Ex-post Equilibrium Research Articles

Double Implementation in Dominant Strategy Equilibria and Ex-Post Equilibria with Private Values

Abstract We consider the implementation problem for incomplete information and private values. We investigate double implementability of social choice functions in dominant strategy equilibria and ex-post equilibria. We define a new strategic axiom that implies “strategy-proofness” and that is implied by “secure strategy-proofness,” but the converse of these relationships does not hold. We call it “weak secure-strategy-proofness.” We show that a social choice function is doubly implementable if and only if it is weakly securely-strategy-proof.

The iterative deferred acceptance mechanism

Lately, there has been an increase in the use of sequential mechanisms, instead of the traditional direct counterparts, in college admissions in many countries, including Germany, Brazil, and China. We describe these mechanisms and identify their shortcomings in terms of incentives and outcome properties. We introduce a new mechanism, which improves upon these shortcomings. Unlike direct mechanisms, which ask students for a full preference ranking over colleges, our mechanism asks students to sequentially make choices or submit partial rankings from sets of colleges. These are used to produce a tentative allocation at each step. If at some point it is determined that a student can no longer be accepted into previous choice, then she is asked to make another choice among colleges that would tentatively accept her. Participants following the simple strategy of choosing the most-preferred college in each step is an ex-post equilibrium that yields the Student-Optimal Stable Matching.

Quasi Ex-Post Equilibrium in Competing Mechanisms

This paper studies competing mechanism games with no restrictions on the complexity of mechanisms where principals can announce mechanisms and agents select and communicate with at most one principal. It proposes the solution concept of robust quasi ex-post equilibrium in which agents' strategies of communicating with a non-deviating principal is ex-post optimal. Two simple revelation principles are established. The Strong Revelation Principle allows us to check if an equilibrium allocation in a specific competition model is a robust quasi ex-post equilibrium allocation. The Weak Revelation Principle leads to the characterization of the set of robust quasi ex-post equilibrium allocations in terms of model primitives.

Real Exchange Rate Overshooting in Large Depreciations

The consequences of large depreciations on economic activity depend on the relative strength of the contractionary balance sheet and expansionary expenditure switching effects. However, the two operate over different time horizons: the balance sheet effect hits almost immediately, while expenditure switching is delayed by nominal rigidities and other frictions. The paper hypothesizes that the overshooting phase—observed early in the depreciation episode and driven by the balance sheet effect—is largely irrelevant for expenditure switching, which is more closely aligned with ex-post equilibrium depreciation. Given this, larger real exchange rate overshooting should signal a relatively stronger balance sheet effect. Empirical findings support this hypothesis: (i) overshooting is driven by factors associated with the balance sheet effect (high external debt, low reserves, low trade openness), (ii) overshooting-based measures of the balance sheet effect foreshadow post-depreciation output losses, and (iii) the balance sheet effect is strongest early on, while expenditure switching strengthens over the medium term.

Ex-Post Equilibrium Market Recommendations: Implications for Indian Agriculture

Ex-Post Equilibrium Market Recommendations: Implications for Indian Agriculture

Real Exchange Rate Overshooting in Large Depreciations: Determinants and Consequences

The consequences of large depreciations on economic activity depend on the relative strength of the contractionary balance sheet and expansionary expenditure switching effects. However, the two operate over different time horizons: the balance sheet effect hits almost immediately, while expenditure switching is delayed by nominal rigidities and other frictions. The paper hypothesizes that the overshooting phase-observed early in the depreciation episode and driven by the balance sheet effect-is largely irrelevant for expenditure switching, which is more closely aligned with ex-post equilibrium depreciation. Given this, larger real exchange rate overshooting should signal a relatively stronger balance sheet effect. Empirical findings support this hypothesis: (i) overshooting is driven by factors associated with the balance sheet effect (high external debt, low reserves, low trade openness), (ii) overshooting-based measures of the balance sheet effect foreshadow post-depreciation output losses, and (iii) the balance sheet effect is strongest early on, while expenditure switching strengthens over the medium term.

Designing mechanisms to focalize welfare-improving strategies

Many institutions use matching algorithms to allocate resources to individuals. Examples include the assignment of doctors, students and military cadets to hospitals, schools and branches, respectively. Oftentimes, agents' ordinal preferences are highly correlated, motivating the use of mechanisms that provide agents with channels through which they can express some cardinal preference information. This paper studies two such mechanisms, one from the field and one we design. In each of the games induced by these algorithms, we identify the strategies that constitute the unique symmetric ex-post equilibrium. Interestingly, when we test the mechanisms in the lab, these equilibrium predictions fail. Subjects nevertheless behave largely in concordance with the mechanisms' intended strategies; the focalization of such strategies lead to greater welfare in relation to a popular existing mechanism.

