payoffs be well-defined before and after the deviation. For instance, consider a single-stage game and suppose that all players are playing stationary strategies which are mutual best responses within the class of stationary strategies. To verify that such a strategy profile is a Bayesian Nash equilibrium, one would like to verify that no player can benefit by deviating, including deviations to nonstationary strategies. While nonstationary strategies (except nearly stationary strategies) do not yield well-defined ex ante payoffs, the limit suprema of payoffs are well-defined; optimality implies these must be bounded above by the (well-defined) payoff of the original, stationary strategy. In this sense, deviations can be ruled out. Hence, while our approach would already rule out such deviations on the basis of admissibility, an equilibrium concept could be defined using a stronger notion of best response which allows deviations outside the admissible class.

Second, in some cases it is possible to obtain well-defined expected payoffs from profiles of strategies which are not nearly stationary. Our admissibility criterion has imposed the inclusion of stationary strategies in an admissible class; we consider this inclusion to be a natural requirement for stationary games. However, dropping this requirement allows one to construct smaller admissible classes. Consider the following example: two players $i=1,2$ obtain perfect signals $s_i=\theta$ and choose actions $a_i\in [s_i,s_i+1]$. Their payoffs are $u_i(a_1,a_2,\theta )=-(a_1-a_2)^2(a_i-\theta )^2$. Consider the (singleton) class of strategy profiles $\{(a_1^*,a_2^*)\}$ where $a_i^*(s_i):=s_i+mathbb{1}_{s_i\ge 0}$. This class is clearly admissible, but the $a_i^*$ are neither stationary nor nearly stationary. On the other hand, the strategy $a_2^*$ does not yield a well-defined payoff against the stationary strategy ${\tilde{a}}_1$ defined by ${\tilde{a}}_1(s_1)=s_1$; player 2’s ex post payoff would be 0 when $\theta <0$, but it would be $-1$ when $\theta \ge 0$.

Third, there may be a “natural” labeling of the state space, or a natural choice of the origin in some games, such as a status quo. In such games, strategies such as $a_i^*(s_i)$ above may be quite reasonable, and one might wish to find an admissible class which includes these. To that end, one could relax our requirement that all diffusing sequences of priors yield the same expected payoffs in the limit and instead focus on a restricted class of sequences of priors, or a particular sequence of priors such as $U([-k,k])$ for $k=1,2,\dots$. Note that even a particular sequence of priors such as this one still disciplines the strategies. For example, consider perfect signals $s_i=\theta$ and payoffs $u_i(a_i,\theta )=-(a_i-\theta )^2$, and define strategies by $a_i(s_i)=s_i+1$ on intervals of the form $[-2^{2n+1},-2^{2n}]$ or $[2^{2n},2^{2n+1}]$ and $a_i(s_i)=s_i$ elsewhere. The expected payoff converges to $-2/3$ along a subsequence $U([-2^{2n+1},2^{2n+1}])$ but converges to $-1/3$ along a subsequence $U([-2^{2n},2^{2n}])$. On the other hand, the strategy $a_i^*(s_i)$ from above, which is not nearly stationary, yields a well-defined expected payoff of 1/2 in this case. Hence, restriction to a smaller class of sequences of priors appropriate for a given problem can admit new strategy profiles while continuing to exclude others.

7 Conclusion

We have presented a formal method for defining ex ante payoffs in games with diffuse prior. The key features of the diffuse prior can be captured using a limit construction, in which sequences of proper priors exhibit these properties in a limiting sense. Under our construction, stationary strategies admit well-defined payoffs in stationary games, and conversely, all strategies admitting well-defined payoffs are nearly stationary in a precise sense.

Our methodology can be readily extended in several directions. Although we have considered a one-dimensional state of the world $\theta$, in many applications, there is uncertainty over multiple dimensions. One could model a diffuse prior over a multi-dimensional state by generalizing our notion of a diffusing sequence. Another, related direction would be to consider renewed uncertainty; in a multistage game, each stage t might introduce more uncertainty through the realization of a state $\theta _{t}$. We leave detailed exploration of these directions to future work.

Although we have considered exogenous uncertainty over the state $\theta$, our construction could also be used to allow a player to choose the diffuse prior as a mixed strategy. That is, suppose an additional player $i=0$ is introduced who chooses an action $\theta$ at the beginning of the game. Under our construction, this player can play the diffuse prior as a mixed strategy with a well-defined payoff.

Proofs

Proof of Lemma 1

We prove the result for informed players; the arguments for uninformed players are essentially a subset of those here. Note that ${\mathcal {M}}_i$ is a metric space, as it is a subspace of the metric space (with the Prokhorov metric) consisting of measures over the complete and separable metric space $(S_i\times X_i,d_{S_i\times X_i})$ where $d_{S_i\times X_i}((s_i,x_i),(s_i',x_i'))=\frac{1}{2}\max \{|s_i-s_i'|,|x_i-x_i'|\}$.¹² To show that ${\mathcal {M}}_i$ is compact it suffices to show that it is relatively compact and that it is closed. We show that ${\mathcal {M}}_i$ is relatively compact; to verify that ${\mathcal {M}}_i$ is also closed is straightforward and only requires the tedious steps of showing that convergent sequences of strategies converge to limits which satisfy the key properties of Definition 1. Given any $\eta >0$, pick any compact interval $T\subset S_i$ such that $\phi (T)>1-\eta$. Define $Z_i\subset S_i\times X_i$ by $Z_i:=\{(s_i,x_i):x_i\in A_i(s_i)\text { and }s_i\in T\}$. Now $Z_i$ is compact as a consequence of Assumption 1, and it satisfies $\mu _i(Z_i)=\phi (T)>1-\eta$. Hence, ${\mathcal {M}}_i$ is tight and by Prokhorov’s theorem,¹³ it is relatively compact. Together with closedness, we conclude that ${\mathcal {M}}_i$ is compact. $\square$

Before proving Lemma 2, we first state and prove Lemmas 5 and 6 (in the context of the baseline model of Sect. 3); this does not involve any circularity.

