Abstract
In this paper, we focus on the domain of decreasing marginal valuations and show that in the presence of budget-constrained bidders there is no multi-unit auction that satisfies the following four properties: incentive compatibility, Pareto optimality, individual rationality and no positive transfers. This result strengthens the impossibility results of two previous studies.
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Notes
Other contributions to this literature include those by Fiat et al. (2011) and Le (2017). Under the condition of bidders’ budgets being public information, Fiat et al. (2011) extends the result of Dobzinski et al. (2012) to single-valued combinatorial auctions and show that there is no combinatorial auction with two items and two bidders that satisfies the above-mentioned four properties. Under the condition of bidders’ budgets being private information, Le (2017) prove that there is no auction defined on the general domain that is incentive compatible, individually rational, symmetric, non-wasteful and non-bossy.
“Theorem 2 (Milgrom and Segal 2002): Suppose that \(f\left( {x, \cdot } \right)\) is absolutely continuous for all \(x \in X\). Suppose also that there exists an integrable fttnction \(b:\left[ {0,1} \right] \to R_{ + }\) such that \(\left| {f_{t} \left( {x,t} \right)} \right| \le b\left( t \right)\) for all \(x \in X\) and almost all \(t \in \left[ {0,1} \right]\). Then \(V\) is absolutely continuous. Suppose, in addition, that \(f\left( {x, \cdot } \right)\) is differentiable for all \(x \in X\), and that \(X^{*} \left( t \right) \ne \emptyset\) almost everywhere on \(\left[ {0,1} \right]\). Then for any selection \(x^{*} \left( t \right) \in X^{*} \left( t \right)\), \(V\left( t \right) = V\left( 0 \right) + \mathop \smallint \limits_{0}^{{\text{t}}} f_{t} \left( {x^{*} \left( s \right),s} \right)ds\).”.
Here \(X\) denotes the choice set, \(f:X \times \left[ {0,1} \right] \to R\), \(V\left( t \right) = \mathop {\sup }\limits_{x \in X} f\left( {x,t} \right)\) and \(X^{*} \left( t \right) = \left\{ {x \in X:f\left( {x,t} \right) = V\left( t \right)} \right\}\).
Lemma 4 is also a corollary of Proposition 2.5 in Dobzinski et al. (2012).
Lemma 5 is proved firstly by Lavi and May (2012) for the case of \(M=2\).
\({p}_{1}\left({v}_{1},{v}_{2}\right)=0\) follows from Lemma 6(ii) and \({d}_{1}\left({v}_{1},{v}_{2}\right)=0\) follows from Lemma 4.
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Yi, J. A note on the impossibility of multi-unit auctions with budget-constrained bidders. Rev Econ Design (2023). https://doi.org/10.1007/s10058-023-00342-w
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DOI: https://doi.org/10.1007/s10058-023-00342-w