We discuss a problem concerning Dasgupta, Hammond, and Maskin''s (1979) definition of a rich domain and a very well-known result they established for these domains: on rich domains, if a social choice function is implementable in Nash strategies, then it is truthfully implementable in dominant strategies Dasgupta, Hammond, and Maskin''s (1979, Theorem 7.2.3). This result is cited many times in later papers, e.g., Laffont and MaskinL(1982, Theorem 4) and Maskin (1985, Theorem 7). Dasgupta, Hammond, and Maskin''s (1979) proof of this result essentially is based on showing that (Maskin) monotonicity implies strategy-proofness (or equivalently independent person-by-person monotonicity IPM).In the sequel we abbreviate Dasgupta, Hammond, and Maskin (1979) by DHM. We describe DHM''s model in Section 2.In Section 3 we first construct an example of a DHM rich domain and a social choice function that is monotonic but not strategy-proof (Example 1). This suggests that DHM''s rich domain definition is not sufficient to show that monotonicity implies strategy-proofness (or that Nash implementability implies truthful implementability in dominant strategies). We then investigate which step in DHM''s proof is problematic - since DHM do not give a direct proof of the result, we reproduce Maskin’s (1985) proof. In Section 4, we consider the presentation of Dasgupta, Hammond, and Maskin''s (1979) definition of a rich domain in the Maskin and Sjöström (2002). It turns out that there definition of a rich domain is different from DHM''s original definition. With this adjusted richness condition the proof that monotonicity implies strategy-proofness is correct.