Lemma 5

For all $i\in {\mathcal {I}}$ and all strategies $\mu _i\in {\mathcal {M}}_i$, the map $\theta \mapsto \mu _i^{\theta }$ from ${\mathbb {R}}$ to ${\mathcal {M}}_i$ is uniformly continuous.

Proof

The result is trivial for uninformed players, since $\mu _i^{\theta }$ is constant in $\theta$. Hence, we prove the result for informed players. We first specify a few preliminaries. We recall the space $S_i\times X_i$ is metrized by $d_{S_i\times X_i}$ defined in the proof of Lemma 1. For any subset $Y\subseteq S_i \times X_i$ and $\eta >0$, let $Y^\eta :=\bigcup _{z\in Y} N_\eta (z)$, the union of all $\eta$-neighborhoods (under the metric $d_{S_i \times X_i}$) centered at points in Y. It follows from these definitions that for any $\eta \in {\mathbb {R}}$ (possibly negative), $Y+\eta \subseteq Y^{|\eta |}$, where by our notational convention $Y+\eta$ denotes the translation of the set Y by $\eta$ (with respect to the standard metric on ${\mathbb {R}}$) in all dimensions. The space of measures on $S_i \times X_{i}$ is metrized by the Prokhorov metric, $d_{P,i}(\mu _i,{\hat{\mu }}_i):=\inf \{\eta >0: \mu _i(Y)\le {\hat{\mu }}_i(Y^\eta )+\eta \text { and } {\hat{\mu }}_i(Y)\le \mu _i(Y^\eta )+\eta \text { for all } Y\subseteq S_i\times X_i\}$. Define a measure $\chi (\mu _i,\theta ,\theta ')$ on $S_i\times X_i$ by $\chi (\mu _i,\theta ,\theta ')(Y):=\mu _i^{\theta }(Y-(\theta '-\theta ))$, which differs from $\mu _i^{\theta '}$ only in its marginal distribution over $S_i$. Define $Y(s_i):=\{x_i\in X_i:(s_i,x_i)\in Y\}$.

To establish uniform continuity, we first show that for all $\eta >0$, if $|\theta -\theta '|<\eta$, then $d_{P,i}(\mu _i^{\theta },\chi (\mu _i,\theta ,\theta '))<\eta$. For $Y\subseteq S_i\times X_i$, we have

$$\begin{aligned} \mu _i^{\theta }(Y)=\chi (\mu _i,\theta ,\theta ')(Y+\theta '-\theta )\le \chi (\mu _i,\theta ,\theta ')(Y^{|\theta '-\theta |})< \chi (\mu _i,\theta ,\theta ')(Y^\eta )+\eta , \end{aligned}$$

and by a symmetric argument $\chi (\mu _i,\theta ,\theta ')(Y)< \mu _i^{\theta }(Y^\eta )+\eta$. By the definition of $d_{P,i}$, $d_{P,i}(\mu _i^\theta ,\chi (\mu _i,\theta ,\theta '))<\eta$.

Next, we show that for sufficiently small $|\theta '-\theta |$, $d_{P,i}(\chi (\mu _i,\theta ,\theta '),\mu _i^{\theta '})<3\eta$. We have

$$\begin{aligned} |\mu _i^{\theta '}(Y)-\chi (\mu _i,\theta ,\theta ')(Y)|&\le \int _{s_i\in S_i} \mu _i^{\theta '}(Y(s_i)|s_i) |g(s_i)-g(s_i-(\theta '-\theta ))|\mathrm{d}s_i\\&\le \int _{s_i\in S_i} |g(s_i)-g(s_i-(\theta '-\theta ))|\mathrm{d}s_i, \end{aligned}$$

The last expression is less than $3\eta$ for sufficiently small $|\theta '-\theta |$. To see this, choose $[{\underline{s}}_i,{\overline{s}}_i]$ such that $\int _{S_i\setminus [{\underline{s}}_i-1,{\overline{s}}_i+1]} g(s_i)\mathrm{d}s_i<\eta$; hence $\int _{S_i\setminus [{\underline{s}}_i,{\overline{s}}_i]} g(s_i-\beta )\mathrm{d}s_i<\eta$ for all $\beta \in [-1,1]$, so by the triangle inequality $\int _{S_i\setminus [{\underline{s}}_i,{\overline{s}}_i]} |g(s_i)-g(s_i-(\theta '-\theta ))|\mathrm{d}s_i<2\eta$ whenever $|\theta '-\theta |<1$. And by the bounded convergence theorem, the integral over $[{\underline{s}}_i,{\overline{s}}_i]$ is less than $\eta$ for sufficiently small $|\theta '-\theta |$. Summing these yields the bound $3\eta$ for the integral over $S_i$, and by the definition of the Prokhorov metric, $d_{P,i}(\chi (\mu _i,\theta ,\theta '),\mu _i^{\theta '})<3\eta$. By another application of the triangle inequality, for sufficiently small $|\theta '-\theta |$, $d_{P,i}(\mu _i^{\theta },\mu _i^{\theta '})<4\eta$, giving us uniform continuity. $\square$

Lemma 6

For all $i\in {\mathcal {I}}$, $u_i(\mu ,\theta )$ is uniformly continuous in $\mu _j$ for each $j\in {\mathcal {I}}$, and in $\theta$.

