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01-10-2014

All-pay-all aspects of political decision making

Authors: Thomas Giebe, Paul Schweinzer

Published in: Public Choice | Issue 1-2/2014

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Abstract

Decision-making processes are studied using non-standard all-pay structures. Our interest is motivated by regulatory, political, legal, military, and economic applications in which individual actions determine the consequences for a larger group or the general public. The common features of these examples are a competitive environment, a winner-takes-all reward structure, and some form of all-pay-all payment rule.

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See, for instance, the examples in Baye et al. (1998, 2012).

In this game, the players’ bids are interpreted as proposals, rather than efforts. Only the winning proposal is ex post payoff relevant.

The EU commission’s recent regulation of mobile phone roaming charges may serve as an example, documented in, for example, King (2011). The payoff from being perceived as consumer friendly may be the motivation behind mobile carrier Vodafone’s announcement of the ‘travel promise’ (the statement of 3-Aug-2005 begins with ‘Commitment to simplify and increase value to Vodafone customers’) and competitor T-Mobile’s ‘Un-carrier’ strategy, both of which can be seen as widely advertised announcements of ‘consumer-friendly’ price reductions in order to generate future payoffs.

Three examples from politics, business, and the public sector, respectively, are Kasperowicz (2013), Chazan (2013), and Foster (2010).

This idea is not new. The two-player complete information case of this setup has recently been analyzed by Baye et al. (2012). In particular, see their section ‘Territorial contests with injuries.’ Moreover, Baye et al. (1998) specify a ‘civil war’ parametrization of their general contest class, which is a special case of our auction model.

Higgins et al. (1985) provide the first analysis of endogenous participation in contests.

The beautiful theory of these games has been developed, among others, by Dasgupta (1986), who discusses R&D races, Hillman and Samet (1987), who introduce rent-seeking and lobbying contests, Hillman and Riley (1989) and Baye et al. (1993), who analyze competitions for monopoly, Clark and Riis (1998), who study the competition for promotions, and Moldovanu and Sela (2001), who discuss optimal prize structures. The availability of recent and comprehensive surveys of the contests literature such as Garfinkel and Skaperdas (2006), Congleton et al. (2007), Corchón (2007), and Konrad (2008) allows us to focus our discussion on only the most directly related parts of the literature.

For further discussions of asymmetric all-pay auctions, see, for instance, Hillman and Riley (1989), Fibich et al. (2006), Parreiras and Rubinchik (2010), or Szech (2011). For a detailed discussion, we appeal, again, to the above-mentioned literature surveys.

A later published variant Goeree et al. (2005), is less related to the present work because, there, the payment rules do not exhibit an ‘all-pay-all’ structure.

The parameters (α,β,δ,θ,γ) appear in the winner’s and loser’s linear payoff functions in Baye et al. (1998, 2012). In Baye et al. (2005), the parameters are essentially the same, with γ=0.

So called after former US Vice President Dan Quayle, who chaired a commission that recommended a reform of the US legal system. According to this proposal, the loser should reimburse the winner’s legal costs up to the amount actually spent by the loser. The Quayle system is one of the legal systems analyzed in Baye et al. (1998, 2005).

Ties have zero probability in the equilibria we derive.

In a simple symmetric oligopoly model, the price cap is proportional to the firm’s profits. In an asymmetric setting, the firms’ bids can be understood as proposals for direct lump-sum reductions of profits.

Given full participation, this boundary condition is intuitive: if the valuation is zero, then winning has no upside. The zero type would focus on minimizing payments: by making the lowest feasible bid, the price reduction in the event of winning is minimized.

The case of taxation is mentioned in Baye et al. (2005) as a way of reducing wasteful legal expenditures.

Baye et al. (2005) show that these performance measures are a function of their β only, with β=1 and β=0 in the American and British systems, respectively, while our example uses β=1/2. Apart from this case, we cannot compare results because, as soon as we deviate from α=1/2 and linear payoffs, we violate assumption (A3) in Baye et al. (2005), which is incorporated in their solution from the beginning. Moreover, note that these results of Baye et al. (2005) are comparable with ours given only that there is litigation, i.e., participation.

