Rationality, aggregate monotonicity and consistency in cooperative games: some (im)possibility results

Abstract

On the domain of cooperative games with transferable utility, we investigate if there are single-valued solutions that reconcile individual rationality, core selection, consistency and monotonicity (with respect to the worth of the grand coalition). This paper states some impossibility results for the combination of core selection with either complement consistency (Moulin, J Econ Theory 36:120–148, 1985) or projected consistency (Funaki, Dual axiomatizations of solutions of cooperative games. Mimeo, Tokyo, 1998), and core selection, max consistency (Davis and Maschler, Naval Res Logist Q 12:223–259, 1965) and monotonicity. By contrast, possibility results are manifest when combining individual rationality, projected consistency and monotonicity.

Appendix

We begin proving lemmas. To give general arguments, Theorem 10 is proved before Theorem 9. Finally, we show the independence of the properties used in the characterization results.

Proof

(Lemma 2) Let $\sigma$ be a single-valued solution satisfying individual rationality and regular aggregate monotonicity on $\varGamma$ . Let $N\in \mathcal {N}$ and $(N,v)\in \varGamma$. Define $(N,v')$ as $v'(S)=v(S)$ for all $S\subset N$, and $v'(N)=\sum _{i\in N}v(i)$. By individual rationality (and feasibility), $\sigma _i(N, v')=v(i)$ for all $i\in N$. By regular aggregate monotonicity, there exists a monotone path $f\in \mathcal {F}_{mon}$ such that, for all $i\in N$,

$$\begin{aligned} \begin{array}{lllll} \displaystyle \sigma _i(N,v)= & {} \displaystyle v(i)+f_i\left( N, v(N)-\sum _{i\in N}v(i)\right) . \end{array} \end{aligned}$$

Let us see that f is additive. Let $N\in \mathcal {N}$ and $t, t'\in \mathbb {R}$. Consider three games $(N,v), (N,v')$ and $(N,v'')$ defined as follows: for all $S\subset N$, $v(S)=v'(S)=v''(S)$, $v(N)-v'(N)= t$ and $v'(N)-v''(N)= t'$. Then,

$$\begin{aligned} \begin{array}{lll} f(N,t+t') &{} = &{} f(N, (v'(N)+t)-(v'(N)-t')) \\ &{} = &{} f(N, v(N)-v''(N)) \\ &{} = &{} \sigma (N,v)-\sigma (N,v'')\\ &{} = &{} \sigma (N,v)-\sigma (N,v')+\sigma (N,v')-\sigma (N,v'')\\ &{} = &{} f(N, v(N)-v'(N))+ f(N, v'(N)-v''(N))\\ &{} = &{} f(N,t)+f(N,t'), \end{array} \end{aligned}$$

where third and fifth equalities follow from $\sigma$ satisfying regular aggregate monotonicity. Hence, $\sigma =CI^f$ where $f\in \mathcal {F}_{mon}$ is additive.

To show the reverse implication, let $\sigma$ be a single-valued solution on $\varGamma$ such that $\sigma =CI^{f}$, for some additive monotone path $f\in \mathcal {F}_{mon}$. Let $N\in \mathcal {N}$ and (N, v) be an essential game. Then, $v(N)-\sum _{i\in N} v(i)\ge 0$ and thus, from the monotonicity of f, $f_i(N, v(N)-\sum _{i\in N} v(i))\ge 0$, for all $i\in N$. Hence, $\sigma _i(N,v)=v(i)+f_i(N, v(N)-\sum _{i\in N} v(i))\ge v(i)$ for all $i\in N$, which proves individual rationality.

To see regular aggregate monotonicity, let $N\in \mathcal {N}$ and consider two games $(N,v), (N,v')$ such that $v(S)=v'(S)$, for all $S\subset N$. Taking into account the additivity of f we have, for all $i\in N$,

$$\begin{aligned} \begin{array}{lll} \displaystyle \sigma _i(N,v)-\sigma _i(N,v')&{} = &{} \displaystyle f_i\left( N, v(N)-\sum _{i\in N} v(i)\right) -f_i\left( N, v'(N)-\sum _{i\in N} v(i)\right) \\ &{} = &{} \displaystyle f_i\left( N, v(N)-\sum _{i\in N} v(i)\right) +f_i\left( N, -v'(N)+\sum _{i\in N} v(i)\right) \\ &{} = &{} \displaystyle f_i\left( N, v(N)-v'(N)\right) . \end{array} \end{aligned}$$

This, together with $f\in \mathcal {F}_{mon}$, proves regular aggregate monotonicity. $\square$

Proof

(Lemma) 3 Let $\sigma$ be a single-valued solution satisfying efficiency, individual rationality and aggregate monotonicity on $\varGamma$.

