Sequential Equilibrium Concept Research Articles

Signaling Service Quality Through Queue Disclosure

Problem definition: We consider a single-server queueing system where service quality is either high or low. The server, who knows its exact quality level, can signal this quality information to customers by revealing or concealing its queue length. Based on this queue disclosure action and the observed queue length in the case of a revealed queue, customers decide whether to join the system. Academic/practical relevance: The queue disclosure action is regarded as a signal indicating the service quality. Methodology: We develop a signaling game and adopt the sequential equilibrium concept to solve it. We further apply the perfect sequential equilibrium as an equilibrium-refinement criterion. Results: In our baseline model, where all of the customers are uninformed of service quality, the pure-strategy perfect sequential equilibrium is always a pooling one, except at several discrete values of market size (measured by the potential arrival rate). When the market size is below a certain threshold, both high- and low-quality servers adopt queue concealment; otherwise, both types of servers adopt queue revelation. We also consider a general scenario in which the market is composed of both quality informed and uninformed customers. Under this setting, when the server conceals the queue, we can fully characterize customers’ equilibrium queueing strategies and the corresponding effective arrival rates. The unique sequential equilibrium outcome is still a pooling one when the market size is either below a lower threshold or above an upper threshold. A separating equilibrium can occur only when the market size falls between two thresholds; under that circumstance, the uninformed customers can infer the server’s quality from its queue disclosure behavior. Managerial implications: Under separating sequential equilibria, uninformed customers can fully infer the quality information and thus behave in an informed way. Unlike studies where queue disclosure is not regarded as a quality signal, our study reveals that the signaling effect of queue disclosure increases (decreases) the effective arrival rate of the high-quality (low-quality) server and also increases the customers’ total utility when the server is of low quality. Funding: P. Guo acknowledges the financial support from the Research Grants Council of Hong Kong [Grant 15502820]. The research of M. Haviv was funded by Israel Science Foundation [Grant 1512/19]. Z. Luo acknowledges the financial support from the Internal Start-up Fund of the Hong Kong Polytechnic University [Grant P0039035] and the National Natural Science Foundation of China [Grant 71971184]. Y. Wang’s work was supported by the Research Grants Council of Hong Kong [Grant 15505019]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/msom.2022.1170 .

The Revelation Principle in Multistage Games

Abstract The communication revelation principle (RP) of mechanism design states that any outcome that can be implemented using any communication system can also be implemented by an incentive-compatible direct mechanism. In multistage games, we show that in general the communication RP fails for the solution concept of sequential equilibrium (SE). However, it holds in important classes of games, including single-agent games, games with pure adverse selection, games with pure moral hazard, and a class of social learning games. For general multistage games, we establish that an outcome is implementable in SE if and only if it is implementable in a canonical Nash equilibrium in which players never take codominated actions. We also prove that the communication RP holds for the more permissive solution concept of conditional probability perfect Bayesian equilibrium.

Perfect Conditional ε‐Equilibria of Multi‐Stage Games With Infinite Sets of Signals and Actions

We extend Kreps and Wilson's concept of sequential equilibrium to games with infinite sets of signals and actions. A strategy profile is a conditional ε‐equilibrium if, for any of a player's positive probability signal events, his conditional expected utility is within ε of the best that he can achieve by deviating. With topologies on action sets, a conditional ε‐equilibrium is full if strategies give every open set of actions positive probability. Such full conditional ε‐equilibria need not be subgame perfect, so we consider a non‐topological approach. Perfect conditional ε‐equilibria are defined by testing conditional ε‐rationality along nets of small perturbations of the players' strategies and of nature's probability function that, for any action and for almost any state, make this action and state eventually (in the net) always have positive probability. Every perfect conditional ε‐equilibrium is a subgame perfect ε‐equilibrium, and, in finite games, limits of perfect conditional ε‐equilibria as ε → 0 are sequential equilibrium strategy profiles. But limit strategies need not exist in infinite games so we consider instead the limit distributions over outcomes. We call such outcome distributions perfect conditional equilibrium distributions and establish their existence for a large class of regular projective games. Nature's perturbations can produce equilibria that seem unintuitive and so we augment the game with a net of permissible perturbations.

