The geometry of implementation: a necessary and sufficient condition for straightforward games

Abstract

We characterize games which induce truthful revelation of the players’ preferences, either as dominant strategies (straightforward games) or in Nash equilibria. Strategies are statements of individual preferences on Rn. Outcomes are social preferences. Preferences over outcomes are defined by a distance from a bliss point. We prove that g is straightforward if and only if g is locally constant or dictatorial (LCD), i.e., coordinate-wise either a constant or a projection map locally for almost all strategy profiles. We also establish that: (i) If a game is straightforward and respects unanimity, then the map g must be continuous, (ii) Straightforwardness is a nowhere dense property, (iii) There exist differentiable straightforward games which are non-dictatorial. (iv) If a social choice rule is Nash implementable, then it is straightforward and locally constant or dictatorial.

The first versions of these results were completed in 1979, and they were then revised and extended in 1980 and 1981. Versions were circulated as Essex working papers under the titles “Incentives to Reveal Preferences”, “Incentive Compatibility and Local Simplicity” and “A Necessary and Sufficient Condition for Straightforwardness”. Research support from NSF Grants. SES 79–14050, 92–16028 and 91–10460 and the United Kingdom S.S.R.C. is gratefully acknowledged.