Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
89,607 characters · 13 sections · 65 citation commands
Equilibrium in Misspecified Markov Decision Processes
}
\thispagestyle{empty}
\thispagestyle{empty}
\setcounter{page}{1}
Early interest on studying the behavior of agents who hold misspecified views of the world (e.g., Arrow-Green, kirman75learning, sobel1984non, kagel1986winner, nyarko1991learning, sargent-book) has recently been renewed by the work of piccione2003modeling, jehiel2005analogy, eyster2005cursed, jehiel2008revisiting, esponda2008behavioral, Esponda and Pouzo (2012, 2016)\nocite{esponda2016berk}\nocite{esponda2012conditional}, eyster2013approach, Spiegler (2013, 2016a, 2016b)\nocite{spiegler2013placebo}\nocite{spiegler2016abayesian}\nocite{spiegler2016bon}, heidhues2016unrealistic, and fudenberg2016active. There are least two reasons for this interest. First, it is natural for agents to be uncertain about their complex environment and to represent this uncertainty with parsimonious parametric models that are likely to be misspecified. Second, endowing agents with misspecified models can explain how certain biases in behavior arise endogenously as a function of the primitives.\footnote{We take the misspecified model as a primitive and assume that agents learn and behave optimally given their model. In contrast, hansen2008robustness study optimal behavior of agents who have a preference for robustness because they are aware of the possibility of model misspecification.}
The previous literature mostly focuses on problems that are intrinsically “static” in the sense that they can be viewed as repetitions of static problems where the only link between periods arises because the agent is learning the parameters of the model. Yet dynamic decision problems, where an agent chooses an action that affects a state variable (other than a belief), are ubiquitous in economics. The main goal of this paper is to provide a tractable framework to study dynamic settings where the agent learns with a possibly misspecified model.
We study a Markov Decision Process where a single agent chooses actions at discrete time intervals. A transition probability function describes how the agent's action and the current state affects next period's state. The current payoff is a function of states and actions. We assume that the agent is uncertain about the true transition probability function and wants to maximize expected discounted payoff. She has a prior belief over a set of possible transition functions, and her model is possibly misspecified, meaning that we do not require the true transition probability function to be in the support of her prior. The agent uses Bayes' rule to update her belief after observing the realized state.
To better illustrate the main question and results, consider a dynamic savings problem with unknown returns, where $s$ is current income, $x$ is the choice of savings, $\pi(s-x)$ is the payoff from current consumption, and next period's income $s'$ is drawn from the distribution $Q(\cdot\mid s,x)$. The agent, however, does not know the return distribution $Q$. She has a parametric model representing the set of possible return distributions $Q_{\theta}$ indexed by a parameter $\theta\in\Theta$. The agent has a prior $\mu$ over $\Theta$, and this belief is updated using Bayes' rule based on current income, the savings decision, and the income realized next period, $\mu'=B(s,x,s',\mu)$, where $B$ denotes the Bayesian operator and $\mu'$ is the posterior belief. The agent is correctly specified if the support of her prior includes the true return distribution $Q$ and is misspecified otherwise. We represent this problem recursively via the following Bellman equation:
The solution to this Bellman equation determines the evolution of states, actions, and beliefs. A large computational literature provides algorithms that agents and researchers can use to approximate the solution to problems such as ((ref)), where a belief is part of the state variable; see powell2007approximate for a textbook treatment.\footnote{Of course, we do not expect less sophisticated agents to apply these numerical methods. But, following the standard view in the literature, the dynamic programming approach is still a useful tool for the researcher to model the behavior of an agent facing intertemporal tradeoffs.} The issue for economists, however, is that these numerical methods do not usually allow us to make general predictions about behavior.
We propose to circumvent this problem by instead characterizing the agent's steady state behavior and beliefs. The main question that we ask is whether we can replace a dynamic programming problem with learning, such as ((ref)), by a problem where beliefs are not being updated, such as
where $\mu^{*}$ is the agent's equilibrium or steady-state belief over $\Theta$ and $\bar{Q}_{\mu^{*}}=\int_{\Theta}Q_{\theta}\mu^{*}(d\theta)$ is the corresponding subjective transition probability function. We refer to this problem as a Markov Decision Process (MDP) with transition probability function $\bar{Q}_{\mu^{*}}$. The main advantage of this approach is that, provided that we can characterize the equilibrium belief $\mu^{*}$, it obviates the need to include beliefs in the state space, thus making the problem much more amenable to analysis. This focus on equilibrium behavior is indeed a distinguishing feature of economics.
We begin by defining a notion of equilibrium to capture the steady state behavior and belief of an agent who does not know the true transition probability function. We call this notion a Berk-Nash equilibrium because, in the special case where the environment is static, it collapses to the single-agent version of Berk-Nash equilibrium, a concept introduced by esponda2016berk to characterize steady state behavior in static environments with misspecified agents. A strategy in an MDP is a mapping from states to actions; recall that beliefs are not included in the state for an MDP. For a given strategy and true transition probability function, the stochastic process for states and actions in an MDP is a Markov chain and has a corresponding stationary distribution that can be interpreted as the steady-state distribution over outcomes. A strategy and corresponding stationary distribution is a Berk-Nash equilibrium if there exists a belief $\mu^{*}$ over the parameter space such that: (i) the strategy is optimal for an MDP with transition probability function $\bar{Q}_{\mu^{*}}$, and (ii) $\mu^{*}$ puts probability one on the set of parameter values that yield transition probability functions that are “closest” to the true transition probability function. The notion of \textquotedblleft closest\textquotedblright is given by a weighted version of the Kullback-Leibler divergence that depends on the equilibrium stationary distribution.
We use the framework to revisit three classic examples. These examples illustrate how our framework makes dynamic environments with uncertainty amenable to analysis and expands the scope of the classical dynamic programming approach. First, we consider the classic problem of a monopolist with unknown demand function. We assume that demand is dynamic, so that a sale in the current period affects the likelihood of a sale the next period. The monopolist, however, has a misspecified model and believes that demand is not dynamic. We show that a monopolist who thinks demand is not dynamic does not necessarily set higher prices.
The second illustrative example is a search model where a worker does not realize that she gets fired with higher probability in times in which it is actually harder to find another job. We show that she becomes pessimistic about the chances of finding a new job and sub-optimally accepts wage offers that are too low.
The final example is a stochastic growth model along the lines of the problem represented by ((ref)). The agent determines how much of her income to invest every period, which determines, together with an unknown productivity process, next period's income. We assume that there are correlated shocks to both the agent's utility and productivity, but the agent believes these shocks to be independent. If the shocks are positively correlated, the misspecified agent invests more of her income when productivity is low. She ends up underestimating productivity and, therefore, underinvesting in equilibrium.
We then turn to providing a foundation for Berk-Nash equilibrium by studying the limiting behavior of a Bayesian agent who takes actions and updates her beliefs about the transition probability function every period. We ask if an equilibrium approach is appropriate in this environment, i.e., “Is it possible to characterize the steady state behavior of a Bayesian agent by reference to a simpler MDP in which the agent has fixed (though possibly incorrect) beliefs about the transition probability function?”
