On the axiomatic characterization of the coalitional multinomial probabilistic values

Abstract

The coalitional multinomial probabilistic values extend the notion of multinomial probabilistic value to games with a coalition structure, in such a way that they generalize the symmetric coalitional binomial semivalues and link and combine the Shapley value and the multinomial probabilistic values. By considering the property of balanced contributions within unions, a new axiomatic characterization is stated for each one of these coalitional values, provided that it is defined by a positive tendency profile, by means of a set of logically independent properties that univocally determine the value. Two applications are also shown: (a) to the Madrid Assembly in Legislature 2015–2019 and (b) to the Parliament of Andalucía in Legislature 2018–2022.

Appendix: logical independence

Proposition A.1

The axiomatic system used in Theorem 3.2 is logically independent.

Proof

We will assume that a player set N (with $n=|N|\ge 2$) and a positive tendency profile $\varvec{p}$ in N are given. We will abbreviate the properties as follows: LI = linearity, DP = dummy player property, BC = balanced contributions within unions, TP = coalitional $\varvec{p}$-multinomial total power property, and WU = property of $\varvec{p}$-weighted payoffs for quotients of unanimity games.

• 1. LI is logically independent of DP, BC, TP and WU. We define a coalitional value g for all $[v;B]\in {\mathcal {G}}_N^{{\text {cs}}}$ as follows.

  • (a) Unanimity games. If $v=u^T$ with $\emptyset \ne T\subseteq N$ then

    $$\begin{aligned} g_i[v;B]=\Lambda _i^{\varvec{p}}[v;B]\quad \text { for each}i\in N. \end{aligned}$$

  • (b) Otherwise, that is, if v is not a unanimity game,

    $$\begin{aligned} g_i[v;B]= {\left\{ \begin{array}{ll} \Lambda _i^{\varvec{p}}[v;B]=v(\{i\})\quad \text { if }i \text { is a dummy in} v,\\ \dfrac{1}{b'_k}\displaystyle \sum _{j\in B'_k}\Lambda _j^{\varvec{p}}[v;B] \quad \text { if} i\in B_k\in B \text { is not a dummy in }v, \end{array}\right. } \end{aligned}$$

    where $B'_k$ denotes the set of players of $B_k$ that are not dummies in v, and $b'_k=|B'_k|$. DP, BC, TP and WU are clearly satisfied by g. However, this value fails to satisfy LI for e.g. game $v=u^{\{1\}}+2u^{\{1,2\}}$ and the trivial coalition structure $B=B^N$.

• 2. DP is logically independent of LI, BC, TP and WU. We define a coalitional value g for all $[v;B]\in {\mathcal {G}}_N^{{\text {cs}}}$ as follows. For each $i\in N$,

  $$\begin{aligned} g_i[v;B]=\dfrac{v(N)}{n}\quad \text { if} \; B=B^N, \end{aligned}$$

  and otherwise

  $$\begin{aligned} g_i[v;B]=\Lambda _i^{\varvec{p}}[v;B]. \end{aligned}$$

  LI, BC, TP and WU are clearly satisfied by g. However, this value fails to satisfy DP for the unanimity game $v=u^{\{1\}}$ and the trivial coalition structure $B=B^N$.

• 3. BC is logically independent of LI, DP, TP and WU. We define a coalitional value g for all $[v;B]\in {\mathcal {G}}_N^{{\text {cs}}}$ as follows.

  • (a) If $n=2$ and $B=B^N$, then

    $$\begin{aligned} {\left\{ \begin{array}{ll} g_1[v;B]=v(\{1\}),\\ g_2[v;B]=v(N)-v(\{1\}).\\ \end{array}\right. } \end{aligned}$$

  • (b) Otherwise, that is, if some of the above conditions does not hold,

    $$\begin{aligned} g_i[v;B]=\Lambda _i^{\varvec{p}}[v;B]\quad \text { for each }i\in N. \end{aligned}$$

    LI, DP, TP and WU are clearly satisfied by g. However, this value fails to satisfy BC for the unanimity game $v=u^{\{1,2\}}$ and the trivial coalition structure $B=B^N$.

• 4. TP is logically independent of LI, DP, BC and WU. We define a coalitional value g for all $[v;B]\in {\mathcal {G}}_N^{{\text {cs}}}$ as follows.

