Correction to: Relationally equal treatment of equals and affine combinations of values for TU games

1 Correction to: Social Choice and Welfare https://doi.org/10.1007/s00355-019-01180-y

We are grateful to Zhengxing Zou for pointing out an insufficiency in the definition of an element of our new basis for the space of TU games, $w_T$, in Sect. 5.1. With keeping the definition of $(N,w_T) \in \Gamma$ as it is for $|T| \ne \frac{n}{2}$, we modify it by

$$\begin{aligned} w_T=u_T-\sum _{\ell \in N\backslash T} u_{N\backslash \ell }. \end{aligned}$$

only for $|T|=\frac{n}{2}$.

According to this modification, the following changes are required.

Proof of Lemma 1

This immediately follows from the fact that $|T| \ge 3$ and $|N| \ge 3$. $\square$

Proof of Lemma 2

For $|T| \ne \frac{n}{2}$, the proof is as it is. For $|T|=\frac{n}{2}$, if $i\in T$, $w_T(N\backslash i)=0-0=0$, and if $i\notin T$, $w_T(N\backslash i)=u_T(N\backslash i)-u_{N\backslash i}(N\backslash i)=1-1=0$. $\square$