The answer is yes if the agent is sufficiently impatient. But, if the agent is sufficiently patient, some subtle issues arise in the dynamic setting that lead to a more nuanced answer: The answer is yes provided that we restrict attention to steady states with a property we call exhaustive learning. Under exhaustive learning, the agent perceives that she has nothing else to learn in steady state. In the context of the previous example, this condition guarantees that optimal actions in problem ((ref)) are also optimal in problem ((ref)). Without exhaustive learning, an action may be optimal in problem ((ref)) because the agent is not updating her beliefs. But the same action could be suboptimal if she were to update beliefs because, as we show in this paper, the value of experimentation can be negative in dynamic settings. This situation is not possible in static settings because the value function is only a function of beliefs and its convexity and the martingale property of Bayesian beliefs imply that the value of experimentation is always nonnegative.
The notion of exhaustive learning motivates a natural refinement of Berk-Nash equilibrium in dynamic settings. This refinement, however, still allows beliefs to be incorrect due to lack of experimentation, which is a hallmark of the bandit (e.g., rothschild1974two, mclennan1984price, easley1988controlling) and self-confirming equilibrium (e.g., battigalli1987compartamento, fudenberg1993self, dekel2004learning, fershtman2012dynamic) literatures. Following selten1975reexamination, we define a further refinement, perfect Berk-Nash equilibrium, to characterize behavior that is robust to experimentation, and provide conditions for its existence.
Our asymptotic characterization of beliefs and actions contributes to the literature that studies asymptotic beliefs and/or behavior under Bayesian learning. Table 1 categorizes some of the more relevant papers in connection to our work. The table on the left includes papers where the agent learns from data that is exogenous in the sense that she does not affect the stochastic properties of the data. This topic has mostly been tackled by statisticians for both correctly-specified and misspecified models and for both i.i.d. and non-i.i.d. data. The table on the right includes papers where the agent learns from data that is endogenous in the sense that it is driven by the agent's actions, a topic that has been studied by economists mostly in static settings. By static we mean that the problem reduces to a static optimization problem if stripped of the learning dynamics.\footnote{Formally, we say a problem is static if, for a fixed strategy and belief over the transition probability function, outcomes (states and actions) are independent across time.}
Table 1 also differentiates between two complementary approaches to studying asymptotic beliefs and/or behavior. The first approach is to focus on specific settings and provide a complete characterization of asymptotic actions and beliefs, including convergence results; these papers are marked with a superscript $\mbox{\^{}}$ in Table 1. Some papers pursue this approach in dynamic and correctly specified stochastic growth models (e.g., freixas1981optimal, koulovatianos2009optimal). In static misspecified settings, nyarko1991learning, esponda2008behavioral, and heidhues2016unrealistic study passive learning problems where there is no experimentation motive. fudenberg2016active is the only paper that provides a complete characterization in a dynamic decision problem with active learning.\footnote{Under active learning, different actions convey different amount of information and a non-myopic agent takes the exploitation vs. experimentation tradeoff into account. There can be passive or active learning in both static and dynamic settings. },\footnote{The environment in fudenberg2016active is dynamic because the agent controls the drift of a Brownian motion, even though the only relevant state variable for optimality ends up being the agent's belief.} The second approach, which we follow in this paper and we followed earlier for the static case (esponda2016berk) is to study general settings and focus on characterizing the set of steady states.\footnote{In macroeconomics there are several models where agents make forecasts using statistical models that are misspecified (e.g., Evans-book Ch. 13, sargent-book Ch. 6).}
The paper is also related to the literature which provides learning foundations for equilibrium concepts, such as Nash or self-confirming equilibrium (see fudenberg1998theory for a survey). In contrast to this literature, we consider Markov decision problems and allow for misspecified models. Particular types of misspecifications have been studied in extensive form games. jehiel1995limited considers the class of repeated alternating-move games and assumes that players only forecast a limited number of time periods into the future; see jehiel1998learning for a learning foundation. We share the feature that the learning process takes place within the play of the game and that beliefs are those that provide the best fit given the data.\footnote{jehiel2007valuation consider the general class of extensive form games with perfect information and assume that players simplify the game by partitioning the nodes into similarity classes.}
The framework and equilibrium notion are presented in Sections 2 and 3. In Section 4, we work through several examples. We provide a foundation for equilibrium in Section 5 and study equilibrium refinements in Section 6.
We begin by describing the environment faced by the agent.
We sometimes use MDP($Q$) to denote an MDP with transition probability function $Q$ and exclude the remaining primitives.
The timing is as follows. At the beginning of each period $t=0,1,2,...$, the agent observes state $s_{t}\in\mathbb{S}$ and chooses a feasible action $x_{t}\in\Gamma(s_{t})\subset\mathbb{X}$. Then a new state $s_{t+1}$ is drawn according to the probability distribution $Q(\cdot\mid s_{t},x_{t})$ and the agent receives payoff $\pi(s_{t},x_{t},s_{t+1})$ in period $t$. The initial state $s_{0}$ is drawn according to the probability distribution $q_{0}$.
The agent facing an MDP chooses a policy rule that specifies at each point in time a (possibly random) action as a function of the history of states and actions observed up to that point. As usual, the objective of the agent is to choose a feasible policy rule to maximize expected discounted utility, $\sum_{t=0}^{\infty}\delta^{t}\pi(s_{t},x_{t},s_{t+1})$.
By the Principle of Optimality, the agent's problem can be cast recursively as
where $V_{Q}:\mathbb{S}\rightarrow\mathbb{R}$ is the (unique) solution to the Bellman equation ((ref)).
Let $\Sigma$ denote the space of all strategies and let $\sigma(x\mid s)$ denote the probability that the agent chooses $x$ when the state is $s$.\footnote{A standard result is the existence of a deterministic optimal strategy. Nevertheless, allowing for randomization will be important in the case where the transition probability function is uncertain.}
Let $\Sigma(Q)$ be the set of all strategies that are optimal for an MDP($Q$).
A strategy determines the transitions in the space of states and actions and, consequently, the set of stationary distributions over states and actions. For any strategy $\sigma$ and transition probability function $Q$, define a transition kernel $M_{\sigma,Q}:Gr(\Gamma)\rightarrow\Delta\left(Gr(\Gamma)\right)$ by letting
for all $(s,x),(s',x')\in Gr(\Gamma)$. The transition kernel $M_{\sigma,Q}$ is the transition probability function over $Gr(\Gamma)$ given strategy $\sigma$ and transition probability function $Q$.
For any $m\in\Delta(Gr(\Gamma))$, let $M_{\sigma,Q}[m]\in\Delta(Gr(\Gamma))$ denote the probability measure \[ \sum_{(s,x)\in Gr(\Gamma)}M_{\sigma,Q}(\cdot,\cdot\mid s,x)m(s,x). \]
A stationary distribution represents the steady-state distribution over outcomes (i.e, states and actions) when the agent follows a given strategy. Let $I_{Q}(\sigma)\equiv\{m\in\Delta(Gr(\Gamma))\mid m=M_{\sigma,Q}[m]\}$ denote the set of stationary distributions given $(\sigma,Q)$.
Our main objective is to study the behavior of an agent who faces an MDP but is uncertain about the transition probability function. We begin by introducing a new object to model the problem with uncertainty, which we call the Subjective Markov decision process (SMDP). We then define the notion of a Berk-Nash equilibrium of an SMDP.
