The role of noise in alliance formation and collusion in conflicts

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.

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

$$\begin{aligned} \frac{\sum _{j\ne i}x_j + 2\alpha }{\left( \sum _{i=1}^3x_i+ 3\alpha \right) ^2}v - 1 = 0. \end{aligned}$$

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

$$\begin{aligned} p_1=p_2=\frac{x_1+x_2+2\alpha }{x_1+x_2+x_3+3\alpha }, \end{aligned}$$

(8)

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:

$$\begin{aligned} \frac{\partial \pi _i}{\partial x_1}= & {} \frac{x_3+\alpha }{\left( \sum _{i=1}^3x_i+3\alpha \right) ^2}\frac{v}{2}-1=0, \\ \frac{\partial \pi _2}{\partial x_2}= & {} \frac{x_3+\alpha }{\left( \sum _{i=1}^3x_i+3\alpha \right) ^2}\frac{v}{2}-1=0, \end{aligned}$$

and

$$\begin{aligned} \frac{\partial \pi _3}{\partial x_3} = \frac{x_1+x_2+2\alpha }{\left( \sum _{i=1}^3x_i+3\alpha \right) ^2}\frac{v}{2}-1=0. \end{aligned}$$

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.