Define a monotone $\varGamma _{I}\;-\,$selection F as follows. Take $f\in \mathcal {F}_{mon}$. For all $N' \in \mathcal {N}$ and all $(N',v)\in \varGamma _{I}$, define

$$\begin{aligned} f^v(N,t):=\left\{ \begin{array}{ll} \sigma (N, v+t\cdot u_N)-\sigma (N,v) &{}\quad \hbox {if\;} N=N', \\ f(N,t) &{} \quad \hbox {if\;} N\not = N',\\ \end{array} \right. \end{aligned}$$

(5)

for all $N\in \mathcal {N}$ and all $t\in \mathbb {R}$.

Let us show that $f^v\in \mathcal {F}_{mon}$. Clearly, $f^v(N,t)\in \mathbb {R}^{N}$, $f^{v}(N,0)=(0,\ldots ,0)\in \mathbb {R}^{N}$ and, by the efficiency of $\sigma$ and the definition of $f\in \mathcal {F}_{mon}$, $\sum _{i\in N}f_{i}^{v}(N,t)=t$, for all $N\in \mathcal {N}$ and $t\in \mathbb {R}$. Moreover, if $t^{\prime }\in \mathbb {R}$ is such that $t^{\prime }>t$ and $N=N'$, then

$$\begin{aligned} \begin{array}{lll} f^{v}(N,t^{\prime }) &{} = &{} \sigma (N,v+t^{\prime }\cdot u_{N})-\sigma (N,v) \\ &{} \ge &{} \sigma (N,v+t\cdot u_{N})-\sigma (N,v) \\ &{} = &{} f^{v}(N,t) \end{array} \end{aligned}$$

where the inequality follows from the aggregate monotonicity of $\sigma$. If $N\not = N'$, we have $f(N,t')\ge f(N,t)$ since $f\in \mathcal {F}_{mon}$. Consequently, $f^v\in \mathcal {F}_{mon}$ for all $(N', v)\in \varGamma _{I}$, and thus F is a monotone $\varGamma _{I}\;-\,$selection.

Let $N\in \mathcal {N}$ and $(N,v)\in \varGamma$. Since (N, v) can be expressed as $v=v^{I}+(v(N)-v^{I}(N))\cdot u_N$, it is easy to see that

$$\begin{aligned}\sigma (N,v)-\sigma (N,v^{I})=f^{v^{I}}\left( N, v(N)-\sum _{i\in N} v(i)\right) ,\end{aligned}$$

being $f^{v^{I}}$ as defined in (5).

By individual rationality (and feasibility), $\sigma _i\left( N, v^{I} \right) =v(i)$ for all $i\in N$, and thus

$$\begin{aligned} \displaystyle \sigma _i\left( N, v\right) =v(i)+f^{v^{I}}_i\left( N, v(N)-\sum _{i\in N} v(i) \right) . \end{aligned}$$

Hence, $\sigma =CI^F$ being F a monotone $\varGamma _I$-selection.

To show the reverse implication, let $\sigma$ be a single-valued solution on $\varGamma$ such that $\sigma =CI^{F}$, for some monotone $\varGamma _{I}\,-\,$selection F. Hence, for all $N\in \mathcal {N}$, all $(N,v)\in \varGamma$ and all $i\in N$ it holds

$$\begin{aligned} \sigma _i(N,v)=v(i)+f_i^{v^{I}}\left( N, v(N)-\sum _{i\in N} v(i) \right) . \end{aligned}$$

(6)

From (6) it is not difficult to check individual rationality by using symmetric arguments as in the proof of Lemma 2. Efficiency and aggregate monotonicity comes from $f^{v^{I}}\in \mathcal {F}_{mon}$. $\square$

Proof

(Lemma) 4 Let $\sigma$ be a single-valued satisfying individual rationality and projected consistency on $\varGamma$. For one person games, efficiency follows directly from individual rationality (and feasibility). Let $N\in \mathcal {N}$ with $|N|\ge 2$, $(N,v)\in \varGamma$ and $i\in N$. Then, efficiency for one person games implies $\sigma _i(\{i\}, r^{\{i\}}_{P, x}(v) )=r^{\{i\}}_{P, x}(v)(i)=v(N)-\sum _{j\in N\setminus \{i\}}\sigma _j(N,v)$, where $x=\sigma (N,v).$ By projected consistency, $\sigma _i(N,v)=\sigma _i(\{i\}, r^{\{i\}}_{P, x}(v) )$ and thus $\sigma _i(N,v)=v(N)-\sum _{j\in N\setminus \{i\}}\sigma _j(N,v)$, which proves efficiency. $\square$