We interpret the set $\mathcal{Q}_{\Theta}$ as the different transition probability functions (or models of the world) that the agent considers possible. We sometimes use SMDP($Q,\mathcal{Q}_{\Theta}$) to denote an SMDP with true transition probability function $Q$ and a family of transition probability functions $\mathcal{Q}_{\Theta}$.
The first two conditions in Definition (ref) place parametric and continuity assumptions on the subjective models.\footnote{Without the assumption of a finite-dimensional parameter space, Bayesian updating need not converge to the truth for most priors and parameter values even in correctly specified statistical settings (freedman1963asymptotic, diaconis1986consistency). Note that the parametric assumption is only a restriction if the set of states or actions is nonfinite, a case we consider in some of the examples.} The last condition plays two roles. First, it rules out a stark form of misspecification by guaranteeing that there exists at least one parameter value that can rationalize every feasible observation. Second, it implies that the correspondence of parameters that are a closest fit to the true model is upper hemicontinuous. esponda2016berk provide a simple (non-dynamic) example where this assumption does not hold and equilibrium fails to exist.
The goal of this section is to define the notion of Berk-Nash equilibrium of an SMDP. The next definition is used to place constraints on the belief $\mu\in\Delta(\Theta)$ that the agent may hold if $m$ is the stationary distribution over outcomes.
The set $\Theta_{Q}(m)$ contains the parameter values constitute the best fit with the true transition probability function $Q$ when outcomes are drawn from the distribution $m$.
We now define equilibrium.
Condition (i) in the definition of Berk-Nash equilibrium requires $\sigma$ to be an optimal strategy in the MDP where the transition probability function is $\int_{\Theta}Q_{\theta}\mu(d\theta)$. Condition (ii) requires that the agent only puts positive probability on the set of closest parameter values given $m$, $\Theta_{Q}(m)$. Finally, condition (iii) requires $m$ to be a stationary distribution given $(\sigma,Q)$.
The next result establishes existence of equilibrium in any regular SMDP.
The standard approach to proving existence begins by defining a “best response correspondence” in the space of strategies. This approach does not work here because the possible non-uniqueness of beliefs implies that the correspondence may not be convex valued. The trick we employ is to define equilibrium via a correspondence on the space of strategies, stationary distributions, and beliefs, and then use Lemmas (ref), (ref) and (ref) to show that this correspondence satisfies the assumptions of a generalized version of Kakutani's fixed point theorem.\footnote{esponda2016berk rely on perturbations to show existence of equilibrium in a static setting. In contrast, our approach does not require the use of perturbations.}
An SMDP is correctly specified if the set of subjective models contains the true model.
In decision problems, data is endogenous and so, following esponda2016berk, it is natural to consider two notions of identification: weak and strong identification. These definitions distinguish between outcomes on and off the equilibrium path. In a dynamic environment, the right object to describe what happens on and off the equilibrium path is not the strategy but rather the stationary distribution over outcomes $m$.
Weak identification implies that, for any equilibrium distribution $m$, the agent has a unique belief along the equilibrium path, i.e., for states and actions that occur with positive probability. It is a condition that turns out to be important for proving the existence of equilibria that are robust to experimentation (see Section (ref)) and is always satisfied in correctly specified SMDPs.\footnote{The following is an example where weak identification fails. Suppose an unbiased coin is tossed every period, but the agent believes that the coin comes up heads with probability 1/4 or 3/4, but not 1/2. Then both 1/4 and 3/4 minimize the Kullback-Leibler divergence, but they imply different distributions over outcomes. Relatedly, Berk (1966) shows that beliefs do not converge.} Strong identification strengthens the condition by requiring that beliefs are unique also off the equilibrium path.
Proposition 1 says that, in environments where the agent is uncertain about the transition probability function but her subjective model is both correctly specified and strongly identified, then Berk-Nash equilibrium corresponds to the solution of the MDP under correct beliefs about the transition probability function. If one drops the assumption that the SMDP is strongly identified, then the “if” part of the proposition continues to hold but the “only if” condition does not hold. In other words, there may be Berk-Nash equilibria of correctly-specified SMDPs in which the agent has incorrect beliefs off the equilibrium path. This feature of equilibrium is analogous to the main ideas of the bandit and self-confirming equilibrium literatures.
We use three classic examples to illustrate how easy it is to use our framework to expand the scope of the classical dynamic programming approach.
The problem of a monopolist facing an unknown, static demand function was first studied by rothschild1974two and nyarko1991learning in correctly and misspecified settings, respectively. In the following example, the monopolist faces a dynamic demand function but incorrectly believes that demand is static.
MDP: In each period $t$, a monopolist chooses price $x_{t}\in\mathbb{X}=\{L,H\}$, where $0<L<H$. It then sells $s_{t+1}\in\mathbb{S}=\{0,1\}$ units at zero cost and obtains profit $\pi(x_{t},s_{t+1})=x_{t}s_{t+1}$. The probability that $s_{t+1}=1$ is $q_{sx}\equiv Q(1\mid s_{t}=s,x_{t}=x)$, where $0<q_{sx}<1$ for all $(s,x)\in Gr(\Gamma)=\mathbb{S}\times\mathbb{X}$.\footnote{The set of feasible actions is independent of the state, i.e., $\Gamma(s)=\mathbb{X}$ for all $s\in\mathbb{S}$.} The monopolist wants to maximize expected discounted profits, with discount factor $\delta\in[0,1)$.
Demand is dynamic in the sense that a sale yesterday increases the probability of a sale today: $q_{1x}>q_{0x}$ for all $x\in\mathbb{X}$. Moreover, a higher price reduces the probability of a sale: $q_{sL}>q_{sH}$ for all $s\in\mathbb{S}$. Finally, for concreteness, we assume that
Expression ((ref)) implies that current-period profits are maximized by choosing price $L$ if there was no sale last period and price $H$ otherwise (i.e., $Lq_{0L}>Hq_{0H}$ and $Hq_{1H}>Lq_{1L}$). Thus, the optimal strategy of a myopic monopolist (i.e., $\delta=0$) who knows the primitives is $\sigma(H\mid0)=0$ and $\sigma(H\mid1)=1$. If, however, the monopolist is sufficiently patient, it is optimal to always choose price $L$.\footnote{Formally, there exists $C_{\delta}\in[q_{1L}/q_{1H},q_{0L}/q_{0H}]$, where $C_{0}=q_{1L}/q_{1H}$ and $\delta\mapsto C_{\delta}$ is increasing, such that, if $H/L<C_{\delta}$, the optimal strategy is $\sigma(H\mid0)=\sigma(H\mid1)=0$.}
SMDP. The monopolist does not know $Q$ and believes, incorrectly, that demand is not dynamic. Formally, $\mathcal{Q}_{\Theta}=\{Q_{\theta}:\theta\in\Theta\}$, where $\Theta=[0,1]^{2}$ and, for all $\theta=(\theta_{L},\theta_{H})\in\Theta$, $Q_{\theta}(1\mid s,L)=\theta_{L}$ and $Q_{\theta}(1\mid s,H)=\theta_{H}$ for all $s\in\mathbb{S}$. In particular, $\theta_{x}$ is the probability that a sale occurs given price $x\in\{L,H\}$, and the agent believes that it does not depend on $s$. Note that this SMDP is regular. For simplicity, we restrict attention to equilibria in which the monopolist does not condition on last period's state, and denote a strategy by $\sigma_{H}$, the probability that price $H$ is chosen.
Equilibrium.
