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Social Choice and Welfare

Erschienen in:

08.05.2021 | Original Paper

Undominated mechanisms and the provision of a pure public good in two agent economies.

verfasst von: Efthymios Athanasiou, Giacomo Valletta

Erschienen in: Social Choice and Welfare | Ausgabe 4/2021

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Abstract

We revisit the problem of providing a pure public good in economies involving two agents, under the assumption that preferences are quasilinear. We characterize the class of strategy-proof, anonymous and feasible mechanisms that are not dominated by another strategy-proof, anonymous and feasible mechanism. Mechanisms that decide, at some profiles of preferences, against the alternative that is preferred by all agents are members of this class. On the contrary, Groves mechanisms, with the sole exception of the Pivotal mechanism, are not. Finally, the only budget-balanced mechanism that belongs to this class is the Unanimity mechanism.

Vorheriger Artikel Voting power on a graph connected political space with an application to decision-making in the Council of the European Union

Nächster Artikel Almost mutually best in matching markets: rank gaps and size of the core

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Guo et al. (2011) and Hashimoto and Shiozaway (2016) also follow this approach.

We normalize utilities in such a way that, for each \(\theta \in \mathbb {R}\), at the bundle \((0,0)\in \{0,1\}\times \mathbb {R}\), \(u((0,0);\theta )=0\). Even if this normalization seems to treat the bundle \((0,0)\in \{0,1\}\times \mathbb {R}\) as the status quo, the choice is entirely arbitrary. Our results would carry over if we were to normalize utilities in such a way that, for each \(\theta \in \mathbb {R}\), given the bundles (0, 0) and \((1,0) \in \{0,1\}\times \mathbb {R}\), one has \(u((0,0);\theta _i)+u((1,0);\theta _i)=0\) [as in Moulin (1988), chapter 8].

Anonymity allows us to rule out some undominated, yet unacceptable, mechanisms such as dictatorship.

A function \(f:\mathbb {R}^{2}\rightarrow \{0,1\}\) is weakly monotonic if and only if for each \((\theta _{1},\theta _{2}),(\theta '_{1},\theta '_{2})\in \mathbb {R}^{2}\), \((\theta _{1},\theta _{2})\le (\theta '_{1},\theta '_{2})\) implies \(f(\theta _{1},\theta _{2})\le f(\theta '_{1},\theta '_{2}).\)

Put differently, \(f(A)\equiv \big \{f(\theta _1,\theta _2)\; :\; (\theta _1,\theta _2)\in A \big \}\).

In order to invoke the result by Holmström (1979) we need to point out that the domain of preferences is convex and thus smoothly connected.

Guo et al. (2013) have shown that, in our setting, the least wasteful Groves mechanism is the Pivotal mechanism. Hence, the least wasteful Groves mechanism is also Undominated.

If it were the case that for each \(\theta \in \mathbb {R}\) and each \(\epsilon >0\), \(f(\theta ,\beta ^{f}-\epsilon )=0\), so that \(p(\beta ^{f})=0\), then for each \(\theta ^{\prime }>\beta ^{f}\), \(p(\theta ^{\prime })=\beta ^{f}.\)

If \(k=1\) were the case, the construction would need to be adapted accordingly. However, the argument underlying the proof would remain the same.

If instead of 6.11 the departing assumption were \(\tilde{\theta }_{1}<\beta ^{f}\) and \(\tilde{\theta }_{2}<p(\tilde{\theta }_{2})\), we would arrive at the same contradiction by a similar argument.

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Titel: Undominated mechanisms and the provision of a pure public good in two agent economies.
verfasst von: Efthymios Athanasiou
Giacomo Valletta
Publikationsdatum: 08.05.2021
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 4/2021
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-021-01336-9

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