Skip to main content
Log in

Reciprocal rent-seeking contests

  • Original Paper
  • Published:
Social Choice and Welfare Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 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 \(r>1\).

  2. 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.

  3. We refer to this equilibrium as the “reciprocity equilibrium” even though its definition is the same as that of Rabin’s (1993) “fairness equilibrium”.

  4. We follow the terms used in Sobel (2005).

  5. Note that \(Nx^{\textit{SNE}}<V\) for any finite \(N\).

  6. 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.

References

  • Anderson LR, Stafford SL (2003) An experimental analysis of rent seeking under varying competitive conditions. Public Choice 115:199–216

    Article  Google Scholar 

  • Corcoran WJ (1984) Long-run equilibrium and total expenditures in rent-seeking. Public Choice 43:89–94

    Article  Google Scholar 

  • Corcoran WJ, Karels GV (1985) Rent-seeking behavior in the long-run. Public Choice 46:227–246

    Article  Google Scholar 

  • Davis DD, Reilly RJ (1998) Do too many cooks always spoil the stew? An experimental analysis of rent-seeking and the role of a strategic buyer. Public Choice 95:89–115

    Article  Google Scholar 

  • Dufwenberg M, Kirchsteiger G (2004) A theory of sequential reciprocity. Games Econ Behav 47:268–298

    Article  Google Scholar 

  • Fehr E, Schmidt KM (1999) The theory of fairness, competition, and cooperation. Q J Econ 114:817–868

    Article  Google Scholar 

  • Fehr E, Schmidt KM (2006) The economics of fairness, reciprocity and altruism—experimental evidence and new theories. In: Kolm S-C, Ythier JM (eds) Handbook of the economics of giving, altruism and reciprocity, vol 1. North-Holland, pp 615–691

  • Hazlett TW, Michaels RJ (1993) The cost of rent-seeking: evidence from cellular telephone license lotteries. South Econ J 59:425–435

    Article  Google Scholar 

  • Hehenkamp B, Leininger W, Possajennikov A (2004) Evolutionary equilibrium in Tullock contests: spite and dissipation. Eur J Political Econ 20:1045–1057

    Article  Google Scholar 

  • Herrmann B, Orzen H (2008) The appearance of homo rivalis: social preferences and the nature of rent seeking. CeDEx Discussion Paper Series No. 2008–2010, University of Nottingham

  • Hillman AL, Katz E (1984) Risk-averse rent seekers and the social cost of monopoly power. Econ J 94:104–110

    Article  Google Scholar 

  • Millner EL, Pratt MD (1989) An experimental investigation of efficient rent-seeking. Public Choice 62:139–151

    Article  Google Scholar 

  • Millner EL, Pratt MD (1991) Risk aversion and rent-seeking: an extension and some experimental evidence. Public Choice 69:81–92

    Article  Google Scholar 

  • Nitzan S (1991) Collective rent dissipation. Econ J 101:1522–1534

    Article  Google Scholar 

  • Potters J, de Vries CG, van Winden F (1998) An experimental examination of rational rent-seeking. Eur J Political Econ 14:783–800

    Article  Google Scholar 

  • Rabin M (1993) Incorporating fairness into game theory and economics. Am Econ Rev 83:1281–1302

    Google Scholar 

  • Riechmann T (2007) An analysis of rent-seeking games with relative-payoff maximizers. Public Choice 133:147–155

    Article  Google Scholar 

  • Sano H (2009) Imitative learning in Tullock contests: does overdissipation prevail in the long run? J Inst Theor Econ 165:365–383

    Article  Google Scholar 

  • Schaffer ME (1988) Evolutionarily stable strategies for a finite population and a variable contest size. J Theor Biol 132:469–478

    Article  Google Scholar 

  • Schmitt P, Shupp R, Swope K, Cadigan J (2004) Multi-period rent-seeking contests with carryover: theory and experimental evidence. Econ Gov 5:187–211

    Article  Google Scholar 

  • Shogren JF, Baik KH (1991) Reexamining efficient rent-seeking in laboratory markets. Public Choice 69:69–79

    Article  Google Scholar 

  • Sobel RS, Garrett TA (2002) On the measurement of rent seeking and its social opportunity cost. Public Choice 112:115–136

    Article  Google Scholar 

  • Sobel J (2005) Interdependent preferences and reciprocity. J Econ Lit 153:392–436

    Article  Google Scholar 

  • Tullock G (1980) Efficient rent seeking. In: Buchanan JM, Tollison RD, Tullock G (eds) Toward a theory of the rent-seeking society. A &M University Press, Texas, pp 97–112

    Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hiroyuki Sano.

