Natural implementation with semi-responsible agents in pure exchange economies

Abstract

We study Nash implementation by natural price–quantity mechanisms in pure exchange economies when agents have intrinsic preferences for responsibility. An agent has an intrinsic preference for responsibility if she cares about truth-telling that is in line with the goal of the mechanism designer besides her material well-being. A semi-responsible agent is an agent who, given what her opponents do, acts in an irresponsible manner when a responsible behavior poses obstacles to her material well-being. The class of efficient allocation rules that are Nash implementable is identified provided that there is at least one agent who is semi-responsible. The Walrasian rule is shown to belong to that class.

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Notes

  1. 1.

    These requirements are formally defined in Sect. 3 below.

  2. 2.

    There are seminal related works such as Glazer and Rubinstein (1998), Eliaz (2002), Corchón and Herrero (2004) and Kartik and Tercieux (2012).

  3. 3.

    Doğan (2014) studies problems of the fair allocation of indivisible goods by employing a similar definition of responsible agents.

  4. 4.

    This informational assumption is used also in other environments; see, e.g., Saporiti (2014).

  5. 5.

    The constrained Walrasian rule is a super-correspondence of the Walrasian rule. The two rules coincide when agents’ preferences are convex and Walrasian equilibrium allocations are interior.

  6. 6.

    What we do in this paper extends to the case where \(U_{i}\) is the class of allowable utility functions which are continuous and quasi-concave on \( {\mathbb {R}}_{+}^{\ell }\), strongly monotonic on the strictly positive orthant of \({\mathbb {R}}^{\ell }\), and satisfy the following boundary condition: the utility of every interior commodity bundle is strictly higher than the utility of any boundary commodity bundle. The Cobb-Douglas utility function is an example of such utility functions.

  7. 7.

    We use the following conventions: For all \(x,y\in {\mathbb {R}}_{+}^{\ell }\), \( x\ge y\) if \(x_{l}\ge y_{l}\) for each \(l\in L\); \(x>y\) if \(x\ge y\) and \( x\ne y\); and \(x\gg y\) if \(x_{l}>y_{l}\) for each \(l\in L\).

  8. 8.

    For a formal definition see Sect. 4.

  9. 9.

    See also Dutta et al. (1995). Similar properties are used also in general environments; see, e.g., Tatamitani (2001) and Lombardi and Yoshihara (2013a).

  10. 10.

    For a set S, we write \(\#S\) to denote the number of elements in S.

  11. 11.

    See Maskin (1999). For the criticism against such construction, see Jackson (1992).

  12. 12.

    Where x was not a totally balanced allocation, we could define the punishment allocation of condition (i) by \({\mathbf {z}}\left( p,x\right) =0\). In that case condition (iv) is satisfied (vacuously) where \(I^{W}\left( p,x\right) =N\).

  13. 13.

    Indeed, noting that \(p^{W}\) is the only Walrasian equilibrium price vector whenever x is a Walrasian equilibrium allocation for some economy, x cannot be a Walrasian allocation at \(u^{*}\). This is because \(x_{1}\) fails to maximize the utility of agent 1 over the budget set defined by the pair \(( p^{W},\omega _{1}) \) at the economy \(u^{*}\), due to the fact that \(u_{1}^{*}\) is differentiable at the quantity \(x_{1}\) and its gradient vector at \(x_{1}\) is equal to \(p_{a}\), which is different from \(p^{W}\).

  14. 14.

    If there is more than one of type \(N_{u}\left( m;\left( x,p\right) \right) \) having the maximal cardinality, fix any one of them.

  15. 15.

    Formal arguments can be found in Lombardi and Yoshihara (2014).

  16. 16.

    Note that since the egalitarian-equivalent and efficient rule is essentially single-valued, and since, moreover, each \(u_{i}^{*}\) is strictly concave, it follows that \(EE\left( u^{*}\right) =\left\{ x^{*}\right\} \). An allocation rule F is essentially single-valued if for every allowable profile u it holds that: \(x,x^{\prime }\in F\left( u\right) \implies u_{i}\left( x_{i}\right) =u_{i}\left( x_{i}^{\prime }\right) \) for every agent \(i\in N\).

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Correspondence to Michele Lombardi.

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We are grateful to an associate editor and two referees of this Journal for their useful comments and suggestions. They are not responsible for remaining deficiencies.

