Robust and secure implementation: equivalence theorems

This paper shows that in private value environments, strategy-proofness and the rectangular property are necessary conditions for (full) robust implementation (Bergemann and Morris, 2011). As corollaries, we obtain the equivalence between robust and secure implementation (Saijo et al., 2007), the revelation principle for robust implementation, and characterization of double implementation in robust and secure implementation.     

Double implementation in Nash and undominated Nash equilibria in social choice environments

Summary Jackson [1] and Yamato [12] constructed game forms which implement social choice correspondences in Nash and undominated Nash equilibria simultaneously in exchange economies. In this paper, I deal with social choice environments and construct a game form which implements social choice correspondences satisfying monotonicity, no veto power and having at least three agents in Nash and undominated Nash equilibria under the existence of an alternative called a holocaust. The game form constructed in this paper includes an integer game but satisfies the boundedness condition introduced by Jackson [1].I am grateful to Professors Tatsuyoshi Saijo and Stephen Turnbull, and to an anonymous referee for their helpful comments and suggestions. However all remaining errors are mine. An earlier version of this paper was presented at the 1991 Annual Meeting of the Japan Association of Economics and Econometrics at Hokkaido University.After this paper was submitted toEconomic Theory I became aware of the work of Jackson, Palfrey, and Srivastava [2] who have obtained a similar result to mine as a by-product of their main result.     

Undominated Nash Implementation in Bounded Mechanisms

We study implementation in undominated Nash equilibrium by bounded mechanisms. (An undominated Nash equilibrium is a Nash equilibrium in which no agent uses a weakly dominated strategy. A mechanism is bounded if every dominated strategy is dominated by some undominated strategy.) We identify necessary conditions and sufficient conditions for such implementation. These conditions are satisfied in virtually all economic environments, and are also satisfied by interesting correspondences from the social choice literature. For economic settings, we provide a particularly simple implementing mechanism for which the undominated equilibrium outcomes coincide with those obtained from the iterative elimination of weakly dominated strategies. Journal of Economic Literature Classification Numbers: 022, 025, 026.     

Ratifiability of efficient collusive mechanisms in second-price auctions with participation costs

We investigate whether efficient collusive bidding mechanisms are affected by potential information leakage from bidders' decisions to participate in them within the independent private values setting. We apply the concept of ratifiability introduced by Cramton and Palfrey [Cramton, P.C., Palfrey, T.R., 1995, Ratifiable mechanisms: Learning from disagreement, Games Econ. Behav. 10 (2), 255–283] and show that when the seller uses a second-price auction with participation costs, the standard efficient cartel mechanisms such as pre-auction knockouts analyzed in the literature will not be ratified by cartel members. A high-value bidder benefits from vetoing the cartel mechanism since doing so sends a credible signal that she has high value, which in turn discourages other bidders from participating in the seller's auction.     

The core of an economy with differential information

Summary We introduce a new core concept for an exchange economy with differential information which is contained in the coarse core concept of Wilson (1978). We prove the existence of: (i) a core allocation for an exchange economy with differential information and; (ii) an -core strategy for a game in normal form with differential information.On different occasions I have benefited from discussions, comments and suggestions by C.D. Aliprantis, Kim Border, Don Brown, Baskar Chacravorti, Mark Feldman, Leo Hurwicz, Charlie Kahn, John Ledyard, Andreu Mas-Colell, Flavio Menezes, Tom Palfrey, Ed Prescott, Aldo Rustichini, David Schmeidler and Sanjay Srivastava. Mark Feldman and Aldo Rustichini both independently brought to my attention the related work of Wilson (1978). My thanks are extended to all the above individuals as well as to a careful referee. Of course, I am responsible for any remaining shortcomings.     

Secure implementation experiments: Do strategy-proof mechanisms really work?

Strategy-proofness, requiring that truth-telling is a dominant strategy, is a standard concept used in social choice theory. Saijo, Sjöström and Yamato [Saijo, T., Sjöström, T., Yamato, T., 2003. Secure implementation: Strategy-proof mechanisms reconsidered. Working paper 4-03-1. Department of Economics, Pennsylvania State University] argue that this concept has serious drawbacks. In particular, many strategy-proof mechanisms have a continuum of Nash equilibria, including equilibria other than dominant strategy equilibria. For only a subset of strategy-proof mechanisms do the set of Nash equilibria and the set of dominant strategy equilibria coincide. For example, this double coincidence occurs in the Groves mechanism when preferences are single-peaked. We report experiments using two strategy-proof mechanisms. One of them has a large number of Nash equilibria, but the other has a unique Nash equilibrium. We found clear differences in the rate of dominant strategy play between the two.     

Secure implementation experiments: Do strategy-proof mechanisms really work?