Optimality. Because the monopolist believes that demand is static, the optimal strategy is to choose the price that maximizes current period's profit. Let \[ \Delta(\theta)\equiv H\theta_{H}-L\theta_{L} \] denote the perceived expected payoff difference of choosing $H$ vs. $L$ under the belief that the parameter value is $\theta=(\theta_{L},\theta_{H})$ with probability 1. If $\Delta(\theta)>0$, $\sigma_{H}=1$ is the unique optimal strategy; if $\Delta(\theta)<0$, $\sigma_{H}=0$ is the unique optimal strategy; and if $\Delta(\theta)=0$, any $\sigma_{H}\in[0,1]$ is optimal.
Beliefs. For any $m\in\Delta(\mathbb{S}\times\mathbb{X})$, the wKLD simplifies to \[ K_{Q}(m,\theta)=\sum_{x\in\{L,H\}}m_{\mathbb{X}}(x)\left\{ \bar{s}_{x}(m)\ln\theta_{x}+(1-\bar{s}_{x}(m))\ln(1-\theta_{x})\right\} +Const, \] where $\bar{s}_{x}(m)=m_{\mathbb{S}\mid\mathbb{X}}(0\mid x)q_{0x}+m_{\mathbb{S}\mid\mathbb{X}}(0\mid x)q_{1x}$ is the probability of a sale given $x$.
If $\sigma_{L}>0$ and $\sigma_{H}>0$, $\theta_{Q}(m)\equiv(\bar{s}_{L}(m),\bar{s}_{H}(m))$ is the unique parameter value that minimizes the wKLD function. If, however, one of the prices is chosen with zero probability, there are no restrictions on beliefs for the corresponding parameter, i.e., the set of minimizers is $\Theta_{Q}(m)=\{(\theta_{L},\theta_{H})\in\Theta:\theta_{H}=\bar{s}_{H}(m)\}$ if $\sigma_{L}=0$ and $\Theta_{Q}(m)=\{(\theta_{L},\theta_{H})\in\Theta:\theta_{L}=\bar{s}_{L}(m)\}$ if $\sigma_{H}=0$.
Stationary distribution. Fix a strategy $\sigma_{H}$ and denote a corresponding stationary distribution by $m(\cdot;\sigma_{H})\in\Delta(\mathbb{S}\times\mathbb{X})$. Since the strategy does not depend on the state, $m_{\mathbb{S}\mid\mathbb{X}}(\cdot\mid x;\sigma_{H})$ does not depend on $x$ and, therefore, coincides with the marginal stationary distribution over $\mathbb{S}$, denoted by $m_{\mathbb{S}}(\cdot;\sigma_{H})\in\Delta(\mathbb{S})$. This distribution is unique and given by the solution to \[ m_{\mathbb{S}}(1;\sigma_{H})=(1-m_{\mathbb{S}}(1;\sigma_{H}))((1-\sigma_{H})q_{0L}+\sigma_{H}q_{0H})+m_{\mathbb{S}}(1;\sigma_{H})((1-\sigma_{H})q_{1L}+\sigma_{H}q_{1H}). \]
Equilibrium. We restrict attention to equilibria that are robust to experimentation (i.e., perfect equilibria; see Section (ref)) by focusing on the belief $\theta(\sigma_{H})=(\theta_{L}(\sigma_{H}),\theta_{H}(\sigma_{H}))\equiv\theta_{Q}(m(\cdot;\sigma_{H}))$ for a given strategy $\sigma_{H}\in[0,1]$.\footnote{Both $\sigma_{H}=0$ and $\sigma_{H}=1$ are Berk-Nash equilibria supported by beliefs $\theta_{H}(0)=0$ and $\theta_{L}(1)=0$, respectively. These outcomes, however, are not robust to experimentation, and are eliminated by requiring $\theta_{H}(0)=\lim_{\sigma_{H}\rightarrow0}\bar{s}_{H}(m(\cdot;\sigma_{H}))=\bar{s}_{H}(m(\cdot;0))$, and similarly for $\theta_{L}(1)$.} Next, let $\Delta(\theta(\sigma_{H}))$ be the perceived expected payoff difference for a given strategy $\sigma_{H}$. Note that $\sigma_{H}\mapsto\Delta(\theta(\sigma_{H}))$ is decreasing\footnote{The reason is that $\frac{d}{d\sigma_{H}}\Delta(\theta(\sigma_{H}))=\frac{d}{d\sigma_{H}}m_{\mathbb{S}}(1;\sigma_{H})\left(H(q_{1H}-q_{0H})+L(q_{1L}-q_{0L})\right)>0$, since $\frac{d}{d\sigma_{H}}m_{\mathbb{S}}(1;\sigma_{H})<0$ and $q_{1x}>q_{0x}$ for all $x\in\{L,H\}$.}, which means that a higher probability of choosing price $H$ leads to more pessimistic beliefs about the benefit of choosing $H$ vs. $L$. Therefore, there exists a unique (perfect) equilibrium strategy. Figure (ref) depicts an example where the equilibrium is in mixed strategies.\footnote{See esponda2016berk for the importance of mixed strategies in misspecified settings.} Since $\Delta(\theta(0))>0$, an agent who always chooses a low price must believe in equilibrium that setting a high price would instead be optimal. Similarly, $\Delta(\theta(1))<0$ implies that an agent who always chooses a high price must believe in equilibrium that settings a low price would instead be optimal. Therefore, in equilibrium, the agent chooses a strictly mixed strategy $\sigma_{H}^{*}\in(0,1)$ such that $\Delta(\theta(\sigma_{H}^{*}))=0$.\footnote{More generally, the unique equilibrium is $\sigma_{H}=0$ if $\Delta(\theta(0))<0$ (i.e., $\frac{H}{L}\leq D_{1}\equiv\frac{q_{0L}}{(1-q_{1L})q_{0H}+q_{1H}q_{0L}}$), $\sigma_{H}=1$ if $\Delta(\theta(1))>0$ (i.e., $\frac{H}{L}\geq D_{2}\equiv(1-q_{1H})\frac{q_{0L}}{q_{0H}}+q_{1L}$), and $\sigma_{H}^{*}\in(0,1)$ the solution to $\Delta(\theta(\sigma_{H}^{*}))=0$ if $D_{1}<\frac{H}{L}<D_{2}$, where $\frac{q_{1L}}{q_{1H}}<D_{1}<D_{2}<\frac{q_{0L}}{q_{1H}}$.}
The misspecified monopolist may end up choosing higher prices than optimal, since she fails to realize that high prices today cost her in the future. But, a bit more surprisingly, she also may end up choosing lower prices for some primitives.\footnote{This happens if $C_{\delta}<H/L<D_{1}$; see footnotes (ref) and (ref). } The reason is that her failure to realize that $H$ does relatively better in state $s=1$ makes $H$ unattractive to her.
Search-theoretic models have been central to understanding labor markets since mccall1970economics. Most of the literature assumes that the worker knows all the primitives. Exceptions include rothschild1974searching and burdett1988declining, wherein the worker does not know the wage distribution but has a correctly-specified model. In contrast, we study a worker or entrepreneur who knows the distribution of wages or returns for new projects but does not know the probability that she would be able to find a new job or fund a new project. The worker or entrepreneur, however, does not realize that she is fired or her project fails with higher probability in times in which it is actually harder to find a new job or fund a new project. We show that the worker or entrepreneur becomes pessimistic about the chances of finding new prospects and sub-optimally accepts prospects with low returns in equilibrium.