Proof

(Theorem 10) Let $\sigma$ be a single-valued solution satisfying individual rationality, aggregate monotonicity and projected consistency on $\varGamma$. By Lemma 4, $\sigma$ satisfies efficiency, and by Lemma 3 we conclude that $\sigma =CI^F$ for some monotone $\varGamma _I$- selection F. To show that F is consistent, let $N\in \mathcal {N}, (N,v)\in \varGamma _I, \emptyset \not = N' \subset N,$ and $t \in \mathbb {R}$. Define the game $(N,v')$ as $v'=v+ t \cdot u_N$. Notice that $\left( v'\right) ^{I}=v$. Let us denote $x=\sigma (N, v')$. Then, for all $j\in N'$ we have

$$\begin{aligned} \sigma _j(N,v') = v'(j)+f_j^{(v')^{I}}\left( N, v'(N)-\sum _{i\in N}v'(i)\right) = v'(j)+f_j^{v}\left( N, t\right) . \end{aligned}$$

(7)

From (7), and taking into account the efficiency of $\sigma$, the definition of projected reduced game and the fact that $\left( N',\left( r^{N'}_{P,x}(v')\right) ^{I}\right) =\left( N',\left( v_{|N'}\right) ^{I}\right)$, we obtain

$$\begin{aligned} \begin{array}{lll} \displaystyle \sigma _j\left( N',r^{N'}_{P, x}(v')\right) &{} = &{} \displaystyle r^{N'}_{P, x}(v')(j)+f_j^{\left( r^{N'}_{P, x}(v')\right) ^{I}}\left( N', r^{N'}_{P, x}(v')(N')- \sum _{i\in N'}r^{N'}_{P, x}(v')(i)\right) \\ &{}=&{} \displaystyle v'(j)+f_j^{\left( v_{|N'}\right) ^{I}}\left( N', \sum _{i\in N'}\sigma _i(N,v')- \sum _{i\in N'}v'(i)\right) \\ &{} = &{} \displaystyle v'(j)+f_j^{\left( v_{|N'}\right) ^{I}}\left( N', \sum _{i\in N'} f_i^{v}\left( N, t \right) \right) . \end{array} \end{aligned}$$

(8)

By projected consistency, (7) and (8) must coincide and thus

$$\begin{aligned} f_j^{v}\left( N, t\right) =f_j^{\left( v_{|N'}\right) ^{I}}\left( N', \sum _{i\in N'} f_i^{v}\left( N, t \right) \right) , \end{aligned}$$

which proves that F is consistent.

Hence, $\sigma =CI^F$ being F a consistent monotone $\varGamma _{I}\,-\,$selection.

To show the reverse implication, let $\sigma$ be a single-valued solution on $\varGamma$ such that $\sigma =CI^{F}$, for some consistent monotone $\varGamma _{I}\,-\,$selection F. From Lemma 3, $\sigma$ satisfies individual rationality and aggregate monotonicity. To check projected consistency, let (N, v) be a game and $\emptyset \not = N'\subset N$. Let us denote $x=\sigma (N,v)$. For all $j\in N'$ we have

$$\begin{aligned} \begin{array}{lll} \displaystyle \sigma _j\left( N',r^{N'}_{P, x}(v)\right) &{} = &{} \displaystyle r^{N'}_{P, x}(v)(j)+f_j^{\left( r^{N'}_{P, x}(v)\right) ^{I}}\left( N', r^{N'}_{P, x}(v)(N')- \sum _{i\in N'}r^{N'}_{P, x}(v)(i)\right) \\ &{}=&{} \displaystyle v(j)+f_j^{\left( v_{|N'}\right) ^{I}}\left( N', \sum _{i\in N'}\sigma _i(N,v)- \sum _{i\in N'}v(i)\right) \\ &{} = &{} \displaystyle v(j)+f_j^{\left( v_{|N'}\right) ^{I}}\left( N', \sum _{i\in N'} f_i^{v^{I}}\left( N, v(N)-\sum _{i\in N} v(i)\right) \right) \\ &{} = &{} \displaystyle v(j)+f_j^{v^{I}}\left( N,v(N)-\sum _{i\in N} v(i)\right) \\ &{} = &{} \sigma _j(N,v), \end{array} \end{aligned}$$

(9)

where the second equality comes from the efficiency of $\sigma$ and the definition of projected reduced game, the third one from $\sigma =CI^F$ for some monotone $\varGamma _{I}$- selection F and the last but one from the consistency of F. This proves projected consistency. $\square$

Proof

(Theorem 9) The proof of this theorem can be obtained following the same lines as in Theorem 10’s proof by using Lemma 2 instead of Lemma 3 and the notion of a consistent monotone path f (Definition 4) instead of the notion of a consistent monotone $\varGamma _{I}$- selection F (Definition 7). $\square$

1.1 Independence of the properties

To see that the properties in both Lemma 2 and Theorem 9 are independent, we introduce the following monotone paths:

• 1. Let $\pi$ be a permutation on $\mathbb {N}$. For all $N\in \mathcal {N}$ and all $t\in \mathbb {R},$ define $f^{\sharp }(N,t)=t \cdot e_{\{j\}}$, being $j\in N$ such that $\pi (j)\le \pi (i)$ for all $i \in N$ if |N| is even, and $\pi (j) \ge \pi (i)$ for all $i\in N$ if |N| is odd. If the cardinality of N is even, $f^{\sharp }$ assigns all of amount t to the first player in N according to $\pi$; otherwise, $f^{\sharp }$ assigns t to the last player in N according to $\pi$.