Appendix

Appendix

Proof of Theorem 1

Let the premises hold. Fix any \(F\in {\mathcal {F}}\). The proof that condition M-PQ is necessary for semi-responsible Nash implementation of F by a natural p-q mechanism is discussed in the main text and so is omitted here (see the Addendum below for formal arguments). In the following, we prove that condition M-PQ is also sufficient for it. We then suppose that F satisfies condition M-PQ.

We first establish some notation and definitions. For any \(x\in Q^{n}\) and any \(p\in P\), define the set \(\partial \Lambda _{i}^{F}\left( x,p\right) \) by \(\partial \Lambda _{i}^{F}\left( x,p\right) \equiv \{y_{i}\in Q|y_{i}\in \Lambda _{i}^{F}\left( x,p\right) \) and \(\not \exists z_{i}\in \Lambda _{i}^{F}\left( x,p\right) \) such that \(z_{i}\gg y_{i}\}\). For each \(i\in N\), let \(M_{i}\equiv P\times Q\) and let \(m_{i}=\left( p^{i},x_{i}^{i}\right) \) denote agent i’s strategy. For any \(m\in M\), \(u\in U_{N}\), \(x\in Q^{n}\) and \(p\in P\), define \(N_{u}\left( m\right) \) by \(N_{u}\left( m\right) \equiv \{i\in N|m_{i}=\left( p^{i},x_{i}^{i}\right) \) is responsible for \(\left( u,F\right) \}\) and \(N_{u}\left( m;\left( x,p\right) \right) \) by \( N_{u}\left( m;\left( x,p\right) \right) \equiv \{i\in N_{u}\left( m\right) |m_{i}=\left( p,x_{i}\right) ,x\in F\left( u\right) ,p\in \pi ^{F}\left( x,u\right) \}\). With these preliminaries, we define the outcome function g for any \(m\in M\) to be:

  • Rule 1: If for all \(j\in N\), \(m_{j}= ( p,x_{j} ) \), \( I^{F}\left( p,x\right) =N\), then \(g\left( m\right) ={\mathbf {z}}\left( p,x\right) \).

  • Rule 2: If for all \(j\in N\), \(m_{j}= ( p,x_{j} ) \) with \( x_{j}\ne 0\), \(1\le \#I^{F}\left( p,x\right) \le n-1\), then \(g\left( m\right) =\bar{\mathbf {z}}\left( p,x\right) \).

  • Rule 3: If for some \(i\in N\), \(m_{j}= ( p,x_{j} ) \) for all \( j\ne i\) and \(m_{i}= ( p^{i},x_{i}^{i}) \), with \(p\ne p^{i}\) and \( x_{i}^{i}\ne 0\), \(i\in I^{F}\left( p,x\right) \), then:

    $$\begin{aligned} g\left( m\right) =\left\{ \begin{array}{ll} ( z_{i} ( ( p^{i},x_{i} ) ; ( p,x_{-i} ) ) ,0_{-i} ) &{} \text { if }z_{i} ( ( p^{i},x_{i} ) ; ( p,x_{-i} ) ) \ne 0; \\ \left( x_{i},0_{-i}\right) &{} \begin{array}{l} \text {if }z_{i}\left( \left( p^{i},x_{i}\right) ;\left( p,x_{-i}\right) \right) =0\text { and} \\ x_{i}\in \Lambda _{i}^{F} (( x_{i}^{\Omega },x_{-i}) ,p) \text {;} \end{array} \\ ( {\hat{x}}_{i},0_{-i}) &{} \text { otherwise,} \end{array} \right. \end{aligned}$$

    where \(\left\{ {\hat{x}}_{i}\right\} \equiv \partial \Lambda _{i}^{F}\left( \left( x_{i}^{\Omega },x_{-i}\right) ,p\right) \bigcap \{y_{i}\in {\mathbb {R}}_{+}^{\ell }|\exists \alpha \in {\mathbb {R}}_{+},y_{i}=\alpha x_{i}\}\).

  • Rule 4: Otherwise:

  • Rule 4.1: If for some i, \(x_{i}^{i}=0\), \(N_{u}\left( m;\left( x,p\right) \right) \ne \varnothing \) for some \(\left( u,x,p\right) \), \( \#N_{u}\left( m;\left( x,p\right) \right) \ge \#N_{{\bar{u}}}\left( m;\left( {\bar{x}},{\bar{p}}\right) \right) \) for all \(\left( {\bar{u}},{\bar{x}},{\bar{p}} \right) \),Footnote 14 then for each \(i\in N\) , \(g_{i}\left( m\right) =\left( \frac{x_{i}}{\left( n+1\right) -\#N_{u}\left( m;\left( x,p\right) \right) }\right) \).

