Partial Ex-Post Verifiability and Unique Implementation of Social Choice Functions

This study investigates the unique implementation of a social choice function in iterative dominance in the ex-post term. We assume partial ex-post verifiability; that is, after determining an allocation, the central planner can only observe partial information about the state as verifiable. We demonstrate a condition of the state space, termed “full detection,” under which any social choice function is uniquely implementable even if the range of the players’ lies, which the ex-post verifiable information directly detects, is quite narrow. To prove this, we construct a dynamic mechanism according to which each player announces his (or her) private signal before the other players observe this signal at an earlier stage, and each player also announces the state at a later stage. In this construction, we can impose several severe restrictions, such as boundedness, permission of only tiny transfers off the equilibrium path, and no permission of transfers on the equilibrium path. This study does not assume either expected utility or quasi-linearity.


Introduction
This study investigates the unique implementation of a social choice function (SCF).
The equilibrium concept we adopt is an iteratively undominated strategy in the ex-post term, which is a very weak notion. Therefore, the uniqueness requirement is demanding.
To achieve the allocation implied by a social choice function, which is contingent on the state, the unaware central planner must require informed players to reveal what they know about the state. Hence, the central planner must construct an appropriate mechanism that incentivizes each player to make the desirable (i.e., truthful) announcements as unique equilibrium behavior. This construction should be generally regarded as a difficult task.
To overcome this difficulty, we assume partial ex-post verifiability, that is, after determining an allocation, the central planner can only observe partial information about the state. This observation is verifiable and contractible ex-post. Hence, to resolve the difficulty presented by this uniqueness issue, the central planner makes ex-post monetary transfers with these players. These transfers are contingent not only on players' announcements, but also on the verifiable information. We thus present the possibility that this partial ex-post verifiability fully solves the unique implementation problem.
We assume complete information about the state just before determining an allocation.
We further assume incomplete information, such that each player observes his (or her) private signal concerning the state earlier than the other players do. We design a dynamic mechanism, such that the central planner requires each player to announce his (or her) private signal at an earlier stage and announce the whole description of the state at a later stage.
Based on this dynamic procedure of information acquisition and revelation, we introduce a condition on the state space, which we term full detection. We show that with full detection, an SCF is uniquely implementable via iterative dominance, even if the range of the players' lies, which the ex-post verifiable information directly detects, is quite narrow.
Full detection requires ex-post verifiable information to detect only a limited class of players' lies. However, the elimination of these detected lies can help in turn detect another class of lies. By using this "chain of detection," we can iteratively detect all possible lies.
Full detection is a necessary condition for the existence of a direct revelation mechanism in which truth-telling is the unique iteratively undominated strategy for any player whose only concerns is his (or her) monetary transfer. Our main theorem states that full detection is generally sufficient for unique implementation via iterative dominance.
The dynamic mechanism in our main theorem satisfies various severe restrictions, such as boundedness (e.g., Jackson, 1992), permission of only tiny monetary transfers off the equilibrium path, and no permission of transfers on the equilibrium path. We do not assume either expected utility or quasi-linearity. In fact, we only make basic assumptions on preferences, such that each player's utility function is continuous in lottery over allocations and is continuous and increasing in monetary transfers.
The literature on implementation already establishes that it is generally impossible to uniquely (or fully) implement an SCF in a Nash equilibrium if there is no such verifiable signal. 3 For instance, Makin-monotonicity is a necessary condition for an SCF to be implementable in Nash equilibrium (e.g., Maskin, 1999). However, Maskin-monotonicity is quite a demanding condition for a deterministic SCF.
Several works attempt to weaken the requirements of unique implementation to derive the corresponding possibility results. For instance, Matsushima (1988) and Abreu and Sen (1991) show permissive results in the Nash equilibrium by considering stochastic SCFs and requiring not exact but virtual (i.e., approximate) implementability. Abreu and Matsushima (1992) strengthens these results in virtual implementation by replacing Nash equilibrium with a much weaker solution concept termed iterative dominance, as well as by utilizing only bounded mechanisms (e.g., Abreu-Matsushima mechanisms). Palfrey and Srivastava (1991) replaces Nash equilibrium with a more restrictive solution concept termed weakly undominated Nash equilibrium, and then derived a possibility result in exact implementation. Abreu and Matsushima (1994) strengthens this result by replacing the uniqueness of weakly undominated Nash equilibrium with a more restrictive 3 For surveys on implementation theory, see Moore (1992), Palfrey (1992), Osborne and Rubinstein (1994, Chapter 10), Jackson (2001), and Maskin and Sjöström (2002). requirement, that is, the unique survival from the iterative elimination of weakly dominated strategies. Moore and Repullo (1988) derive a possibility result in exact implementation by replacing the Nash equilibrium with a refinement, that is, subgame perfect equilibrium. Matsushima (2008aMatsushima ( , 2008b  However, the argument in Matsushima (2018) relies on the full verifiability of the state. It is generally unrealistic to assume that the state is fully ex-post verifiable, even if the technology to properly process data will substantially improve in the future. This study aims to extend the permissive result in Matsushima (2018) to the more general case of partial ex-post verifiability. This extension is by no means an easy task: without full verifiability, in a simple majority rule, players can successfully coordinate to communicate the same lie about the unverifiable parts of the state. Hence, we must develop a different mechanism design to detect such lies, and then clarify a condition under which this design method functions. This study proposes a new method for mechanism design and shows that full detection is a sufficient condition to guarantee the same result as in Matsushima (2018), even if we replace full ex-post verifiability with partial ex-post verifiability. This paper is organized as follows. Section 2 describes the basic model. Section 3 defines iterative dominance and unique implementation. Section 4 introduces full detection. Section 5 argues for the necessity of full detection. Section 6 shows the main theorem, and Section 7 concludes.

