Abstract
We construct a model in which all rational rent-seekers may be willing to accept a negative payoff in a Tullock contest in which the rent-seekers are driven by reciprocity motives. We show that in the reciprocal Tullock contest, a unique reciprocity equilibrium exists if the reciprocal concerns of rent-seekers are sufficiently small relative to their material concerns, and that otherwise, there are two reciprocity equilibria: a destructive equilibrium and a constructive equilibrium. The individual rent-seeking expenditure in the former equilibrium is more than that in the Nash equilibrium in the original Tullock contest; moreover, over-dissipation can occur in a destructive equilibrium even in the case of constant returns to expenditure. These results derived from our reciprocal contest model are consistent with observations in most existing experimental studies on the Tullock contest.
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Notes
In a contest in which the number of homogeneous rent-seekers is a finite \(N\), a pure-strategy Nash equilibrium exists for any \(r\le N/(N-1)\) but cannot be established for any \(r>N/(N-1) \). Therefore, as \(N\rightarrow \infty \) such that the contest is perfectly competitive, no pure-strategy Nash equilibrium can be established for \(r>1\).
Sano (2009) constructed explicit evolutionary dynamics that are driven by players’ imitation and experimentation. The paper shows that full dissipation may prevail in the long run as long as there are increasing returns, assuming that each player does not imitate any negative payoff even though it is the highest individual payoff because he/she can obtain a zero payoff by choosing a zero bid.
We refer to this equilibrium as the “reciprocity equilibrium” even though its definition is the same as that of Rabin’s (1993) “fairness equilibrium”.
We follow the terms used in Sobel (2005).
Note that \(Nx^{\textit{SNE}}<V\) for any finite \(N\).
On the basis of our theoretical results, we predict that either underspending or overspending accompanied by over-dissipation will be observed in each of experimental contests with a sufficiently small value of the prize. Therefore, for the purpose of testing our theoretical prediction, it would be informative to examine frequency distribution of total expenditures observed in the experiment in which many contests are conducted for each of various values of the prize. If relatively small and large total expenditures exhibit significantly high frequency compared with other levels of total expenditures, the experimental evidence might support our result that two reciprocity equilibria exist for any sufficiently small value of the prize.
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Acknowledgments
I wish to thank Atsushi Kajii, Takashi Sekiguchi, and participants at the Summer Workshop on Economic Theory (SWET) 2010. My special thanks are due to the associate editor and two anonymous referees of this journal for helpful suggestions and comments. I am also grateful to Jun-ichi Itaya, Hideki Konishi, and Yoichi Hizen. However, all errors remain my own responsibility.
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Appendices
Appendices
1.1 Appendix A: Proof of Proposition 1
First, suppose that both \(y_{j}\) and \(z_{i}\) are greater than \(V\). In this case, there cannot exist any constructive reciprocity-equilibrium since \( \lambda _{j}\left( y_{j},z_{i}\right) =-y_{j}/\left( z_{i}+y_{j}\right) <0\) from (15). From (14) and (15), the first-order condition to maximize \(U^{i}\left( x_{i},y_{j},z_{i}\right) \) in (13) is
It is clear that \(U^{i}\left( x_{i},\cdot , \cdot \right) \) is strictly concave; thus, there exists only a single global maximizer for \(U^{i}\left( x_{i},\cdot , \cdot \right) \) given \(y_{j}\) and \(z_{i}\). Substituting \( z_{i} =y_{j} =x_{i} \equiv x\) into (24) yields \(x =\left( 2V+\omega \right) /8\), which is greater than \(V\) if and only if \(V/\omega <1/6\). That is, when \(V/\omega <1/6\), we have the destructive reciprocity-equilibrium expenditure of \(x^{\textit{SRE}} =\left( 2V+\omega \right) /8 >V \).
