The projective core of symmetric games with externalities

Abstract

The purpose of this paper is to study which coalition structures have stable distributions. We employ the projective core as a stability concept. Although the projective core is often defined only for the grand coalition, we define it for every coalition structure. We apply the core notion to a variety of economic models including the public goods game, the Cournot and Bertrand competition, and the common pool resource game. We use a partition function to formulate these models. We argue that symmetry is a common property of these models in terms of a partition function. We offer some general results that hold for all symmetric partition function form games and discuss their implications in the economic models. We also provide necessary and sufficient conditions for the projective core of the models to be nonempty. In addition, we show that our results hold even in the presence of small perturbations of symmetry.

Appendix

Proof of Lemma 3.1

First, we have $\sigma ^{S^*}({\mathcal {P}}^S)=\sigma ^{S^*}(\{S\}\cup {\mathcal {P}}_{N{\setminus } S})= \{\sigma ^{S^*}(S)\}\cup \sigma ^{S^*}({\mathcal {P}}_{N{\setminus } S})$. We now focus on $\sigma ^{S^*}({\mathcal {P}}_{N{\setminus } S})$. Then, we have

$$\begin{aligned} \sigma ^{S^*}({\mathcal {P}}_{N{\setminus } S})= & {} \sigma ^{S^*}(\{(N{\setminus } S)\cap C|C\in {\mathcal {P}}\})\\= & {} \{\sigma ^{S^*}((N{\setminus } S)\cap C)|C\in {\mathcal {P}}\}\\= & {} \{\sigma ^{S^*}(N{\setminus } S)\cap \sigma ^{S^*}(C)|C\in {\mathcal {P}}\}\\= & {} \{(N{\setminus } \sigma ^{S^*}(S))\cap \sigma ^{S^*}(C)|C\in {\mathcal {P}}\}\\&\mathop = \limits ^{{S^*\in {\mathcal {P}}}}&\{(N{\setminus } \sigma ^{S^*}(S))\cap C|C\in {\mathcal {P}}\} ={\mathcal {P}}_{N{\setminus } \sigma ^{S^*}(S)}. \end{aligned}$$

Therefore, we obtain $\{\sigma ^{S^*}(S)\}\cup {\mathcal {P}}_{N{\setminus } \sigma ^{S^*}(S)}={\mathcal {P}}^{\sigma ^{S^*}(S)}$. $\square$

Proof of Proposition 3.2

The proof of $\Leftarrow$ is clear. Below, we show $\Rightarrow$. Let ${\mathcal {P}}=\{S_1,\ldots ,S_{|{\mathcal {P}}|}\}$. For every $S\in {\mathcal {P}}$, let $\sigma ^S$ satisfy $\sigma ^S(i)=i$ for every $i\in N{\setminus } S$, i.e., $\sigma ^S$ denotes a permutation only for the members in S. Let $x\in C^{\text {proj}}(v,{\mathcal {P}})$. From the definition of $C^{\text {proj}}$, it follows that $\sum _{j\in S}x_j\ge v(S,{\mathcal {P}}^S)$ for any $S\subseteq N$, and $\sum _{j\in S}x_j= v(S,{\mathcal {P}})$ for any $S\in {\mathcal {P}}$. For any $\sigma$, we define $\sigma (x)_i:=x_{\sigma (i)}$ for any $i\in N$. Then, for any $S\subseteq N$, we have $\sum _{j\in S}\sigma ^{S_1}(x)_j =\sum _{j\in S}x_{\sigma ^{S_1}(j)} =\sum _{j\in \sigma ^{S_1}(S)}x_{j} \ge v(\sigma ^{S_1}(S),{\mathcal {P}}^{\sigma ^{S_1}(S)}),$ by Lemma 3.1, which equals $v(\sigma ^{S_1}(S),\sigma ^{S_1}({\mathcal {P}}^S))=v(S,{\mathcal {P}}^S).$ Similarly, it follows from Lemma 3.1, and $v\in {\mathcal {G}}_N^{S}$ that for every $S\in {\mathcal {P}}$, $\sum _{j\in S}\sigma ^{S_1}(x)_j=v(S,{\mathcal {P}})$. Hence, $\sigma ^{S_1}(x)\in C^{\text {proj}}(v,{\mathcal {P}})$. This holds for every permutation $\sigma ^{S_1}$. Note that there are $|S_1|!$ permutations that arrange the members in $S_1$. We denote the set of the $|S_1|!$ permutations by $A^{S_1}$. We define $y:=\frac{1}{|S_1|!}\sum _{\sigma ^{S_1}\in A^{S_1}}\sigma ^{S_1}(x)$. For any player $i\in S_1$, we have $y_i=\frac{1}{|S_1|!}\sum _{\sigma ^{S_1}\in A^{S_1}}\sigma ^{S_1}(x)_i =\frac{1}{|S_1|!}(|S_1|-1)!\left( \sum _{j\in S_1}x_j\right)$. This is equivalent to $\frac{1}{|S_1|!}(|S_1|-1)!v(S_1,{\mathcal {P}})$, and we obtain $\frac{v(S_1,{\mathcal {P}})}{|S_1|}$. Hence, allocation y is given as follows: $y_i:=\frac{v(S_1,{\mathcal {P}})}{|S_1|}$ for every $i\in S_1$, $y_i:=x_i$ for every $i\in N{\setminus } S_1$; and, in view of the convexity of the core, belongs to $C^{\text {proj}}(v,{\mathcal {P}})$, namely, with slightly abusing the notation, $y=(e^{S_1},x^{S_2},\ldots ,x^{S_{|{\mathcal {P}}|}})\in C^{\text {proj}}(v,{\mathcal {P}})$. We repeat this process for each $S_2,\ldots ,S_{|{\mathcal {P}}|}$ and obtain $e(v,{\mathcal {P}})\in C^{\text {proj}}(v,{\mathcal {P}})$. $\square$

