The $\chi$ value and team games

Abstract

In this paper we employ the $\chi$ value (Casajus, Games Econ Behav 65(1): 49–61, 2009)—a coalition structure value—to analyse team games (Hernández and Sánchez-Sánchez, Int J Games Theory 39(3): 319–350, 2010) . We answer two questions for two special cases: first, which components are stable and second, how is the worth of a component divided among the members of the component.

Appendix

Comment to Corollary 2

The players $i\in N\backslash C_{1}$ get $\chi _{i}(N,v,\mathcal {P} )=\frac{v\left( C\right) }{t}>0.$ If $\left| C_{1}\right| =t,$ the players $i\in C_{1}$ get the same payoffs$.$ It is not possible to raise these payoffs. If $\left| C_{1}\right| <t$ the $\chi$-payoffs of the players in $C_{1}$ are zero. It is not possible to form an alternative partition $\mathcal {P}^{\prime }=\{K,N\backslash K\}$ with $\left| K\right| =t$ and $C_{1}\cap K\ne \varnothing$ without at least one player from $C_{2},\dots ,C_{m}.$ The $\chi$-payoff of this player is unchanged, hence, $\mathcal {P}$ is $\chi$-stable.

Proof of Theorem 3

The players in components with $\left| C_{i}\right| \ne t$ get the $\chi$-payoff zero. An alternative partition $\mathcal {P}^{\prime }=\{K,N\backslash K\}$ with $\left| K\right| =t$ and $K\subseteq \bigcup \nolimits _{\left| C_{i}\right| \ne t}C_{i}$ raises the payoffs of all $i\in K,$ $\chi _{i}(N,v,\mathcal {P})=\frac{v\left( C\right) }{t}>0,$ hence, $\mathcal {P}$ is not $\chi$-stable. $\square$

Proof of Theorem 5

The bounds with respect to zero are deduced from Hernández-Lamoneda and Sánchez-Sánchez (2010). They state: to every team game is associated a unique (up to sign) power ranking of its players. In the case of $t<n,$ they compute player’s i ranking by:

$$\begin{aligned} I_{i}(N,v)=\frac{1}{\left( \begin{array}{l} n-2 \\ t-1 \end{array} \right) }\sum \nolimits _{i\in T}v\left( T\right) . \end{aligned}$$

This payoff agrees with $\mathrm{{Sh}}_{i}\left( N,v\right)$ up to a constant. For $\left| R\right| <\left| L\right|$ we have $I_{i}(N,v)<I_{j}(N,v);$ $j\in R$ and $i\in L.$

The other bounds are based on the following considerations. The marginal contributions of the players in any rank order are only non-zero at positions t and $t+1; \mathrm{MC}_{i}^{\rho }\left( v\right) =0$ if $i\ne \rho \left( t\right)$ or $i\ne \rho \left( t+1\right) .$ At position $t,$ a player can create a positive worth (number of gloves pairs created with him). At position $t+1$ a player gets non-positive marginal contributions since the worth of the coalition with the player is zero. At position t, the highest marginal contribution a player could achieve is $\frac{t}{2}$. Assume a player $i\in R$ gets always this marginal contribution at position $t$ (and marginal contributions zero at position $t+1)$. The probability for $i=\rho \left( t\right)$ is $\frac{1}{n}.$ Then the player’s Shapley payoff is $\frac{t}{2}\cdot \frac{1}{n}<\frac{1}{2}.$ Analogously we can argue for the lower bound of any player $j\in L.$

Proof of Theorem 6

First we look at the players in components $C_{h+1},\dots ,C_{m}.$ We have to check, whether the $\chi$-payoffs are positive or not. Since $\mathrm{{Sh}}_{i}\left( N,v\right) >\mathrm{{Sh}}_{j}\left( N,v\right) ,$ $i\in R$ and $j\in L,$ we have $\chi _{i}(N,v,\mathcal {P} )>\chi _{j}(N,v,\mathcal {P}),$ $i\in \mathcal {P}\left( j\right) .$ We have:

$$\begin{aligned} \chi _{j}(N,v,\mathcal {P})&= \mathrm{{Sh}}_{j}\left( N,v\right) + \frac{\frac{t}{2}-\frac{t}{2}\cdot \mathrm{{Sh}}_{j}\left( N,v\right) -\frac{t}{2}\cdot \mathrm{{Sh}}_{i}\left( N,v\right) }{t}\end{aligned}$$

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$$\begin{aligned}&= \frac{\frac{t}{2}+\frac{t}{2}\cdot \mathrm{{Sh}}_{j}\left( N,v\right) -\frac{t}{2}\cdot \mathrm{{Sh}}_{i}\left( N,v\right) }{t} \nonumber \\&= \frac{\frac{t}{2}\cdot \underset{>0}{\underbrace{\left( 1+\underset{- \frac{1}{2}<\dots <0}{\underbrace{\mathrm{{Sh}}_{j}\left( N,v\right) }}-\underset{\frac{1}{2}>\dots >0}{\underbrace{\mathrm{{Sh}} _{i}\left( N,v\right) }}\right) }}}{t}>0 \end{aligned}$$

