Perfect Bayesian Equilibrium Research Articles

Informational Hold Up and Intermediaries

Why do some incomplete information markets feature intermediaries while others do not? I study the allocation of two goods in an incomplete information setting with a single principal, multiple agents with unit demand, and interdependent valuations. I construct a novel dynamic mechanism implemented by a principal who faces a set of intermediaries, each of whom represents an ex ante identical set of agents. This mechanism has a unique (up to permutation) weak perfect Bayesian equilibrium. The dynamic mechanism is inefficient with positive probability. Nevertheless, under mild conditions the agents are ex ante better off under the dynamic mechanism relative to a Vickrey-like auction because the intermediaries are more able to exploit information asymmetries in the dynamic mechanism than agents are able to exploit information asymmetries in the Vickrey-like auction. Finally, I show that in large markets the dynamic mechanism and Vickrey-like auction have the same expected total surplus. The comparison between the two mechanisms gives a stylized intuition for the hierarchical structure of larger markets and institutions.

Informational Cascades With Nonmyopic Agents

We consider an environmentwhere players need to decide whether to buy a certain product (or adopt a technology) or not. The product is either good or bad, but its true value is unknown to the players. Instead, each player has her own private information on its quality. Each player can observe the previous actions of other players and estimate the quality of the product. A classic result in the literature shows that in similar settings, informational cascades occur, where learning stops for the whole network and players repeat the actions of their predecessors. In contrast to this literature, in this paper, players get more than one opportunity to act. In each turn, a player is chosen uniformly at random from all the players and can decide to buy the product and leave the market or wait. Her utility is the total expected discounted reward, and thus, myopic strategies may not constitute equilibria. We provide a characterization of perfect Bayesian equilibria (PBEs) with forward-looking strategies through a fixed-point equation of dimensionality that grows only quadratically with the number of players. Using this tractable fixed-point equation, we show the existence of a PBE and characterize PBEs with threshold strategies. Based on this characterization, we study informational cascades in two regimes. First, we show that for a discount factor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula> strictly smaller than 1, informational cascades happen with high probability as the number of players <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> increases. Furthermore, only a small portion of the total information in the system is revealed before a cascade occurs. Second, and more surprisingly, we show that for a fixed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> , and for a sufficiently large <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta &lt; 1$</tex-math></inline-formula> , when the product is bad, there exists an equilibrium where an informational cascade can happen only after at least half of the players revealed their private information, and consequently, the probability for a “bad cascade” where all the players buy the product vanishes exponentially with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> . Finally, when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta =1$</tex-math></inline-formula> and the product is bad, there exists an equilibrium where informational cascades do not happen at all.

Designing incentive mechanism in contract farming considering reciprocity preference.

Contract farming is a growing practice in agricultural economy. A well-designed crop-planting and buyout contract, offered by an enterprise, to a fraction of contract farmers brings benefit to farmers as well as to the enterprise itself. However, in the process of contract fulfillment, farmers possess private information about the degree of effort on fulfilling the contract of themselves. Thus, the more informed farmers may not work hard in the process of planting crops. This opportunistic behavior of farmers caused by asymmetric information has seriously affected the sustainability of contract farming. An enterprise with reciprocity preference tends to make a contract both improves farmers’ welfare and brings enough profit to sustain its own operations, while a farmer with reciprocity will work hard in return for the enterprise’s reward. In this paper, we develop a non-profit index principal-agent model between enterprises and farmers, assuming both have reciprocity preference, to investigate how to design an incentive mechanism in contract farming. We begin our analysis by establishing a non-profit index evaluation system to evaluate farmers’ effort degree in contract fulfillment. Then we solve principal-agent problem with the assumption that farmers’ expected certainty income premium (ECIP) is constant. We find that in the perfect Bayesian equilibrium, farmers with higher degree of reciprocity preference require less ECIP, and will improve efforts to complete contract tasks, even actively sacrifice their own interests to repay extra rewards from enterprises. Furthermore, we explore our model to the scenario in which farmers’ ECIP is a function of enterprise payment difference (EPD). We find that the higher the degree of reciprocity preference of farmers, the greater the probability of enterprises to increase income. Numerical simulations are conducted to verify the validity of the conclusions. Our study shows that the reciprocity preference behavior of enterprises and farmers improves the fulfillment rate of contract farming, which contributes to the realization of the incentive mechanism of contract farming.