  • Rule 4.2: Otherwise, \(g\left( m\right) =0\).

Note that \(\gamma \equiv \left( M,g\right) \) is a natural p-q mechanism. Note also that in Rule 3 agent i can realize any element of \( \partial \Lambda _{i}^{F}\left( \left( x_{i}^{\Omega },x_{-i}\right) ,p\right) \) by a suitable choice of \(m_{i}=\left( p^{i},x_{i}^{i}\right) \).

Let \(\left( u^{*},H\right) \) be the true state of the world.

We show that \(F\left( u^{*}\right) =NA\left( \gamma ,R^{\gamma }\left[ u^{*},F,H\right] \right) \). Since it is a routine exercise to prove \(F\left( u^{*}\right) \subseteq NA\left( \gamma ,R^{\gamma }\left[ u^{*},F,H \right] \right) \), we shall omit the proof here. Conversely, fix any \(m\in NE\left( \gamma ,R^{\gamma }\left[ u^{*},F,H\right] \right) \). Note, m cannot correspond to Rule 2 nor to Rule 3 nor to Rule 4.Footnote 15 Suppose therefore m falls into Rule 1.

Then, \(g\left( m\right) ={\mathbf {z}}\left( p,x\right) \). Note that each i can induce Rule 3 and attain any quantity \({\hat{x}}_{i}\in \partial \Lambda _{i}^{F}( ( x_{i}^{\Omega },x_{-i} ) ,p) \) by suitably selecting \(m_{i}^{\prime }= ( p^{\prime },x_{i}^{\prime }) \), where \(p^{\prime }\) is a boundary point of P and \( x_{i}^{\prime }\in Q\backslash \Lambda _{i}^{F} ( ( x_{i}^{\Omega },x_{-i} ) ,p ) \). Then, we have that \(\partial \Lambda _{i}^{F} ( ( x_{i}^{\Omega },x_{-i} ) ,p ) \subseteq g_{i}\left( M_{i},m_{-i}\right) \). It follows from \(m\in NE ( \gamma ,R^{\gamma } [ u^{*},F,H ] ) \) that \(\partial \Lambda _{i}^{F} ( ( x_{i}^{\Omega },x_{-i} ) ,p ) \subseteq L ( {\mathbf {z}}_{i}\left( p,x\right) ,u_{i}^{*} ) \). Since \( u_{i}^{*}\) is strongly monotonic on \({\mathbb {R}}_{+}^{\ell }\), it can be seen that \(\Lambda _{i}^{F} ( ( x_{i}^{\Omega },x_{-i} ) ,p ) \subseteq L ( {\mathbf {z}}_{i}\left( p,x\right) ,u_{i}^{*} ) \). Note that since \(i\in N\) was arbitrary, the latter set inclusion holds for any \(i\in N\).

Further, suppose \({\mathbf {z}}\left( p,x\right) \notin F\left( u^{*}\right) \). Condition M-PQ implies that there exists \(i\in H\) such that \(m_{i}\) is not responsible for \(\left( u^{*},F\right) \) and \( u_{i}^{*} ( z_{i} ( ( p^{\prime },x_{i}^{\prime }) ;\left( p,x_{-i}\right) ) ) =u_{i}^{*}\left( {\mathbf {z}}_{i}\left( p,x\right) \right) \) for a strategy \(\left( p^{\prime },x_{i}^{\prime }\right) \) that is responsible for \(\left( u^{*},F\right) \). Agent i can change \(m_{i}=\left( p,x_{i}\right) \) into \( m_{i}^{\prime }= ( p^{\prime },x_{i}^{\prime }) \). If \(p=p^{\prime }\), then she obtains \(g_{i} ( m_{i}^{\prime },m_{-i} ) =z_{i} ( ( p^{\prime },x_{i}^{\prime }) ;\left( p,x_{-i}\right) ) \) by either Rule 1 or Rule 2. On the other hand, if \(p\ne p^{\prime }\), then agent i obtains \(g_{i} ( m_{i}^{\prime },m_{-i} ) =z_{i} ( ( p^{\prime },x_{i}^{\prime } ) ;\left( p,x_{-i}\right) ) \) by Rule 3. In either case, agent i obtains a profitable deviation, in violation of \(m\in NE ( \gamma ,R^{\gamma } [ u^{*},F,H]) \). We conclude from this that \({\mathbf {z}}\left( p,x\right) \in F\left( u^{*}\right) \). Since \( \left( u^{*},H\right) \) was arbitrary, the proof is completed. \(\square \)

Proof of Theorem 2

Let the premises hold. Recall Definition 4. Fix any \(p\in P\) and \(x\in Q^{n}\) . Recall \(x_{i}^{\Omega }\) denotes \(\Omega -\sum _{j\ne i}x_{j}\).

