Constrained dictatorial rules are subject to variable-population paradoxes

Abstract

In the context of classical exchange economies, we study four ways in which agents can strategically take advantage of allocation rules by affecting who participates and on what terms (Thomson in Soc Choice Welf 42:289–311, 2014). (1) An agent transfers their endowment to someone else and withdraws. The two of them may end up controlling resources that allow them to simultaneously reach higher welfare levels than they otherwise would. (2) An agent invites someone in and let their guest use some of their (the host’s) endowment. The guest transfers back to them what they are assigned over their endowment. The host may benefit. (3) An agent withdraws with their endowment. As in (1), they and someone who stays may end up controlling resources that allow the two of them to simultaneously reach higher welfare levels than they otherwise would. (4) An agent pre-delivers to someone else the net trade that the rule would assign to that agent had the agent participated. The second agent withdraws. The first agent participates with a modified endowment. The first agent may benefit. We ask whether “the constrained priority rules”, defined by maximizing the welfare of a particular agent subject to each of the others finding their assignment at least as desirable as their endowment satisfy these various requirements. The answers are all negative. Because these types of rules are often better behaved than rules that attempt some fairness in distributing gains from trade, these results strengthen the negative conclusions reached in Thomson (2014), and they may provide the key to identifying circumstances in which rules exist that satisfy the axioms, or to proving general impossibility results.

Appendix

In this appendix, we prove the propositions.

Proof of Proposition 1

(a) Homothetic domain. Let $N\equiv \{1,2,3\}$. We construct an economy $(R,\omega ) \in {{\mathcal {E}}}^N_{hom}$ in steps, as follows. First, let $\omega \equiv ((100, 100), (100, 100), (100, 100))$. Let $x\in X(R,\omega )$ be equal to ((150, 60), (10, 140), (140, 100)).

Next, let agent 2 merge their endowment with agent 1’s endowment and withdraw. Let $\omega '_1\equiv \omega _1+\omega _2$. Let $x'\in X(N{\setminus }\{2\}, \omega '_1, \omega _3)$ be equal to ((200, 200), (100, 100)).

We now specify $R\in {\mathcal {R}}^N_{hom}$ so that $x=CD^\prec (R,\omega )$ and $x'=CD^\prec (R_{-2}, \omega '_1, \omega _3)$.

Let $p\equiv (1,2)$ and $p'\equiv (1,1)$. Note that (i) $\frac{x'_{12}}{x'_{11}}=1>.25 = \frac{x_{12}}{x_{11}}$, and $px_1 <p\omega _1$; (ii) $px_2< p\omega _2$; (iii) $\frac{x'_{32}}{x'_{31}}= 1 >.7\simeq \frac{x_{32}}{x_{31}}$; and (iv) $\frac{p'_2}{p'_1}=1 <2=\frac{p_2}{p_1}$

Because of (i), there is $R_1 \in {{\mathcal {R}}}_{hom}$ such that $p\in \textrm{supp}(R_1,x_1)$ and $p' \in \textrm{supp}(R_1,x'_1)$, and $\omega _1 \mathrel {I_1} x_1$. Because of (ii), there is $R_2\in {{\mathcal {R}}}_{hom}$ such that $\omega _2 \mathrel {I_2} x_2$. Because of (iii), there is $R_3\in {{\mathcal {R}}}_{hom}$ such that $p\in \textrm{supp}(R_3,x_3)$ and $p'\in \textrm{supp}(R_3,x'_3)$.

It remains to verify that indeed $x\in CD^\prec (R, \omega )$ and $x'\in CD^\prec (R_{-2}, \omega '_1, \omega _3)$. Finally, we observe that $x'_1=(200, 200)=\omega _1 + \omega _2$. Because of (ii), $p \notin \textrm{supp}(R_2, \omega _2)$. This, together with (iv), implies that the lines of support to agents 1 and 2’s upper-contour sets at $\omega _1$ and $\omega _2$, respectively, have different slopes, and thus $(\omega _1, \omega _2)\notin P(R_1, R_2, \omega _1, \omega _2)$: these agents can achieve a distribution of $x'_1=\omega _1 + \omega _2$ that make them better off than at $x_1$ and $x_2$, respectively. Thus, endowments-merging-proofness is violated. (The violation is in physical terms.)