MDP. At the beginning of each period $t$, a worker (or entrepreneur) faces a wage offer (or a project with returns) $w_{t}\in\mathbb{S}=[0,1]$ and decides whether to reject or accept it, $x_{t}\in\mathbb{X}=\{0,1\}$.\footnote{The set of feasible actions is independent of the state, i.e., $\Gamma(w)=\mathbb{X}$ for all $w\in\mathbb{S}$.} Her payoff in period $t$ is $\pi(w_{t},x_{t})=w_{t}x_{t}$; i.e, she earns $w_{t}$ if she accepts and zero otherwise. After making her decision, an economic fundamental $z_{t}\in\mathbb{Z}$ is drawn from an i.i.d. distribution $G$.\footnote{To simplify the notation, we assume the fundamental is unobserved, although the results are identical if it is observed, since it is i.i.d. and it is realized after the worker makes her decision.} If the worker is employed, she is fired (or the project fails) with probability $\gamma(z_{t})$. If the worker is unemployed (either because she was employed and then fired or because she did not accept employment at the beginning of the period), then with probability $\lambda(z_{t})$ she draws a new wage $w_{t+1}\in[0,1]$ according to some absolutely continuous distribution $F$ with density $f$; wages are independent and identically distributed across time. With probability $1-\lambda(z_{t})$, the unemployed worker receives no wage offer, and we denote the corresponding state by $w_{t+1}=0$ without loss of generality. The worker will have to decide whether to accept or reject $w_{t+1}$ at the beginning of next period. If the worker accepted employment at wage $w_{t}$ at the beginning of time $t$ and was not fired, then she starts next period with wage offer $w_{t+1}=w_{t}$ and will again have to decide whether to quit or remain in her job at that offer.\footnote{Formally, $Q(w'\mid w,x)$ is as follows: If $x=1$, then $w'=w$ with probability $1-\overline{\gamma}$, $w'$ is a draw from $F$ with probability $E[\gamma(Z)\lambda(Z)]$, and $w'=0$ with probability $E[\gamma(Z)(1-\lambda(Z))]$; If $x=0$, then $w'$ is a draw from $F$ with probability $\overline{\lambda}$ and $w'=0$ with probability $1-\overline{\lambda}$.} The agent wants to maximize discounted expected utility with discount factor $\delta\in[0,1)$. Suppose that $\overline{\gamma}\equiv E[\gamma(Z)]>0$ and $\overline{\lambda}\equiv E[\lambda(Z)]>0$.
We assume that $Cov(\gamma(Z),\lambda(Z))<0$; for example, the worker is more likely to get fired and less likely to receive an offer when economic fundamentals are strong, and the opposite holds when fundamentals are weak.
SMDP. The worker knows all the primitives except $\lambda(\cdot)$, which determines the probability of receiving an offer. The worker has a misspecified model of the world and believes $\lambda(\cdot)$ does not depend on the economic fundamental, i.e., $\lambda(z)=\theta$ for all $z\in\mathbb{Z}$ , where $\theta\in[0,1]$ is the unknown parameter.\footnote{The results are identical if the agent is also uncertain of $\gamma(\cdot)$; given the current misspecification, the agent only cares about the expectation of $\gamma$ and will have correct beliefs about it.} The transition probability function $Q_{\theta}(w'\mid w,x)$ is as follows: If $x=1$, then $w'=w$ with probability $1-\theta$, $w'$ is a draw from $F$ with probability $\theta\overline{\gamma}$, and $w'=0$ with probability $(1-\theta)\overline{\gamma}$; If $x=0$, then $w'$ is a draw from $F$ with probability $\theta$ and $w'=0$ with probability $1-\theta$.
Equilibrium. Optimality. Suppose that the worker believes that the true parameter is $\theta$ with probability 1. The value of receiving wage offer $w\in\mathbb{S}$ is
By standard arguments, her optimal strategy is a stationary reservation wage strategy $w(\theta)$ that solves the following equation:
The worker accepts wages above the reservation wage and rejects wages below it. Also, $\theta\mapsto w(\theta)$ is increasing: The higher is the probability of receiving a wage offer, then the more she is willing to wait for a better offer in the future. Figure (ref) depicts an example.
Beliefs. For any $m\in\Delta(\mathbb{S}\times\mathbb{X})$, the wKLD simplifies to
where the density of $W'$ cancels out because the workers knows it and where $m_{\mathbb{X}}$ is the marginal distribution over $\mathbb{X}$. In the Online Appendix, we show that the unique parameter that minimizes $K_{Q}(m,\cdot)$ is
To see the intuition behind equation ((ref)), note that the agent only observes the realization of $\lambda$, i.e., whether she receives a wage offer, when she is unemployed. Unemployment can be voluntary or involuntary. In the first case, the agent rejects the offer and, since this decision happens before the fundamental is realized, it is independent of getting or not an offer. Thus, with conditional on unemployment being voluntary, the agent will observe an unbiased average probability of getting an offer, $\overline{\lambda}$ (see the first term in the RHS of ((ref))). In the second case, the agent accepts the offer but is then fired. Since $Cov(\gamma,\lambda)<0$, she is less likely to get an offer in periods in which she is fired and, because she does not account for this correlation, she will have a more pessimistic view about the probability of receiving a wage offer relative to the average probability $\overline{\lambda}$ (the second term in the RHS of ((ref)) captures this bias).
Stationary distribution. Fix a reservation wage strategy $w$ and denote the marginal over $\mathbb{X}$ of the corresponding stationary distribution by $m_{\mathbb{X}}(\cdot;w)\in\Delta(\mathbb{X})$. In the Online Appendix, we characterize $m_{\mathbb{X}}(\cdot;w)$ and show that $w\mapsto m_{\mathbb{X}}(0;w)$ is increasing. Intuitively, the more selective the worker, the higher the chance of being unemployed.
Equilibrium. Let $\theta(\omega)\equiv\theta_{Q}(m(\cdot;w))$ denote the equilibrium belief for an agent following reservation wage strategy $w$. The weight on $\overline{\lambda}$ in equation ((ref)) represents the probability of voluntary unemployment conditional on unemployment. This weight is increasing in $\omega$ because $w\mapsto m_{\mathbb{X}}(0;w)$ is increasing. Therefore, $w\mapsto\theta(w)$ is increasing. In the extreme case in which $w=1$, the worker rejects all offers, unemployment is always voluntary, and the bias disappears, $\theta(1)=\overline{\lambda}$. An example of the schedule $\theta(\cdot)$ is depicted in Figure (ref). The set of Berk-Nash equilibria is given by the intersection of $w(\cdot)$ and $\theta(\cdot)$. In the example depicted in Figure (ref), there is a unique equilibrium strategy $w^{M}=w(\theta^{M})$, where $\theta^{M}<\overline{\lambda}$.