The Model
We consider a situation in which the central planner determines an allocation and makes monetary transfers. Let denote the finite set of all states, that is, the state space. An SCF is defined as The state-contingent utility function for each player i N  is defined as: Specifically, we describe a state as

Iterative Dominance
We define iterative dominance as the following ex-post term. For every i N  , let Recursively, for each exists, such that for every Recursively, for each We define Definition 4: A dynamic mechanism  is said to uniquely implement an SCF f in iterative dominance if the unique iteratively undominated strategy profile s S  exists in  , and this profile induces the value of the SCF, that is,

Full Detection
This section demonstrates a condition on the state space  , which we term full detection. For each i N  , consider an arbitrary function : The interpretation of i   is as follows. Consider a direct revelation at stage 1. All players, including player 0, are asked to announce their respective private signals. Here, we regard player 0 as the dummy player who always announces 0  truthfully. In this direct revelation, i   describes a pattern of announcements by all players except for player i ; they announce a profile that belongs to ( ) We introduce a notion on i   , termed detection, as follows.

Definition 5: A function
Suppose that i  is correct, but player i announces i i     . Suppose also that for every i   , the other players announce according to Note that if player ' i s announcement i  is correct, the other players announce , that is, they announce a profile and for every ( 1)( ) implies the set of all announcements by all players except for player i that can survive through the h -round iterative removal of detected lies.

Importantly, ( )( )
Since  is finite, it follows that ( )( ) Full detection implies that the iterative removal of detected lies eventually eliminates all lies; truth-telling is therefore the only announcement that survives through this removal procedure.
The following example satisfies full detection: which implies full detection.

Necessity of Full Detection
To understand the role of full detection, we now consider a hypothetical direct Since the announcement by any other player In contrast, he (or she) can avoid this fine by announcing i  truthfully. Since 1 ( ) h  is large enough to satisfy (2), player i never announces any element that does not belong to ( )( ) From the above arguments, we have proved that if 1 i s is strictly iteratively undominated, then which, along with full detection, implies that The proof of the second half of this theorem is as follows. Consider an arbitrary direct revelation mechanism x , and suppose 1* s is the unique iteratively undominated strategy profile in x . Note that We fix 2 h  arbitrarily, and suppose that Hence, we have proved that if 1* i s is strictly iteratively undominated, then implying full detection. Q.E.D.

Unique Implementation
To uniquely implement an SCF f in iterative dominance, we consider the following manner of designing a dynamic mechanism. We first fix the arbitrary positive real numbers, According to the basic method explored by Abreu and Matsushima (1992a, 1992b, we construct a dynamic mechanism *  Because of continuity, we can select a sufficiently large K , such that whenever Following the same logic as Theorem 1, we can prove that players' truth-telling at stage 1 is the unique equilibrium behavior. By using their truthful revelations at stage 1 and the ex-post verifiable information as the reference, we apply the same logic as Abreu and Matsushima (1992a, 1992b to prove that players are willing to make a truthful profile of announcements at stage 2. According to i z , any first deviant from the combination of the profile of first announcements and the verified information

Concluding Remarks
This study investigated the unique implementation of an SCF in iterative dominance in ex-post terms. We assumed partial ex-post verifiability; that is, after determining an allocation, the central planner can observe partial but verifiable information about the state and make ex-post monetary transfers contingent on this information. We demonstrated a condition on the state space ("full detection"), which we prove is a sufficient condition for unique implementation. We constructed a dynamic mechanism with boundedness, in which each player announces his (or her) private signal at stage 1 before the other players observe it. In this construction, we only used small monetary transfers off the equilibrium path, but no transfers on the equilibrium path.
Our study contributes to the field of dynamic mechanism design (e.g., Krähmer and Strausz, 2015). Penta (2015) investigates full implementation in incomplete information environments, where players receive information over time. Unlike this study, the author does not consider verifiability and adopts interim solution concepts such as perfect Bayesian equilibrium, instead of ex-post-term concepts. Hence, it is important for future research to investigate a more dynamic situation than this study considered, in which stage 1 is divided into multiple sub-stages through which each player gradually acquires the whole picture of his (or her) private signals. Here, the central planner requires each player to make an announcement at each sub-stage. By considering these sub-stages, we expect to demonstrate weaker sufficient conditions than full detection.
This study is related to the works by Kartik and Tercieux (2012) and Ben-Porath and Lipman (2012), which investigate full implementation with hard evidence. These works assume that each player can receive verifiable information at early stages as hard evidence, and state that the degree to which hard evidence in the ex-ante and interim terms directly proves players' announcements to be correct is crucial in implementing a wide variety of SCFs. In contrast, this study emphasizes that a wide variety of SCFs are implementable even if the information is only verifiable ex-post and to a limited extent. To deepen our understanding of the role of verifiability in implementation, future research could explore integrating our study with Kartik and Tercieux (2012) and Ben-Porath and Lipman (2012).