(b) Quasi-linear domain. The proof is obtained by modifying the example used to prove (a) as follows. We increase $x_{22}$ by an amount $a>0$ such that we still have $px_2<p\omega _2$ and so that $x_{32}-a>0$, and we decrease $x_{32}$ by a. Let $p'' \equiv (2, 1)$. Because $x_{12}<x'_{12}$, there is $R_1 \in {\mathcal {R}}_{ql}$ such that $p \in \textrm{supp}(R_1, x_1)$, $p''\in \textrm{supp}(R_1,x'_1)$, and $\omega _1 \mathrel {I_1} x_1$. We specify $R_2$ as before. Because $x_{32}< x'_{32}$, we can choose $R_3 \in {\mathcal {R}}_{ql}$ so that $p \in \textrm{supp}(R_3, x_3)$ and $p''\in \textrm{supp}(R_3, \omega _3)$. $\square$

Proof of Proposition 2

(a) Homothetic domain. Let $N\equiv \{1,3\}$. We construct an economy $(R,\omega ) \in {{\mathcal {E}}}^N_{hom}$ in steps, as follows. First, let $\omega \equiv ((160, 160), (160, 160))$. Let $x\in X(N,\omega )$ be ((240, 80), (80, 240)).

Next, let agent 1 split their endowment with some new agent, agent 2. We will specify $R_2$ to be the same as $R_3$. Let $\omega '_1\equiv \frac{\omega _1}{2}= (80, 80)$ and $\omega '_2\equiv \omega '_1$. Let $x'\in X(R_1,R_2, R_3,\omega '_1, \omega '_2,\omega _3)$ be equal to ((181.4, 15), (38.6, 85), (100, 220)).

We now specify $R\in {\mathcal {R}}^N_{hom}$ so that $x=CD^\prec (R,\omega )$ and $x'=CD^\prec (R_1,R_2, R_3, \omega '_1, \omega '_2, \omega _3)$.

Let $p\equiv (2,3)$ and $p'\equiv (1,3)$. Note that (i) $\frac{x'_{12}}{x'_{11}}=.088<.33\simeq \frac{x_{12}}{x_{11}}$, $px_1< p\omega _1$, and $p'\frac{x_1}{2}< p'x'_1$; (ii) $p'x'_2< p'\omega '_2$, (iii) $\frac{x_{32}}{x_{31}}= 3 > 2.2\simeq \frac{x'_{32}}{x'_{31}}$. Because of (i), there is a strictly convex relation $R_1 \in {{\mathcal {R}}}_{hom}$ such that $p\in \textrm{supp}(R_1,\frac{x_1}{2})$, and therefore $p\in \textrm{supp}(R_1,x_1)$, $p' \in \textrm{supp}(R_1,x'_1)$, and $\omega '_1 \mathrel {I_1} \frac{x_1}{2}\mathrel {I_1} x'_1$, and therefore, by homotheticity, $\omega _1 \mathrel {I_1} x_1$. Because of (ii), (iii), and the fact that $x'_2$ and $x'_3$ are proportional, there is $R_2=R_3$ such that $\omega '_2 \mathrel {I_2} x'_2$, $p\in \textrm{supp}(R_3,x_3)$, $p'\in \textrm{supp}(R_2,x'_2)$, and $p'\in \textrm{supp}(R_3,x'_3)$.

It remains to verify that indeed $x\in CD^\prec (R, \omega )$ and $x'\in CD^\prec (R_1,R_2, R_3, \omega '_1, \omega '_2, \omega _3))$. Finally, because agent 1 ends up with $x'_1+x'_2$, a bundle that lies on the segment connecting $\omega _1$ and $x_1$, two points between which agent 1 is indifferent, and agent 1’s preferences are strictly convex, agent 1 prefers their final bundle to the bundle they would have been assigned if they had not invited agent 2. Thus endowments-splitting-proofness is violated.

(b) Quasi-linear domain. Let $N\equiv \{1,3\}$. We construct an economy $(R,\omega ) \in {{\mathcal {E}}}^N_{ql}$ in steps, as follows. First, let $\omega \equiv ((160, 160), (160, 160))$. Let $x\in X(N,\omega )$ be equal to ((200, 0), (120, 320)).

Next, let agent 1 split their endowment with some new agent, agent 2. We will specify $R_2$ to be equal to $R_1$. Let $\omega '_1\equiv \frac{\omega _1}{2}= (80, 80)$ and $\omega '_2\equiv \omega '_1$. Let $x'\in X(R_1,R_2, R_3, \omega '_1, \omega '_2, \omega _3)$ be equal to ((110, 0), (110, 0), (100, 320)).

We now specify $R\in {\mathcal {R}}^N_{ql}$ so that $x=CD^\prec (R,\omega )$ and $x'=CD^\prec (R_1,R_2, R_3, \omega '_1, \omega '_2, \omega _3)$.