Next, suppose that both \(y_{j}\) and \(z_{i}\) are lying in the interval \( \left( 0,V\right] \). Then, from (14) and (15), we get that the first-order condition for an interior solution maximizing \(U^{i}\left( x_{i},y_{j},z_{i}\right) \) in (13) is
Furthermore, the second-order derivative is given by
Therefore, \(U^{i}\left( x_{i},\cdot , \cdot \right) \) is strictly concave unless \(\partial U^{i}/\partial x_{i} \le -1\). This implies that \(x_{i}\) satisfying (25) is the only global maximizer for \(U^{i}\left( x_{i},y_{j},z_{i}\right) \) given any beliefs \(y_{j}\) and \(z_{i}\). Substituting \(z_{i} =y_{j} =x_{i} \equiv x\) into (25), we obtain
From (26), \(V/\omega \ge 1/6\) if and only if \(\varphi \left( V\right) \le 0\). If \(2V-\left( 1-2\theta \right) \omega \ge 0\) or equivalently \(V/\omega \ge 1/2-\theta \), we obtain \(\varphi ^{\prime }\left( x\right) <0\) for any \( x \le V\) and \(\lim _{x\rightarrow 0}\varphi \left( x\right) =\infty \). Hence, a unique symmetric reciprocity-equilibrium exists if \(V/\omega \ge \max \left\{ 1/6,1/2-\theta \right\} \) since there exists only one \(x <V\) such that \(\varphi \left( x\right) =0\) in that case. Moreover, we can verify that \( x^{\textit{SRE}} >x^{\textit{SNE}}\) since \(\varphi \left( x^{\textit{SNE}}\right) =\theta \omega /2V >0\), which implies that the reciprocity equilibrium is destructive. Suppose that \(V/\omega <1/2-\theta \) for \(\theta <1/2\). We can then verify that \( \lim _{x\rightarrow 0}\varphi \left( x\right) =-\infty ,\,\lim _{x\rightarrow 0}\varphi ^{\prime }\left( x\right) =+\infty \), and \(\varphi \left( V\right) \le 0\) when \(V/\omega \ge 1/6\); moreover, the function \(\varphi \left( x\right) \) is strictly concave unless it decreases with \(x\). Together with the fact that \( \varphi \left( x^{\textit{SNE}}\right) >0\), these imply that the global maximum of \(\varphi \left( x\right) \) is strictly positive; thus, \(\varphi \left( x\right) =0\) has only two positive real roots. Since \( \varphi \left( x^{\textit{SNE}}\right) >0\), one of the two real roots is the destructive-equilibrium expenditure \(x^{\textit{SRE}} >x^{\textit{SNE}}\), while the other is the constructive-equilibrium expenditure \( x^{\textit{SRE}} <x^{\textit{SNE}}\) if \(V/\omega <1/2-\theta \) for \( \theta <1/2\). The black curve drawn in Figure 1 is a configuration of the function \(\varphi \left( x\right) \) when \(V/\omega <1/2-\theta \). The curve intersects the \(x\)-axis at two points: the larger point is the destructive-equilibrium expenditure and the smaller is the constructive-equilibrium expenditure. At the same time, if \(V/\omega \ge \max \left\{ 1/6,1/2-\theta \right\} \), the function \( \varphi \left( x\right) \) is depicted as the gray curve in the same figure. The gray curve intersects the \(x\)-axis at only one point: the only reciprocity equilibrium expenditure.