Proof

To establish that $u_i(\mu ,\theta )$ is continuous in each $\mu _j$, for all strategy profiles $\mu$ and all $\theta$, define ${\tilde{\mu }}$ is the product measure over $Z:=\left( \times _{i\in {\mathcal {I}}} X_i\right) \times S$ induced by the strategy profile $\mu$, modified so that the marginal distribution over $S_i$ has density $f_i(s_i-\theta )$. We can then write $u(\mu ,\theta )=\int _{(a,s)\in Z} u_i(a,\theta ) \mathrm{d}{\tilde{\mu }}(a,s)$. Let $\epsilon >0$ and $\mu \in {\mathcal {M}}$ be arbitrary. Fix any compact set ${\overline{S}}\subseteq {\mathbb {R}}^{|{\mathcal {I}}^{\textit{inf}}|}$ of the form ${\overline{S}}=\times _{i\in {\mathcal {I}}^{\textit{inf}}}[{\underline{s}}_i,{\overline{s}}_i]$ large enough so that $\int _{{\hat{s}} \notin {\overline{S}}} \sup _{a \in A({\hat{s}}+\theta )} |u(a_i,a_{-i},\theta )| \mathrm{d}F({\hat{s}})<\epsilon$, where we have used Assumption 2. Let ${{\overline{Z}}}:=\left( \times _{i\in {\mathcal {I}}} X_i\right) \times {{\overline{S}}}$, which is compact and has zero-measure boundary under ${\tilde{\mu }}$. Suppose a sequence $(\mu ^n)_{n \in {\mathbb {N}}}$ converges to $\mu$; then ${\tilde{\mu }}^n\rightarrow {\tilde{\mu }}$, and thus for sufficiently large N, for all $n\ge N$,

$$\begin{aligned}&|\int _{(a,s)\in Z}u_i(a,\theta )\mathrm{d}{\tilde{\mu }}^n(a,s)-\int _{(a,s)\in Z}u_i(a,\theta )\mathrm{d}{\tilde{\mu }}(a,s)|\\&\quad \le |\int _{(a,s)\in {\overline{Z}}}u_i(a,\theta )\mathrm{d}{\tilde{\mu }}^n(a,s)-\int _{(a,s)\in {\overline{Z}}}u_i(a,\theta )\mathrm{d}{\tilde{\mu }}(a,s)|\\&\qquad +|\int _{(a,s)\in Z\setminus {\overline{Z}}}u_i(a,\theta )\mathrm{d}{\tilde{\mu }}^n(a,s)|+|\int _{(a,s)\in Z\setminus {{\overline{Z}}}}u_i(a,\theta )\mathrm{d}{\tilde{\mu }}(a,s)|\\&\quad <3\epsilon . \end{aligned}$$

Hence $u_i(\mu ,\theta )$ is continuous in each $\mu _j$, and since the $\mu _j$ lie in a compact domain, this continuity is uniform. Next, since $u_i(\mu ,\theta )=u_i(\mu ^{-\theta },0)$ and by Lemma 5 each $\mu _j^{-\theta }$ term is uniformly continuous in $\theta$, we have uniform continuity in $\theta$. $\square$

Proof of Lemma 2

For the “only if” direction, note that by the definition of a limit, $\lim _{\theta \rightarrow +\infty } \mu _i^{\theta }=\lim _{\theta \rightarrow -\infty } \mu _i^{\theta }=\mu _i^*$ implies that for all $\eta >0$, there exists $M>0$ such that for all $\theta \le -M$ and all $\theta \ge M$, $d_{P,i}(\mu _i^{\theta },\mu _i^*)<\eta$. Hence $\{\theta :d_{P,i}(\mu _i^{\theta },\mu _i^*)\ge \eta \}\subseteq [-M,M]$ and thus $\lambda \left( \{\theta :d_{P,i}(\mu _i^{\theta },\mu _i^*)\ge \eta \}\right) <\infty$. For the “if” direction, we prove the contrapositive. Suppose that some arbitrary strategy $\mu _i^*$ is not a limit strategy for $\mu _i$. Then given any $\eta >0$, it is possible to construct a sequence $(\theta _j)_{j\in {\mathbb {N}}}$ such that (i) for all $j,k\in {\mathbb {N}}$ with $j\ne k$, $|\theta _j-\theta _k|>1$ and (ii) for all $j\in {\mathbb {N}}$, $d_{P,i}(\mu ^{\theta _j}_i,\mu _i^*)\ge 2\eta$. Since the map $\theta \mapsto \mu _i^{\theta }$ is uniformly continuous (Lemma 5), there exists $\delta >0$ such that whenever $|\theta -\theta '|<\delta$, $d_{P,i}(\mu _i^{\theta },\mu _i^{\theta '})<\eta$. It follows that for all $k\in {\mathbb {N}}$ and all $\theta '\in (\theta _k-\delta ,\theta _k+\delta )$, $d_{P,i}(\mu _i^{\theta '},\mu _i^*)\ge d_{P,i}(\mu ^{\theta _k},\mu _i^*)-d_{P,i}(\mu _i^{\theta _k},\mu _i^{\theta '})>2\eta -\eta =\eta$. Hence $\bigcup _{k\in {\mathbb {N}}} (\theta _k-\delta ,\theta _k+\delta )\subseteq \{\theta :d_{P,i}(\mu _i^{\theta },\mu _i^*)\ge \eta \}$, and since $\lambda \left( \bigcup _{k\in {\mathbb {N}}} (\theta _k-\delta ,\theta _k+\delta )\right) =\infty$, we have $\lambda \left( \{\theta :d_{P,i}(\mu _i^{\theta },\mu _i^*)\ge \eta \}\right) =\infty$, concluding the proof of the contrapositive. $\square$

For the proof of the main result, we make use of a weaker property than that of Definition 3.