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and, hence, $\chi _{i}(N,v,\mathcal {P})>0.$ Deviating to components $C_{j}$ with $\left| C_{j}\right| =t$ and $G\left( C_{j}\right) <\frac{t}{2}$ reduces the second term of the $\chi$-formula (Eq. 3) and reduces the $\chi$-payoffs of the players. For example, replacing one right glove owner by a left glove owner changes the $\chi$-payoffs by

$$\begin{aligned} \frac{\overset{\text {reducing the number of pairs by one }}{\overbrace{-1}}+ \underset{<1}{\underbrace{\underset{\frac{1}{2}>\dots >0}{\underbrace{ \mathrm{{Sh}}_{i}\left( N,v\right) }}-\underset{-\frac{1}{2}<\dots <0 }{\underbrace{\mathrm{{Sh}}_{j}\left( N,v\right) }}}}}{t}<0, \end{aligned}$$

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$i\in R$ and $j\in L$.

Also, deviating to components $C_{j}$ with $\left| C_{j}\right| \ne t$ is not reasonable for the players. These components get the worth zero. Hence, if some players get positive payoffs some other players get negative payoffs. The last ones would not accept this and could form single-components with players’ payoffs zero.

In a second step we look at the players in components $C_{1}\dots C_{h}.$ They get the payoff zero. Since $\left| \bigcup \nolimits _{C_{1}\dots C_{h}}C_{i}\right| <t,$ amalgamating these components leads to the same problem as described above. The case of forming an alternative partition $\mathcal {P}^{\prime }=\{K,N\backslash K\}$ with $\left| K\right| =t$ and$\left( \bigcup \nolimits _{C_{1}\dots C_{h}}C_{i}\right) \cap K\ne \varnothing$ is discussed in Corollary 2.

Proof of Theorem 8

Again, our first look is on the players in components $C_{h+1},\dots ,$ $C_{m}.$ Analogous to Theorem 6, we check, whether the $\chi$-payoffs are positive or not. For $j\in L,$ we have:

$$\begin{aligned}&\left. \chi _{j}(N,v,{\mathcal {P}})\right. \nonumber \\&= \mathop {\mathrm{Sh}}\nolimits _{j}\left( N,v\right) +\frac{\frac{t}{2}-\frac{1}{2} -\left( \frac{t}{2}+\frac{1}{2}\right) \cdot \mathop {\mathrm{Sh}}\nolimits _{j}\left( N,v\right) -\left( \frac{t}{2}-\frac{1}{2}\right) \cdot \mathop {\mathrm{Sh}} \nolimits _{i}\left( N,v\right) }{t} \nonumber \nonumber \\&= \frac{\frac{t}{2}-\frac{1}{2}+\frac{t}{2}\cdot \mathop {\mathrm{Sh}} \nolimits _{j}\left( N,v\right) -\frac{1}{2}\cdot \mathop {\mathrm{Sh}} \nolimits _{j}\left( N,v\right) -\frac{t}{2}\cdot \mathop {\mathrm{Sh}} \nolimits _{i}\left( N,v\right) +\frac{1}{2}\cdot \mathop {\mathrm{Sh}} \nolimits _{i}\left( N,v\right) }{t} \nonumber \nonumber \\&= \frac{\frac{t}{2}\cdot \left( 1+\mathop {\mathrm{Sh}}\nolimits _{j}\left( N,v\right) -\mathop {\mathrm{Sh}}\nolimits _{i}\left( N,v\right) \right) -\frac{1}{2} \cdot \left( 1+\mathop {\mathrm{Sh}}\nolimits _{j}\left( N,v\right) -\mathop {\mathrm{Sh}} \nolimits _{i}\left( N,v\right) \right) }{t} \nonumber \nonumber \\&= \frac{\underset{>0}{\underbrace{\left( \frac{t}{2}-\frac{1}{2}\right) }} \cdot \underset{>0}{\underbrace{\left( 1+\underset{-\frac{1}{2}<\dots <0}{ \underbrace{\mathop {\mathrm{Sh}}\nolimits _{j}\left( N,v\right) }}-\underset{\frac{1}{ 2}>\dots >0}{\underbrace{\mathop {\mathrm{Sh}}\nolimits _{i}\left( N,v\right) }} \right) }}}{t}>0 \end{aligned}$$

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and $\chi _{i}(N,v,\mathcal {P})>0,$ $i\in R.$ Deviating to components $C_{j}$ with $\left| C_{j}\right| =t$ and $\left| C_{s}\cap R\right| =\left| C_{s}\cap L\right| +1$ changes the second term of the $\chi$ -formula (equation 3) by

$$\begin{aligned} \frac{\underset{-\frac{1}{2}<\dots <0}{\underbrace{\mathop {\mathrm{Sh}} \nolimits _{j}\left( N,v\right) }}-\underset{\frac{1}{2}>\dots >0}{ \underbrace{\mathop {\mathrm{Sh}}\nolimits _{i}\left( N,v\right) }}}{t}<0 \end{aligned}$$

(12)

and hence reduces the $\chi$-payoffs of the players. The next steps are similar to Theorem 6. $\square$