Finally, suppose that player \(i\) has the beliefs \(y_{j} =z_{i} =0\) regarding the opponent. We then have \(\kappa _{i}\left( x_{i},0\right) =\lambda _{j}\left( 0,0\right) =0\); thus, from (13), player \(i\)’s utility is \(U^{i}\left( x_{i},0,0\right) =\pi _{i}\left( x_{i},0\right) =V-x_{i}\) choosing any \(x_{i} >0\) while \(U^{i}\left( x_{i},0,0\right) =0\) choosing \(x_{i} =0\). Hence, player \(i\) never chooses \(x_{i} =0\), which implies that nobody chooses a zero bid in a symmetric reciprocity equilibrium.\(\square \)
1.2 Appendix B: Proof of Proposition 2
Substituting \(V/2\) for \(x\) in (26) yields
if and only if \(V/\omega \lesseqqgtr \left( \sqrt{2}-1\right) /2 +\left( 2-\sqrt{2}\right) \theta /2\). In the destructive equilibrium, the rent-dissipation rate \(2x^{\textit{SRE}}/V\) is greater (smaller) than \(1\) if \( 1/6 \le V/\omega <\left( \sqrt{2}-1\right) /2 +\left( 2-\sqrt{2}\right) \theta /2\) (\(V/\omega >\left( \sqrt{2}-1\right) /2 +\left( 2-\sqrt{2}\right) \theta /2\) ). Clearly, if \(V/\omega <1/6\), then the total expenditure \( 2x^{\textit{SRE}}\) is greater than \(V\); that is, over-dissipation occurs when \( V/\omega <1/6\).\(\square \)
1.3 Appendix C: Proof of Proposition 3
If player \(i\)’s first- and second-order beliefs are all \(0\), then player \(i\) does not choose \(x_{i} =0\), for the same reason as in the two-player contest. Therefore, no player chooses a zero bid in a symmetric reciprocity-equilibrium.
Given that player \(i\)’s first- and second-order beliefs are all positive, from (19) and (20), the first-order condition maximizing \(U^{i}\left( x_{i},y_{-i},\left\{ z_{-j}\right\} _{j\in J\backslash \left\{ i\right\} }\right) \) in (21) is
Suppose that \(Y_{-i} >V\) and \(Z_{-j} >V\). We then easily see from (27) that \(U^{i}\left( x_{i},\cdot ,\cdot \right) \) is strictly concave whenever \(\partial U^{i}/\partial x_{i} >-1\), and hence, there exists only a global maximizer for \(U^{i}\left( x_{i},\cdot , \cdot \right) \) given \(y_{-i}\) and \(z_{-j}\), and no local minimum exists. Since \(x_{i} =x^{\textit{SRE}},\,y_{-i} =\left\{ x^{\textit{SRE}},\ldots , x^{\textit{SRE}}\right\} \) and \(z_{-j} =\left\{ x^{\textit{SRE}},\ldots , x^{\textit{SRE}}\right\} \) in the symmetric reciprocity-equilibrium, from (27) we obtain
which is greater than \(V/\left( N-1\right) \) if and only if \(V/\omega <\left( N-1\right) ^{2}/N\left( 2N-1\right) \). Therefore, when \( V/\omega <\left( N-1\right) ^{2}/N\left( 2N-1\right) ,\,x^{\textit{SRE}}\) in (28) is a reciprocity-equilibrium expenditure. Moreover, this equilibrium is destructive since \(x^{\textit{SRE}} >x^{\textit{SNE}} =\left( N-1\right) V/N^{2}\).
Next, suppose that \(Y_{-i} \le V\) and \(Z_{-j} \le V\). Again, the first-order condition to maximize \(U^{i}\left( x_{i},y_{-i},\left\{ z_{-j}\right\} _{j\in J\backslash \left\{ i\right\} }\right) \) in (21) is given by (27). Substituting \(x\) for \(x_{i}\), all \( y_{j} \in y_{-i}\) and all \(z_{k} \in \left\{ z_{-j}\right\} _{j\in J\backslash \left\{ i\right\} }\) in (27) and using (19) and (20), we obtain