Lemma 7

For any $i\in {\mathcal {I}}$ and strategy $\mu _i \in {\mathcal {M}}_i$, there exists $\mu _i^*\in {\mathcal {M}}_i$ with the property that for all $\eta >0$,

$\lambda \left( \{ \theta \in {\mathbb {R}}:d_{P,i}(\mu _i^{\theta },\mu _i^*)<\eta \}\right) =\infty$, where $\lambda$ denotes the Lebesgue measure and $d_{P,i}$ denotes the Prokhorov distance defined on ${\mathcal {M}}_i$. We say that any such $\mu _i^*$ is an attraction for $\mu _i$.

Proof

We prove a more general result. Let (C, d) be a compact metric space, $B\subseteq {\mathbb {R}}$ with $\lambda (B)=\infty$, and $\pi :B\rightarrow C$ a Lebesgue measurable function. We show there exists $c\in C$ with the property that for all $\eta >0$, $\lambda \left( \{b\in B:d(\pi (b),c)<\eta \}\right) =\infty$. Suppose on the contrary that for each $c\in C$, there exists $\eta _c>0$ such that $\lambda \left( \{b\in B:d(\pi (b),c)<\eta _c\}\right) <\infty$. The collection $\{N_{\eta _c}(c):c\in C\}$ is an open covering of C, and by compactness, it has a finite subcovering denoted $\{N_{\eta _i}(c_i)\}_{i=1}^n$ for some $n\in {\mathbb {N}}$. It follows that $B\subseteq \cup _{i=1}^n\pi ^{-1}(N_{\eta _i}(c_i))$ and thus $\lambda (B)\le \sum _{i=1}^n \lambda \left( \pi ^{-1}(N_{\eta _i}(c_i))\right) < \infty$, a contradiction. Thus there exists c such that the property holds. To conclude, note that this result specializes to the lemma statement by setting $C={\mathcal {M}}_i$, $d=d_{P,i}$, $B=\varTheta ={\mathbb {R}}$ and $\pi$ to be the map $\theta \mapsto \mu _i^{\theta }$; given the existence of c as above, we set $\mu _i^*=c$. $\square$

Lemma 7 says that every strategy has at least one attraction. Note that the example immediately following Definition 3 has two attractions, $\kappa _0$ and $\kappa _1$, but as shown in that example, it is not nearly stationary.

Proof of Theorem 1

We begin with sufficiency.

Sufficiency: Let ${\mathcal {M}}^*_i$ be the class of nearly stationary strategies for each player i, and consider any profile $(\mu _1,\mu _2,\dots ,\mu _N)$ of such strategies.

By definition, for each $\mu _i\in {\mathcal {M}}^*_i$, there exists a strategy $\mu _i^*$ such that $\mu _i^*$ is a limit strategy for $\mu _i$; by Lemma 3, this $\mu _i^*$ is stationary and by Lemma 2, for all $\eta >0$,

$$\begin{aligned} \lambda \left( \{\theta :d_{P,i}(\mu _i^{\theta },\mu _i^*)\ge \eta \}\right) <\infty . \end{aligned}$$

(7)

Next, for all $\eta >0$, define $\varTheta _{\ge \eta }=\{\theta \in {\mathbb {R}}:\max _{i\in {\mathcal {I}}} d_{P,i}(\mu _i^{\theta },\mu _i^*)\ge \eta \}$, which has finite measure by (7) and the fact that there are finitely many players. Let ${\mathbb {P}}_n$ be any diffusing sequence. For all n, by definition of the recentering function,

$$\begin{aligned} \int _{\theta \in {\mathbb {R}}} u_i(\mu _i,\mu _{-i},\theta )\, \mathrm{d}{\mathbb {P}}_n(\theta )=\int _{\theta \in {\mathbb {R}}}u_i(\mu _i^{-\theta },\mu _{-i}^{-\theta },0)\,\mathrm{d}{\mathbb {P}}_n(\theta ). \end{aligned}$$

(8)

We can write the RHS of (8) as

$$\begin{aligned} \int _{\theta \in \varTheta _{\ge \eta }}u_i(\mu _i^{-\theta },\mu _{-i}^{-\theta },0)\,\mathrm{d}{\mathbb {P}}_n(\theta )+\int _{\theta \in {\mathbb {R}}\setminus \varTheta _{\ge \eta }}u_i(\mu _i^{-\theta },\mu _{-i}^{-\theta },0)\,\mathrm{d}{\mathbb {P}}_n(\theta ). \end{aligned}$$

(9)

In the first term of (9), by Assumption 2, the integrand $u_i(\mu _i^{-\theta },\mu _{-i}^{-\theta },0)$ is bounded in absolute value by some quantity, call it M. For the second term of (9), note that by uniform continuity of ex interim payoffs, for any $\epsilon >0$, there exists $\eta >0$ such that for any profile of strategies $(\tilde{\mu }_1,\tilde{\mu }_2,\dots ,\tilde{\mu }_N)$, if $d_{P,i}(\tilde{\mu }_i,\mu _i^*)<\eta$ for all $i\in {\mathcal {I}}$, then

$$\begin{aligned} |u_i(\tilde{\mu }_i,\tilde{\mu }_{-i},0)-u_i(\mu _i^*,\mu _{-i}^*,0)|<\epsilon . \end{aligned}$$

