Strategy-proof Cardinal Decision Schemes

We construct an example to show that the random dictatorship result no longer holds when individual utility functions satisfy this additional restriction. We then analyze the consequences of gradually reducing the grid size. That is, we consider an arbitrary sequence of strategy-proof and unanimous decision schemes defined on a sequence of decreasing grid sizes approaching zero. We obtain a ‘limit’ random dictatorship result in the sense that the sequence of such decision schemes must converge to a random dictatorship for all profiles for which the limit exists.