A constrained egalitarian solution for convex multi-choice games

Abstract

This paper deals with a constrained egalitarian solution for convex multi-choice games named the d value. It is proved that the d value of a convex multi-choice game belongs to the precore, Lorenz dominates each other element of the precore and possesses a population monotonicity property regarding players’ participation levels. Furthermore, an axiomatic characterization is given where a specific consistency property plays an important role.

Appendix

Proof of Lemma 5.1

By BMP

$$ \sum_{i\in {\rm car}(s^{\max })}\sum_{j\leq s_{i}^{\max }}\varphi _{ij}(N,m,v)\leq v(s^{\max }). $$

Suppose

$$ \sum_{i\in {\rm car}(s^{\max })}\sum_{j\leq s_{i}^{\max }}\varphi _{ij}(N,m,v)<v(s^{\max }). $$

Since φ satisfies LPM, we know that for all i ∈ car(s ^max) and $j\leq s_{i}^{\max }, ij\in \arg \max \{\varphi _{kl}(N,m,v)\left| k\in N,l\in M_{k}\right. \}.$ On the other hand, since all such ij received the same payoff, we obtain

$$ \left|| s^{\max }\right|| \varphi _{ij}(N,m,v)<v(s^{\max })=\left|| s^{\max }\right|| \alpha (s^{\max },v). $$

Hence, we have

$$ \varphi _{ij}(N,m,v)<\alpha (s^{\max },v),\quad{\text{ for all }}i\in {\rm car}(s^{\max }) \hbox{ and }j\leq s_{i}^{\max }, $$

which is a direct contradiction with φ satisfying EDS. $\square$

Proof of Lemma 5.2

(i) Since φ satisfies EDS there exist i′ ∈ car(s ¹) and j′ ≤ s ¹ _i with $\varphi _{i^{\prime }j^{\prime }}(N,m,v)\geq \alpha (s^{1},v).$ From the proof of Lemma 5.1, we have that

$$ \alpha (s^{\max },v)=\varphi _{kl}(N,m,v)\geq \varphi _{i^{\prime }j^{\prime }}(N,m,v)\geq \alpha (s^{1},v), $$

for all $kl\in \arg \max \{\varphi _{i^{\prime }j^{\prime }}(N,m,v)\}.$ Since α(s ¹, v) ≥ α(s, v) for all s, we obtain α(s ^max, v) = α(s ¹, v).

(ii) Since s ≤ s ^max by (i) we have

$$ \begin{aligned} \sum_{i\in {\rm car}(s)}\sum_{j\leq s_{i}}\varphi _{ij}(N,m,v) =&\left|| s\right|| \alpha (s^{\max },v)=\left|| s\right|| \alpha (s^{1},v)\\ \geq &\left|| s\right|| \alpha (s,v)=v(s).\square \end{aligned} $$

Proof of Lemma 5.3

If t = 0, then $v_{-s^{\max }}({\bf 0})=0=v(s^{\max })-v(s^{\max }).$

If t = m − s ^max, by Lemma 5.1 and the definition of the reduced game, we can derive that

$$ \begin{aligned} v_{-s^{\max }}(m-s^{\max})\,=&\,v(m)-\sum_{i\in {\rm car}(s^{\max })}\sum_{j\leq s_{i}^{\max }}\varphi _{ij}(N,m,v)\\ =&\,v(m-s^{\max }+s^{\max })-v(s^{\max }). \end{aligned} $$

If ${\bf 0}<t<m-s^{\max },$ from Lemma 5.2 (ii) we have for each s ≤ s ^max, 

$$ v(s+t)-\sum_{i\in {\rm car}(s)}\sum_{j\leq s_{i}}\varphi _{ij}(N,m,v)\leq v(s+t)-v(s). $$

In particular, for s = s ^max, we obtain $v_{-s^{\max }}(t)\leq v(s^{\max }+t)-v(s^{\max }).$ The reverse inequality follows in a similar way from BMP. Therefore, we conclude that the result holds. $\square$