Define $u_i^*:=u_i(\mu _i^*,\mu _{-i}^*,0)$ and note that by the definition of $\varTheta _{\ge \eta }$, for $\theta \in {\mathbb {R}}\setminus \varTheta _{\ge \eta }$ we have $\max _{i\in {\mathcal {I}}} d_{P,i}(\mu _i^{\theta },\mu _i^*)< \eta$ and thus $|u_i(\mu _i^{-\theta },\mu _{-i}^{-\theta },0)-u_i(\mu _i^*,\mu _{-i}^*,0)|<\epsilon$. Putting these together,

$$\begin{aligned} \left| \int _{\theta \in {\mathbb {R}}} u_i(\mu _i,\mu _{-i},\theta )\, \mathrm{d}{\mathbb {P}}_n(\theta )-u_i^*\right| \le&\int _{\theta \in \varTheta _{\ge \eta }}|u_i(\mu _i^{-\theta },\mu _{-i}^{-\theta },0)-u_i^*|\,\mathrm{d}{\mathbb {P}}_n(\theta )\nonumber \\&+\int _{\theta \in {\mathbb {R}}\setminus \varTheta _{\ge \eta }}|u_i(\mu _i^{-\theta },\mu _{-i}^{-\theta },0)-u_i^*|\,\mathrm{d}{\mathbb {P}}_n(\theta )\nonumber \\ \le&{\mathbb {P}}_n(\varTheta _{\ge \eta })(M+|u_i^*|)+\epsilon \cdot {\mathbb {P}}_n({\mathbb {R}}\setminus \varTheta _{\ge \eta })\nonumber \\ \le&{\mathbb {P}}_n(\varTheta _{\ge \eta })(M+|u_i^*|)+\epsilon . \end{aligned}$$

(10)

Since $\varTheta _{\ge \eta }$ has finite Lebesgue measure, by Lemma 4 there exists K such that for $n\ge K$, ${\mathbb {P}}_n(\varTheta _{\ge \eta })<\frac{\epsilon }{M+|u_i^*|}$, and thus the RHS of (10) is less than $2\epsilon$. Since $\epsilon$ is arbitrary, we have shown that

$\lim _{n\rightarrow \infty }\int _{\theta \in {\mathbb {R}}}u_i(\mu _i,\mu _{-i},\theta )\, \mathrm{d}{\mathbb {P}}_n(\theta )=u_i^*$, so by definition, the class of nearly stationary strategies is admissible.

Necessity: We prove by contradiction that every strategy $\mu _i\in {\mathcal {M}}^0_i$ is nearly stationary. Since strategies for uninformed players are stationary (and nearly stationary), assume i is an informed player. Suppose toward a contradiction that $\mu _i\in {\mathcal {M}}^0_i$ but $\mu _i$ is not nearly stationary. By Lemma 7, there exists an attraction $\mu _i^*\in {\mathcal {M}}_i$ for $\mu _i$.

Next, given the existence of an attraction $\mu _i^*$, we establish uniqueness. Suppose there also exists an attraction ${\hat{\mu }}_i\in {\mathcal {M}}_i$ for $\mu _i$ with ${\hat{\mu }}_i\ne \mu _i^*$. We show that there exists a profile of stationary strategies of the rivals such that if player i plays $\mu _i$, there is not a well-defined expected payoff in the limit. By irreducibility of payoffs (Assumption 3), there exists a state $\theta _0\in {\mathbb {R}}$ and a profile $\mu _{-i}$ of stationary rival strategies such that $u(\mu _i^*,\mu _{-i},\theta _0)\ne u({\hat{\mu }}_i,\mu _{-i},\theta _0)$. We show that this contradicts admissibility.

Note that by Lemma 6, for any $\epsilon >0$, there exists $\eta >0$ such that if $d_{P,i}(\mu ^{\theta }_i,\mu _i^*)<\eta$, then

$$\begin{aligned} \epsilon&>|u_i(\mu ^{\theta }_i,\mu _{-i},\theta _0)-u_i(\mu ^*_i,\mu _{-i},\theta _0)|. \end{aligned}$$

(11)

An analogous statement holds for ${\hat{\mu }}_i$, so let us redefine $\eta$ so that for $\tilde{\mu }_i\in \{\mu ^*_i,{\hat{\mu }}_i\}$, $d_{P,i}(\mu ^{\theta }_i,\tilde{\mu }_i)<\eta$ implies $|u_i(\mu _i^{\theta },\mu _{-i},\theta _0)-u_i(\tilde{\mu }_i,\mu _{-i},\theta _0)|<\epsilon$.

Recall that by admissibility, there is some $u_i^*$ such that $\lim _{n\rightarrow \infty }\int u_i(\mu _i,\mu _{-i},\theta ) \mathrm{d}{\mathbb {P}}_n(\theta )=u_i^*$ for all diffusing sequences $({\mathbb {P}}_n)$. For the contradiction, we construct two sequence of measures $({\mathbb {P}}^1_n)$ and $({\mathbb {P}}^2_n)$ along which the limits

$\lim _{n\rightarrow \infty }\int u_i(\mu _i,\mu _{-i},\theta ) \mathrm{d}{\mathbb {P}}^1_n(\theta )$ and $\lim _{n\rightarrow \infty }\int u_i(\mu _i,\mu _{-i},\theta ) \mathrm{d}{\mathbb {P}}^2_n(\theta )$ differ. Let $v^*=u_i(\mu ^*_i,\mu _{-i},\theta _0)$ and ${\hat{v}}=u_i({\hat{\mu }}_i,\mu _{-i},\theta _0)$, and recall from above that $v^*\ne {\hat{v}}$. Since $\mu _i^*$ is an attraction for $\mu _i$ and $\lambda \left( \{\theta :d_{P,i}(\mu _i^{\theta _0-\theta },\mu _i^*)<\eta \}\right) =\infty$, for each $n\in {\mathbb {N}}$, there exists $C^1_n\subset \{\theta :d_{P,i}(\mu _i^{\theta _0-\theta },\mu _i^*)<\eta \}\setminus [-n,n]$ with $\lambda (C^1_n)=2n^2$. Define $B^1_n:=[-n,n]\cup C^1_n$, and define ${\mathbb {P}}^1_{n}(\theta ):=mathbb{1}_{B^1_n}(\theta )/\lambda (B^1_n)$. By construction, $({\mathbb {P}}^1_n)_{n\in {\mathbb {N}}}$ is a diffusing sequence of measures, and by the assumption that $\mu _i$ is admissible, we must have $\lim _{n\rightarrow \infty }\int u_i(\mu _i,\mu _{-i},\theta ) \mathrm{d}{\mathbb {P}}^1_n=u_i^*$. Pick any $C>|v^*|$ such that C is an upper bound, over all $\theta$, on the magnitude of $u_i(\mu _i,\mu _{-i},\theta )=u_i(\mu _i^{\theta _0-\theta },\mu _{-i},\theta _0)$.¹⁴ We have

