Behavioral Implementation : Supplemental Appendix

A first natural idea to extend Maskin monotonicity into a property that would be applicable on any domain of individual choice functions is to use Bernheim and Rangel’s (2009) extended revealed preference, as defined at the beginning of Section 4 of the main paper. For each individual i, each state θ, and each option x, the lower contour set LCi(x, θ) of x given θ is now the set of options y such that x unambiguously dominates y given Ci(·, θ). The property of Maskin monotonicity can then be reproduced as stated on page 11 of the main article. Unfortunately, a look at the SCR fij from Proposition 7 in the main text shows that Nash-implementable SCRs may violate this extension of Maskin monotonicity. To see this, consider θ ∈ Θ such that θi ∈ Ai and θj ∈ Θj \ Aj ranks x as the best element of Z. In that case, fij(θ) = x, and the extended lower contour set of x for i is empty. Changing i’s type to θ′ i ∈ Θi \ Ai such that x is second best expands that extended lower contour set to a non-empty set. The extension of Maskin monotonicity would then require fij to pick x at ∗Brown University, Department of Economics, Providence, Rhode Island [email protected].