Common Learning and Cooperation in Repeated Games

We study repeated games in which players learn the unknown state of the world by observing a sequence of noisy private signals. We find that for generic signal distributions, the folk theorem obtains using ex-post equilibria. In our equilibria, players commonly learn the state, that is, the state becomes asymptotic common knowledge.

On mechanisms eliciting ordinal preferences

When is a mechanism designer justified in only asking for ordinal information about preferences? Simple examples show that, even if the planner's goal (expressed by a social choice correspondence, or SCC) depends only on ordinal information, eliciting cardinal information may help with incentives. However, if agents may be uncertain about their own cardinal preferences, then a strong robustness requirement can justify the focus on ordinal mechanisms. Specifically, when agents' preferences over pure outcomes are strict, if a planner is able to implement an SCC (in ex-post equilibrium) using a mechanism that is robust to interdependence of arbitrary form in cardinal preferences, then there must exist such a mechanism that elicits only ordinal preferences. The strictness assumption can be dropped if we further allow the possibility of non-expected-utility preferences.

We Can Cooperate Even When the Monitoring Structure Will Never Be Known

This paper considers infinite-horizon stochastic games with hidden states and hidden actions. The state changes over time, players observe only a noisy public signal about the state each period, and actions are private information. In this model, uncertainty about the monitoring structure does not disappear. We show how to construct an approximately efficient equilibrium in a repeated Cournot game. Then we extend it to a general case and obtain the folk theorem using ex-post equilibria under a mild condition.

Approximation in mechanism design with interdependent values

This paper studies the revenue maximization problem in environments wherein buyers have interdependent values and correlated types. We show that (1) when the system of feasible sets is a matroid and buyer valuations satisfy a single-crossing condition, the generalized Vickrey–Clarke–Groves mechanisms with lazy reserves (VCG-L) are ex-post incentive compatible and ex-post individually rational; (2) if, in addition, the valuation distribution satisfies a generalized monotone hazard rate condition, the VCG-L mechanism with conditional monopoly reserves is approximately optimal. Then we construct an ascending auction that implements the truth-telling equilibrium of a VCG-L mechanism in ex-post equilibrium. Finally, we discuss the connection between the VCG-L mechanisms and greedy algorithms studied in Lehmann et al. (2002) and deferred-acceptance auctions studied in Milgrom and Segal (2014), and the impact of competition by proving a Bulow and Klemperer (1996) type result.

Long-Term Capital Budgeting Mechanism in Multidivision Firms

We study the optimal dynamic mechanism for capital budgeting and managerial compensation in a firm that consists of two divisions. Division managers privately observe the characteristics of their own project, which govern the evolution of the project types, and the project’s subsequent profitability. At the same time, managers supply costly efforts to run the projects. The optimal mechanism incorporates the distortion driven by persistent private information and the production complementarity among the divisions. The headquarters should commit to investment policy that excludes or terminates projects, and controls the growth rate and volatility of project scales. With proportional distortions, the investment for good projects grows faster but is more volatile. In addition, we show that a linear contract implements the optimal investment and effort policy in an ex-post equilibrium, and the evolution of the power of incentives depends on the nature of initial private information. When a manager's private information generates decreasing distortion, her pay-performance sensitivity grows stronger over time, and the headquarters provides incentives through firm-level pay instead of divisional-level pay.

Quality-adjusted productivity gain in the propagation of innovation

We introduce a class of production function whose inputs and outputs constitute multiples of quality and quantity. Under the efficiency unit approach, we precisely reduce innovation regarding qualitative and quantitative improvements of production to the measurement of quality-adjusted productivity gain. We then consider a system of compatible unit cost functions inclusive of such productivity improvements in any industry, for which we can solve for the ex-post equilibrium to examine the technological structural propagation. In this way, we can evaluate any given innovation with respect to its social welfare gain. We use this framework of multi-industry multi-factor production to study effective industry-wise research and development investment allocations.

Deposit Contract Design with Preferences for Honesty

This study proposes a new selection mechanism to identify the efficient allocation of a deposit contract model by using agents' preference for honesty. The main result is that the efficient allocation is uniquely implementable through iterated elimination of strictly dominated strategies, while it is never implementable in ex-post equilibrium under a canonical environment without preferences for honesty. This result is obtained under small and weak preferences for honesty. Furthermore, a mechanism designer requires no information on whose preference it is.