Bargaining with Mechanisms

Two players bargain over a single indivisible good and a transfer, with one-sided incomplete information about preferences. Both players can offer arbitrary mechanisms to determine the allocation. We show that there is a unique perfect Bayesian equilibrium outcome. In the equilibrium, one of the players proposes a menu that is optimal for the uninformed player among all menus, such that each type of the informed player receives at least her payoff under complete information. The optimal menu can be implemented with at most three allocations. Under a natural assumption on the uninformed player’s beliefs, the optimal menu coincides with the Myerson’s neutral solution to the bargaining problem in this environment. (JEL C78, D82, D83)

Oligopoly and Collusion in the Colombian Electricity Market

The Wholesale Electricity Market –WEM– in Colombia was created with Laws 142 and 143 of 1994 with the aim of promoting and preserving competition among agents and guaranteeing an efficient energy spot price. However, in practice the market works as an oligopoly in competition; evidence indicates there is market concentration, without reaching abuse of dominant position or price collusion, at least not explicitly. This paper develops a game model with incomplete information, in which there is at least a Perfect Bayesian Equilibrium with implicit collusion among agents. That is, it is highly likely that, under certain conditions, some generators tacitly follow the price strategies of agents with greater installed generation capacity (pivotal) and hence propose similar bid prices. The empirical analysis shows that scenarios of radical and similar increases in bid prices, for different generators, are a pattern present in the WEM.

Strategic information transmission with sender's approval: The single-crossing case

We consider games in which an informed sender first talks at no cost to a receiver; then, the latter proposes a decision and, finally, the sender accepts the proposal or “exits”. We make the following assumptions: the sender has finitely many types, the receiver's decision is real-valued, utility functions over decisions are concave, single-peaked and single-crossing, exit is damaging to the receiver. In this setup, it may happen that babbling equilibria necessarily involve exit. We nevertheless propose a constructive algorithm that achieves a pure perfect Bayesian equilibrium without exit in every game of the class considered.

Communication and cooperation in Prisoner's Dilemma games

We consider a Prisoner's Dilemma model with a pre-play communication stage. Players get a disutility from deceiving others after sending a message of conditional cooperation. Each player's realized lying aversion cost is private information. We prove existence and characterize the (unique) symmetric perfect Bayesian equilibrium that maximizes cooperation. In this distinguished equilibrium, the frequency of cooperation decreases as material gains from lying (or losses from being deceived) increase. We introduce an iterated procedure allowing to characterize the set of rationalizable strategies. We find that cooperation may decrease as a population starts to value honesty more. We extend the model to account for the possibility of a benefit obtained when lying or the effect of the audience size.

Equilibrium and Welfare Analysis in Second‐Price Auctions with Resale and Costly Entry

This study investigates the effects of resale allowance on entry strategies, seller’s expected revenue, and social welfare in a second‐price auction with two‐dimensional private information on values and participation costs. We characterize the perfect Bayesian equilibrium in cutoff strategies and identify sufficient conditions under which the equilibrium is unique. Our analysis suggests that resale allowance leads the low‐value bidder to become more aggressive on entry, while high‐value bidder has a lower incentive to enter. Furthermore, the allowance of resale can increase the social welfare under a sufficient condition, and its effect on expected revenue is ambiguous.

Dynamic signaling with stochastic stakes

We study dynamic signaling in a game of stochastic stakes. Each period, a privately informed agent of binary type chooses whether to continue receiving a return that is an increasing function of both her reputation and an exogenous public stakes variable or to irreversibly exit the game. A strong type has a dominant strategy to continue. In the unique perfect Bayesian equilibrium, the weak type plays a mixed strategy that depends only on current stakes and her historical minimum and she builds a reputation by continuing when the stakes reach a new minimum. We discuss applications to corporate reputation management, online vendor reputation, and limit pricing with stochastic demand.

Compromise, Don't Optimize: Generalizing Perfect Bayesian Equilibrium to Allow for Ambiguity

We introduce a solution concept for extensive-form games of incomplete information in which players need not assign likelihoods to what they do not know about the game. This is embedded in a model in which players can hold multiple priors. Players make choices by looking for compromises that yield a good performance under each of their updated priors. Our solution concept is called perfect compromise equilibrium. It generalizes perfect Bayesian equilibrium. We show how it deals with ambiguity in Cournot and Bertrand markets, public good provision, Spence's job market signaling, bilateral trade with common value, and forecasting.