• 2. Let $\pi$ be a permutation on $\mathbb {N}$. For all $N\in \mathcal {N}$ and all $t\in \mathbb {R},$ define

  $$\begin{aligned} \widehat{f^{\pi }}(N,t):= \left\{ \begin{array}{ll} \left\lfloor \frac{t}{|N|}\right\rfloor \cdot e_{N}+ \sum _{i\in S^{*}}e_{\{i\}}\\ \quad +\left( t\,\text {mod}\,|N|- \left\lfloor t\,\text {mod}\,|N|\right\rfloor \right) \cdot e_{\{k\}} &{} \quad \hbox {if\;} t\ge 0 , \\ \\ -\widehat{f^{\pi }}(N,- t) &{}\quad \hbox {if\;} t < 0, \end{array} \right. \end{aligned}$$

  where $S^{*}\subset N$ is formed by the first $\left\lfloor t\,\text {mod}\,|N|\right\rfloor$ players according to $\pi$ (if any) and $k\in N\setminus S^{*}$ with $\pi (k)\le \pi (j)$, for all $j\in N\setminus S^{*}$ ¹². The interpretation of $\widehat{f^{\pi }}$ when $t\ge 0$ is as follows: if $0\le t \le 1$, then $\widehat{f^{\pi }}$ assigns amount t to the first player in N according to $\pi$. If $1 < t \le 2$, the first player receives a unit of t and the second player $t-1$, etc. If $t > |N|$, then after distributing one unit of t to every player, one additional unit is again assigned to the first player, and so on until the amount t is exhausted.

It is not difficult to check that $f^{\sharp }$ is an additive but not consistent monotone path. Thus, $CI^{f^{\sharp }}$ satisfies individual rationality and regular aggregate monotonicity but not projected consistency. In contrast, $\widehat{f^{\pi }}$ is consistent but not additive, which means that $CI^{\widehat{f^{\pi }}}$ satisfies individual rationality and projected consistency but not regular aggregate monotonicity. Finally, ED satisfies regular aggregate monotonicity and projected consistency but not individual rationality.

To see that the properties in both Lemma 3 and Theorem 10 are independent, we introduce two single-valued solutions:

• 1. Let $\sigma ^{*}$ be the single-valued solution on $\varGamma$ defined as follows: for all $N\in \mathcal {N}$, all (N, v) and all $i\in N$, $\sigma ^{*}_i(N,v):=CI_i(N,v)$ if $(N,v)\in \varGamma _E$, and $\sigma ^{*}_i(N,v):=CI_i(N,v)-1$ otherwise.

• 2. For all $N\in \mathcal {N}$ and all $t\in \mathbb {R}$, let $S^N(t)=\{i\in N\,|\, i \ge |t|\}$, where $|t|=t$ if $t\ge 0$ and $|t|=-t$ otherwise. If $S^N(t)\not = \emptyset$, choose $i^{*}\in S^N(t)$ to be such that $i^{*}\le j$, for all $j\in S^N(t)$. If $S^N(t)= \emptyset$, choose $i^{*}\in N$ to be such that $i^{*}\ge j$, for all $j\in N$. Let $g:\mathcal {N}\times \mathbb {R}\rightarrow \bigcup _{N\in \mathcal {N}} \mathbb {R}^N$ be a function defined as $g(N,t)= t \cdot e_{\{i^{*}\}}$. Then, define $\sigma ^g$ to be the single-valued solution on $\varGamma$ defined as follows: for all $N\in \mathcal {N}$, all $(N,v)\in \varGamma$ and all $i\in N$,

  $$\begin{aligned}\sigma ^g_i(N,v):=v(i)+g_i\left( N, v(N)-\sum _{i\in N} v(i)\right) .\end{aligned}$$

The single-valued solution $\sigma ^{*}$ satisfies individual rationality and aggregate monotonicity but neither efficiency nor projected consistency. The function g satisfies conditions (i) and (ii) in the definition of a monotone path (Definition 1), but not condition (iii). Moreover, g satisfies expression () in the definition of consistent monotone path (Definition 4). Then, $\sigma ^g$ satisfies individual rationality and projected consistency (and thus efficiency) but not aggregate monotonicity. Finally, ED satisfies efficiency, aggregate monotonicity and projected consistency but not individual rationality.