Correction to: Hart–Mas-Colell consistency and the core in convex games (International Journal of Game Theory, (2022), 51, 2, (413-429), 10.1007/s00182-021-00798-6)

Theorem 5.1 of the original article is incorrect. In particular, the proofs of weak and converse HM-consistency are wrong. Indeed, the following example shows that, for 0<δ<1, neither the homothetic image of the core with the Shapley value as center and δ as ratio, C, nor its relative interior, riC, satisfies weak HM-consistency or converse HM-consistency. In the following example, we use the notation of the original article. We assume that N={1,2,3}⊆U and consider the unanimity game (N, v) on N defined by, for all S⊆N,v(S)=1ifS=N,0otherwise. Let 0<δ<1, put x=2δ+13,1-δ3,1-δ3, and note that x∈C(N,v). Let T={1,2} and assume that x=(xS) is an allocation scheme with x=x and x∈C(S,v) for all S∈2\{∅} such that x∈C(T,v). Then v({i})=0 for i=1,2, and v(T)=2+δ3. Therefore, (Formula presented.) However, x=1-δ3<(2+δ)(1-δ)6 so that x∉C(T,v). Similarly, it is shown that, for ε>0 small enough, y:=x+(-2ε,ε,ε)∈riC(N,v) and y∉riC(T,v) for each allocation scheme y=(yS) with y=y and y∈riC(S,v) for all S∈2\{∅}. Theorem 5.1 can be corrected by changing the definition of C into a recursive definition. Let δ∈[0,1] and (N,v)∈Γ. If |N|≤2, then define C(N,v)=δC(N,v)+(1-δ)ϕ(N,v) (as before). Let k≥3 and assume that C(N,v) has been defined for |N|