Generalized three-sided assignment markets: core consistency and competitive prices

Abstract

A generalization of the classical three-sided assignment market is considered, where value is generated by pairs or triplets of agents belonging to different sectors, as well as by individuals. For these markets we represent the situation that arises when some agents leave the market with some payoff by means of a generalization of Owen (Ann Econ Stat 25–26:71–79, 1992) derived market. Consistency with respect to the derived market, together with singleness best and individual anti-monotonicity, axiomatically characterize the core for these generalized three-sided assignment markets. When one sector is formed by buyers and the other by two different type of sellers, we show that the core coincides with the set of competitive equilibrium payoff vectors.

Appendix

Proof of Proposition 2

Let us write ${\hat{w}}=w_{{\hat{\gamma }}^{S,x}}$. We have to show that ${\hat{w}}$ is superadditive, ${\hat{w}}\ge w_{\gamma }^{S,x}$ and ${\hat{w}}$ is minimal with these two properties.

By definition, ${\hat{w}}$ is superadditive. Now, we show that ${\hat{w}}(T)\ge w_{\gamma }^{S,x}(T)$ for all $T\subseteq S$. Notice that, for all $T\subseteq S$ there exists $Q\subseteq N{\setminus } S$ such that

$$\begin{aligned} w_{\gamma }^{S,x}(T)=w_{\gamma }(T\cup Q)-\sum \limits _{l\in Q} x_{l}. \end{aligned}$$

(4)

Let $\mu$ be a matching on $T\cup Q$ such that $w_{\gamma }(T\cup Q)=\sum _{E\in \mu }v(E)$. We introduce the following partition of the set of basic coalitions in $\mu$:

$$\begin{aligned} I_{1}= & {} \{\{i,j,k\}\in \mu \mid i\in T, j\in T, k\in T\}\\ I_{2}= & {} \{\{i,j,k\}\in \mu \mid i\not \in T, j\not \in T, k\not \in T\}\\ I_{3}= & {} \{\{ i,j,k\} \in \mu \mid i\in T, j\in T, k\notin T\} \\ I_{4}= & {} \{ \{ i,j,k\} \in \mu \mid i \in T, j\notin T, k\notin T\} \\ I_{5}= & {} \{ \{ i,j\} \in \mu \mid i\in T, j\in T\} \\ I_{6}= & {} \{\{ i,j\} \in \mu \mid i\notin T, j\notin T\} \\ I_{7}= & {} \{\{ i,j\} \in \mu \mid i \in T, j\notin T\}\\ I_{8}= & {} \{\{ i\}\in \mu \mid i\in T\}. \\ I_{9}= & {} \{\{ i\}\in \mu \mid i\notin T\}. \end{aligned}$$

We write $w_{\gamma }(T\cup Q)$ in terms of the above partition.

$$\begin{aligned} w_{\gamma }(T\cup Q)&=\sum \limits _{\{i,j,k\}\in I_{1}}v(\{ i,j,k\})+\sum \limits _{\{i,j,k\}\in I_{2}}v(\{i,j,k\})+\sum \limits _{\{i,j,k\} \in I_{3}}v(\{i,j,k\}) \nonumber \\&\quad \,\,+\sum \limits _{\{i,j,k\} \in I_{4}}v(\{i,j,k\})+\sum \limits _{\{i,j\}\in I_{5}} v(\{i,j\})+\sum \limits _{\{i,j\}\in I_{6}} v(\{i,j\})\nonumber \\&\quad \,\,+\sum \limits _{\{i,j\}\in I_{7}} v(\{i,j\})+\sum \limits _{\{i\}\in I_{8}} v(\{i\})+\sum \limits _{\{i\}\in I_{9}} v(\{i\}). \end{aligned}$$

(5)

Then, substitute (5) in Eq. (4) and distribute $\sum \nolimits _{l\in Q}x_{l}$ among the sets of the partition.

$$\begin{aligned} w_{\gamma }^{S,x}(T)&=w_{\gamma }(T\cup Q)-\sum \limits _{i\in Q}x_{i} \\&\quad =\sum \limits _{\{i,j,k\}\in I_{1}}v(\{i,j,k\})+\sum \limits _{\{i,j,k\}\in I_{2}}v(\{i,j,k\})-x_{i}-x_{j}-x_{k} \\&\quad \,\,+\sum \limits _{\{i,j,k\}\in I_{3}}v(\{i,j,k\})-x_{k} +\sum \limits _{\{i,j,k\}\in I_{4}}v(\{i,j,k\})-x_{j}-x_{k}\\&\quad \,\,+\sum \limits _{\{i,j\}\in I_{5}}v(\{i,j\})+\sum \limits _{\{i,j\}\in I_{6}}v(\{i,j\})-x_{i}-x_{j}+\sum \limits _{\{i,j\}\in I_{7}}v(\{i,j\})-x_{j}\\&\quad \,\,+\sum \limits _{\{i\}\in I_{8}}v(\{i\})+\sum \limits _{\{i\}\in I_{9}}v(\{i\})-x_{i}. \end{aligned}$$

Since $x\in C(\gamma )$, the second, the sixth and the last term are non-positive.

