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Published in: Public Choice 1-2/2023

26-08-2023

Rent dissipation in large population Tullock contests

Authors: Ratul Lahkar, Rezina Sultana

Published in: Public Choice | Issue 1-2/2023

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Abstract

Tullock contests model rent seeking behavior where agents exert unproductive effort to probabilistically win a fixed prize. Rent dissipation measures the social loss involved in effort exertion in such contests. Tullock contests are characterized by an impact function, which measures how effort impacts success, and a cost of effort function. If these functions are asymmetric and non-linear, then the contest cannot be solved in closed form. Hence, we approximate such contests with a large population contest, for which Nash equilibria and rent dissipation can be explicitly calculated. Rent dissipation is then the ratio of the effort elasticity of impact to the effort elasticity of cost. Greater elasticity of impact incentivizes more exertion of unproductive effort generating higher social loss.

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Appendix
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Footnotes
1
See Congleton et al. (2008) for an extensive literature review.
 
2
There are certain models like R &D and patents races which are not strictly rent seeking contests as in such models, the value of the prize increases with the effort exerted. Nevertheless, such models do share certain strategic similarities with pure rent-seeking contests (Baye et al., 1993) Hence, the insight from rent-seeking contests can be useful in understanding such models as well.
 
3
The power in both impact and cost functions are assumed to be the same for all agents. Asymmetry arises through the presence of different multiplicative constants. The multiplicative constant in the impact function is called the bias parameter and captures the idea that the same level of effort can have different impact for different players.
 
4
See Sect. 6 for those simulations.
 
5
In a polymorphic equilibrium, different agents of the same type play different strategies.
 
6
The next section will clarify why we are using \({\hat{x}}_i\) instead of \(x_i\) to denote effort. We will use \(x_i\) to denote a transformation of variable that we will interpret as effort in the large population contest.
 
7
Heterogeneity in cost functions is equivalent to heterogeneity in prize valuation in our model. To see this, we can divide (7) by \(\kappa _p\). All agents would have the identical cost function \({\hat{x}}^{\gamma }\) but type specific valuations \(\frac{V}{\kappa _p}\).
 
8
Thus, \(\frac{1}{N}x_i=1\times {\hat{x}}_i\).
 
9
Here, \(\delta _{x_p}\) is the Dirac distribution with probability 1 on \(x_p\).
 
10
Notice that in the second payoff of (10), we write \(V-k_px^{\gamma }\) whereas in the second payoff of (8), we simply write V. This is because in (8), due to the finite number of players, \(\sum _{q\in \mathcal {P}}\sum _{j\in q}\theta _q x_j^r=0\) only if \(x_i=0\) for all players. Hence, writing \(V-k_px_i^{\gamma }\) is redundant. But that is not so in (10). A single measure zero agent cannot affect \(A(\mu )\). Therefore, it is possible that \(A(\mu )=0\) but a single agent is playing \(x>0\). That player’s payoff will be \(V-k_px^{\gamma }<V\).
 
11
Suppose at \(\alpha \), \(b_p(\alpha )<{\bar{x}}\) for all \(p\in \mathcal {P}\). Then, from (18), we obtain \(b_p(\alpha )=\left( \frac{\theta _p Vr}{k_p\alpha \gamma }\right) ^{\frac{1}{\gamma -r}}\). Hence, \(\sum _{p\in \mathcal {P}}m_p\theta _pb_p^{r}(\alpha )=\left( \frac{Vr}{\alpha \gamma }\right) ^{\frac{r}{\gamma -r}} \sum _{p\in \mathcal {P}}m_pk_p^{\frac{-r}{\gamma -r}}\theta _p^{\frac{\gamma }{\gamma -r}}\). Solving (19), we then obtain (20).
 
12
Nor does, in the case of an asymmetric contest, the type distribution matter for the limiting value of \(\frac{r}{\gamma }\).
 
13
Section 2.1 in Nitzan (1994) presents a similar result for symmetric agents.
 
14
By a symmetry argument, all players of the same type will play the same strategy in equilibrium. Hence, with just three types, we have only three first order conditions to solve to find the Nash equilibrium. This is irrespective of the number of players. The additional assumption that the number of players of each type is the same further simplifies the numerical exercise.
 
15
One such Nash equilibrium in monomorphic states is where every agent of every type \(p\in \mathcal {P}^{*}\) plays strategy \(\alpha _p^{*}=\left[ \frac{V\theta _p}{k_p}\left( \frac{1}{\sum _{q\in \mathcal {P}^{*}}m_q\theta _q}\right) \right] ^{\frac{1}{r}}\). The resulting value of \(A(\mu )\) is \(\sum _{p\in \mathcal {P}^{*}}m_p\theta _p(\alpha ^{*})^r=\frac{V\theta _p}{k_p}\), as in (30). Notice that if \(p,q\in \mathcal {P}^{*}\), \(\alpha _p^{*}=\alpha _q^{*}\) because by (29), \(\frac{\theta _p}{k_p}=\frac{\theta _q}{k_q}\). On the other hand, a polymorphic Nash equilibrium will be where for every \( p \in {\mathcal{P}}^{*} \), half the agents play \(x=0\) and the other half play \(2\alpha _p^{*}\).
 
16
For example, the expression \(\left( \frac{\theta _p Vr}{k_p\alpha \gamma }\right) ^{\frac{1}{\gamma -r}}\) in (18) would then be the minimizer instead of the maximizer.
 
17
In fact, this can be directly verified by applying (22) to \(\sum _{p\in \mathcal {P}}m_p\theta _pb_p^r(\alpha ^{*})\).
 
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Metadata
Title
Rent dissipation in large population Tullock contests
Authors
Ratul Lahkar
Rezina Sultana
Publication date
26-08-2023
Publisher
Springer US
Published in
Public Choice / Issue 1-2/2023
Print ISSN: 0048-5829
Electronic ISSN: 1573-7101
DOI
https://doi.org/10.1007/s11127-023-01103-7

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