Constrained Allocation of Projects to Heterogeneous Workers with Preferences over Peers

Flip Klijn

doi:10.1515/bejte-2017-0038

Article

Constrained Allocation of Projects to Heterogeneous Workers with Preferences over Peers

Flip Klijn

Published/Copyright: September 14, 2017

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From the journal The B.E. Journal of Theoretical Economics Volume 19 Issue 1

Abstract

We study the problem of allocating projects to heterogeneous workers. The simultaneous execution of multiple projects imposes constraints across project teams. Each worker has preferences over the combinations of projects in which he can potentially participate and his team members in any of these projects. We propose a revelation mechanism that is Pareto-efficient and group strategy-proof (Theorem 1). We also identify two preference domains on which the mechanism is strongly group strategy-proof (Theorem 2). Our results subsume results by Monte and Tumennasan (2013) and Kamiyama (2013) .

Keywords: matching; allocation; heterogeneous agents; preferences over peers; efficiency; (group) strategy-proofness

JEL Classification: C78; D61; D78; I20

Funding statement: Agència de Gestió d’Ajuts Universitaris i de Recerca, (Grant / Award Number: ’2014-SGR-1064’) Aix-Marseille School of Economics, (Grant / Award Number:) Spanish Ministry of Economy and Competitiveness, (Grant / Award Number: ’ECO2014-59302-P’) Severo Ochoa Programme for Centres of Excellence in R&D, (Grant / Award Number: ’SEV-2015-0563’)

Acknowledgements

I thank Ronald Peeters and a reviewer for useful comments and suggestions. I gratefully acknowledge financial support from the Generalitat de Catalunya (2014-SGR-1064), the Spanish Ministry of Economy and Competitiveness through Plan Nacional I+D+i (ECO2014-59302-P), and the Severo Ochoa Programme for Centres of Excellence in R & D (SEV-2015-0563). A first draft was written while I was visiting the Aix-Marseille School of Economics. Their hospitality and financial support is gratefully acknowledged.

Appendix

Kamiyama (2013, Section 4.2) shows that the class of matching markets with quorums is a special class of abstract matching markets and that for matching with quorums the SDPC mechanism coincides with the GSDPC mechanism. Hence, it is sufficient to establish the equivalence of the GSDPC mechanism with the serial shrink project allocation mechanism for abstract matching markets. We first introduce an equivalent way to specify feasible project allocations.

Let Pˉ=P∪{∅}. A matching is a correspondence μ:Pˉ→I, i.e., for each p∈Pˉ, μ(p)⊆I, such that ⋃p∈Pˉμ(p)=I and for all p,p′∈Pˉ with p≠p′, μ(p)∩μ(p′)=∅. Let α∈A be a feasible project allocation. Then, μ defined by

μ(p)=Tif p∈P and T⊆I is such that (p,T)∈α;∅if p∈P and there is no T⊆I with (p,T)∈α

and μ(∅)=I∖⋃p∈Pμ(p) is a matching. Matchings that are thus obtained from feasible project allocations are called feasible matchings.

Lemma 2

There is a one-to-one correspondence between feasible matchings and feasible project allocations.

Proof

The statement follows from the observation that if μ is a feasible matching, then

α=⋃p∈P:μ(p)≠∅{(p,μ(p))}

is the feasible project allocation that induces μ.

Let an abstract matching market (Kamiyama 2013) have feasible project allocations Aa, i.e., as specified in eq. (2). One easily verifies that a matching ν is feasible if and only if

(3)⋃p∈P:ν(p)≠∅⋃i∈ν(p){{i,p}}∈F.

In abstract matching markets, workers are indifferent with respect to peers only. So, all allocation problems are in R=. Then, for convenience, we can omit co-workers from the description of each worker’s assignment (at any allocation). Similarly, with slight abuse of notation, we can use ⪰i to denote worker i’s linear order over projects (and being unassigned). In particular, for all p,p′∈P with p≠p′ we will write p⪰ip′ if and only if for all/some {(p,T)},{(p′,T′)}∈A(i), {(p,T)}⪰i{(p′,T′)}. We similarly slightly abuse the notation ≻i.

Fix the order of workers as 1,…,n. The GSDPC algorithm can be described as follows.

GSDPC algorithm (Kamiyama 2013)

Input: ⪰∈R=.

– Step 0 (initialization): for each p∈Pˉ, set μ(p)≡∅.

– Step i=1,…,n:

Compute^[7]
Pi={p∈P:for someF∈F,[{{i,p}}∪⋃p′∈P:μ(p′)≠∅⋃j∈μ(p′){{j,p′}}]⊆F}∪{∅}.
Let p∗∈Pi such that for each p∈Pi∖{p∗}, p∗≻ip.

Adjust μ(p∗)≡μ(p∗)∪{i}.

Output: a feasible matching μ⪰≡μ.

The GSDPC mechanism yields for each ⪰∈R= the feasible matching μ⪰ obtained by applying the GSDPC algorithm to ⪰.

Proposition 1

In abstract matching markets, the GSDPC mechanism coincides with the unique SSPA mechanism. Formally, for each ⪰∈R=, each worker is assigned to the same project (or no project) at μ⪰ and σ(⪰).

Proof

Let ⪰∈R=. From eq. (3), it follows that for each i=1,…,n,

Pi={p∈P:there is a feasible matching νsuch that[{{i,p}}∪⋃p′∈P:μ(p′)≠∅⋃j∈μ(p′){{j,p′}}]⊆⋃p′∈P:ν(p′)≠∅⋃j∈ν(p′){{j,p′}}}∪{∅}.

Using Lemma 2 one can now easily prove by induction that for each i=1,…,n, if p∗∈Pi is such that p∗≠∅ and for each p∈Pi∖{p∗}, p∗≻ip, then for each α∈Σi in the shrink algorithm for ⪰, p∗ is the unique project in P(α(i)). Similarly, if for each p∈Pi∖{∅}, ∅≻ip, then for each α∈Σi in the shrink algorithm for ⪰, P(α(i))=∅. Hence, each worker is assigned to the same project (or no project) at μ⪰ and σ(⪰).

The following result is an immediate corollary to Proposition 1 and Kamiyama (2013, Section 4.2).

Proposition 2

In matching with quorums, the SDPC mechanism coincides with the unique SSPA mechanism.

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Published Online: 2017-09-14

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Articles in the same Issue

https://doi.org/10.1515/bejte-2017-0038

Keywords for this article

matching; allocation; heterogeneous agents; preferences over peers; efficiency; (group) strategy-proofness