Strategy-proofness, requiring that truth-telling is a dominant strategy, is a standard concept used in social choice theory. Saijo, Sjöström and Yamato [Saijo, T., Sjöström, T., Yamato, T., 2003. Secure implementation: Strategy-proof mechanisms reconsidered. Working paper 4-03-1. Department of Economics, Pennsylvania State University] argue that this concept has serious drawbacks. In particular, many strategy-proof mechanisms have a continuum of Nash equilibria, including equilibria other than dominant strategy equilibria. For only a subset of strategy-proof mechanisms do the set of Nash equilibria and the set of dominant strategy equilibria coincide. For example, this double coincidence occurs in the Groves mechanism when preferences are single-peaked. We report experiments using two strategy-proof mechanisms. One of them has a large number of Nash equilibria, but the other has a unique Nash equilibrium. We found clear differences in the rate of dominant strategy play between the two.     

Not quite the best response: truth-telling,strategy-proof matching,and the manipulation of others

Following the advice of economists, school choice programs around the world have lately been adopting strategy-proof mechanisms. However, experimental evidence presents a high variation of truth-telling rates for strategy-proof mechanisms. We crash test the connection between the strategy-proofness of the mechanism and truth-telling. We employ a within-subjects design by making subjects take two simultaneous decisions: one with no strategic uncertainty and one with some uncertainty and partial information about the strategies of other players. We find that providing information about the out-of-equilibrium strategies played by others has a negative and significant effect on truth-telling rates. That is, most participants in our within-subjects design try and fail to best-respond to changes in the environment. We also find that more sophisticated subjects are more likely to play the dominant strategy (truth-telling) across all the treatments. These results have potentially important implications for the design of markets based on strategy-proof matching mechanisms.     

Privacy concerns,voluntary disclosure of information,and unraveling: An experiment

We study the voluntary revelation of private information in a labor-market experiment where workers can reveal their productivity at a cost. While rational revelation improves a worker׳s payoff, it imposes a negative externality on others and may trigger further revelation. Such unraveling can be observed frequently in our data although less often than predicted. Equilibrium play is more likely when subjects are predicted to conceal their productivity than when they should reveal. This tendency of under-revelation, especially of low-productivity workers, is consistent with the level-k model. A loaded frame where the private information concerns the workers׳ health status leads to less revelation than a neutral frame.     

Perfect Bayesian implementation

Summary We study Social Choice Sets (SCS) implementable as perfect Bayesian equilibria of some incomplete information extensive form game. We provide a necessary condition which we callcondition . The condition is analogous tocondition C that Moore and Repullo [1988] show to be necessary for subgame perfect implementation in games of complete information, and it is weaker than the Bayesian Monotonicity condition stated in Jackson [1991]. Our first theorem establishes that Incentive Compatibility, Closure and Condition are necessary for implementation.Our second theorem establishes sufficient conditions. We show that any SCS which satisfies Incentive Compatibility, Closure and a condition called Sequential Monotonicity No Veto (SMNV) is implementable. SMNV is similar in spirit but weaker than the Monotonicity No Veto condition stated in Jackson [1991]. It is also similar to a combination of condition and No Veto Power, which Abreau and Sen show to be sufficient for implementation in subgame perfect equilibrium.This paper is a revised version of a chapter of my dissertation at Stanford University, Graduate School of Business. I would like to thank, without implicating, Faruk Gul, Bob Wilson and especially John Roberts for their constant advice and encouragement. I would also like to thank participants at the 1993 Summer Meeting of the Econometric Society and a referee for comments leading to substantial improvement in the paper. Financial support from Bocconi University is gratefully acknowledged.     

An augmented Becker-DeGroot-Marschak mechanism for transaction cycles

We introduce an augmented Becker-DeGroot-Marschak mechanism for the revelation of willingness-to-accept and willingness-to-pay in transaction cycles. The mechanism can be used to test for a behavioral anomaly.     

A Characterization for Dominant Strategy Implementation

We introduce a perfect price discriminating mechanism for allocation problems with private information. A perfect price discriminating mechanism treats a seller, for example, as a perfect price discriminating monopolist who faces a price schedule that does not depend on her report. In any perfect price discriminating mechanism, every player has a dominant strategy to truthfully report her private information.We establish a characterization for dominant strategy implementation: Any outcome that can be dominant strategy implemented can also be dominant strategy implemented using a perfect price discriminating mechanism. We apply this characterization to derive the optimal, budget-balanced, dominant strategy mechanisms for public good provision and bilateral bargaining.     

Ex-post implementation and preference aggregation via potentials

We introduce several notions of potentials for mechanism design problems with interdependent values, and relate them to implementation in ex-post equilibrium. Whereas ex-post implementation is closely linked to the ordinal concept of best-alternative potentials, the celebrated Vickrey–Clarke–Groves mechanism corresponds to the stronger notion of a cardinal potential: agents agree not only on the best alternative, but also on the quantitative differences between all alternatives. We characterize all valuations that allow for cardinal potentials, and use this characterization for: (1) Identifying valuations for which ex-post implementation is possible; (2) Identifying classes of valuations for which all ex-post implementable choice rules correspond to cardinal potentials. The latter allows us to extend to interdependent valuations a result for dominant strategy implementation in private values settings, due to Roberts (The Characterization of Implementable Choice Rules, North-Holland, Amsterdam, 1979). We wish to thank an associate editor, two anonymous referees, Paul Milgrom and Burkhard Schipper for helpful comments and stimulating remarks.     