Who is Really Managing Reputations?

I consider many principals who interact sequentially against the same agent. The agent hides his disutility for efforts from the principals, and tries to use this asymmetric information to extract some surplus from the principals. Principals are isolated, and can coordinate their actions only through direct communication. I show that communication is systematically manipulated by the principals, and that it leads to inefficiency. The result holds not only with Bayesian principals, but also for most ways of updating beliefs. I introduce a new concept of equilibrium to take account of non-Bayesian updating; this concept generalizes the concept of Sequential Equilibrium introduced in Kreps and Wilson \cite{Kr-Wi}. I characterize all the equilibria. The inefficiency generated by this form of direct communication is made clear by the above characterization.

A Model of Competitive Signaling

Multiple candidates (senders) compete over an exogenous number of jobs. There are different tasks in which the candidates' unobservable ability determines their probability of success. We study a signaling game with multiple senders each choosing one task to perform, and one receiver who observes all task choices and performances (success or failure) and matches the senders to jobs. In order to analyze the effects of different levels of competition we consider two refinements of the concept of sequential equilibrium: (i) sequential equilibria that survive when varying the number of senders; (ii) sequential equilibria that are supported by out-of-the-equilibrium-path beliefs satisfying a monotonicity condition (implied by Banks and Sobel's divinity refinement). We show that the set of sequential equilibria includes simple pooling equilibria where all senders choose the same task, and these simple pooling equilibria are the only type of sequential equilibria that satisfies (i). The unique sequential equilibrium under both (i) and (ii) is a simple pooling equilibrium with every sender choosing the most informative task. If senders have a lower overall likelihood of success in more informative tasks, this unraveling towards conspicuousness is inefficient.

Weak isomorphism of extensive games

Based on the Selten [Mathematical Social Sciences 5 (1983) 269] and Oh [Three Essays on Equilibrium Selection in Games, PhD Thesis, (1995) Graduate College of The University of Iowa] symmetries, we introduce weak isomorphism of extensive games that, in contrast to the Peleg et al. [The canonical extensive form of a game form: Symmetries. In: Alkan, A., Aliprantis, C., Yannelis, N. (Eds.), Current Trends in Economics: Theory and Applications, Springer, (1999) pp. 367–387] isomorphism, is compatible with the traditional extensive representation of strategic games. While being sufficiently ‘weak’ to ignore the order of moves to some extent, weak isomorphism is ‘strong’ enough not to violate sequential rationality considerations as incorporated in the concept of sequential equilibrium. In addition, there is some generic equivalence between weak isomorphism and isomorphism of the agent normal form.

Modifying Customer Expectations of Price Decreases for a Durable Product

We study the introductory signalling strategy for a durable product that faces optimistic expectations among customers about price declines over time. The firm introducing the product knows that experiential learning is low for the product. However, customers, being uncertain about the extent of experiential learning, assign a nonzero probability that the firm's new product will enjoy a high cost reduction with cumulative experience. The optimistic expectations of customers reduce their willingness to pay a high price at the product's introduction while predisposing them to buying later. The challenge facing the low-experience firm is to choose an introductory strategy that will credibly convey the low experience-curve effect to customers. We use the sequential equilibrium concept in a game-theoretic framework to identify the firm's signalling strategy. We identify the unique separating equilibrium of the game after refining the set of separating equilibria. We demonstrate that a high introductory price credibly signals the low experiential learning to customers. We also show that signalling causes an artificial learning-curve effect.

The Never-a-Weak-Best-Response Test in Infinite Signaling Games

This essay considers a strong refinement of the sequential equilibrium concept, the Never-a-Weak-Best-Response test, and identifies conditions under which the limit of the equilibrium outcomes of finite approximating games is an equilibrium outcome of the limit infinite game. As a consequence of this result, the existence of equilibria in a class of infinite signaling games is established. Since equilibria satisfying the Never-a-Weak-Best-Response test also satisfy various other refinements, e.g., the Intuitive Criterion and the Universal Divinity test, the existence results in this essay hold for those refinements as well.Journal of Economic LiteratureClassification Numbers: C72, D82.