We conclude by comparing Berk-Nash equilibria to the optimal strategy of a worker who knows the primitives, $w^{*}$. By standard arguments, $w^{*}$ is the unique solution to
The only difference between equations ((ref)) and ((ref)) appears in the term multiplying the RHS, which captures the cost of accepting a wage offer. In the misspecified case, this term is $\delta\theta(1-\overline{\gamma})$; in the correct case, it is $\delta(\overline{\lambda}-E[\gamma\lambda])=\delta\overline{\lambda}(1-\overline{\gamma})-\delta Cov(\gamma,\lambda)$. The misspecification affects the optimal threshold in two ways. First, the misspecified agent estimates the mean of $\lambda$ incorrectly, i.e., $\theta<\overline{\lambda}$; therefore, she (incorrectly) believes that, in expectation, offers arrive with lower probability. Second, she does not realize that, because $Cov(\gamma,\lambda)<0$, she is less likely to receive an offer when fired. Both effects go in the same direction and make the option to reject and wait for the possibility of drawing a new wage offer next period less attractive for the misspecified worker. Formally, $\theta\delta(1-\overline{\gamma})<\delta\overline{\lambda}(1-\overline{\gamma})-\delta Cov(\gamma,\lambda)$ and so $w^{M}<w^{*}$.
Stochastic growth models have been central to studying optimal intertemporal allocation of capital and consumption since the work of brock1972optimal. freixas1981optimal and koulovatianos2009optimal assume that agents learn the distribution over productivity shocks with correctly specified models. We follow hall1997macroeconomic and subsequent literature in incorporating shocks to both preferences and productivity, but assume that these shocks are (positively) correlated. We show that agents who fail to account for the correlation of shocks underinvest in equilibrium.
MDP. In each period $t$, an agent observes $s_{t}=(y_{t},z_{t})\in\mathbb{S}=\mathbb{R}_{+}\times\{L,H\}$, where $y_{t}$ is income from the previous period and $z_{t}$ is a current utility shock, and chooses how much income to save, $x_{t}\in\Gamma(y_{t},z_{t})=[0,y_{t}]\subseteq\mathbb{X}=\mathbb{R}_{+}$, consuming the rest. Current period utility is $\pi(y_{t},z_{t},x_{t})=z_{t}\ln(y_{t}-x_{t})$. Income next period, $y_{t+1}$, is given by
where $\varepsilon_{t}=\gamma^{*}z_{t}+\xi_{t}$ is an unobserved productivity shock, $\xi_{t}\sim N(0,1)$, and $0<\delta\beta^{*}<1$, where $\delta\in[0,1)$ is the discount factor. We assume that $\gamma^{*}>0$, so that the utility and productivity shocks are positively correlated. Let $0<L<H$ and let $q\in(0,1)$ be the probability that the shock is $H$.\footnote{Formally, $Q(y',z'\mid y,z,x)$ is such that $y'$ and $z'$ are independent, $y'$ has a log-normal distribution with mean $\alpha^{*}+\beta^{*}\ln x+\gamma^{*}z$ and unit variance, and $z'=H$ with probability $q$.}
SMDP. The agent believes that
where $\varepsilon_{t}\sim N(0,1)$ and is independent of the utility shock. For simplicity, we assume that the agent knows the distribution of the utility shock, and is uncertain about $\theta=(\alpha,\beta)\in\Theta=\mathbb{R}^{2}$. The subjective transition probability function $Q_{\theta}(y',z'\mid y,z,x)$ is such that $y'$ and $z'$ are independent, $y'$ has a log-normal distribution with mean $\alpha+\beta\ln x$ and unit variance, and and $z'=H$ with probability $q$. The agent has a misspecified model because she believes that the productivity and utility shocks are independent when in fact $\gamma^{*}\neq0$.
Equilibrium. Optimality. The Bellman equation for the agent is \[ V(y,z)=\max_{0\leq x\leq y}z\ln(y-x)+\delta E\left[V(Y',Z')\mid x\right] \] and it is straightforward to verify that the optimal strategy is to invest a fraction of income that depends on the utility shock and the unknown parameter $\beta$, i.e., $x=A_{z}(\beta)\cdot y$, where $A_{L}(\beta)=\frac{\delta\beta((1-q)L+qH)}{(1-\delta\beta(1-q))H+\delta\beta(1-q)}$ and $A_{H}(\beta)=\frac{\delta\beta((1-q)L+qH)}{\delta\beta qH+(1-\delta\beta q)L}<A_{L}(\beta)$. For the agent who knows the primitives, the optimal strategy is to invest fractions $A_{L}(\beta^{*})$ and $A_{H}(\beta^{*})$ in the low and high state, respectively. Since $\beta\mapsto A_{z}(\beta)$ is increasing, the equilibrium strategy of a misspecified agent can be compared to the optimal strategy by comparing the equilibrium belief about $\beta$ with the true $\beta^{*}$.
Beliefs and stationary distribution. Let $A=(A_{L},A_{H})$, with $A_{H}<A_{L}$, represent a strategy, where $A_{z}$ is the proportion of income invested given utility shock $z$. Because the agent believes that $\varepsilon_{t}$ is independent of the utility shock and normally distributed, minimizing the wKLD function is equivalent to performing an OLS regression of equation ((ref)). Thus, for a strategy represented by $A=(A_{L},A_{H})$, the parameter value $\hat{\beta}(A)$ that minimizes wKLD is
where $Cov$ and $Var$ are taken with respect to the (true) stationary distribution of $(Y,Z)$. Since $A_{H}<A_{L}$, then $Cov(Z,\ln A_{Z})<0$. Therefore, the assumption that $\gamma^{*}>0$ implies that the bias $\hat{\beta}(A)-\beta^{*}$ is negative and its magnitude depends on the strategy $A$. Intuitively, the agent invests a larger fraction of income when $z$ is low, which happens to be during times when $\varepsilon$ is also low.
Equilibrium. We establish that there exists at least one equilibrium with positive investment by showing that there is at least one fixed point of the function $\hat{\beta}(A_{L}(\beta),A_{H}(\beta))$.\footnote{Our existence theorem is not directly applicable because we have assumed, for convenience, nonfinite state and action spaces.} The function is continuous in $\beta$ and satisfies $\hat{\beta}(A_{L}(0),A_{H}(0))=\hat{\beta}(A_{L}(1/\delta),A_{H}(1/\delta))=\beta^{*}$ and $\hat{\beta}(A_{L}(\beta),A_{H}(\beta))<\beta^{*}$ for all $\beta\in(0,1/\delta)$. Then, since $\delta\beta^{*}<1$, there is at least one fixed point $\beta^{M}$, and any fixed point satisfies $\beta^{M}\in(0,\beta^{*})$. Thus, the misspecified agent underinvests in equilibrium compared to the optimal strategy.\footnote{It is also an equilibrium not to invest, $A=(0,0)$, supported by the belief $\beta^{*}=0$, which cannot be disconfirmed since investment does not take place. But this equilibrium is not robust to experimentation (i.e., it is not perfect; see Section (ref)).} The conclusion is reversed if $\gamma^{*}<0$, illustrating how the framework provides predictions about beliefs and behavior that depend on the primitives (as opposed to simply postulating that the agent is over or under-confident about productivity).
In this section, we provide a learning foundation for the notion of Berk-Nash equilibrium of SMDPs. We fix an SMDP and assume that the agent is Bayesian and starts with a prior $\mu_{0}\in\Delta(\Theta)$ over her set of models of the world. She observes past actions and states and uses this information to update her beliefs about $\Theta$ in every period.