$$\begin{aligned} \left| \int _{\theta \in {\mathbb {R}}} u_i(\mu _i,\mu _{-i},\theta ) \mathrm{d}{\mathbb {P}}^1_n(\theta ) - v^*\right| =&\left| \int _{\theta \in {\mathbb {R}}} u_i(\mu _i^{\theta _0-\theta },\mu _{-i},\theta _0) \mathrm{d}{\mathbb {P}}^1_n(\theta ) - v^*\right| \\ \le&\int _{\theta \in {\mathbb {R}}} \left| u_i(\mu _i^{\theta _0-\theta },\mu _{-i},\theta _0) - v^* \right| \mathrm{d}{\mathbb {P}}^1_n(\theta )\\ =&\int _{\theta \in B_n^1} \left| u_i(\mu _i^{\theta _0-\theta },\mu _{-i},\theta _0) - v^* \right| \mathrm{d}{\mathbb {P}}^1_n(\theta )\\ =&\int _{\theta \in C_n^1} \left| u_i(\mu _i^{\theta _0-\theta },\mu _{-i},\theta _0) - v^* \right| \mathrm{d}{\mathbb {P}}^1_n(\theta )\\&+\int _{\theta \in [-n,n]} \left| u_i(\mu _i^{\theta _0-\theta },\mu _{-i},\theta _0) - v^* \right| \mathrm{d}{\mathbb {P}}^1_n(\theta )\\ \le&\frac{\epsilon \cdot 2n^2+2C\cdot 2n}{2n^2+2n}\rightarrow \epsilon , \end{aligned}$$

where the final inequality uses (i) that, by the earlier construction, $\theta \in C_n^1$ implies $d_{P,i}(\mu _i^{\theta _0-\theta },\mu _i^*)<\eta$ which implies $|u_i(\mu _i^{\theta _0-\theta },\mu _{-i},\theta _0)-v^*|<\epsilon$ and (ii) that for all $\theta$, $C>\max \{|v^*|$, $|u_i(\mu _i^{\theta _0-\theta }, \mu _{-i},\theta _0)|\}$. It follows that $u_i^*\in [v^*-\epsilon ,v^*+\epsilon ].$

Likewise, ${\hat{\mu }}_i$ is an attraction, so $\lambda \left( \{\theta :d_{P,i}(\mu _i^{\theta _0-\theta },{\hat{\mu }}_i)<\eta \}\right) =\infty$, and choose $C^2_n\subset \{\theta :d_{P,i}(\mu _i^{\theta _0-\theta },{\hat{\mu }}_i)<\eta \}\setminus [-n,n]$ with $\lambda (C^2_n)=2n^2$, $B^2_n:=[-n,n]\cup C^2_n$, and ${\mathbb {P}}^2_{n}(\theta ):=mathbb{1}_{B^2_n}(\theta )/\lambda (B^2_n)$, such that $|\int u_i(\mu _i,\mu _{-i},\theta ) \mathrm{d}{\mathbb {P}}^2_n-{\hat{v}}|\le \frac{\epsilon \cdot 2n^2+2C\cdot 2n}{2n^2+2n} \rightarrow \epsilon$, and thus $u_i^*\in [{\hat{v}}-\epsilon ,{\hat{v}}+\epsilon ]$. Since $\eta$ is arbitrary, we choose $\epsilon <\frac{|v^*-{\hat{v}}|}{2}$ and obtain a contradiction of the fact that $v^*\ne {\hat{v}}$. Hence we conclude that there is a unique attraction $\mu _i^*$.

We now prove that $\mu _i^*$ is a limit strategy for $\mu _i$ in the sense of Definition 3. We derive a contradiction by showing that otherwise, the uniqueness result above would be violated. Suppose by way of contradiction that $\mu _i^*$ is not a limit strategy of $\mu _i$, and hence by Lemma 2, for some $\eta >0$, $\lambda ( \varTheta _{\ge \eta })=\infty$ where $\varTheta _{\ge \eta }:=\{\theta :d_{P,i}(\mu _i^{\theta },\mu _i^*)\ge \eta \}.$ Let ${\mathcal {Q}}:=\{\tilde{\mu }_i\in {\mathcal {M}}_i: \exists \theta \in \varTheta _{\ge \eta } \text { s.t. } {\tilde{\mu }}_i=\mu _i^\theta \}$. By (the more general result shown in the proof of) Lemma 7, there exists an attraction ${\hat{\mu }}_i\in {\mathcal {Q}}$ for $\mu _i$. By construction, ${\hat{\mu }}_i\ne \mu _i^*$. This contradicts the uniqueness of the attraction $\mu _i^*$ as argued previously, so $\mu _i^*$ must be a limit strategy for $\mu _i$, as desired.