Let $p\equiv (1,1)$ and $p'\equiv (1,1)$. Note that (i) $px_1< p\omega _1$ and that (170, 80) is below the segment with endpoints $\omega _1$ and $x_1$. Thus, there is $R_1 \in {{\mathcal {R}}}_{ql}$ such that (iii) $p\in \textrm{supp}(R_1,x_1)$, and $\omega _1 \mathrel {I_1} x_1 \mathrel {I_1} (170, 80)$.

Now, by quasi-linearity and the last part of (iii), $\omega '_1 \mathrel {I_1} x'_1$ and since $R_1=R_2$, $\omega '_2 \mathrel {I_2} x'_2$. Now it is easy to verify that $x\in CD^\prec (R, \omega )$ and $x'\in CD^\prec (R_1,R_2, R_3, \omega '_1, \omega '_2, \omega _3)$.

We conclude by noting that $x'_1+x'_2=(110,0)+ (110,0)\ge x_1=(200,0)$, so that agent 1 prefers their final bundle to the bundle they would have been assigned if they had not invited agent 2. Thus endowments-splitting-proofness is violated. (The violation is in physical terms.) $\square$

Proof of Proposition 3

(a) Homothetic domain. Let $N\equiv \{1,2,3\}$. We construct an economy $(R,\omega ) \in {{\mathcal {E}}}^N_{hom}$ in steps, as follows. First, let $\omega \equiv ((100, 100), (100, 100), (100, 100))$. Let $x\in X(N,\omega )$ be equal to ((130, 70), (130, 70), (40, 160)).

Next, let agent 2 withdraw with their endowment. Let $x'\in X(N\backslash \{2\}, \omega _{-2})$ be equal to ((160, 65), (40, 135)).

We now specify $R\in {\mathcal {R}}^N_{hom}$ so that $x=CD^\prec (R,\omega )$ and $x'=CD^\prec (R_{-2}, \omega _{-2})$.

Let $p\equiv (1,2)$ and $p'\equiv (1,10)$. Note that (i)  $\frac{x'_{12}}{x'_{11}}= \frac{65}{160}\simeq 0.406<0.538\simeq \frac{70}{130}=\frac{x_{12}}{x_{11}}$, $\omega _{11}< x_{11} <x'_{11}$, $p\omega _1> px_1$, $px'_1> px_1$, and $p'x_1> p'x'_1$; (ii)  $\frac{x_{32}}{x_{31}} = 4 > 2.96 = \frac{x'_{32}}{x'_{31}}$; and (iii) $\frac{p'_2}{p'_1} =10 > 2= \frac{p_2}{p_1}$. Because of (i) and (iii), there is $R_1 \in {{\mathcal {R}}}_{hom}$ such that $\omega _1 \mathrel {I_1} x_1 \mathrel {I_1} x'_1$, $p\in \textrm{supp}(R_1,x_1)$, and $p' \in \textrm{supp}(R_1,x'_1)$. We set $R_2\equiv R_1$. Because of (ii) and (iii), there is $R_3\in {{\mathcal {R}}}_{hom}$ so that $p\in \textrm{supp}(R_3,x_3)$ and $p'\in \textrm{supp}(R_3,x'_3)$.

It remains to verify that indeed $x\in CD^\prec (R, \omega )$ and $x'\in CD^\prec (N\backslash \{2\}, \omega _{-2})$, and to observe that $x'_1+ \omega _2=(260, 165)\ge (260, 140)=x_1+x_2$. Thus, withdrawal-proofness is violated. (The violation is in physical terms.)

(b) Quasi-linear domain. We can use the same example. Indeed, for agents 1 and 2, we had to specify only one indifference curve satisfying certain constraints from which we could derive a homothetic map by homothetic expansions and contractions. One could also specify an indifference curve satisfying the same constraints from which a quasi-linear map could be derived by horizontal translations. The reason is that the lines of support to agent 1’s upper-contour sets at $x'_1$, $x_1$, and $\omega _1$ are steeper and steeper and the ordinates of these three points are in increasing order. For agent 3, it suffices to observe that $x'_{32} < x_{32}$ and that $\frac{p'_2}{p'_1}=10 > 2=\frac{p_2}{p_1}$. Again, a quasi-linear relation can be specified for which the required slopes of the supporting lines at $x_3$ and $x'_3$ are obtained. $\square$

Proof of Proposition 4

(a) Homothetic domain. Let $N\equiv \{1,2,3\}$. We construct an economy $(R,\omega ) \in {{\mathcal {E}}}^N_{hom}$ in steps, as follows. First, let $\omega \equiv ((100, 100), (100, 100),(100, 100))$. Let $x\in X(N,\omega )$ be equal to ((120, 20), (40, 150), (140, 130)).