Note that if \(x\in W\left( u\right) \) for some \(u\in U\) and \(p\in \pi ^{W}\left( x,u\right) \), then p is a supporting price for x and for W; that is, \(p\in \Pi ^{W}\left( x,u\right) \) (simply because W is defined on U). Further, if \(x\in W\left( u\right) \) for some \(u\in U\) and \(p\in \pi ^{W}\left( x,u\right) \), then for all \(i\in N\), we have \(\Lambda _{i}^{W}\left( x,p\right) =\left\{ y_{i}\in Q\mid p\cdot y_{i}\le p\cdot \omega _{i}\right\} \). With these preliminaries, let us first verify conditions (i)–(iii).

Suppose that \(I^{W}\left( p,x\right) =N\). Define \({\mathbf {z}}\left( p,x\right) \) as follows: (a) \({\mathbf {z}}\left( p,x\right) =x\) if \(\sum _{j\in N}x_{j}=\Omega \), and (b) \({\mathbf {z}}\left( p,x\right) ={\mathbf {0}}\) if \( \sum _{j\in N}x_{j}\ne \Omega \). Then for all \(i\in N\), we have \({\mathbf {z}}_{i}\left( p,x\right) \in \Lambda _{i}^{W} ( ( x_{i}^{\Omega },x) ,p) \). This verifies condition (i).

Suppose that \(1\le \#I^{W}\left( p,x\right) \le n-1\). Define \(\bar{\mathbf {z}}\left( p,x\right) \) to be \(\bar{\mathbf {z}}\left( p,x\right) ={\mathbf {0}}\) . Then, condition (ii) is satisfied.

Fix any \(i\in N\). Define \(z_{i}\left( \cdot ;\left( p,x_{-i}\right) \right) \) for any \( ( p^{\prime },x_{i}^{\prime }) \) to be (a) \(z_{i}(\left( p^{\prime },x_{i}^{\prime }\right) ; \left( p,x_{-i}\right) )=x_{i}^{\Omega }\) if \(( p^{\prime },x_{i}^{\prime }) \) is responsible for some allowable \(\left( u,W\right) \), \(p^{\prime }\ne p\) and \(i\in I^{W}( p,( x_{i}^{\prime },x_{-i}) ) \); (b) \( z_{i}(( p^{\prime },x_{i}^{\prime } ) ; ( p,x_{-i} )) ={\mathbf {z}}_{i}\left( p,x\right) \) if \(( p^{\prime },x_{i}^{\prime }) \) is responsible for some allowable \( \left( u,W\right) \), \(p^{\prime }=p\) and \(I^{W}( p,( x_{i}^{\prime },x_{-i})) =N\); (c) \(z_{i}( ( p^{\prime },x_{i}^{\prime }) ;\left( p,x_{-i}\right) ) =\bar{\mathbf {z}}_{i}\left( p,x\right) \) if \(\left( p^{\prime },x_{i}^{\prime }\right) \) is responsible for some allowable \(\left( u,W\right) \), \(p^{\prime }=p\), \(i\in I^{W}( p,( x_{i}^{\prime },x_{-i}) ) \) and \(1\le \#I^{W}( p, ( x_{i}^{\prime },x_{-i}) ) \le n-1\); (d) \( z_{i} ( ( p^{\prime },x_{i}^{\prime }) ;\left( p,x_{-i}\right) ) =0\), otherwise. We then have condition (iii) satisfied.