Information revelation in auctions

Auction theory has emphasized the importance of private information to the profits of bidders. However, the theory has failed to consider to what extent initially private information will remain private. We show that in a variety of contexts bidders will reveal their information, even if this information revelation is (ex ante) detrimental to them. Similarly, a seller may reveal her information although this revelation lowers revenues. We also show that bidders may be harmed by private information, even in contexts where more information has traditionally been presumed to be beneficial.     

Implementing efficient allocations in a model of financial intermediation

In a finite-trader version of the Diamond and Dybvig (J. Polit. Econ. 91 (1983) 401) model, the ex ante efficient allocation is implementable by a direct mechanism (i.e., each trader announces the type of his own ex post preference) in which truthful revelation is the strictly dominant strategy for each trader. When the model is modified by formalizing the sequential-service constraint (cf. Wallace (Fed. Reserve Bank Minneapolis Quart. Rev. 12 (1988) 3)), the truth-telling equilibrium implements the symmetric, ex ante efficient allocation with respect to iterated elimination of strictly dominated strategies.     

Finite supermodular design with interdependent valuations

This paper studies supermodular mechanism design in environments with arbitrary (finite) type spaces and interdependent valuations. In these environments, the designer may have to use Bayesian equilibrium as a solution concept, because ex-post implementation may not be possible. We propose direct (Bayesian) mechanisms that are robust to certain forms of bounded rationality while controlling for equilibrium multiplicity. In quasi-linear environments with informational and allocative externalities, we show that any Bayesian mechanism that implements a social choice function can be converted into a supermodular mechanism that also implements the original decision rule. The proposed supermodular mechanism can be chosen in a way that minimizes the size of the equilibrium set, and we provide two sets of sufficient conditions to this effect. This is followed by conditions for supermodular implementation in unique equilibrium.     

Top dominance and the possibility of strategy-proof stable solutions to matching problems

Summary This paper explores the possibility of designing strategy-proof mechanisms yielding satisfactory solutions to the marriage and to the college admissions problem. Our first result is negative. We prove that no strategy-proof mechanism can always choose marriages that are individually rational and Pareto efficient. This strengthens a result by Roth (1982) showing that strategy-proof mechanisms cannot always select stable marriages. The result also applies, a fortiori, to college admissions. Since finding difficulties with strategy-proofness is quite an expected result, we then address a second question which is classical within the incentives literature. Are there restrictions on the preferences of agents under which strategy-proof and stable mechanisms do exist? We identify a nontrivial restriction on the domain of preferences, to be called top dominance, under which there exist strategy-proof and stable mechanisms for both types of matching problems. The mechanisms turn out to be exactly those that derive from the most classical algorithms in the literature; namely, the women's optimal, the men's optimal and the student's optimal. Finally, top dominance is shown to be essentially necessary, as well as sufficient, for the existence of strategy-proof stable matching mechanisms.This work is partially supported by grant PB 89-0294, from the Directión General de Investigatión Ciencia y Tecnología of the Spanish Ministerio de Educación y Ciencia. Salvador Barberà is also grateful to the Instituto de Estudios Fiscales. This research was initiated while both authors were visting GREMAQ, Université des Sciencies Sociales, Toulouse, whose hospitality is gratefully acknowledged. The paper extends results that were circulated as GREMAQ W.P. 91.22.232. We are grateful to Matthew Jackson and Marilda Sotomayor for their comments.     

Ex post implementation

We analyze the problem of fully implementing a social choice set in ex post equilibrium. We identify an ex post monotonicity condition that is necessary and—in economic environments—sufficient for full implementation in ex post equilibrium. We also identify an ex post monotonicity no veto condition that is sufficient. Ex post monotonicity is satisfied in all single crossing environments with strict ex post incentive constraints.We show by means of two classic examples that ex post monotonicity does not imply nor is it implied by Maskin monotonicity. The single unit auction with interdependent valuations is shown to satisfy ex post monotonicity but not Maskin monotonicity. We further describe a Pareto correspondence that fails ex post monotonicity but satisfies Maskin monotonicity.     

Choosing and sharing

Implementing a project, like a nationwide nuclear waste disposal, which benefits all involved agents but brings major costs only to the host is often problematic. In practice, revelation issues and redistributional concerns are significant obstacles to achieving stable agreements. We address these issues by proposing the first mechanism to select the efficient site (the host with the lowest cost) while simultaneously allowing for the implementation of any individually rational division of the whole hosting cost. The subgame perfect Nash equilibria of our simple mechanism are efficient, budget-balanced and individually rational. Moreover, the truthful equilibrium is immune to coalitional deviations.     

The robustness of robust implementation

We show that a mechanism that robustly implements optimal outcomes in a one-dimensional supermodular environment continues to robustly implement ε-optimal outcomes in all close-by environments. Robust implementation of ε-optimal outcomes is thus robust to small perturbations of the environment. This is in contrast to ex-post implementation which is not robust in this sense as only trivial social choice functions are ex-post implementable in generic environments.