Let agent 1 pre-deliver their net trade of $(-60, 50)$ to agent 2. Agent 1’s revised endowment is $\omega '_1\equiv \omega _1 -x_2+\omega _2= (160, 50)$. Let $x' \in X(N\backslash \{2\}, \omega '_1, \omega _3)$ be equal to ((160, 50)), (100, 100)).

Next, we specify $R\in {\mathcal {R}}^N_{hom}$ so that $x=CD^\prec (R,\omega )$ and $x'=CD^\prec (R_1,R_3, \omega '_1, \omega _3)$.

Let $p\equiv (1,1)$ and $p' \equiv (2,1)$. Note that (i) $\frac{x'_{12}}{x'_{11}}\simeq .35>.166\simeq \frac{x_{12}}{x_{11}}$ and that $px_1 < p\omega _1$, (ii) $px_2<p\omega _2$, and (ii) $\frac{x'_{32}}{x'_{31}}= 1 >.92\simeq \frac{x_{32}}{x_{31}}$. Because of (i) there is $R_1 \in {{\mathcal {R}}}_{hom}$ such that $p\in \textrm{supp}(R_1,x_1)$, $p' \in \textrm{supp}(R_1,x'_1)$, $x_1\mathrel {I_1} \omega _1$. Because of (ii), there is $R_2 \in {{\mathcal {R}}}_{hom}$ such that $p\in \textrm{supp}(R_2,x_2)$ and $x_2 \mathrel {I_2} \omega _2$. Finally, because of (iii), there is $R_3\in {{\mathcal {R}}}_{hom}$ such that $p\in \textrm{supp}(R_3,x_3)$, and $p'\in \textrm{supp}(R_3,x'_3)$.

It remains to verify that indeed $x\in CD^\prec (R, \omega )$ and that $x'\in CD^\prec (R_{-2}, \omega '_1, \omega _3)$. (The second statement follows from the fact that $x'\in P(R_{-2}, \omega '_1, \omega _3)$.) Finally, we observe that $x'_1>x_1$. Thus pre-delivery-proofness is violated. (The violation is in physical terms.)

(b) Quasi-linear domain. Let $N\equiv \{1,2,3\}$. We construct an economy $(R,\omega ) \in {{\mathcal {E}}}^N_{ql}$ in steps, as follows. First, let $\omega \equiv ((100, 100), (100, 100),(100, 100))$. Let $x\in X(N,\omega )$ be equal to ((140, 50), (110, 50), (50, 200)).

Let agent 1 pre-deliver their net trade to agent 2. Agent 1’s revised endowment is $\omega '_1\equiv \omega _1 -x_2+\omega _2= (90, 150)$. Let $x' \equiv ((150, 50), (40, 200))$.

We now specify $R \in {\mathcal {R}}^N_{ql}$ so that $x' = CD^\prec (R_1,R_3,\omega '_1,\omega_3 )$ and $x=CD^\prec (R,\omega )$.

Let $p\equiv (1,1)$. Note that (i) $px_1<p\omega _1$, $px'_1<p\omega '_1$ and $\omega _1$ lies below the line passing through $x_1$ and $\omega '_1$ and (ii) $px_2<p\omega _2$.

Because of (i), there is $R_1 \in {\mathcal {R}}_{ql}$ such that $p\in \textrm{supp}(R_1,x_1)$, $x_1 \mathrel {I_1} \omega _1$ and $x'_1 \mathrel {I_1} \omega '_1$. Then, by quasi-linearity, $p\in \textrm{supp}(R_1,x'_1)$.

Because of (ii), there is $R_2 \in {\mathcal {R}}_{ql}$ such that $p\in \textrm{supp}(R_2,x_2)$ and $x_2 \mathrel {I_2} \omega _2$. Finally, there is $R_3 \in {\mathcal {R}}_{ql}$ such that $p\in \textrm{supp}(R_3,x_3)$. Then, by quasi-linearity, $p\in \textrm{supp}(R_3,x'_3)$.

It remains to verify that indeed $x = CD^\prec (R, \omega )$ and $x' = CD^\prec (R_1,R_3,\omega '_1,\omega _3)$. Thus pre-delivery-proofness is violated. (The violation is in physical terms.) $\square$