Finally, we show that W satisfies condition (iv). Let \(I^{W}\left( p,x\right) =N\). If \(\sum _{j\in N}x_{j}\ne \Omega \), then \( {\mathbf {z}}\left( p,x\right) ={\mathbf {0}}\) and so condition (iv) is vacuously satisfied. Therefore, let \(\sum _{j\in N}x_{j}=\Omega \) and so \( {\mathbf {z}}\left( p,x\right) =x\). Suppose that for some \(u^{*}\in U\), \( \Lambda _{j}^{W}\left( x,p\right) \subseteq L ( x_{j},u_{j}^{*} ) \) for all \(j\in N\) and \(x\notin W\left( u^{*}\right) \). Since \( W^{-1}\left( x,p\right) \) is non-empty, it follows that there is \(u\in U\) such that \(x\in W\left( u\right) \) and \(p\in \pi ^{W}\left( x,u\right) \), and therefore \(p\in \Pi ^{W}\left( x,u\right) \). Since \(x\notin W\left( u^{*}\right) \), there is an agent \(j\in N\) such that \(x_{j}\notin \arg \max _{y_{j}\in {\mathbb {R}}_{+}^{\ell }:p\cdot y_{j}\le p\cdot \omega _{j}}u_{j}^{*}( y_{j}) \). Moreover, since \(p\in \pi ^{W}\left( x,u\right) \) and \(u\in U\) imply that \(p\gg 0\), there exists \( x_{j}^{*}\) such that \(x_{j}^{*}\in \arg \max _{y_{j}\in {\mathbb {R}}_{+}^{\ell }:~p\cdot y_{j}\le p\cdot \omega _{j}}u_{j}^{*}( y_{j}) \). However, by \(\Lambda _{j}^{W}\left( x,p\right) \subseteq L ( x_{j},u_{j}^{*}) \), it follows that \(x_{j}^{*}\notin Q\) . We then have that p cannot be a competitive equilibrium price for \( u^{*}\) and so for all \(i\in N\), \(\left( p,x_{i}\right) \) is not responsible for \(\left( u^{*},W\right) \). Let \(H\in {\mathcal {H}}\) and \( i\in H\) be given. Let \(( p^{\prime },x_{i}^{\prime }) \) be any responsible pair for \(\left( u^{*},W\right) \). Then, \(p^{\prime }\ne p\) . By \(I^{W}\left( p,x\right) =N\), it follows that \(i\in I^{W}( p,( x_{i}^{\prime },x_{-i})) \). By definition of \(z_{i}\left( \cdot ;\left( p,x_{-i}\right) \right) \), we have \(z_{i} ( ( p^{\prime },x_{i}^{\prime }) ;(p,x_{-i} )) =x_{i}\). This completes the proof of condition (iv). \(\square \)

Proof of Claim 1

Let the premises hold. Take any profile \(u\in U\) such that \(u_{i}\) is strictly concave and differentiable for every agent \(i\in N\). Furthermore, fix any interior allocation \(x\in EE\left( u\right) \cap {\mathbb {R}}_{++}^{n\ell }\).

Note that there exists a unique supporting price p for u and EE at x , that is, \(\pi ^{EE}\left( x,u\right) =\left\{ p\right\} \), and that the set \(\Lambda _{i}^{EE}\left( x,p\right) \) is a well-defined set for every agent i. By definition of the rule, it follows that for every \(v\in EE^{-1}\left( x,p\right) \) it holds that there exists a unique real number \( \lambda _{v}\) in the open interval \(\left( 0,1\right) \) such that \( v_{i}\left( x_{i}\right) =v_{i}\left( \lambda _{v}\Omega \right) \) for every agent i.

Write \(\lambda _{\min }\) for the solution to the problem:

$$\begin{aligned} \min \left\{ \lambda _{v}|\text {for }v\in EE^{-1}\left( x,p\right) :v_{i}\left( x_{i}\right) =v_{i}\left( \lambda _{v}\Omega \right) \text { for every }i\in N\right\} \text {,} \end{aligned}$$

and write \(\partial \Lambda _{i}^{EE}\left( x,p\right) \) for the upper boundary of \(\Lambda _{i}^{EE}\left( x,p\right) \), that is:

$$\begin{aligned} \partial \Lambda _{i}^{EE}\left( x,p\right) \equiv \{y_{i}\in Q|y_{i}\in \Lambda _{i}^{EE}\left( x,p\right) \text { and }\not \exists z_{i}\in \Lambda _{i}^{EE}\left( x,p\right) \text { such that }z_{i}\gg y_{i}\}\text {.} \end{aligned}$$

By construction, it follows that the common quantity \(\lambda _{\min }\Omega \) is an element of \(\partial \Lambda _{i}^{EE}\left( x,p\right) \) for every agent i. Moreover, for every agent i it also holds that there is a proper subset \(B\left( x_{i}\right) \) of \(\partial \Lambda _{i}^{EE}\left( x,p\right) \), which constitutes a neighborhood of \(x_{i}\) in \(\partial \Lambda _{i}^{EE}\left( x,p\right) \). By taking such an neighborhood sufficiently small, it follows from the definition of \(\partial \Lambda _{i}^{EE}\left( x,p\right) \) that \(p\cdot y_{i}=p\cdot x_{i}\) for every quantity \(y_{i}\in B\left( x_{i}\right) \), since \(x_{i}\) is an interior consumption bundle.