Epsilon-Efficiency in a Dynamic Partnership with Adverse Selection and Moral Hazard

Abstract For a dynamic partnership with adverse selection and moral hazard, we design a direct profit division mechanism that satisfies ϵ-efficiency, periodic Bayesian incentive compatibility, interim individual rationality, and ex-post budget balance. In addition, we design a voting mechanism that implements the profit division rule associated with this direct mechanism in perfect Bayesian equilibrium. For establishing these possibility results, we assume that the partnership exhibits intertemporal complementarities instead of contemporaneous complementarities; equivalently, an agent’s current effort affects other agents’ future optimal efforts instead of current optimal efforts. This modelling assumption fits a wide range of economic settings.

Simpler and better: Supply chain contracting in the presence of contract unobservability and upstream competition

A retail platform or intermediary can purchase products from multiple upstream manufacturers through different types of contracts. Traditional studies assume that the contracts and their terms are common knowledge; that is, each party of the supply chain can observe the contracts reached. In practice, however, this is not necessarily the case. Taking the result under contract observability as a benchmark, this paper investigates the contracting of the supply chain comprised of one retail platform and two upstream manufacturers when contracts are unobservable. Two representative contracts are considered: the traditional and succinct wholesale price contract and the nonlinear, more complicated, two-part tariff contract. We apply the framework of perfect Bayesian equilibrium and also use passive beliefs as an equilibrium refinement criterion to analyze the model. Our study shows that the classical result – that the more complicated contract can increase channel profitability – may no longer be valid when contracts are unobservable to competing manufacturers. On the contrary, the manufacturers and the whole supply chain can benefit from adopting wholesale price contracts rather than two-part tariff contracts. To examine the robustness of our findings, we consider a commission fee contract and another classic demand function in an extended analysis, and the results show that the main findings and key insights remain valid. The intuition behind these findings is that the unobservability of contracts affects the opportunistic behavior of competing manufacturers. The practical value of these findings is mainly reflected in the evidence that supply chain members must carefully consider the impact of contract (un)observability when agreeing on contracts.

Justified Communication Equilibrium

Justified communication equilibrium (JCE) is an equilibrium refinement for signaling games with cheap-talk communication. A strategy profile must be a JCE to be a stable outcome of nonequilibrium learning when receivers are initially trusting and senders play many more times than receivers. In the learning model, the counterfactual “speeches” that have been informally used to motivate past refinements are messages that are actually sent. Stable profiles need not be perfect Bayesian equilibria, so JCE sometimes preserves equilibria that existing refinements eliminate. Despite this, it resembles the earlier refinements D1 and NWBR, and it coincides with them in co-monotonic signaling games. (JEL C70, D82, D83, J23, M51)

Mode selection and risk estimation of financing in new energy automobile enterprises

Financing is a necessary economic activity of the new energy automobile enterprises, and it has an important function to push the enterprise to expand reproduction and promote the development speed of the enterprise. At the same time, there are many kinds of financing modes, such as internal financing, external debt financing and equity financing. If the financing mode is matched with the enterprise status, the financing activity is easier to succeed and the risk is also smaller. Otherwise, improper financing mode will not only affect the profit of enterprises, but also affect the survival of enterprises. We suppose that, after financing, the business manager of the enterprise can make a choice to invest either hydrogen-fueled vehicle project or lithium-ion battery vehicle project, in this case, we made an equilibrium analysis on the choice of investing which one new energy vehicle project and corresponding financing mode, according to the incomplete information dynamic game model, it is concluded that the equilibrium point (perfect Bayesian equilibrium refinement) of financing is most likely to appear in the state of equity financing; when the rate of return on assets of investment projects is within a certain range, both the financing enterprise and the investor can obtain the bigger profit. In addition, the risk value of the project also is calculated by using VaR model.