Proof of Proposition 3.3

We show that $e(v,{\mathcal {P}})\not \in C^{\text {proj}}(v,{\mathcal {P}})$. Let S and $S'$ in ${\mathcal {P}}$ satisfy $|S|>|S'|$ and $\frac{v(S,{\mathcal {P}})}{|S|}>\frac{v(S',{\mathcal {P}})}{|S'|}$. There exists a coalition $T\subseteq N$ such that $|T|=|S|$ and $T=S' \cup R$ for some $\emptyset \ne R\subsetneq S$. We have $|S{\setminus } T|=|S'|$. Since $v\in {\mathcal {G}}_N^{S}$, we have $v(T,\{T\}\cup {\mathcal {P}}_{N{\setminus } T}) =v(T,\{T,S{\setminus } T\}\cup {\mathcal {P}}_{N{\setminus } (T\cup S)}\}) =v(S,\{S,S'\}\cup {\mathcal {P}}_{N{\setminus } (S\cup S')}) =v(S,{\mathcal {P}}).$ Hence, we obtain

$$\begin{aligned} \sum _{j\in T}e_j(v,{\mathcal {P}})= & {} |R|\frac{v(S,{\mathcal {P}})}{|S|}+|S'|\frac{v(S',{\mathcal {P}})}{|S'|} <|R|\frac{v(S,{\mathcal {P}})}{|S|}+|S'|\frac{v(S,{\mathcal {P}})}{|S|}\\= & {} |T|\frac{v(S,{\mathcal {P}})}{|S|}\\= & {} |T|\frac{v(T,\{T\}\cup {\mathcal {P}}_{N{\setminus } T})}{|T|} =v(T,\{T\}\cup {\mathcal {P}}_{N{\setminus } T}) =v(T,{\mathcal {P}}^T). \end{aligned}$$

Thus, $e(v,{\mathcal {P}})\not \in C^{\text {proj}}(v,{\mathcal {P}})$. $\square$

Proof of Proposition 3.5

We first show that $C^{\text {proj}}(v,{\mathcal {P}})= \emptyset$ for any ${\mathcal {P}}\in \varPi (N){\setminus } \{N\}$. In view of Lemma 3.4, consider ${\mathcal {P}}\ne \{N\}$ satisfying $k^*:= |S'|$ for every $S'\in {\mathcal {P}}$. We have $e_i(v,{\mathcal {P}})=I(k^*)+(|{\mathcal {P}}|-1)E(k^*)$ for every $i\in N$. Consider $S\subseteq N$ with $|S|=k^*+1$. We have

$$\begin{aligned}&v(S,{\mathcal {P}}^S\})-\sum _{j\in S}e_j(v,{\mathcal {P}})\\&\quad =s[I(k^*+1)+E(k^*-1)+(|{\mathcal {P}}|-2)E(k^*)]-s[I(k^*)+(|{\mathcal {P}}|-1)E(k^*)]\\&\quad = s[I(k^*+1)-I(k^*)-(E(k^*)-E(k^*-1))] =s[\varDelta ^I(k^*+1)-\varDelta ^E(k^*)]\\&\quad >s[\varDelta ^I(k^*+1)-\varDelta ^I(k^*)] \ge 0. \end{aligned}$$

Hence, $e(v,{\mathcal {P}})\not \in C^{\text {proj}}(v,{\mathcal {P}})$. From Proposition 3.2, $C^{\text {proj}}(v,{\mathcal {P}})=\emptyset$ follows.

