Abstract
Scoring auctions are important mechanisms for procurement in both developed and developing countries. Till date the literature has mainly dealt with cases where the scoring rule is quasilinear. Very few papers in the literature have dealt with non-quasilinear scoring rules. In chapter 4 we fill this gap. Under some conditions we derive a clear ranking of the expected scores in first-score and second-score auctions. We show how in many cases second price auctions lead to higher expected score. We show with the help of two examples, that while expected score may be higher with second-score auctions, total expected welfare need not be higher.
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Notes
- 1.
See Koning and van de Meerendonk (2014).
- 2.
See Krishna (2010) for all the standard results around the benchmark model.
- 3.
We provide the following example to illustrate the above two auctions. Let the scoring rule be \(S\left( p, q\right) =2q-p\). Suppose two firms A and B offer \(\left( 5,7\right) \) and \(\left( 3,5\right) \) as their \(\left( p, q\right) \) pairs. We have \(S\left( 5,7\right) =9\) and \(S\left( 3,5\right) =7\). Under both auction formats (first-score and second-score) firm A is declared the winner. The final contract awarded to firm A is \(\left( 5,7\right) \) under the first-score auction and any \(\left( p, q\right) \) satisfying \(S\left( p, q\right) =7\) under the second-score auction.
- 4.
See Asker and Cantillon (2008) and Che (1993) for other examples.
- 5.
See Nakabayashi and Hirose (2014) for other details.
- 6.
Hanazono (2010) uses the quality-to-price scoring rule. That is, here \( S\left( p, q\right) =\frac{q}{p}\). It may be noted that this short note is written in Japanese. I am grateful to Masaki Aoyagi for helping me understand the results of this paper.
- 7.
In Wang and Liu (2014) the scoring rule is as follows: \(S\left( p, q\right) =\omega _{1}\frac{\bar{p}}{p}+\omega _{2}\frac{q}{\underline{q}}\), where weights \(\omega _{1},\omega _{2}\) satisfy \(\omega _{1}+\omega _{2}=1\), \(\bar{ p}\) is the highest acceptable bidding price and \(\underline{q}\) is the lowest acceptable quality.
- 8.
This paper avoids specific functional forms but instead imposes some restrictions on the induced utility.
- 9.
In a second-price auction of the canonical model bidders bid their valuations. In a first-price auction bids are strictly less than valuations.
- 10.
In Che (1993) we have \(C_{q\theta }\left( .\right) >0\) and in Branco (1997) we have \(C_{q\theta }<0\). In Nishimura (2015) \(C_{\theta }\) has strictly increasing differences in \(\left( q,\theta \right) \).
- 11.
It is possible to have \(A\left( p, q\right) =0\ \ \forall \left( p, q\right) \in \mathbb {R} _{++}^{2}\) even with non-quasilinear rules (for example, take \(S\left( p, q\right) =e^{q-p}\)). Proposition 4 shows that when \(A\left( p, q\right) =0\) then \(q^{I}\left( \theta \right) =q^{II}\left( \theta \right) \). In this case also it can be shown that \(S^{I}\left( \theta \right) <S^{II}\left( \theta \right) \) \(\forall \theta \in \left[ \underline{\theta },\bar{\theta } \right) \).
- 12.
See Chap. 5 of Mood et al. (1974) for a nice discussion on expectation of functions of random variables.
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Dastidar, K.G. (2017). On Some Aspects of Scoring Auctions. In: Oligopoly, Auctions and Market Quality. Economics, Law, and Institutions in Asia Pacific. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55396-0_4
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