## Abstract

Many real-world conflicts are to some extent determined randomly by noise, and many also depend critically on the formation of alliances or long-run cooperative relationships. In this paper, we emphasize that the specific manner by which noise is modeled in contest success functions (CSFs) has implications for both the possibility of forming cooperative relationships and the features of such relationships. The key issue is that there are two distinct approaches to modeling noise in CSFs, each with their own merits and each leading to different results depending on which type of alliance formation is under consideration. In a one-shot conflict, we find that when noise is modeled as an exponential parameter in the CSF, there is a range of values for which an alliance between two parties can be beneficial; that is not the case for models with an additive noise parameter. In an infinitely repeated conflict setting, we again find discrepant results: with additive noise, sustaining collusion via Nash reversion strategies is easier the more noise there is and more difficult the larger the contest’s prize value, while an increase in the contest’s number of players can make sustaining collusion either more or less difficult. This is all in marked contrast to the case of an exponential noise parameter, when noise plays no impact on the sustainability of collusion. Given that alliances do occur in both scenarios in the real world, this contrast could be seen as supporting the importance of both specifications.

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## Notes

Other applications of contest models include political lobbying, electoral competition, litigation, advertising competition, R&D competition, and sporting competition.

The restriction on \(\gamma\) is sufficient to ensure an interior pure-strategy Nash equilibrium.

Dasgupta and Nti (1998) also use a similar CSF specification in their study of optimal contest design, but interpret their parameterization as the probability that the contest does not award the prize, which is more like the contests with the possibility of a draw studied by Blavatskyy (2010) and Jia (2012).

We go through the first order conditions for a more complicated version of the model shortly in the paper and the maximization process itself is fairly well-known so we omit the specifics of this version here.

More generally, in a standard

*n*-player contest with a Tullock CSF as in (1) with all parties maximizing their payoffs individually, \(x_i^T=\frac{\gamma (n-1)}{n^2}v\) and \(\pi _i^T=\frac{n-\gamma (n-1)}{n^2}v\) for all \(i\in {I}\).For a more specific illustration of the derivation of these results we refer readers to the paper’s “Appendix”.

Splitting the prize via second-stage intra-alliance conflict only harms the allies relative to the unallied party, as the additional conflict further dissipates the prize value for the allies.

A Wolfram Alpha link to a graph illustrating this relationship (with

*v*normalized to 1) can be found at https://tinyurl.com/yczg5beb.A Wolfram Alpha code to illustrate this result, with

*v*normalized to 1 can be found at https://tinyurl.com/y73upr5m.Similar logic explains why the expected payoff to an allied party \(i\in \{1,2\}\) gets closer to that of the unallied party 3 as noise increases, though the unallied party remains advantaged due its lack of collective action problem, which is somewhat of a paradox.

Wolfram Alpha link here: https://tinyurl.com/y8nmygf2.

There also exist a number of studies that analyze explicit collusion in one-shot contests (e.g., Alexeev and Leitzel 1991, 1996; Huck et al. 2002) and that develop models of infinitely repeated contests to analyze non-collusive behavior (e.g., Itaya and Sano 2003; Mehlum and Moene 2006; Krähmer 2007; Eggert et al. 2011; Grossmann et al. 2011).

It is straightforward to show that the first derivative of (5) with respect to \(x_{it}\) is positive when \(x_{jt}=0\) for all \(j\in {I}{\setminus }{\{i\}}\) and \(\alpha <(n-1)v/n^{2}\), ruling out all players making 0 expenditures as a Nash equilibrium. It is also straightforward to show that (5) is strictly concave in \(x_{it}\).

Numerous studies of collusion in repeated contests follow a similar approach; see, for example, Linster (1994), Amegashie (2006a), Amegashie (2011), Shaffer and Shogren (2008), and Cheikbossian (2012). Therefore, we adopt this approach so that our results on incentives for collusion are comparable to ones already existing in the literature.

It is straightforward to show that (6) is strictly concave in \(x_{it}\).

Shaffer and Shogren (2008) analyze the critical discount rate (\(r^{*}\)) sustaining collusion, which relates to the critical discount factor (\(\delta ^{*}\)) we analyze as \(\delta ^{*}=1/(1+r^{*})\).

Thanks very much to an anonymous reviewer for this interpretation.

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## Appendix: Comparative statics for the additive noise CSF

### Appendix: Comparative statics for the additive noise CSF

In a one-shot, three-party contest with the additive noise CSF specified in (2), each player *i* maximizes \(\pi _i(x_1,x_2,x_3) = p_i(x_1,x_2 ,x_3)v - x_i\) with respect to their own expenditure \(x_i\), leading to three first order conditions of the form

Solving those three first order conditions for a symmetric equilibrium leads to equilibrium expenditures of \(x_i^*=\frac{2}{9}v-\alpha\), thus equal (\(\frac{1}{3}\)) probabilities of victory, and expected payoffs of \(\pi _i^*=\frac{1}{9}+\alpha\). Note that this is extremely similar to the results from the standard Tullock contest without any alliances, with the noise factor \(\alpha\) modifying things in a very straightforward manner. Also, as with exponential noise, noise decreases expenditures since the CSF is less sensitive to effort and thus increases expected payoffs.

To complete the comparison results from Sect. 2, we now consider a one-shot contest with a modified additive noise CSF so that the two parties who ally have the probabilities of victory equal to

with a complementary unallied \(p_3\). With each party maximizing their corresponding payoff functions, and assuming that the two allies simply agree to split the prize evenly in the event of victory, we then have the first order conditions:

and

The fact that the first two first order conditions are identical is what leads to the inability to solve for \(x_1^*\) or \(x_2^*\) uniquely. This also illustrates what leads to the alliance puzzle results in the alliance model with a Tullock CSF and no noise (\(\gamma =1\)), as in Ke et al. (2013). The first order conditions in that case are the same as those above but without the \(\alpha\) parameters, leading to the equilibrium values in *(i)*, *(ii)*, and *(iii)*.

In the additive noise version, the presence of the additive parameter means solving for analytical solutions leads to quadratic expressions, but the same qualitative results as those in *(i)*, *(ii)*, and *(iii)* hold in terms of the allies remaining unable to find unique expenditure levels, each therefore having the incentive to free-ride off of the other, and at best ending up the same as in the unallied case.

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Boudreau, J.W., Sanders, S. & Shunda, N. The role of noise in alliance formation and collusion in conflicts.
*Public Choice* **179**, 249–266 (2019). https://doi.org/10.1007/s11127-018-0564-y

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DOI: https://doi.org/10.1007/s11127-018-0564-y