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Social Choice and Welfare

Erschienen in:

27.09.2016 | Original Paper

Axiomatization of reverse nested lottery contests

verfasst von: Jingfeng Lu, Zhewei Wang

Erschienen in: Social Choice and Welfare | Ausgabe 4/2016

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Abstract

The reverse nested lottery contest proposed by Fu et al. (2014) is the “mirror image” of the classical nested lottery contest of Clark and Riis (1996a), which has been axiomatized by Lu and Wang (2015). In this paper, we close the gap and provide an axiomatic underpinning for the reverse nested lottery contest by identifying a set of six necessary and sufficient axioms. These axioms proposed specify the properties of contestants’ probabilities of being ranked the lowest among all players or within subgroups, while the axiomatization of the classical nested lottery contest by Lu and Wang (2015) relies on axioms on contestants’ probabilities of being ranked the highest among all players or within subgroups.

Vorheriger Artikel A contest success function for rankings

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Namely, axioms: (A1) Imperfect Discrimination; (A2) Monotonicity; (A3) Anonymity; (A4) Sub-contest Consistency; and (A5) Independence from Irrelevant Contestants (IIC).

For instance, the model has been adopted in the studies of Clark and Riis (1998b), Amegashie (2000), Yates and Heckelman (2001), Szymanski and Valletti (2005), Fu and Lu (2009), Fu and Lu (2012a), Fu and Lu (2012b), Schweinzer and Segev (2012), etc.

Fu et al. (2014) also provide a micro foundation for the reverse nested lottery contest by showing that the model is uniquely underpinned by a noisy-performance-ranking model. This noisy performance model with Weibull (minimum) distribution for the shocks was first proposed by Hirshleifer and Riley (1992) in a two-player environment.

In the literature on perfectly discriminating contests, the most “classical” model on multi-prize contests might be the all-pay auction/contest model where players are ranked in a deterministic fashion. In this paper we focus on the literature on imperfectly discriminating contests where the nested lottery contest model is widely adopted—we call it the “classical” nested lottery contest model to distinguish it from the reverse nested lottery contest model.

In the noisy performance ranking contest, each contestant’s observable output is the sum of a deterministic component, which increases with his effort, and a random noise term. Clark and Riis (1996b) first point out the equivalence between a noisy-ranking-based discrete random choice model (McFadden 1973, 1974) and the single-prize lottery contest model. Fu and Lu (2012b) further establish a unique stochastic equivalence between a noisy performance ranking contest and the classical (multi-prize) nested lottery contest.

The performance of each experiment could follow any distribution.

Real-world examples include competitions in acrobatics, diving, and many skilled competitions across different professions.

As mentioned earlier, the existing literature has shown that the classical and reverse models can be interpreted as best-shot contests and worst-shot contests, respectively.

Before that, Fu et al. (2014) find that with a common single prize and identical impact function of power function form across the two models, the reverse model always induces more effort whenever a pure-strategy equilibrium exists. However, when the number of (common) prizes gets larger, the reverse contest loses its advantage gradually and eventually is dominated by the classical contest.

The lottery-form CSF is called the ratio-form CSF in Hwang (2012).

As shown in Fu et al. (2014), the classical and reverse nested lottery contests converge if and only if \(n=2\). In this paper we focus on the multi-prize case which requires at least three players.

Here “complete” means all players are ranked and “strict” means they are ranked without ties.

One can find that the two sets of axioms that underpin the classical and reverse contests [Lu and Wang (2015) and this paper] are parallel, so we use the same set of names of axioms for brevity. But one should bear in mind that those axioms in the two studies differ in whether they are concerned with the players’ probabilities of being ranked the highest or lowest in subgroups.

Some of the axioms may seem to be a bit stronger, this is because [differing from Skaperdas (1996)] the axioms in this paper explicitly allow zero effort in our framework.

Note that this result implies that \(\eta _{\mathbf {N}}^{i}(\mathbf {0})=1/| \mathbf {N}|=1/n\), which is consistent with Axiom R1. One may find that Axioms R1 and R3 are not independent as the second requirement of part (iii) of Axiom R1, which demands \(\eta _{\mathbf {N}}^{i}(Y_{\mathbf {N }})=1\mathbf {/|N}_{\mathbf {0}}(Y_{\mathbf {N}})|\), easily follows from other parts of Axiom R1 combined with Axiom R3. However, we do not weaken part (iii) of Axiom R1 as in later analysis (Proposition 2) it is indeed needed when Axiom R3 is dropped.

For instance, in a contest with \(\mathbf {N}=\{1,2,3\}\) players and effort entries \(Y_{\mathbf {N}}=(y_{1},y_{2},y_{3})\), let \(\mathbf {M}=\{1,2\}\), if \( \eta _{\mathbf {N}}^{1}(Y_{\mathbf {N}})=1/6\) and \(\eta _{\mathbf {N}}^{2}(Y_{ \mathbf {N}})=1/3\), then by Axiom R4, we have \(\eta _{\mathbf {M}}^{1}(Y_{ \mathbf {N}})=(1/6)/(1/6+1/3)=1/3\) and \(\eta _{\mathbf {M}}^{2}(Y_{\mathbf {N} })=(1/3)/(1/6+1/3)=2/3\), thus \(\eta _{\mathbf {M}}^{1}(Y_{\mathbf {N}})/\eta _{ \mathbf {M}}^{2}(Y_{\mathbf {N}})=\eta _{\mathbf {N}}^{1}(Y_{\mathbf {N}})/\eta _{\mathbf {N}}^{2}(Y_{\mathbf {N}})=1/2.\)

It was initially called the “ratio-form CEF” in Sect. 2.1 of Fu et al. (2014).

The detailed proof can be obtained following a very similar pattern in Appendix A.3.2 of Lu and Wang (2015), which is omitted for brevity in this study.

This differs from the classical nested lottery contest model of Clark and Riis (1996a) where the draws for higher prizes are conducted earlier.

When \(\exists j\in \Omega _{k}\), \(y_{j}=0\), the probability that player i \( \in \Omega _{k}\) wins prize \(v_{k}\) is \(1/[\#(j|f(y_{j})=0,j\in \Omega _{k})] \), where \(\#(j|f(y_{j})=0,j\in \Omega _{k})\) is the count of zero efforts among \(\Omega _{k}\).

It is straightforward to extend the result to the cases where \(\mathbf {N}_{ \mathbf {0}}\ne \mathbf {\emptyset }\) by allowing efforts of some players to be zero. The proof is omitted to save space.

It can be shown that the reverse nested lottery contest model also satisfies Axioms 1–3 and 5 of Lu and Wang (2015). Similarly, Axioms 4 and 6 in Lu and Wang (2015) characterize distinctive properties that only the classical nested lottery contest model satisfies, among the three models compared.

Lu, Lu et al. (2016) show that the existing results on the optimal design of multi-stage elimination contests, which are obtained under classical nested lottery technology [e.g. Gradstein and Konrad (1999), and Fu and Lu (2012a)], may vary when being reexamined under the reverse nested lottery technology.

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Titel: Axiomatization of reverse nested lottery contests
verfasst von: Jingfeng Lu
Zhewei Wang
Publikationsdatum: 27.09.2016
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 4/2016
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-016-0998-4

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