Abstract
We completely characterize the class of fair and group strategy-proof mechanisms. We consider two notions of fairness, anonymity in welfare and no-envy. Both fairness axioms, when applied with strategy-proofness, imply decision efficiency, and lead to the same class of group strategy-proof mechanisms (where the group size is restricted to two). We find that the only feasible mechanism satisfying a mild zero transfer axiom, in this class, is the Pivotal mechanism.
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Notes
For any \(i\ne j, i \succ j\) means that the tie is broken in favour of agent \(i\). That is, if \(M(b) = b_3=b_7\) and \(3 \succ 7\), then \(b(1)=b_3\).
A mechanism \((d,\tau )\) satisfies anonymity in bundle if for any bijection \(\pi : N \mapsto N, \forall \, b\in \mathbb R _+^n, \forall \, i \in N, (d_i(b),\tau _i(b))= (d_{\pi (i)}(\hat{b}),\tau _{\pi (i)}(\hat{b}))\) where \(b_i=\hat{b}_{\pi (i)}, \forall \, i\in N\).
As mentioned in Ohseto (2006), when there are multiple homogeneous goods to be allocated, NE also implies that any two agents getting the good must get the same transfer.
The same holds true in the homogeneous good setting with unit demand.
This is because Proposition 1, essentially, underlines the incompatibility of EFF (implied by AN and SP) with SPGS.
To check for NE, w.l.o.g. fix a profile \(b\) such that \(b_1\ge b_2\ge \ldots \ge b_n\). Therefore \(d^e_1(b) = 1\) and \(d^e_j(b) = 0, \forall \, j\ne 1\) and \(\tau _j(b) = \tau _{j^{\prime }}(b), \forall \, j,j^{\prime } \ne 1\). Hence, NE is trivially satisfied for all such pair of agents. Now, choose any \(i \ne 1\) and consider the pair of agents \(1\) and \(i\). Define \(D_i^1:=b_i-b_2 + \min \{b_2,\eta \} - \min \{b_1,\eta \}\) and \(D_1^i:=\min \{b_1,\eta \} - [b_1-b_2 + \min \{b_2,\eta \}]\). Now, since \(b_i < b_1, D_i^1 \le 0\) and \(D_1^i\le 0\) irrespective of the ordering amongst \(b_1,b_2,\eta \). Hence NE holds.
The results of this paper can also be readily extended to a setting where a costly object is to be distributed.
Consider two VCG mechanisms (as described in Corollary 1) satisfying AN, with \(g(b_{-i}) = \frac{M(b_{-i})}{2}\) and \(g(b_{-i}) =M(b_{-i})\) for all possible \(b_{-i}\), respectively. Observe that only the second mechanism is WPGS. The second mechanism also, is WGS. In fact, any WPGS VCG mechanism satisfying AN (or NE) must also be WGS.
The technique of proof is taken from Mishra and Mitra (2010).
If the other combination of inequalities \([\Delta _2(b,b^{\prime })>0, \Delta _1(b^{\prime },b)>0]\) were to be chosen, the direction of the inequalities in (5.3) and (5.4) would get reversed. However, even with this different combination of inequalities; using the same techniques, we could show that \(M(b_{-\{1,2\}}) \le b^{\prime }_2 \le b^{\prime }_1 < b_2 \le b_1\) and derive similar results to arrive at the same \(g(.)\) maps.
In (5.5), the left hand inequality implies that \(\eta > b_1\) while the right hand inequality implies that \(\eta < b^{\prime }_2\).
Note that both NE and AN require that the \(g(.)\) function be independent of agent labels. Hence, \(\eta \) cannot depend on agent labels.
The same continuity is implied by Corollary 2, in case of WPGS mechanisms that satisfy NE.
With a slight abuse of notation.
Note that for case (ii), \(G_{ij} = M(b_{-\{i,j\}}) - \underline{C}(b_{-\{i,j\}})\).
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I thank Satya R. Chakravarty, Bhaskar Dutta, Manipushpak Mitra, Hervé Moulin, Suresh Mutuswami, Debasis Mishra, Arunava Sen, participants of Society for Social Choice and Welfare Conference (2012), two anonymous referees and handling editor, for their comments and suggestions. I also thank Rajit Biswas for the invaluable discussions. The usual disclaimer holds.
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Mukherjee, C. Fair and group strategy-proof good allocation with money. Soc Choice Welf 42, 289–311 (2014). https://doi.org/10.1007/s00355-013-0733-3
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DOI: https://doi.org/10.1007/s00355-013-0733-3