Mechanism Design for Scheduling with Uncertain Execution Time

We study the problem where a task (or multiple unrelated tasks) must be executed, there are multiple machines/agents that can potentially perform the task, and our objective is to minimize the expected sum of the agents' processing times. Each agent does not know exactly how long it will take him to finish the task; he only knows the distribution from which this time is drawn. These times are independent across agents and the distributions fulfill the monotone hazard rate condition. Agents are selfish and will lie about their distributions if this increases their expected utility. We study different variations of the Vickrey mechanism that take as input the agents' reported distributions and the players' realized running times and that output a schedule that minimizes the expected sum of processing times, as well as payments that make it an ex-post equilibrium for the agents to both truthfully report their distributions and exert full effort to complete the task. We devise the ChPE mechanism, which is uniquely tailored to our problem, and has many desirable properties including: not rewarding agents that fail to finish the task and having non-negative payments.

Ex-post regret heuristics under private values (I): Fixed and random matching

In contexts in which players have no priors, we analyze a heuristic process based on ex-post regret as a guide to understand how to play games of incomplete information under private values. The conclusions depend on whether players interact within a fixed set (fixed matching) or they are randomly matched to play the game (random matching). The relevant long run predictions are minimal sets that are “closed under same or better reply” operations. Under additional assumptions in each case, the predictions boil down to Nash equilibria, ex-post equilibria or minimax regret equilibria. These three paradigms exhibit nice robustness properties in the sense that they are independent of beliefs about the exogenous uncertainty of type spaces. The results are illustrated in several applications, including second-price auctions, first-price auctions and Bertrand duopolies.

Ex-Post Equilibrium and VCG Mechanisms

Consider an abstract social choice setting with incomplete information, where the number of alternatives is large. Albeit natural, implementing VCG mechanisms is infeasible due to the prohibitive communication constraints. However, if players restrict attention to a subset of the alternatives, feasibility may be recovered. This article characterizes the class of subsets that induce an ex-post equilibrium in the original game. It turns out that a crucial condition for such subsets to exist is the availability of a type-independent optimal social alternative for each player. We further analyze the welfare implications of these restrictions. This work follows that of Holzman et al. [2004] and Holzman and Monderer [2004] where similar analysis is done for combinatorial auctions.

A Clarke tax tâtonnement that converges to the Lindahl allocation

This paper uses the Clarke mechanism to construct an incentive compatible t âtonnement process which converges to the Lindahl allocation of a stylized public good economy when consumers have quadratic preferences. We show truth-telling to be an ex-post perfect equilibrium in the infinite horizon game induced by the tâtonnement.

Pre-marital confinement of women: A signaling and matching approach

Parents in several cultures ‘discipline’ their daughters to inculcate the supposedly feminine virtues and improve their prospects in the marriage market. This process invariably involves imposing restrictions on their behavior, movement, and social relations. We refer to such practices as pre-marital confinement and provide a unified game-theoretic framework which encompasses the different arguments that have been advanced by social scientists to explain pre-marital confinement. The proposed framework views confinement by parents as an intrinsically valueless or valuable signal of some unobservable characteristic of their daughters which is valued by men in the marriage market. We focus on identifying the conditions that lead to a societal norm of confinement. We also show that, behind a veil of ignorance where gender is unknown, agents will choose to prohibit rather than permit confinement if it is intrinsically valueless and the parameters characterizing the society are such that the ex-post equilibrium would involve pooling.

Optimal Keyword Auctions with Shadow Costs

Poor user experiences with search advertisements can lead to ad avoidance thus reduce search engine’s long-term revenue. We capture the effect of negative user experiences on search engine’s future revenue in a new variable called “shadow costs” and examine the optimal keyword auction mechanisms (KAMs) in a general model that takes into account advertiser-specific and positionspecific shadow costs. We show that the optimal KAMs can be implemented in an ex-post equilibrium with a “progressive second price” payment rule. Furthermore, under a few special but practically relevant cases, the optimal KAM takes the form of relatively simple scoring auctions. We show that minimum bids in these scoring auctions may be advertiser- or position- specific and the allocation rule may or may not be greedy. Our results highlight impact of shadow costs on keyword auction designs and hold implications for search engines, advertisers, and Internet users.