Abstract
We apply the idea of partial honesty, first introduced by Dutta and Sen (Game Econ Behav 74:154–169, 2012) in a complete information setting, to environments with incomplete information. We show that with private values and at least three individuals, incentive compatibility and no veto power are together sufficient for full implementation without any further restrictions if all individuals are partially honest. With common values, however, an additional assumption called private best alternatives is needed. This condition holds, for example, in most standard resource allocation problems
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Notes
To be exact, it is not IC that is necessary for implementation, rather it is the existence of an equivalent SCF that is IC. However, at this point we use these interchangeably and postpone the exact definition to Sect. 2.
See also Doghmi and Ziad (2013).
Schweinzer and Shimoji (2011) gives some specific examples where IC is not necessary for implementation under incomplete information. However, their solution concept is rationalizable strategies.
This assumption holds, for example, if \(A\) is finite or if \(U_{i}(x;t)\) is concave and \(A\) a compact set. I thank the referee for pointing out that our main theorem would not hold without this assumption.
We only look at pure strategies in this paper.
Here \(\mathrm{Supp}(T)=\{t \in T \mid p(t)>0\}\).
Dutta and Sen (2012) use this assumption when they study implementation in strictly dominant strategies.
The assumption used in Matsushima (2008b) is not the same that is used in Dutta and Sen (2012) and Lombardi and Yoshihara (2011). He assumes that all individuals incur a cost of \(\epsilon \) from lying, this being a small positive number. Therefore, IC is not a necessary condition in the model of Matsushima (2008b), but the assumption is stronger since it is not lexicographic in nature.
No veto power was first used in implementation theory by Eric Maskin in a working paper that circulated at the end of the 1970s and was later published as Maskin (1999).
I thank the referee for pointing out that the mechanism in the following Theorem does not work if some types are known and also for suggesting how to deal with this.
See Jackson (1991) for an exact argument in the standard case.
The idea to use this recoding function came from Holden et al. (2013).
Here we need the assumption that \(|T_{i}|>1\) for all \(i \in I\). Otherwise there may exist individuals that cannot break the equality \(\sigma _{i}(t_{i})_{1} = \psi _{i}\big (\sigma _{i}(t_{i})_{2}\big )\) and deviate to rule (2).
This construction was used in Dutta and Sen (1991).
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Acknowledgments
Early version of this paper was presented at the 23rd Stony Brook Summer Festival on Game Theory. I thank the participants for comments. I also wish to thank the referee and the editor for detecting few errors and for comments that have greatly improved the quality of this paper. Financial support from the Academy of Finland, Yrjö Jahnsson Foundation, OP-Pohjola Group Research Foundation and Emil Aaltonen Foundation is gratefully acknowledged