A model of electoral competition with incomplete information

A model of two-candidate electoral competition is developed in which voters are uncertain about the policy either candidate would implement if elected. Candidates simultaneously announce policy positions, from which voters attempt to infer the true positions the candidates would adopt. Announcing a position different from the true position is costly to the winning candidate, with these costs increasing as the difference between the true policy and the announced policy increases. A refinement of the sequential equilibrium concept is used to describe the behavior of candidates and voters.

Agency Budgets, Cost Information, and Auditing

A model of legislature-agency interaction is analyzed where the agency possesses an informational advantage in that only it knows the cost of its services. The legislature has the ability to audit the agency, where auditing is a costly means of verifying the agency's information. Two different procedures are analyzed for determining the agency's budget: in one, the agency makes a budget request, after which the legislature can either accept or reject the request, or audit the agency and impose a budget equal to the true cost of services. In the other procedure, the legislature can follow a request with a counterproposal to the agency, which can then either accept or reject. Since under both procedures auditing is costly, it will be optimal for the legislature to refrain from auditing a request if the perceived benefits do not outweigh this cost. At issue is the ability of the legislature to impose discipline on the agency's request and final budget through an optimal choice of audit and counterproposal strategies, the extent of the information transmitted through the agency's budget request, and the efficiency of the resulting outcomes. A refinement of the sequential equilibrium concept provides the behavioral predictions for the procedures.

Consistent conditional systems in noncooperative game theory

A space ofconsistent conditional systems is obtained by combining concepts introduced by Kreps and Wilson (1982) and Myerson (1986). The sequential equilibrium concept is displayed as the set of fixed points of a best response correspondence from this space to itself. A thorough exploration of the space shows that the best response correspondence satisfies the hypotheses of the Eilenberg-Montgomery (1946) fixed point theorem, and this allows notions of stability analogous to those introduced by Kohlberg and Mertens (1986) to be defined.

Refinancement, liquidation et rationnement du crédit: Une approche bayésienne

We examine the relationship between borrowers (firms) and lenders (banks), in a dynamic context, with asymmetric information. In particular, we take into account the fact that firms which apply for a grant have a past. They have liability and the banks will take it into consideration when making their decisions, for extending or not the credit of a defaulting firm. Using the concept of sequential equilibrium, we prove the existence of several types of equilibria. An extreme form of credit rationing can appear in some equilibria, because banks can play mixed strategies, and liquidate some defaulting firms. At last, we compare the efficiency of the different equilibria, and the interest factors which prevail in each of them.

Advertising and Limit Pricing

We enrich Milgrom and Roberts' (1982) limit-pricing model to allow an incumbent to signal his costs with both price and advertisements. Our fundamental result is that a cost-reducing distortion occurs, in that the incumbent behaves as if there were complete information but his costs were lower than they are. Preentry price is therefore distorted downward, and demand-enhancing advertising is distorted upward, as a consequence of signalling. If advertising is a purely dissipative signal, it is not used, nor therefore distorted. Recent refinements of the sequential equilibrium concept are featured.

Introductory Price as a Signal of Cost in a Model of Repeat Business

A two-period game between firms and consumers is considered. Firms are privately informed about their individual costs, and consumers must pay a search cost in order to learn a firm's current price. Consumers thus have incentive to use introductory price as a signal of cost and, hence, second period price. Recent refinements of the sequential equilibrium concept are employed, and the resulting equilibria involve low introductory prices (introductory sales).

Subgame-perfect equilibria of finite- and infinite-horizon games

We show that subgame-perfect equilibria of infinite-horizon games arise as limits, as the horizon grows long and epsilon small, of subgame-perfect epsilon-equilibria of games which are truncated after a finite horizon. A number of applications show that this result provides a useful technique for analyzing the existence and uniqueness of infinite-horizon equilibria. We extend our result to the sequential equilibrium concept.