Now, we show that $C^{\text {proj}}(v,\{N\})\ne \emptyset$. For any $i\in N$, $e_i(v,\{N\})=I(n)$. For any $S\subseteq N$, $v(S,\{S,N{\setminus } S\})=s[I(s)+E(n-s)]$. Hence, for any $S\subseteq N$, we have

$$\begin{aligned}&\sum _{j\in S}e_j(v,{\mathcal {P}})-v(S,\{S,N{\setminus } S\})=s[I(n)-I(s)-E(n-s)]\\&\quad =s\left[ \sum _{k=s+1}^n\varDelta ^I(k) - \sum _{k=1}^{n-s}\varDelta ^E(k)\right] \ge s\left[ \sum _{k=1}^{n-s}\varDelta ^I(k) - \sum _{k=1}^{n-s}\varDelta ^E(k)\right] >0, \end{aligned}$$

which implies $C^{\text {proj}}(v,\{N\})\ne \emptyset$. $\square$

Proof of Proposition 4.3

If-part: For any coalition S with $|S|=:s$, we have

$$\begin{aligned} v(S,\{S,N{\setminus } S\})= \left\{ \begin{array}{ll} \max _{p\le c(n-s)}q(p)(p-c(s)) &{}\quad \text { for }h\le s\le n-1,\\ 0 &{}\quad \text{ for } 1\le s\le h-1. \end{array} \right. \end{aligned}$$

(1)

Hence, for any $S\subseteq N$ with $h\le s\le n-1$, we obtain $\sum _{j\in S}e_j(v,N)=s\frac{\pi ^\text {m}}{n} \ge d^{c,q}(s) \overset{(1)}{=}v(S,\{S,N{\setminus } S\}).$ For any $S\subseteq N$ with $1\le s\le h-1$, we have $\sum _{j\in S}e_j(v,N)=s\frac{\pi ^\text {m}}{n}\ge 0 \overset{(1)}{=}v(S,\{S,N{\setminus } S\}).$ Thus, $e(v,N)\in C^{\text {proj}}(v,\{N\})$.

Only-if-part: From Proposition 3.2, it follows that $e(v,N)\in C^{\text {proj}}(v,\{N\})$. We have $v(S,\{S,N{\setminus } S\}) \le |S|\frac{\pi ^\text {m}}{n}$ for every $S\subseteq N$. Hence, in view of (1), for every $s=h,\ldots ,n-1$, $d^{c,q}(s)\le s\frac{\pi ^\text {m}}{n}$. $\square$

Proof of Proposition 5.1

We fix a PCP game f and a partition ${\mathcal {P}}\in \varPi (N)$. For convenience, we offer the definition of $g^{f,{\mathcal {P}}}$ again:

$$\begin{aligned} g^{f,{\mathcal {P}}}(k)= \left\{ \begin{array}{ll} (|{\mathcal {P}}|-k+1)f(|{\mathcal {P}}|) &{}\quad k\le |{\mathcal {P}}|,\\ \frac{1}{|S_{\max }({\mathcal {P}})|}f(|{\mathcal {P}}|) &{}\quad k>|{\mathcal {P}}|. \end{array}\right. \end{aligned}$$

(2)

Proof of  $\Leftarrow$: Assume that $C^{\text {proj}}(v,{\mathcal {P}})= \emptyset$. Since $f(|{\mathcal {P}}|)\ge 0$, $X_+(v,{\mathcal {P}}):=\{x\in X(v,{\mathcal {P}})|x_j\ge 0\text { for any }j\in N\}$ is not empty. Let $x\in X_+(v,{\mathcal {P}})$. As the projective core for ${\mathcal {P}}$ is empty, there exists a coalition $S\subseteq N$ such that

$$\begin{aligned} \sum _{j\in S}x_j< v(S,{\mathcal {P}}^S) = f(k), \end{aligned}$$