From (29), \(V/\omega \ge \left( N-1\right) ^{2}/N\left( 2N-1\right) \) if and only if \(\psi \left( V/\left( N-1\right) \right) \le 0\). If \(NV-\left( N-1-N\theta \right) \omega \ge 0\) or equivalently \(V/\omega \ge \left( N-1\right) /N-\theta \), from (29) we obtain \(\psi ^{\prime }\left( x\right) <0\) for any positive \(x\) and \(\lim _{x\rightarrow 0}\psi \left( x\right) =\infty \). Hence, a unique symmetric reciprocity-equilibrium exists if \(V/\omega \ge \max \big \{ \left( N-1\right) ^{2}/N\left( 2N-1\right) , \left( N-1\right) / N-\theta \big \} \) since there exists only one \(x^{\textit{SRE}} \le V/\left( N-1\right) \) such that \(\psi \left( x^{\textit{SRE}}\right) =0\). Moreover, we can verify that \(x^{\textit{SRE}} >x^{\textit{SNE}}\) since \(\psi \left( x^{\textit{SNE}}\right) =\theta \omega /NV >0\) which implies that the reciprocity equilibrium is destructive. Suppose that \( V/\omega <\left( N-1\right) /N-\theta \) for any \(\theta <\left( N-1\right) /N\). We see that \(\lim _{x\rightarrow 0}\psi \left( x\right) =-\infty ,\,\lim _{x\rightarrow 0}\psi ^{\prime }\left( x\right) =+\infty \), and \(\psi \left( V/\left( N-1\right) \right) \le 0\) when \(V/\omega \ge \left( N-1\right) ^{2}/N\left( 2N-1\right) \); moreover, the function \(\psi \left( x\right) \) is strictly concave unless it decreases with \(x\). Therefore, the global maximum of \(\psi \left( x\right) \) is strictly positive since \(\psi \left( x^{\textit{SNE}}\right) >0\), and as such, equation \(\psi \left( x\right) =0\) has just two positive real roots. Accordingly, the larger of the two real roots is the destructive-equilibrium expenditure, while the other is the constructive-equilibrium expenditure if \(V/\omega <\left( N-1\right) /N-\theta \) for any \(\theta <\left( N-1\right) /N\) since \(\psi \left( x^{\textit{SNE}}\right) >0\).\(\square \)
1.4 Appendix D: Proof of Proposition 4
It is clear that when \(V/\omega <\left( N-1\right) ^{2}/N\left( 2N-1\right) \), the total expenditure \(Nx^{\textit{SRE}}\) is greater than \(V\) since \( x^{\textit{SRE}} >V/\left( N-1\right) \), which yields \(Nx^{\textit{SRE}} >\frac{N}{N-1}V >V\); in other words, over-dissipation occurs in this case.
Next, consider the destructive equilibrium in the case that \(V/\omega \ge \left( N-1\right) ^{2} /N\left( 2N-1\right) \). Substituting \( V/N\) for \(x\) in (29) yields
where
Therefore, for any finite \(N\ge 2\), we have
In the destructive reciprocity-equilibrium, the rent-dissipation rate \( Nx^{\textit{SRE}}/V\) is greater (smaller) than 1 if \(\left( N-1\right) ^{2}/N\left( 2N-1\right) \le V/\omega <\varsigma \left( N\right) (V/\omega >\varsigma \left( N\right) )\).
Differentiating \(\varsigma \left( N\right) \) in (31) with respect to \(N\), we obtain
Hence, if there exists \(N =\tilde{N}\) such that \(V/\omega <\varsigma \left( N\right) \) or equivalently \(\psi \left( V/N\right) >0\), then, for any finite \(N >\tilde{N}\), we have \(\psi \left( V/N\right) >0\); that is, over-dissipation.\(\square \)
1.5 Appendix E: Proof of Proposition 5
In the destructive equilibrium, when \(V/\omega <\left( N-1\right) ^{2}/N\left( 2N-1\right) \), we can verify from (29) that the total expenditure \(X^{\textit{SRE}} \equiv Nx^{\textit{SRE}} =\left( N-1\right) /NV+\left( N-1\right) /N^{2}\omega \) approaches \(V\) as \( N \rightarrow \infty \); that is, the rent-dissipation rate \( X^{\textit{SRE}}/V\) converges to \(1\) when \(V/\omega <\lim _{N\rightarrow \infty }\left( N-1\right) ^{2}/N\left( 2N-1\right) =1/2\). Next, consider the case that \(V/\omega \ge 1/2\) in the destructive equilibrium. Replacing \(Nx\) with \(X\) in (29) yields