Information Redundancy Neglect versus Overconfidence: A Social Learning Experiment

We study social learning in a continuous action space experiment. Subjects, acting in sequence, state their beliefs about the value of a good after observing their predecessors’ statements and a private signal. We compare the behavior in the laboratory with the Perfect Bayesian Equilibrium prediction and the predictions of bounded rationality models of decision-making: the redundancy of information neglect model and the overconfidence model. The results of our experiment are in line with the predictions of the overconfidence model and at odds with the others’. (JEL C91, D12, D82, D83)

Bargains, price signaling, and efficiency in markets with asymmetric information

We experimentally investigate bargains, price signaling, and efficiency in markets with asymmetric information where some buyers are informed and some sellers are informed. We show that all perfect Bayesian equilibria where transactions occur under known gains from trade fall into one of two categories. In some cases, uninformed sellers charge high prices but uninformed buyers are only willing to pay low prices. Otherwise, uninformed sellers charge low prices and uninformed buyers are willing to pay high prices. This latter case is shown to exhibit more bargains and greater efficiency. Consistent with equilibrium predictions, we observe significantly fewer transactions and bargains in our first treatment. In contrast to equilibrium predictions, uninformed buyers selected far lower reservation prices than informed high quality buyers in the second treatment as predicted by adaptive models.

Trust and reputation under asymmetric information

We study the role of information about the multiplier in a finitely repeated investment game. A high multiplier increases the reputational incentives of a trustee, leading to more repayments. Our perfect Bayesian equilibrium analysis shows that if the trustee is privately informed about the multiplier, both the expected frequency of investments and repayments as well as the expected payoffs of both players are higher compared to a situation where the multiplier is public knowledge. We test this result in a laboratory experiment. The data cannot confirm the predicted welfare dominance of private information about the multiplier. We discuss potential reasons for the deviation between theory and experimental data.

Signaling Equilibria for Dynamic LQG Games With Asymmetric Information

In this article, we consider a finite-horizon dynamic game with two players who observe their types privately and take actions that are publicly observed. Players' types evolve as conditionally independent, controlled linear Gaussian processes, and players incur quadratic instantaneous costs. This forms a dynamic linear quadratic Gaussian game with asymmetric information. We show that under certain conditions, players' strategies that are linear in their private types, together with Gaussian beliefs, form a perfect Bayesian equilibrium (PBE) of the game. This is a signaling equilibrium due to the fact that future beliefs on players' types are affected by the equilibrium strategies. We provide a backward-forward algorithm to find the PBE. Each step of the backward algorithm reduces to solving an algebraic matrix equation for every possible realization of the state estimate covariance matrix. The forward algorithm consists of Kalman filter recursions, where state estimate covariance matrices depend on equilibrium strategies. We provide necessary and sufficient conditions for the existence of the solution of the algebraic matrix equation for scalar actions. Finally, we present numerical examples.

Worker-firm relational contracts in the time of shutdowns: experimental evidence

Exogeneous disruptions in labor demand have become more frequent in recent times. The COVID-19 pandemic has resulted in millions of workers being repeatedly laid off and rehired according to local public health conditions. This may be bad news for market efficiency. Typical employment relations—which resemble non-enforceable (implicit) contracts—rely on reciprocity (Brown et al. in Econometrica 72:747–780, 2004), and hence could be harmed when workers’ efforts no longer guarantee reemployment in the next period. In this paper we extend the BFF paradigm to include a per-period probability (0%, 10%, 50%) of publicly observable “shutdown”, where a specific firm cannot contract with any workers for several periods. A Perfect Bayesian Equilibrium exists in which these shutdowns destabilize relationships, but do not harm efficiency. Our experiment shows that, remarkably, market efficiency can be maintained even with very frequent stochastic shutdowns. However, the dynamic of relational contracts changes from one where a worker finds stable employment to one where she juggles multiple employers, laying the burden of maintaining productivity upon workers and worsening worker-side inequality.Supplementary informationThe online version contains supplementary material available at (10.1007/s10683-020-09697-1).

Risk aversion, informative noise trading, and long-lived information

We build a dynamic strategic trading model in which noise trading demand may be correlated with an asset's fundamental value. This correlation shapes the temporal properties of a perfect Bayesian equilibrium. The more positive the correlation coefficient, the faster prices incorporate private information. In contrast, when the correlation is negative and only one risk-neutral insider exists, an abundant amount of private information cannot be revealed after the final trade, and market liquidity deteriorates over time. Moreover, information is revealed faster during earlier periods if positively correlated noise demand is concentrated early, whereas more information remains hidden after the final trade if negatively correlated noise demand is concentrated later. The inefficiency caused by negatively correlated noise demand can be resolved if insiders are competitive or risk averse.