Fix any two distinct agents j and k. Take any profile \(u^{*}\in U\) such that \(u_{i}^{*}\) is strictly concave and differentiable for every agent i, that \(\Lambda _{i}^{EE}\left( x,p\right) \subseteq L\left( x_{i},u_{i}^{*}\right) \) for every agent i and that for a sufficiently small real number \(\epsilon >0\) and for a real number \(\lambda >\lambda _{\min }\) it holds that:

$$\begin{aligned} u_{i}^{*}\left( x_{i}\right)= & {} u_{i}^{*}\left( \lambda \Omega \right) \text { for every agent }i\ne j,k\text {,} \\ u_{j}^{*} ( x_{j}+ ( \epsilon ,0 ) )= & {} u_{j}^{*}\left( \lambda \Omega \right) \text {,} \\ u_{k}^{*}\left( x_{k}-\left( \epsilon ,0\right) \right)= & {} u_{k}^{*}\left( \lambda \Omega \right) \text { and} \\ Du_{j}^{*} ( x_{j}+ ( \epsilon ,0))= & {} \left\{ p\right\} =Du_{k}^{*}\left( x_{k}-\left( \epsilon ,0\right) \right) \text {,} \end{aligned}$$

where 0 is the \(\ell -1\)-th dimensional zero vector and where \( Du_{j}^{*}\left( x_{j}+\left( \epsilon ,0\right) \right) \) and \( Du_{k}^{*}\left( x_{k}-\left( \epsilon ,0\right) \right) \) denote the gradient vector respectively at the quantity \(x_{j}+\left( \epsilon ,0\right) \) and at \(x_{k}-\left( \epsilon ,0\right) \). This profile \(u^{*}\) exists because of our domain assumption. By the construction of \(u^{*} \), \(B\left( x_{i}\right) \subseteq L ( x_{i},u_{i}^{*}) \) for every agent i, which implies that \(\left\{ p\right\} =\Pi \left( x,u^{*}\right) \). Thus, \(x\in PO\left( u^{*}\right) \).

By the above discussion, there exist a real number \(\lambda _{j}\) in the open interval \(\left( \lambda _{\min },\lambda \right) \) and a real number \( \lambda _{k}>\lambda \) such that \(u_{j}^{*} ( x_{j} ) =u_{j}^{*} ( \lambda _{j}\Omega ) \) and that \(u_{k}^{*}\left( x_{k}\right) =u_{k}^{*}\left( \lambda _{k}\Omega \right) \). Therefore, since \(\lambda _{k}>\lambda >\lambda _{j}\), x is not egalitarian-equivalent at the profile \(u^{*}\). Thus, \(x\in PO\left( u^{*}\right) \backslash EE\left( u^{*}\right) \). The allocation \( x^{*}\), which assigns the quantity \(x_{i}^{*}=x_{i}\) to every agent \( i\ne j,k\), the quantity \(x_{j}^{*}=x_{j}+\left( \epsilon ,0\right) \) to agent j and the quantity \(x_{k}^{*}=x_{k}-\left( \epsilon ,0\right) \) to agent k, is an egalitarian-equivalent and efficient allocation at \( u^{*}\) and, moreover, the price vector p is a supporting price for \( u^{*}\) and possibly for EE at this \(x^{*}\), that is, \(p\in \pi ^{EE}\left( x^{*},u^{*}\right) \).Footnote 16

Take any agent \(i\ne j,k\). The singleton \(\left\{ i\right\} \) is an element of the collection \({\mathcal {H}}\) since this collection has as elements all singletons of the set N and since, moreover, there are \(n\ge 3\) agents in N. We have that the pair \(\left( p,x_{i}\right) \) is responsible for \( \left( u^{*},EE\right) \), in violation of part (iv) of condition M-PQ. \(\square \)

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Lombardi, M., Yoshihara, N. Natural implementation with semi-responsible agents in pure exchange economies. Int J Game Theory 46, 1015–1036 (2017). https://doi.org/10.1007/s00182-017-0568-8

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Keywords

  • Nash equilibrium
  • Exchange economies
  • Intrinsic preferences for responsibility
  • Boundary problem
  • Price–quantity mechanism

JEL Classification

  • C72
  • D71