By the Principle of Optimality, the agent's problem in a Bayesian-SMDP can be cast recursively as
where $\bar{Q}_{\mu}=\int_{\Theta}Q_{\theta}\mu(d\theta)$, $\mu'=B(s,x,s',\mu)$ is next period's belief, updated using Bayes' rule, and $W:\mathbb{S}\times\Delta(\Theta)\rightarrow\mathbb{R}$ is the (unique) solution to the Bellman equation ((ref)). Compared to the case where the agent knows the transition probability function, the agent's belief about $\Theta$ is now part of the state space.
For each $\mu\in\Delta(\Theta)$, let $\bar{\Sigma}(\mu)\subseteq\Sigma$ denote the set of all strategies that are induced by a policy that is optimal, i.e., \[ \bar{\Sigma}(\mu)=\bigl\{\sigma\in\Sigma:\exists\,\,\mbox{optimal }f\,\,\mbox{such that \ensuremath{\sigma(\cdot\mid s)=f(\cdot\mid s,\mu)}for all \ensuremath{s\in\mathbb{S}}}\bigr\}. \]
Let $h^{\infty}=(s_{0},x_{0},...,s_{t},x_{t},...)$ represent the infinite history or outcome path of the dynamic optimization problem and let $\mathbb{H}^{\infty}\equiv(Gr(\Gamma))^{\infty}$ represent the space of infinite histories. For every $t$, let $\mu_{t}:\mathbb{H}^{\infty}\rightarrow\Delta(\Theta)$ denote the agent's Bayesian beliefs, defined recursively by $\mu_{t}=B(s_{t-1},x_{t-1},s_{t},\mu_{t-1})$ whenever $\mu_{t-1}\in D_{s_{t-1},x_{t-1},s_{t}}$ (see Definition (ref)), and arbitrary otherwise. We assume that the agent follows some policy function $f$.
In each period $t$, there is a state $s_{t}$ and a belief $\mu_{t}$, and the agent chooses a (possibly mixed) action $f(\cdot\mid s_{t},\mu_{t})\in\Delta(\mathbb{X})$. After an action $x_{t}$ is realized, the state $s_{t+1}$ is drawn from the true transition probability. The agent observes the realized action and the new state and updates her beliefs to $\mu_{t+1}$ using Bayes' rule. The primitives of the Bayesian-SMDP (including the initial distribution over states, $q_{0}$, and the prior, $\mu_{0}\in\Delta(\Theta)$) and a policy function $f$ induce a probability distribution over $\mathbb{H}^{\infty}$ that is defined in a standard way; let $\boldsymbol{P}^{f}$ denote this probability distribution over $\mathbb{H}^{\infty}$.
We now define strategies and outcomes as random variables. For a fixed policy function $f$ and for every $t$, let $\sigma_{t}:\mathbb{H}^{\infty}\rightarrow\Sigma$ denote the strategy of the agent, defined by setting \[ \sigma_{t}(h^{\infty})=f(\cdot\mid\cdot,\mu_{t}(h^{\infty}))\in\Sigma. \] Finally, for every $t$, let $m_{t}:\mathbb{H}^{\infty}\rightarrow\Delta(Gr(\Gamma))$ be such that, for all $t$, $h^{\infty}$, and $(s,x)\in Gr(\Gamma)$, \[ m_{t}(s,x\mid h^{\infty})=\frac{1}{t}\sum_{\tau=0}^{t}\mathbf{1}_{(s,x)}(s_{\tau},x_{\tau}) \] is the frequency of times that the outcome $(s,x)$ occurs up to time $t$.
One reasonable criteria to claim that the agent has reached a steady-state is that her strategy and the time average of outcomes converge.
Condition ((ref)) requires that strategies and the time frequency of outcomes stabilize.
By compactness, there exists a subsequence of beliefs that converges. The additional requirement of exhaustive learning says that the limit point of one of the subsequences, $\mu^{*}$, is perceived to be a fixed point of the Bayesian operator, implying that no matter what state and strategy the agent contemplates, she does not expect her belief to change. Thus, the agent believes that all learning possibilities are exhausted under $\mu^{*}$. The condition, however, does not imply that the agent has correct beliefs in steady state.
The next result establishes that, if the time average of outcomes stabilize to $m$, then beliefs become increasingly concentrated on $\Theta_{Q}(m)$.
The proof of Lemma (ref) clarifies the origin of the wKLD function in the definition of Berk-Nash equilibrium. The proof adapts the proof of Lemma 2 by esponda2016berk to dynamic environments. Lemma (ref) extends results from the statistics of misspecified learning (berk1966limiting, bunke1998asymptotic, shalizi2009dynamics) by considering a setting where agents learn from data that is endogenously generated by their own actions in a Markovian setting.
The following result provides a learning foundation for the notion of Berk-Nash equilibrium of an SMDP.
Theorem (ref) provides a learning justification for Berk-Nash equilibrium. The main idea behind the proof is as follows. We can always find a subsequence of posteriors that converges to some $\mu^{*}$ and, by Lemma (ref) and the fact that behavior converges to $\sigma$, it follows that $\sigma$ must solve the dynamic optimization problem for beliefs converging to $\mu^{*}\in\Theta_{Q}(m)$. In addition, by convergence of $\sigma_{t}$ to $\sigma$ and continuity of the transition kernel $\sigma\mapsto M_{\sigma,Q}$, an application of the martingale convergence theorem implies that $m_{t}$ is asymptotically equal to $M_{\sigma,Q}[m_{t}]$. This fact, linearity of the operator $M_{\sigma,Q}[\cdot]$, and convergence of $m_{t}$ to $m$ then imply that $m$ is an invariant distribution given $\sigma$.
The proof concludes by showing that $\sigma$ not only solves the optimization problem for beliefs converging to $\mu^{*}$ but also solves the MDP, where the belief is forever fixed at $\mu^{*}$. This is true, of course, if the agent is sufficiently impatient, which explains why part (i) of Theorem (ref) holds. For sufficiently patient agents, the result relies on the assumption that the steady state satisfies exhaustive learning. We now illustrate and discuss the role of this assumption.
$\textsc{example}$. At the initial period, a risk-neutral agent has four investment choices: A, B, S, and O. Action A pays $1-\theta^{*}$, action B pays $\theta^{*}$, and action S pays a safe payoff of 2/3 in the initial period, where $\theta^{*}\in\{0,1\}$. For any of these three choices, the decision problem ends there and the agent makes a payoff of zero in all future periods. Action O gives the agent a payoff of $-1/3$ in the initial period and the option to make an investment next period, where there are two possible states, $s_{A}$ and $s_{B}$. State $s_{A}$ is realized if $\theta^{*}=1$ and state $s_{B}$ is realized if $\theta^{*}=0$. In each of these states, the agent can choose to make a risky investment or a safe investment. The safe investment gives a payoff of 2/3 in both states, and a subsequent payoff of zero in all future periods. The risky investment gives the agent a payoff that is thrice the payoff she would have gotten from choice A, that is, $3(1-\theta^{*})$, if the state is $s_{A}$, and it gives the agent thrice the payoff she would have gotten from choice B, that is, $3\theta^{*}$, if the state is $s_{B}$; the payoff is zero is all future periods.