Similarly, replace f(x)x/F(x) in (9) by α/(n−1), simplify, and solve $\bar{u}=\varPi(0,0)$ to obtain the required upper bound of the outside option.

We cannot compare results with Baye et al. (2005), however, because we violate their assumption (A3) ‘Internalized legal cost,’ on which their relevant results rest.

Recall that g(x)=G′(x)=(n−1)F ⁿ⁻²(x)f(x).

Consider full participation and F(θ)=θ ^t, for some t>0, and for which the auction parameters are k=t(n−1)+1 and α=(k−1)/k, then the equilibrium exhibits truth-telling β(θ _i)=θ _i: F(x)=x ^t implies that g(x)x=t(n−1)x ^t(n−1). Inserting this into (12) and setting the term equal to θ _i (truthful bidding), we get

$$ {\theta_i = \biggl( \frac{1}{\alpha} \int _0^{\theta_i} \bigl(x^{t(n-1)} \bigr)'x dx \biggr)^{\frac{1}{k}}\quad \Rightarrow\quad \alpha \theta_i^k = \frac{t(n-1)}{t(n-1)+1}\theta_i^{t(n-1)+1}.} $$

(13)

This equation is satisfied if we insert the proposed α=(k−1)/k and k=t(n−1)+1. This result is nongeneric, in the sense that it relies on a special distributional assumption.

Recall that the equilibrium bid function of the FP-APA under our assumptions is $\beta^{\text{FP-APA}}(\theta_{i}) = \int_{0}^{\theta_{i}} g(x) x dx$. The corresponding equilibrium payoff is $G(\theta _{i})\theta_{i} - \beta^{\text{FP-APA}}(\theta_{i}) = \int_{0}^{\theta_{i}} G(x) dx$.

Consider (11). Under full participation ($\hat{\theta}=0$), the first term is zero, the second term is the FP-APA-payoff, and the last term is negative. Therefore, the payoff in our auction is lower than under the FP-APA.

For instance, consider the uniform distribution and full participation ($\hat{\theta}=0$). Then (9) is equal to $-(n-2+1/n)+ \theta_{i}^{2}/n$, where the first term is negative and the second term is the FP-APA-payoff. The payoff to our auction thus is smaller.

For a brief discussion of the state of the literature, see Chawla and Hartline (2013). Note that a large part of the all-pay auction literature deals with the complete-information case, whereas we are concerned with the standard private-values setting.

Negative equilibrium payoffs are discussed in, e.g., Krishna and Morgan (1997).

Amann, E., & Leininger, W. (1996). Asymmetric all-pay auctions with incomplete information: the two-player case. Games and Economic Behavior, 14, 1–18.CrossRef

Baye, M. R., Kovenock, D., & de Vries, C. (1993). Rigging the lobbying process: an application of all-pay auctions. The American Economic Review, 83, 289–294.

Baye, M. R., Kovenock, D., & de Vries, C. (1998). A general linear model of contests. Mimeo, Purdue University.

Baye, M. R., Kovenock, D., & de Vries, C. (2005). Comparative analysis of litigation systems: an auction-theoretic approach. The Economic Journal, 115(505), 583–603.CrossRef

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Chazan, G. (2013). BG executive directors to forgo bonuses. Financial Times, 4-Apr-2013.

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Garfinkel, M. R., & Skaperdas, S. (2006). Economics of conflict: an overview. In T. Sandler & K. Hartley (Eds.), Handbook of defense economics (Vol. 2. Amsterdam: Elsevier. Chap. 3

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Title: All-pay-all aspects of political decision making
Authors: Thomas Giebe
Paul Schweinzer
Publication date: 01-10-2014
Publisher: Springer US
Published in: Public Choice / Issue 1-2/2014
Print ISSN: 0048-5829
Electronic ISSN: 1573-7101
DOI: https://doi.org/10.1007/s11127-013-0127-1

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