(3)

where $k=|{\mathcal {P}}^S|$. For the coalition S, define ${\mathcal {T}}^S=\{T\in {\mathcal {P}}|\ T\in {\mathcal {P}}_S\}$. Note that $|{\mathcal {T}}^S|=|{\mathcal {P}}|-k+1$. If $|{\mathcal {T}}^S|\ge 1$, then $k\le |{\mathcal {P}}|$. We have $\sum _{j\in S}x_j = |{\mathcal {T}}^S|\cdot f(|{\mathcal {P}}|)+\sum _{j\in S{\setminus } (\cup _{T\in {\mathcal {T}}^S}T)}x_j \ge |{\mathcal {T}}^S|\cdot f(|{\mathcal {P}}|) =(|{\mathcal {P}}|-k+1)f(|{\mathcal {P}}|).$ Hence, from (3), it follows that $f(k)> (|{\mathcal {P}}|-k+1)f(|{\mathcal {P}}|)$. However, in view of (2), for any $k\le |{\mathcal {P}}|$, $f(k)\le g^{f,{\mathcal {P}}}(k)$ implies that $f(k)\le (|{\mathcal {P}}|-k+1)f(|{\mathcal {P}}|)$. This is a contradiction.

Next, if $|{\mathcal {T}}^S|= 0$, then $k=|{\mathcal {P}}|+1$. Since for any $x\in X_+(v,{\mathcal {P}})$ there exists $S\subseteq N$ satisfying (3), such a coalition S also exists for the equal division $e(f,{\mathcal {P}})\in X_+(v,{\mathcal {P}})$, i.e.,  $e_j(f,{\mathcal {P}})=\frac{f(|{\mathcal {P}}|)}{|{\mathcal {P}}(j)|}$ for every $j\in N$. Note that $e_j(f,{\mathcal {P}})\ge 0$ for every $j\in N$ as $f(|{\mathcal {P}}|)\ge 0$. Hence, there exists a player $i\in S$ such that $e_i(f,{\mathcal {P}})\le \sum _{j\in S}e_j(f,{\mathcal {P}})<f(k)$ by (3). Moreover, $e_i(f,{\mathcal {P}})=\frac{f(|{\mathcal {P}}|)}{|{\mathcal {P}}(i)|}\ge \frac{f(|{\mathcal {P}}|)}{|S_{\max }({\mathcal {P}})|}$. Hence, we have $\frac{f(|{\mathcal {P}}|)}{|S_{\max }({\mathcal {P}})|}< f(k)$. However, in view of (2), for any $k> |{\mathcal {P}}|$, $f(k)\le g^{f,{\mathcal {P}}}(k)$ implies that $f(k)\le \frac{f(|{\mathcal {P}}|)}{|S_{\max }({\mathcal {P}})|}$. This is a contradiction.

Proof of $\Rightarrow$: We show that if there exists $k\in \{1,\ldots ,|{\mathcal {P}}|+1\}$ such that $f(k)>g^{f,{\mathcal {P}}}(k)$, then $C^{\text {proj}}(v,{\mathcal {P}})= \emptyset$. Assume $x\in C^{\text {proj}}(v,{\mathcal {P}})$. If $k\le |{\mathcal {P}}|$, we have $f(k)>(|{\mathcal {P}}|-k+1)f(|{\mathcal {P}}|)=\sum _{a=1}^{|{\mathcal {P}}|-k+1}\sum _{j\in S_a}x_j$, where $S_1,\ldots ,S_{|{\mathcal {P}}|-k+1}$ are arbitrary $|{\mathcal {P}}|-k+1$ coalitions in ${\mathcal {P}}$. Hence, $|{\mathcal {P}}|-k+1$ coalitions in ${\mathcal {P}}$ have an incentive to jointly deviate by merging and obtain f(k) in total after the deviation. If $k> |{\mathcal {P}}|$, then (2) implies $f(k)>\frac{1}{|S_{\max }({\mathcal {P}})|}f(|{\mathcal {P}}|)$. Moreover, there exists $i\in S_{\max }({\mathcal {P}})$ such that $x_i<f(k)$, because otherwise for any $j\in S_{\max }({\mathcal {P}})$ we have $x_j\ge f(k)$, which implies $\sum _{j\in S_{\max }({\mathcal {P}})}x_j\ge |S_{\max }({\mathcal {P}})|f(k)$ and $\sum _{j\in S_{\max }({\mathcal {P}})}x_j=f(|{\mathcal {P}}|)$: a contradiction. Hence, there exists $i\in S_{\max }({\mathcal {P}})$ such that $x_i<f(k)$, and the player i has an incentive to deviate. $\square$