Let us consider ${\hat{v}}={\hat{v}}^{S,x}$ (see Definition 3). For all $t,r,s\in \{1,2,3\}$ such that $r\ne s$, $r\ne t$, $s\ne t$ and all $i\in M_{r}\cap T$, $j\in M_{s}\cap T$,

$$\begin{aligned} {\hat{v}}(\{i,j\})=\max \limits _{k\in Q\cap M_{t}}\{v(\{i,j,k\})-x_{k},v(\{i,j\})\}. \end{aligned}$$

As a consequence, for all $\{i,j,k\}\in I_{3}$, $v(\{i,j,k\})-x_{k}\le {\hat{v}}(\{i,j\})$ and for all $\{i,j\}\in I_{5}$, $v(\{i,j\})\le {\hat{v}}(\{i,j\})$.

Also, for all $t\in \{1,2,3\}$ and $l\in M_{t}\cap T$, if r, s are such that $r\ne s$, $s\ne t$ and $t\ne r$, then,

$$\begin{aligned} {\hat{v}}(\{l\})=\max \limits _{\begin{array}{c} i \in M_{r}\cap Q \\ j\in M_{s}\cap Q \end{array}}\{v(\{i,j,l\})-x_{i}-x_{j},v(\{i,l\})-x_{i},v(\{j,l\})-x_{j}, v(\{l\})\}. \end{aligned}$$

As a consequence, for all $\{i,j,k\}\in I_{4}$, $v(\{i,j,k\})-x_{j}-x_{k}\le {\hat{v}}(\{i\})$; for all $\{i,j\}\in I_{7}$, $v(\{i,j\})-x_{j}\le {\hat{v}}(\{i\})$ and trivially $v(\{i\})\le {\hat{v}}(\{i\})$ for all $\{i\}\in I_{8}$.

To sum up, taking into account that ${\hat{w}}$ is superadditive by definition,

$$\begin{aligned} w_{\gamma }^{S,x}(T)\le \sum \limits _{\{i,j,k\}\in I_{1}}{\hat{v}}(\{i,j,k\})+\sum \limits _{\begin{array}{c} \{i,j,k\}\in I_{3} \\ \{i,j\} \in I_{5} \end{array}}{\hat{v}}(\{i,j\})+\sum \limits _{\begin{array}{c} \{i,j,k\} \in I_{4} \\ \{i,j\} \in I_{7} \\ \{i\}\in I_{8} \end{array}}{\hat{v}}(\{i\}) \le {\hat{w}}(T). \end{aligned}$$

Now, we only need to show that ${\hat{w}}$ is the minimal superadditive game satisfying the above inequality. First, consider $\{k\}\in {\mathcal {B}}^{S}$. Then,

$$\begin{aligned} w_{\gamma }^{S,x}(\{k\})&=\max \limits _{Q\subseteq N{\setminus } S}\{w_{\gamma }(\{k\}\cup Q)-x(Q)\} \nonumber \\&\ge \max \limits _{\begin{array}{c} Q\subseteq N{\setminus } S \\ \{k\}\cup Q\in {\mathcal {B}} \end{array}}\{w_{\gamma }(\{k\}\cup Q)-x(Q)\} \nonumber \\&\ge \max \limits _{\begin{array}{c} Q\subseteq N{\setminus } S \\ \{k\}\cup Q\in {\mathcal {B}} \end{array}}\{v(\{k\}\cup Q)-x(Q)\} \nonumber \\&={\hat{v}}(\{k\}). \end{aligned}$$

(6)

Similarly, we obtain

$$\begin{aligned} w_{\gamma }^{S,x}\{i,j\}\ge & {} {\hat{v}}(\{i,j\})\quad \text{ for } \text{ all } \{i,j\}\in {\mathcal {B}}^{S},\end{aligned}$$

(7)

$$\begin{aligned} w_{\gamma }^{S,x}(\{i,j,k\})\ge & {} {\hat{v}}(\{i,j,k\}) \quad \text{ for } \text{ all } \{i,j,k\}\in {\mathcal {B}}^{S}. \end{aligned}$$

(8)

