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^δ, nor its relative interior, riC^δ, 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^δ(N,v). Let T={1,2} and assume that x=(xS)_S∈2N\{∅} is an allocation scheme with x^N=x and x^S∈C^δ(S,v_S) for all S∈2^N\{∅} such that x_T∈C^δ(T,v_T^x). Then v_T^x({i})=0 for i=1,2, and v_T^x(T)=2+δ3. Therefore, (Formula presented.) However, x₂=1-δ3<(2+δ)(1-δ)6 so that x_T∉C^δ(T,v_T^x). Similarly, it is shown that, for ε>0 small enough, y:=x+(-2ε,ε,ε)∈riC^δ(N,v) and y_T∉riC^δ(T,v_T^y) for each allocation scheme y=(yS)_S∈2N\{∅} with y^N=y and y^S∈riC^δ(S,v_S) for all S∈2^N\{∅}. Theorem 5.1 can be corrected by changing the definition of C^δ into a recursive definition. Let δ∈[0,1] and (N,v)∈Γ_vex. If |N|≤2, then define C^δ(N,v)=δC(N,v)+(1-δ)ϕ(N,v) (as before). Let k≥3 and assume that C^δ(N,v) has been defined for |N|<k. If |N|=k, then define (Formula presented.) Note that C^δ is the unique smallest subsolution (subject to inclusion) of the core that coincides with C^δ for one- and two-person games and satisfies converse HM-consistency. This definition guarantees that it automatically satisfies weak HM-consistency as well. Moreover, riC^δ(N,v) should be replaced by C̲^δ(N,v), which is defined as follows. Let δ∈[0,1] and (N,v)∈Γ_vex. If |N|≤2, then define C̲^δ(N,v)=δriC(N,v)+(1-δ)ϕ(N,v). Let k≥3 and assume that C̲^δ(N,v) has been defined for |N|<k. If |N|=k, then define (Formula presented.) Similarly, C̲^δ is the unique smallest subsolution of the core that coincides with C̲^δ for one- and two-person games and satisfies converse HM-consistency. Hence, it satisfies weak HM-consistency as well. Let δ∈[0,1] and (N,v)∈Γ_vex. Then C̲⁰(N,v)=C⁰(N,v)={ϕ(N,v)},C̲¹(N,v)=riC¹(N,v), and C¹(N,v)=C(N,v). C^δ(N,v) and C̲^δ(N,v) are bounded (as subsets of the core), contain ϕ(N,v), and are convex (which can be shown by induction on |N|). C^δ(N,v) is closed, hence compact. C^δ and C̲^δ satisfy nonemptiness, individual rationality, scale covariance, and superadditivity (which follows again by induction on |N|). C̲⁰(N,v)=C⁰(N,v)={ϕ(N,v)},C̲¹(N,v)=riC¹(N,v), and C¹(N,v)=C(N,v). C^δ(N,v) and C̲^δ(N,v) are bounded (as subsets of the core), contain ϕ(N,v), and are convex (which can be shown by induction on |N|). C^δ(N,v) is closed, hence compact. C^δ and C̲^δ satisfy nonemptiness, individual rationality, scale covariance, and superadditivity (which follows again by induction on |N|). Remark 2 together with a careful inspection of the proof of Theorem 5.1 except the paragraphs on weak and converse HM-consistency show that Theorem 5.1 is correct and the uniqueness part of the proof is left unchanged with the new definition of C^δ and when replacing riC^δ by the new C̲^δ. Therefore, with the aforementioned new definitions, all statements in Sect. 5 of the original article remain valid. In particular, the examples showing that each of the axioms employed in Theorem 5.1 is logically independent of the remaining axioms can still be used.