  • (a) If $n=2$ and $B=B^n$, then

    $$\begin{aligned} {\left\{ \begin{array}{ll} g_1[v;B]=v(\{1\}),\\ g_2[v;B]=v(\{2\}).\\ \end{array}\right. } \end{aligned}$$

  • (b) Otherwise, that is, if some of the above conditions does not hold,

    $$\begin{aligned} g_i[v;B]=\Lambda _i^{\varvec{p}}[v;B]\quad \text { for each }i\in N. \end{aligned}$$

    LI, DP, BC and WU are clearly satisfied by g. However, this value fails to satisfy TP for the unanimity game $v=u^{\{1,2\}}$ and the trivial coalition structure $B=B^n$.

• 5. WU is logically independent of LI, DP, BC and TP. Let $\alpha ,\beta$ be real numbers such that $\alpha +\beta =p_1+p_2$ and $\alpha p_1\ne \beta p_2$. We define a coalitional value g for all $[v;B]\in {\mathcal {G}}_N^{{\text {cs}}}$ as follows.

  • (a) If $n=2$ and $B=B^n$, then

    $$\begin{aligned} {\left\{ \begin{array}{ll} g_1[v;B]=v(\{1\})+\alpha [v(N)-v(\{1\})-v(\{2\})],\\ g_2[v;B]=v(\{2\})+\beta [v(N)-v(\{1\})-v(\{2\})].\\ \end{array}\right. } \end{aligned}$$

  • (b) Otherwise, that is, if some of the above conditions does not hold,

    $$\begin{aligned} g_i[v;B]=\Lambda _i^{\varvec{p}}[v;B]\quad \text { for each }i\in N. \end{aligned}$$

    LI, DP, BC and TP are clearly satisfied by g. However, this value fails to satisfy WU for the unanimity game $v=u^{\{1,2\}}$ and the trivial coalition structure $B=B^n$.$\square$

Proposition A.2

The axiomatic system used in Theorem 3.4is logically independent.

Proof

We continue to assume that a player set N (with $n=|N|\ge 2$) and a positive tendency profile $\varvec{p}$ in N are given. We will continue to abbreviate the properties as follows: LI = linearity, DP = dummy player property, BC = balanced contributions within unions, TP = coalitional $\varvec{p}$-multinomial total power property, and WU = property of $\varvec{p}$-weighted payoffs for quotients of unanimity games. We will also abbreviate TR = transfer property.

Let us consider now the restriction to simple games of each coalitional value g defined for Proposition A.1 in parts 2, 3, 4 and 5. In each case, this restriction is a coalitional power index that satisfies the desired properties with the sole exception of linearity, which does not make sense for simple games. But the transfer property is also satisfied by all these coalitional values. The argument is as follows.

The composition laws $\vee$ and $\wedge$ make sense and satisfy

$$\begin{aligned} u\vee v + u\wedge v = u + v, \end{aligned}$$

for all cooperative games. If u and v are simple games and, for a while, we think of them as cooperative games then, using linearity, we obtain for each g used in Proposition A.1

$$\begin{aligned} g[u\vee v;B] + g[u\wedge v;B] = g[u;B] + g[v;B]\quad \text { for all }B, \end{aligned}$$

which is precisely TR.

Moreover, all counterexamples provided in cases 2, 3, 4 and 5 for Proposition A.1 are simple games, so they can be used also for this Proposition A.2. Thus, it only remains to check the following.

1 bis. TR is logically independent of DP, BC, TP and WU.

We define a coalitional power index g for all $[v;B]\in \mathcal {SG}_N^{{\text {cs}}}$ as follows.

• (a) If $n=2$, $v=u^{\{1\}}\vee u^{\{2\}}$ and $B=B^n$, then

  $$\begin{aligned} {\left\{ \begin{array}{ll} g_1[v;B]=2,\\ g_2[v;B]=-p_1-p_2.\\ \end{array}\right. } \end{aligned}$$

• (b) Otherwise

  $$\begin{aligned} g_i[v;B]=\Lambda _i^{{\varvec{p}}}[v;B]\quad \text { for each }i\in N. \end{aligned}$$

  DP, BC, TP and WU are clearly satisfied by g. However, this power index fails to satisfy TR for game $v=u^{\{1\}}\vee u^{\{2\}}$ and the trivial coalition structure $B=B^n$.

$\square$