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

$$\begin{aligned} \frac{\partial U^{i}}{\partial x_{i}}=\frac{y_{j}}{\left( x_{i}+y_{j}\right) ^{2}}V-1+\frac{\omega }{z_{i}+y_{j}}\left( \frac{y_{j}}{x_{i}+y_{j}}\right) ^{2}=0. \end{aligned}$$
(24)

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

$$\begin{aligned} \frac{\partial U^{i}}{\partial x_{i}}=\frac{y_{j}}{\left( x_{i}+y_{j}\right) ^{2}}V-1-\omega \left[ \frac{z_{i}}{z_{i}+y_{j}}-\theta -\left( 1-\theta \right) \left( \frac{z_{i}}{V}\right) ^{1/2}\right] \frac{y_{j}}{\left( x_{i}+y_{j}\right) ^{2}}=0.\nonumber \\ \end{aligned}$$
(25)

Furthermore, the second-order derivative is given by

$$\begin{aligned} \frac{\partial ^{2}U^{i}}{\partial x_{i}^{2}}=-\frac{2}{x_{i}+y_{j}}\left( \frac{\partial U^{i}}{\partial x_{i}}+1\right) . \end{aligned}$$

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

$$\begin{aligned} \varphi \left( x\right) \equiv \frac{2V-\left( 1-2\theta \right) \omega }{8x} +\frac{\left( 1-\theta \right) \omega }{4\left( Vx\right) ^{1/2}}-1=0 \quad \text{ for } \quad x\le V. \end{aligned}$$
(26)

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.

Fig. 1
figure 1

Possible reciprocity equilibria in a two-player contest

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

$$\begin{aligned} \varphi \left( \frac{V}{2}\right) =-\frac{1}{2}+\frac{\sqrt{2}-1+\left( 2- \sqrt{2}\right) \theta }{4}\frac{\omega }{V}\gtreqqless 0, \end{aligned}$$

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

$$\begin{aligned} \frac{\partial U^{i}}{\partial x_{i}}=\frac{Y_{-i}}{\left( x_{i}+Y_{-i}\right) ^{2}}V-1-\frac{Y_{-i}}{\left( x_{i}+Y_{-i}\right) ^{2}} \frac{\omega }{N-1}\sum _{j\in J\backslash \left\{ i\right\} }\lambda _{j}\left( y_{j},z_{-j}\right) =0.\quad \end{aligned}$$
(27)

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

$$\begin{aligned} x^{\textit{SRE}}=\frac{N-1}{N^{2}}V+\frac{N-1}{N^{3}}\omega , \end{aligned}$$
(28)

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

$$\begin{aligned} \psi \left( x\right) \equiv \frac{\left( N-1\right) \left[ NV-\left( N-1-N\theta \right) \omega \right] }{N^{3}x}+\frac{\left( N-1\right) ^{3/2}\left( 1-\theta \right) \omega }{N^{2}\left( Vx\right) ^{1/2}}-1=0.\qquad \end{aligned}$$
(29)

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

$$\begin{aligned} \psi \left( \frac{V}{N}\right) =-\frac{\omega }{NV}\left[ \frac{V}{\omega } -\varsigma \left( N\right) \right] , \end{aligned}$$
(30)

where

$$\begin{aligned} \varsigma \left( N\right) \equiv \frac{N-1}{N}\left[ N^{1/2}\theta +\left( N-1\right) ^{1/2}\right] \left[ N^{1/2}-\left( N-1\right) ^{1/2}\right] . \end{aligned}$$
(31)

Therefore, for any finite \(N\ge 2\), we have

$$\begin{aligned} \psi \left( \frac{V}{N}\right) \gtreqqless 0\Leftrightarrow \frac{V}{\omega } \lesseqqgtr \varsigma \left( N\right) . \end{aligned}$$

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

$$\begin{aligned} \varsigma ^{\prime }\left( N\right) =\frac{\varsigma \left( N\right) }{ N\left( N-1\right) }+\frac{\left( 1-\theta \right) \left( N-1\right) }{2N} \left[ \left( \frac{N}{N-1}\right) ^{\frac{1}{4}}-\left( \frac{N-1}{N} \right) ^{\frac{1}{4}}\right] ^{2}>0. \nonumber \\ \end{aligned}$$
(32)

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

$$\begin{aligned} \varPsi \left( X\right) \equiv \frac{N-1}{N}\frac{V}{X}-\left( \frac{N-1}{N} \right) ^{2}\frac{\omega }{X}+\frac{N-1}{N}\frac{\theta \omega }{X}+\left( \frac{N-1}{N}\right) ^{\frac{3}{2}}\frac{\left( 1-\theta \right) \omega }{ V^{1/2}X^{1/2}}-1, \nonumber \\ \end{aligned}$$
(33)

where \(X\) denotes the sum of the rent-seeking expenditure. Letting \(\bar{X} \equiv \lim _{N\rightarrow \infty }X\), the function (33) reduces to

$$\begin{aligned} \varPsi \left( \bar{X}\right) =\frac{V-\left( 1-\theta \right) \omega }{\bar{X}} +\frac{\left( 1-\theta \right) \omega }{V^{1/2}\bar{X}^{1/2}}-1, \end{aligned}$$
(34)

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 \(q\):

$$\begin{aligned} \varPhi \left( q\right) \equiv \left[ V-\left( 1-\theta \right) \omega \right] q^{2}+\frac{\left( 1-\theta \right) \omega }{V^{1/2}}q-1=0. \end{aligned}$$

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 \(V <V-\left( 1-\theta \right) \omega \left[ 2V-\left( 1-\theta \right) \omega \right] V^{-1}\).\(\square \)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sano, H. Reciprocal rent-seeking contests. Soc Choice Welf 42, 575–596 (2014). https://doi.org/10.1007/s00355-013-0742-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00355-013-0742-2

Navigation