Assume now (N, w) is superadditive and $w\ge w_{\gamma }^{S,x}$. For all $T\subseteq S$, let $\mu$ be an optimal matching for ${\hat{\gamma }}^{S,x}_{\mid T}$. Then,

$$\begin{aligned} w(T)&\ge \sum \limits _{\{i,j,k\}\in \mu } w(\{i,j,k\})+\sum \limits _{\{i,j\}\in \mu } w(\{i,j\})+\sum \limits _{\{k\}\in \mu } w(\{k\}) \\&\ge \sum \limits _{\{i,j,k\}\in \mu } w_{\gamma }^{S,x}(\{i,j,k\})+\sum \limits _{\{i,j\}\in \mu } w_{\gamma }^{S,x}(\{i,j\})+\sum \limits _{\{ k\}\in \mu } w_{\gamma }^{S,x}(\{k\}) \\&\ge \sum \limits _{\{i,j,k\}\in \mu }{\hat{v}}(\{i,j,k\})+\sum \limits _{\{i,j\}\in \mu } {\hat{v}}(\{i,j\})+\sum \limits _{\{k\}\in \mu }{\hat{v}}(\{k\}) \\&={\hat{w}}(T), \end{aligned}$$

where the last inequality follows from (6) to (8).

This shows that ${\hat{w}}$ is the minimal superadditive game such that ${\hat{w}}\ge w_{\gamma }^{S,x}$, which implies that ${\hat{w}}$ is the superadditive cover of $w_{\gamma }^{S,x}$. $\square$

Proof of Lemma 1

In order to see this, we need to show that if $(p,\mu )$ is a competitive equilibrium, then the matching $\mu$ is a partition of maximal value. Consider a competitive equilibrium $(p,\mu )$ and another matching $\mu '\in {\mathcal {M}}(M_{1},M_{2},M_{3})$. Then,

$$\begin{aligned}&\sum \limits _{E\in \mu }v^{w,c}(E)=\sum \limits _{k\in M_{3}}w^{k}(\mu (k))-c(\mu (k){\setminus }\{k\})\\&\quad \ge \sum \limits _{k\in M_{3}}w^{k}(\mu '(k))-c(\mu (k){\setminus }\{k\})-p(\mu '(k){\setminus }\{k\})+p(\mu (k){\setminus }\{k\})\\&\quad =\sum \limits _{k\in M_{3}}w^{k}(\mu '(k))-c(\mu (k){\setminus }\{k\})-p\left( \bigcup _{k\in M_3}\mu '(k){\setminus } M_3\right) \\&\qquad +p\left( \bigcup _{k\in M_3}\mu (k){\setminus } M_3\right) \\&\quad =\sum \limits _{k\in M_{3}}w^{k}(\mu '(k))-c\left( \bigcup _{k\in M_3}\mu (k){\setminus } M_3\right) \\&\qquad -p\left( \left( \bigcup _{k\in M_3}\mu '(k){{\setminus }}\bigcup _{k\in M_3}\mu (k)\right) {\setminus } M_3\right) \\&\qquad + p\left( \left( \bigcup _{k\in M_3}\mu (k){\setminus }\bigcup _{k\in M_3}\mu '(k)\right) {\setminus } M_3\right) \\&\quad =\sum \limits _{k\in M_{3}}w^{k}(\mu '(k))-c\left( \bigcup _{k\in M_3}\mu (k){\setminus } M_3\right) \\&\qquad -c\left( \left( \bigcup _{k\in M_3}\mu '(k){\setminus }\bigcup _{k\in M_3}\mu (k)\right) {\setminus } M_3\right) \\&\qquad + p\left( \left( \bigcup _{k\in M_3}\mu (k){\setminus }\bigcup _{k\in M_3}\mu '(k)\right) {\setminus } M_3\right) \\&\quad =\sum \limits _{k\in M_{3}}w^{k}(\mu '(k))\\&\qquad -c\left( \bigcup _{k\in M_3}\mu '(k){\setminus } M_3\right) -c\left( \left( \bigcup _{k\in M_3}\mu (k){\setminus }\bigcup _{k\in M_3}\mu '(k)\right) {\setminus } M_3\right) \\&\qquad + p\left( \left( \bigcup _{k\in M_3}\mu (k){\setminus }\bigcup _{k\in M_3}\mu '(k)\right) {\setminus } M_3\right) \\&\quad \ge \sum \limits _{k\in M_{3}}w^{k}(\mu '(k))-c(\mu '(k){\setminus }\{k\})=\sum \limits _{E\in \mu '}v^{w,c}(E), \end{aligned}$$

where the first inequality follows from the definition of the demand set and the fact that $(p,\mu )$ is a competitive equilibrium: $w^{k}(\mu (k))\ge w^{k}(\mu '(k))-p(\mu '(k){\setminus }\{k\})+p(\mu (k){\setminus }\{k\})$. The fourth equality follows from the fact that for all $l\in (\bigcup _{k\in M_3}\mu '(k){\setminus } \bigcup _{k\in M_3}\mu (k)){\setminus } M_3$, $p_{l}=c_{l}$, and the last inequality follows from the feasibility of the price vector p. $\square$