Finally, $\mu _i^*$ is stationary by Lemma 3, and $\mu _i$ is nearly stationary. Payoff equivalence between $\mu _i$ and $\mu _i^*$ follows from a straightforward argument similar to the one given for the sufficiency part of the proof. $\square$

Proof of Theorem 2

For sufficiency, consider any profile of nearly stationary strategies $\mu \in {\mathcal {M}}$. Lemma 3 applies, and hence there exists a unique profile of stationary strategies $(\mu _1^*,\mu _2^*,\dots ,\mu _N^*)$ such that $\mu _i^*$ is a limit strategy of $\mu _i$ for each $i\in {\mathcal {I}}$. Since ${\mathcal {M}}$ is closed, $\mu ^*\in {\mathcal {M}}$. By continuity of each $u_i$ in strategies, as $\theta \rightarrow \pm \infty$, we have $u_i(\mu ,\theta )=u_i(\mu ^{-\theta },0)\rightarrow u_i^*:=u_i(\mu ^*,0)$. Now for any $\epsilon >0$, choose ${\underline{\theta }},{\overline{\theta }}\in {\mathbb {R}}$ with ${\underline{\theta }}<{\overline{\theta }}$ such that $\theta \notin [{\underline{\theta }},{\overline{\theta }}]\implies |u_i(\mu ^{-\theta },0)-u_i^*|<\epsilon$ for all $i\in {\mathcal {I}}$. For any diffusing sequence, ${\mathbb {P}}_n([{\underline{\theta }},{\overline{\theta }}])\rightarrow 0$. Since $u_i(\cdot ,0)$ is bounded in magnitude by some M, the rest of the argument for sufficiency in the proof of Theorem 1 applies.

For necessity, suppose that there exists $i\in {\mathcal {I}}$ and $\mu _i\in {\mathcal {M}}_i^0$ such that $\mu _i$ is not nearly stationary. Note that by Assumption 4, an extension of the arguments used in the proof of Lemma 1 shows that each space of measures satisfying Definition 7 is compact, and since each ${\mathcal {M}}_i$ is a closed subspace, each ${\mathcal {M}}_i$ is also compact. Hence, for each i there exists an attraction $\mu _i^*\in {\mathcal {M}}_i$ in the sense of Lemma 7. If $\mu _i$ is not nearly stationary, then by definition $\mu _i^*$ is not a limit strategy. Hence, there exists $\eta >0$ such that the set $\varTheta _{\ge \eta }:=\{\theta \in {\mathbb {R}}:d_{p,i}(\mu _i^{\theta },\mu _i^*)\ge \eta \}$ satisfies $\lambda (\varTheta _{\ge \eta })=\infty$. Since the set ${\mathcal {M}}_{i,\ge \eta }:=\{{\tilde{\mu }}_i\in {\mathcal {M}}_i: d_{p,i} ({\tilde{\mu }}_i,\mu _i^*)\ge \eta \}$ is a closed subspace of ${\mathcal {M}}_i$, it is compact, and there exists an attraction ${{\hat{\mu }}}_i\in {\mathcal {M}}_{i,\ge \eta }$ for $\mu _i$. Since $\mu _i^*\notin {\mathcal {M}}_{i,\ge \eta }$ by construction, $\mu _i^*\ne {{\hat{\mu }}}_i$. Hence, there exists $\theta _0\in {\mathbb {R}}$ and a profile of stationary strategies $\mu _{-i}\in {\mathcal {M}}_{-i}^0$ for the remaining players such that $v^*:=u_i(\mu _i^*,\mu _{-i},\theta _0)\ne {{\hat{v}}}:=u_i({{\hat{\mu }}}_i,\mu _{-i},\theta _0)$. Now choose any $\epsilon >0$ such that $\epsilon <|v^*-{{\hat{v}}}|/2$. Since payoffs are continuous in strategies, and the space ${\mathcal {M}}_i$ is compact, this continuity is uniform, so we can choose $\eta '>0$ sufficiently small that for $\tilde{\mu }_i\in \{\mu ^*_i,{\hat{\mu }}_i\}$, $d_{P,i}(\mu ^{\theta }_i,\tilde{\mu }_i)<\eta '$ implies $|u_i(\mu _i^{\theta },\mu _{-i},\theta _0)-u_i(\tilde{\mu }_i,\mu _{-i},\theta _0)|<\epsilon$. Then, by the same construction as in the proof of Theorem 1, there exist two diffusing sequences $({\mathbb {P}}_n^1)$ and $({\mathbb {P}}_n^2)$ along which ex ante payoffs for player i from playing $\mu _i$ against $\mu _{-i}$ converge to limits in $(v^*-\epsilon ,v^*+\epsilon )$ and $({{\hat{v}}}-\epsilon ,{{\hat{v}}}+\epsilon )$, respectively. As these are disjoint intervals, the limits are distinct, and hence $\mu _i$ is not part of an admissible class. $\square$