Suppose that the agent knows all the primitives except the value of $\theta^{*}$. Let $\Theta=\{0,1\}$; in particular, the SMDP is correctly specified. We now show that, in any Berk-Nash equilibrium, a sufficiently patient agent never chooses the safe action S: Let $\mu\in[0,1]$ denote the agent's equilibrium belief about the probability that $\theta^{*}=1$. For action S to be preferred to A and B, it must be the case that $\mu\in[1/3,2/3]$. But, for a fixed $\mu$, the perceived benefit from action O is
which is strictly higher than 2/3, the payoff from action S, for all $\mu\in[1/3,2/3]$ provided that $\delta>\bar{\delta}=3/4$. Thus, for a sufficiently patient agent, there is no belief that makes action S optimal and, therefore, S is not chosen in any Berk-Nash equilibrium.
Now consider a Bayesian agent who starts with a prior $\mu_{0}=\Pr(\theta=1)\in(0,1)$ and updates her belief. The value of action O is \[ -\frac{1}{3}+\delta\left(\mu_{0}W(s_{A},1)+(1-\mu_{0})W(s_{B},0)\right)=-\frac{1}{3}+\delta\frac{2}{3}<\frac{2}{3} \] because $W(s_{A},1)=W(s_{B},0)=2/3$. In other words, the agent realizes that if the state $s_{A}$ is realized, then she will update her belief to $\mu_{1}=1$, which implies that the safe investment is optimal in state $s_{A}$; a similar argument holds for state $s_{B}$. She then finds it optimal to choose action A if $\mu_{0}\leq1/3$, B if $\mu_{0}\geq2/3$, and S if $\mu_{0}\in[1/3,2/3]$. In particular, choosing S is a steady state outcome for some priors, although it is not chosen in any Berk-Nash equilibrium if the agent is sufficiently patient. The belief supporting S, however, does not satisfy exhaustive learning, since the agent believes that any other action would completely reveal all uncertainty. $\square$
More generally, the failure of a steady state to be a Berk-Nash equilibrium if the agent is sufficiently patient occurs because the value of experimentation can be negative. To see this point, let the value of experimentation for action $x$ at state $s$ when the agent's belief is $\mu$ be \[ ValueExp(s,x;\mu)\equiv E_{\bar{Q}_{\mu}(\cdot\mid s,x)}\left[W(S',B(s,x,S',\mu))\right]-E_{\bar{Q}_{\mu}(\cdot\mid s,x)}\bigl[V_{\bar{Q}_{\mu}}(S')\bigr]. \] This expression is the difference between the value when the agent updates her prior $\mu$ and the value when the agent has a fixed belief $\mu$. An agent who does not account for future changes in beliefs may end up choosing an action with a negative value of experimentation that is actually suboptimal when accounting for changes in beliefs.
In the previous example, the value of experimentation for action O given $\mu$ is
\[ \left(\mu W(s_{A},1)+(1-\mu)W(s_{B},0)\right)-\left(\mu V_{\bar{Q}_{\mu}}(s_{A})+(1-\mu)V_{\bar{Q}_{\mu}}(s_{B})\right), \] which reduces to $2/3-6\mu(1-\mu)$ and is negative for the values of $\mu$ that make S better than A and B. Thus, it is possible for action O to be optimal if the agent does not account for changes in beliefs, but suboptimal if she does.
We now discuss specifically how the property of exhaustive learning is used in the proof of Theorem (ref). We call an action a steady-state action if it is in the support of a stable strategy and we call it a non steady-state action otherwise. A key step is to show that, if a steady-state action is better than a non steady-state action when beliefs are updated, it will also be better when beliefs are fixed. This is true provided that there is zero value of experimenting in steady state, which is guaranteed by exhaustive learning. If instead of exhaustive learning we were to simply require weak identification, there would be no value of experimentation for steady-state actions. The concern, illustrated by the previous example, is that the value of experimentation can be negative for a non steady-state action. Therefore, a non steady-state action could be suboptimal in the problem where the belief is updated but optimal in the problem where the belief is not updated (and so the negative value of experimentation is not taken into account). As shown by esponda2016berk, this concern does not arise in static settings, where the only state variable is a belief. The reason is that the convexity of the value function and the martingale property of Bayesian beliefs imply that the value of experimentation is always nonnegative.
We conclude with additional remarks about Theorem (ref).
Theorem (ref) implies that, for sufficiently patient players, we should be interested in the following refinement of Berk-Nash equilibrium.
In an equilibrium with exhaustive learning, there is a supporting belief that is perceived to be a fixed point of the Bayesian operator, implying that no matter what state and strategy the agent contemplates, she does not expect her belief to change. The requirement of exhaustive learning does not imply robustness to experimentation. For example, in the monopoly problem studied in Section (ref), choosing low price with probability 1 is an equilibrium with exhausted learning which is supported by the belief that, with probability 1, $\theta_{L}^{*}=0$. We rule out equilibria that are not robust to experimentation by introducing a further refinement.
selten1975reexamination introduced the idea of perfection in extensive-form games. By itself, however, perfection does not guarantee that all $(s,x)\in Gr(\Gamma)$ are reached in an MDP. The next property guarantees that all states can be reached when the agent chooses all strategies with positive probability.
Full communication is standard in the theory of MDPs and holds in all of the examples in Section (ref). It guarantees that there is a single recurrent class of states for all $\varepsilon$-perturbed environments. In cases where it does not hold and there is more than one recurrent class of states, one can still apply the following results by focusing on one of the recurrent classes and ignoring the rest as long as the agent correctly believes that she cannot go from one recurrent class to the other.
Full communication guarantees that there are no off-equilibrium outcomes in a perturbed SMDP. It does not, however, rule out the desire for experimentation on the equilibrium path. We rule out the latter by requiring weak identification.
Proposition (ref) provides conditions such that a steady state satisfies exhaustive learning and a Berk-Nash equilibrium can be supported by a belief that satisfies the exhaustive learning condition. Under these conditions, we can find equilibria that are robust to experimentation, i.e., perfect equilibria, by considering perturbed environments and taking the perturbations to zero (see the examples in Section (ref)).
The next proposition shows that perfect Berk-Nash is a refinement of Berk-Nash with exhaustive learning. As illustrated by the monopoly example in Section (ref), it is a strict refinement.
We conclude by showing existence of perfect Berk-Nash equilibrium (hence, of Berk-Nash equilibrium with exhaustive learning, by Proposition (ref)).
We studied Markov decision processes where the agent has a prior over a set of possible transition probability functions and updates her beliefs using Bayes' rule. This problem is relevant in many economic settings but usually not amenable to analysis. We propose to make it more tractable by studying asymptotic beliefs and behavior. The answer to the question “Can the steady state of a Bayesian-SMDP be characterized by reference to an MDP with fixed beliefs?” is a qualified yes. If the agent is sufficiently impatient, it suffices to focus on the set of Berk-Nash equilibria. If, on the other hand, the agent is sufficiently patient and we are interested in steady states with exhaustive learning, then these steady states are characterized by the notion of Berk-Nash equilibrium with exhaustive learning. Finally, if we are interested on equilibria that are robust to experimentation, we can restrict attention to the set of perfect Berk-Nash equilibria.
Our results hold for both the correctly-specified and misspecified cases, and we are not aware of any prior general results for either of these cases. For the correctly-specified case, our results can justify the common assumption in the literature that the agent knows the transition probability function provided that strong identification holds (or that there is weak identification and one is interested in equilibria that are robust to experimentation). In the misspecified case, our results significantly expand the range of possible applications.
\addcontentsline{toc}{section}{References}