where \(X\) denotes the sum of the rent-seeking expenditure. Letting \(\bar{X} \equiv \lim _{N\rightarrow \infty }X\), the function (33) reduces to
as \(N \rightarrow \infty \). The total expenditure in a reciprocity equilibrium of the perfectly competitive contest is a positive \( \bar{X}\) such that \(\varPsi \left( \bar{X}\right) =0\). To solve \( \varPsi \left( \bar{X}\right) =0\), we define \(q \equiv \bar{X}^{-1/2};\) using (34), we set the following quadratic equation with respect to \(q\):
If \(V-\left( 1-\theta \right) \omega =0\) or equivalently \( V/\omega =1-\theta \), then \(q =V^{-1/2}\) is the only real root for \(\varPhi \left( q\right) =0\). Therefore, we obtain the only solution \(\bar{X} =V\) such that \(\varPsi \left( \bar{X} \right) =0\) from the definition of \(q \equiv \bar{X} ^{-1/2}\). If \(V-\left( 1-\theta \right) \omega \ne 0\), then \( \varPhi \left( q\right) =0\) has two real roots: \(q =V^{-1/2}\) and \(q =-V^{1/2}\left[ V-\left( 1-\theta \right) \omega \right] ^{-1}\). However, if \(V-\left( 1-\theta \right) \omega >0\) or equivalently \(V/\omega >1-\theta \), there does not exist any \(\bar{X}\) such that \(\bar{X}^{-1/2} =-V^{1/2} \left[ V-\left( 1-\theta \right) \omega \right] ^{-1} <0\). Thus, the only solution for \(\varPsi \left( \bar{X}\right) =0\) is \(\bar{X} =V\) since \(\bar{X}^{-1/2} =V^{-1/2}\). At the same time, if \(V-\left( 1-\theta \right) \omega <0\) or equivalently \( V/\omega <1-\theta ,\,\varPsi \left( \bar{X}\right) =0\) has two solutions: \(\bar{X} =V\) and \(\bar{X} =V-\left( 1-\theta \right) \omega \left[ 2V-\left( 1-\theta \right) \omega \right] V^{-1}\).
If \(V/\omega \ge \max \left\{ \left( N-1\right) ^{2}/N\left( 2N-1\right) , \left( N-1\right) /N-\theta \right\} \) for any finite \(N\), as verified in Appendix 7.3, the only reciprocity equilibrium is destructive. As \(N \rightarrow \infty \), the only solution for \( \varPsi \left( \bar{X}\right) =0\) is \(V\) if \(V/\omega \ge \max \left\{ 1/2,1-\theta \right\} \) as shown above. Therefore, we can regard \(V\) as the limit of the total destructive equilibrium expenditure. On the other hand, if \(V/\omega <\left( N-1\right) /N-\theta \), then the constructive equilibrium also exists. When \(N\) approaches infinity, the total constructive equilibrium expenditure cannot exceed the total destructive equilibrium expenditure; thus, \(\bar{X} =\min \left\{ V,V-\left( 1-\theta \right) \omega \left[ 2V-\left( 1-\theta \right) \omega \right] V^{-1}\right\} \) in the constructive equilibrium. Hence, when \( V/\omega \ge \max \left\{ 1/2,1-\theta \right\} \), we have \( \bar{X} =V\) in the destructive equilibrium, while the total constructive equilibrium expenditure is \(V -\left( 1-\theta \right) \omega \left[ 2V-\left( 1-\theta \right) \omega \right] V^{-1}\) and lower than \(V\). At the same time, when \(V/\omega \le \left( 1-\theta \right) /2\), the sum of the constructive equilibrium expenditure amounts to \(V\) since \(V <V-\left( 1-\theta \right) \omega \left[ 2V-\left( 1-\theta \right) \omega \right] V^{-1}\).\(\square \)
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Sano, H. Reciprocal rent-seeking contests. Soc Choice Welf 42, 575–596 (2014). https://doi.org/10.1007/s00355-013-0742-2
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DOI: https://doi.org/10.1007/s00355-013-0742-2