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Social Choice and Welfare

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27.09.2016 | Original Paper

A contest success function for rankings

verfasst von: Alberto Vesperoni

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

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Abstract

In a contest, several candidates compete for winning prizes by expending costly efforts. We assume that the outcome of a contest is a ranking, which is an ordered partition of the set of candidates. We consider rankings of any type (i.e., with any number of candidates at each rank), and define a class of success functions that assign probabilities to all rankings of a given type for each configuration of candidates’ efforts. Our framework can be interpreted as the probabilistic choice of a committee that allocates a given set of prizes following a mix of objective and subjective criteria, where the former depend on candidates’ efforts while the latter are random. We axiomatically characterize our class of success functions and illustrate its relevant features to contest games.

Vorheriger Artikel Incomplete information, proportional representation and strategic voting

Nächster Artikel Axiomatization of reverse nested lottery contests

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A notable exception is Blavatskyy (2010), which considers contests whose outcome can be any ranking with at most two levels. Although we do not consider such contests in this paper, our framework can be easily extended following Blavatskyy (2010)’s approach for the two-level case. See Sect. 2 for a discussion.

By equivalent outcomes we mean allocations that differ only by permuting prizes of equal value across candidates.

The class of success functions in Skaperdas (1996) includes the widely applied ratio-form and difference-form, also known as the Tullock (1975) and Hirshleifer (1989) models respectively.

We follow Fu and Lu (2012a) and Fu et al. (2014) in the use of the terms “best-shot” and “worst-shot”. While the worst-shot success function is relatively new (see Fu et al. 2014), the best-shot success function dates back to Clark and Riis (1996) and it is widely applied, see e.g. Clark and Riis (1996, (1998c), Amegashie (2000), Szymanski and Valletti (2005), Fu and Lu (2009, (2012b) and Schweinzer and Segev (2012).

A previous version of this paper characterizes our class of success functions as a consistent extension of the single-winner model characterized in Skaperdas (1996) via a unique axiom named pair-swap consistency. We thank an anonymous referee for suggesting to derive pair-swap consistency from more basic axioms, which led to this paper in the current form.

While these terms are not used in Skaperdas (1996), we follow the terminology in Wärneryd (2001) which became customary in the literature.

This is a specific case of the general model axiomatized in Blavatskyy (2010), which we discuss below.

This success function is widely applied to model contests with the possibility of a draw. Two alternative success functions for contests with the possibility of a draw are characterized in Jia (2012) stochastically and in Vesperoni and Yildizparlak (2016) axiomatically. The analysis of contests with draws is fairly common in the literature on tournaments and patent races (see, e.g., Nalebuff and Stiglitz 1983; Imhof and Kräkel 2014, 2016). For the analysis of all-pay auctions with the possibility of a draw, see Eden (2006), Cohen and Sela (2007) and Gelder et al. (2015).

For each dimension (objective or subjective), the score of an allocation is a cardinal representation of the committee’s preferences for that dimension. To define a complete preference relation over allocations, one should also specify the “rate of substitution” across dimensions. See Luce (1958) on the representation of a preference relation in a probabilistic setting.

In the analysis below, the cases with inactive candidates can be easily accommodated as limit cases by assuming \(\lim \nolimits _{x_i\rightarrow 0}u(x_i)=-\infty \). For example, this is true with the form \(u(x_i)=\alpha \ln (x_i)\), where \(\alpha >0\).

See Gaertner (2009) for an introduction to the social choice literature.

Note that in Moldovanu et al. (2007) the pure status of an individual depends on the rank of its class, but it is independent of the number of individuals assigned to the same class or to other classes. Similarly, in our model the meritocratic value \(w(v_i)\) depends on the number of prize values that are lower/higher than \(v_i\), but it is independent of the number of candidates that receive prizes of value lower/higher than \(v_i\).

To see an example, consider the standard payoff function \(U_i(x):={g_i(x_i)}/{\sum _{j\in N}g_j(x_j)}- x_i\), where \(g_i:{\mathbb {R}}_{++}\rightarrow {\mathbb {R}}_{++}\) is positive and increasing. With the change of variable \(y_i:=g_i(x_i)\), we obtain \(U_i(y)={y_i}/{\sum _{j\in N}y_j}- g_i^{-1}(y_i)\), so that the success function satisfies anonymity.

Formally, given a type \(t\in T\), for any ranking \(r\in R(t)\) and pair of candidates \(i,j\in N\) the pair-swap ranking \(r_{i,j}\) is defined as the ranking in R(t) where (1) \(r_{i,j}(j)=r(i) \text { and } r_{i,j}(i)=r(j)\); (2) \(\forall \text { } k\in N\backslash \{i,j\}:\text { } r_{i,j}(k)=r(k)\).

We formalize the argument in Appendix, linking effort-independence to sub-contest independence and sub-contest consistency in Skaperdas (1996).

These results directly follow from intermediate steps of the proof of Theorem 1.

In a previous version of this paper an axiom named pair-swap consistency demanded (7) to hold for the case \(r(j)-r(i)=1\), that is, when i and j are ranked at adjacent levels.

We show in Appendix that, for the single-winner type, effort-independence can be derived by a combination of two axioms in Skaperdas (1996): sub-contest independence and sub-contest consistency.

See Mattsson and Weibull (2002) and Voorneveld (2006) for related models of inattentive choice. For alternative approaches, see Manzini and Mariotti (2014) and references therein.

This assumption essentially corresponds to the axiom of ranking-independence.

Axiomatic characterizations of single-winner success functions where f fulfills these criteria are provided in Skaperdas (1996) and Hwang (2012).

Take any \(t\in T\), \(x\in X\) and \(i\in A_x\). Let Q(i, x) be the set of all rankings \(r\in S_{t,x}\) such that \(r(j)\ne r(i)\) for some \(j\in A_x\) and candidate i’s level is strictly above its expected level, i.e., \(r(i)<\rho _t(i,x)\). Take any \(r\in Q(i,x)\). By Point 3 \(p_t(r,x)\) increases in \(x_i\). By Point 4, \(p_t(r,x)\) is concave in \(x_i\) if the distance \((r(i)-\rho _t(i,x))^2\) is smaller than the expected distance. It can be shown that the expected distance is the largest when \(x_i=x_j\) for all \(i,j\in A_x\).

This restriction is for simplicity of exposition only. If some prizes have equal value, by the same arguments below one can show that the stochastic model defined in this section leads to probabilities of allocations that are equal to the ones given by the best-shot and worst-shot success functions.

The proof of the case \(r(h)=r(i)\) is very similar, therefore it is omitted.

We thank an anonymous referee for suggesting this.

As \(r\in S_{t,x}\), if there are active candidates that are not ranked at r(i) or r(j) there must be an active candidate h ranked at \(r(h)=r(j)+1\) or \(r(h)=r(i)-1\). We focus on the case \(r(h)=r(j)+1\), while we omit the proof for the second case that is essentially the same.

We thank an anonymous referee for suggesting to organize the proof in this fashion.

Concavity is implied by Proposition 4 if \(r(i)=2\), and it directly follows from (21) if \(r(i)=1\).

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Titel: A contest success function for rankings
verfasst von: Alberto Vesperoni
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/s00355-016-0997-5

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