Proof of Proposition 1

Stationarity of signals and payoffs is immediate from the definition of the game, as is compactness (Assumption 4), and finiteness of payoffs follows from the quadratic payoff structure with normally distributed signals. For concreteness, let the marginal G in the senders’ strategies be simply $N(0,\sigma ^2)$, the distribution of signals conditional on $\theta =0$. Suppose the class ${\mathcal {M}}$ satisfies parts (i)–(iv) of Assumption 5; we establish irreducibility. Irreducibility for the principal is straightforward, since for two distinct principal strategies, one can marginally adjust one or both of the sender strategies and change the principal’s payoffs under those two strategies by different amounts. We show irreducibility for sender 1, which by symmetry implies irreducibility for sender 2. Suppose the principal always chooses the lower offer: given offers $a_1$ and $a_2$, she chooses $C(a_1,a_2)=\arg \min \{a_1,a_2\}$. Consider two distinct strategies of expert 1, $\mu _1$ and ${\hat{\mu }}_1$. We consider only $b_1=0$; similar arguments apply for any $b_1$. We show that there exists a constant markup strategy $\kappa _m$ for player 2 such that $u_1(\mu _1,\kappa _m,C,0)\ne u_1({\hat{\mu }}_1,\kappa _{m},C,0)$. By the definition of payoffs and signals in the model, $u_1(\mu _1,\kappa _m,C,0)=-\int _{{\mathbb {R}}^2}(\min \{a_1,s_2+m\})^2 (Q\otimes \varPhi )(d(a_1,s_2))$, where Q is the CDF over player 1’s action induced by the strategy $\mu _1$, defined by $Q(x)=\mu _1(\{a_1:a_1\le x\})$, and where $\varPhi$ is the CDF of $N(0,\sigma ^2)$. Define ${\hat{Q}}(x)$ for ${\hat{\mu }}_1$ likewise. Next, we show that we can differentiate w.r.t. m under the integral. Write ${\mathbf {a}}:=(a_1,s_2), r({\mathbf {a}},m):=(\min \{a_1,s_2+m\})^2$, and $\nu = Q\otimes \varPhi$. Note that for all ${\mathbf {a}}\in {\mathbb {R}}^2$, $r({\mathbf {a}},m)$ is absolutely continuous in m on bounded intervals. We have

$$\begin{aligned} a(m)&:=\int _{{\mathbb {R}}^2} r({\mathbf {a}},m)\nu (\mathrm{d}{\mathbf {a}}) \end{aligned}$$

(12)

$$\begin{aligned}&=\int _{{\mathbb {R}}^2} \left[ r({\mathbf {a}},m_0)+\int _{m_0}^{m}r_m({\mathbf {a}},z)\mathrm{d}z\right] \nu (\mathrm{d}{\mathbf {a}}) \end{aligned}$$

(13)

$$\begin{aligned}&=\int _{{\mathbb {R}}^2} r({\mathbf {a}},m_0)\nu (\mathrm{d}{\mathbf {a}})+\int _{m_0}^{m}\int _{{\mathbb {R}}^2} r_m({\mathbf {a}},z)\nu (\mathrm{d}{\mathbf {a}})\mathrm{d}z \end{aligned}$$

(14)

$$\begin{aligned} \implies a'(m)&:=\int _{{\mathbb {R}}^2} r_m({\mathbf {a}},m)\nu (\mathrm{d}{\mathbf {a}})\qquad \text { a.e. { m},} \end{aligned}$$

(15)

where $m_0<m$ can be chosen arbitrarily. To obtain (13) we have used the fact that absolutely continuous functions are the integral of their derivatives.¹⁵ Fubini’s theorem is used to obtain (14). Differentiability-a.e. of the integral yields (15).¹⁶ By the same arguments, we obtain $a''(m)=\int r_{mm}({\mathbf {a}},m)\nu (\mathrm{d}{\mathbf {a}})=2\int _{\{{\mathbf {a}}:a_1>s_2+m\}} \nu (\mathrm{d}{\mathbf {a}})$ a.e. m. Applying this to $\nu =Q\otimes F_2$ and ${\hat{\nu }}={\hat{Q}}\otimes F_2$, it follows that if $u_1(\mu _1,\kappa _m,C,0)=u_1({\hat{\mu }}_1,\kappa _m,C,0)$ for all $m\in {\mathbb {R}}$, then $\int _{\{{\mathbf {a}}:a_1>s_2+m\}}(Q\otimes F_2)(d(a_1,s_2))=\int _{\{{\mathbf {a}}:a_1>s_2+m\}}({\hat{Q}}\otimes F_2)(d(a_1,s_2))$ a.e. m. Rearranging, we have

$$\begin{aligned} \int _{S_2}(1-Q(s_2+m))f_2(s_2)\mathrm{d}s_2&=\int _{S_2}(1-{\hat{Q}}(s_2+m))f_2(s_2)\mathrm{d}s_2\nonumber \\ \implies 0&=\int _{S_2}(Q(m-s_2)-{\hat{Q}}(m-s_2))f_2(s_2)\mathrm{d}s_2, \end{aligned}$$

(16)

by the evenness of the normal distribution, where $f_2$ is the PDF of $N(0,\sigma ^2)$, the noise distribution for player 2. Letting $K:=Q-{\hat{Q}}$, (16) is a convolution equation:

$$\begin{aligned}{}[K*f_2](m)=0\qquad \text { a.e. } m. \end{aligned}$$

Let ${\mathcal {F}}$ denote the normalized Fourier transform, ${\mathcal {F}}(g)(z):=\int _{-\infty }^{\infty }e^{-2\pi i z x}g(x)\mathrm{d}x$. Both K and $f_2$ are Lebesgue integrable functions, and thus by the convolution theorem,¹⁷${\mathcal {F}}(K*f_2)={\mathcal {F}}(K)\cdot {\mathcal {F}}(f_2)$. But since $K *f_2\equiv 0$, ${\mathcal {F}}(K*f_2)\equiv 0$. Since ${\mathcal {F}}(f_2)(z)=e^{-2(\pi \sigma z)^2}>0$ for all z, we must have ${\mathcal {F}}(K)\equiv 0$. Applying the inverse Fourier transform, we have $K=0$ almost everywhere. This contradicts the assumption $Q\ne {\hat{Q}}$, so it must be that $u_1(\mu _1,\kappa _m,C,0)\ne u_1({\hat{\mu }}_1,\kappa _m,C,0)$ for some m.

$\square$