We begin with the original contest of Tullock (1980) as the baseline contest in Sect. 4.1, and describe contests from literature that are strategically equivalent to the Tullock contest. In Sect. 4.2, we introduce a set of modified Tullock-type contests that are strategically equivalent to each other.

Original Tullock contest

In the standard rent-seeking contest, introduced by Tullock (1980), there is no losing prize and regardless of the outcome of the contest, both players forgo their efforts. In such a case, the winning prize value \(W>0\), \(\alpha _1 =\alpha _2 =-1\), and the other parameters in \(\Omega \) are zero. The payoff for player \(i\) in case of winning or losing is

$$\begin{aligned} \pi _i (x_i ,x_j )=\left\{ {{\begin{array}{ll} W-x_i \,&{}\quad \hbox {with probability}\,p_i (x_i ,x_j ) \\ -x_i \,&{}\quad \hbox {with probability}\,1-p_i (x_i ,x_j ) \\ \end{array} }} \right. \end{aligned}$$ (11)

Using our notation, the Tullock contest is defined as \(\Gamma (i,j,\left\{ {W,0,-1,-1,0,0} \right\} )\). The resulting best response function in such a contest for player \(i\) is

$$\begin{aligned} x_i =-x_j +\sqrt{Wx_j }. \end{aligned}$$ (12)

For a generic contest \(\Gamma (i,j,\left\{ {W,L,\alpha _1 ,\alpha _2 ,\beta _1 ,\beta _2 } \right\} )\) to be strategically equivalent to contest \(\Gamma (i,j,\left\{ {W,0,-1,-1,0,0} \right\} )\), according to condition (10), we need to impose the following restrictions: \(\beta _2 -\beta _1 -\alpha _2 =1\), \(\alpha _1 =-1\), and \(L=0\). Such restrictions guarantee that the best response function (4) is exactly the same as the best response function (12). Therefore, by definition these contests are strategically equivalent.

One particularly interesting case arises when we put further restrictions \(\beta _1 =-1\) and \(\alpha _2 =\beta _2 =0\). In such a contest, \(\Gamma (i,j,\left\{ {W,0,-1,0,-1,0} \right\} )\), the new payoff function is:

$$\begin{aligned} \pi _i (x_i ,x_j )=\left\{ {{\begin{array}{l@{\quad }l} W-x_i -x_j \,&{}\hbox {with probability}\,p_i (x_i ,x_j )\\ 0 \,&{}\hbox {with probability}\,1-p_i (x_i ,x_j )\\ \end{array} }} \right. \end{aligned}$$ (13)

Note that in (13), the winner fully reimburses the loser. This can be interpreted as the “Marshall system of litigation” (Baye et al. 2005) in which the winner pays his own legal costs and also reimburses all of the legal costs of the loser, whereas the standard Tullock contest can be interpreted as the “American system of litigation” in which each litigant pays its own legal expenses.Footnote 8 It can easily be shown that the unique equilibrium for contests defined by (11) and (13) is the symmetric equilibrium with \(x_i^*=x_j^*=W/4\). Moreover, the expected payoff in both contests is exactly the same, \(E^{*}\left( \pi \right) =W/4\). Therefore, contests (11) and (13) are strategically, effort and payoff equivalent. This equivalence is surprising, since the two contests are intuitively and structurally very different. However, it has been also shown in an all-pay auction setting under incomplete information (Baye et al. 2005). Therefore, our results provide further evidence that Marshall and American systems of litigation are revenue (in our case, effort) and payoff equivalent.Footnote 9 Furthermore, the robustness of the result shows that the seemingly unfair Marshall system of litigation indeed result in the same effort and payoff to litigants as the standard American system. Hence, it may not be necessary (under the restriction of risk neutrality) for an institution to shift from one to another system of litigation.

It is also straightforward to show that the “input spillover” contest of Chowdhury and Sheremeta (2011a) and Baye et al. (2012), where the effort expended by player \(j\) partially affects player \(i\) and vice versa, is strategically equivalent to the original Tullock contest. The spillover contest can be defined as \(\Gamma (i,j,\left\{ {W,0,-1,-1,\beta ,\beta } \right\} )\), where \(\beta \in (-1,1)\) is the input spillover parameter. This type of contest is motivated by spillover effects in R&D innovation (D’Aspremont and Jacquemin 1988; Kamien et al. 1992). From strategic equivalence condition (10), one can see that for any value of \(\beta \), the resulting best response function is exactly the same as in (11). Hence, the input spillover contest \(\Gamma (i,j,\left\{ {W,0,-1,-1,\beta ,\beta } \right\} )\) is strategically equivalent to the original Tullock contest \(\Gamma (i,j,\left\{ {W,0,-1,-1,0,0} \right\} )\). This result suggests that if an R&D competition is modeled as a lottery contest, then the existence of symmetric spillovers may not affect the equilibrium. However, the “input spillover” contest is not payoff equivalent to the original Tullock contest, since condition (10) is not satisfied. It can be easily shown that a positive (negative) spillover provides a higher (lower) payoff to the players than the Tullock contest. Hence, our analysis gives further support, from a benevolent policymaker’s point of view, to encourage contests with positive spillover and discourage contests with negative spillovers. Since in the R&D contests positive spillovers are often related to the Intellectual Property Rights (IPR) issues, a designer might try to manipulate the spillover parameters such that the positive spillovers are high enough without damaging the IPR issues.

Modified Tullock-type contests

Researchers often use modified versions of the original Tullock contests in order to address specific questions such as taxes, subsidies, externalities, effort-dependent valuations, cost differences, etc. There are instances in the literature where two different Tullock-type contests are strategically equivalent to each other. Here, we briefly discuss some of these examples.

Chung (1996) assumes that the value of the winning prize depends on the total effort expenditures in the contest. A simple linear version of the Chung (1996) model would generate the following payoff function:

$$\begin{aligned} \pi _i (x_i ,x_j )=\left\{ {{\begin{array}{ll} W+a(x_i +x_j )-x_i \,&{}\quad \hbox {with probability}\,p_i (x_i ,x_j ) \\ -x_i \,&{}\quad \hbox {with probability}\,1-p_i (x_i ,x_j ) \\ \end{array} }} \right. \end{aligned}$$ (14)

Hence, (14) can be described as \(\Gamma (i,j,\left\{ {W,0,a-1,-1,a,0} \right\} )\), where \(a\in \left( {0,1} \right) \), and the best response function is

$$\begin{aligned} x_i =-x_j +\sqrt{Wx_j /(1-a)} \end{aligned}$$ (15)

Lee and Kang (1998) study a contest with externalities. In their model, the cost of effort decreases with the total effort expenditures. This contest can be captured by

$$\begin{aligned} \pi _i (x_i ,x_j )=\left\{ {{\begin{array}{ll} W-x_i +b(x_i +x_j )\,&{}\quad \hbox {with probability}\,p_i (x_i ,x_j )\\ -x_i +b(x_i +x_j )\, &{}\quad \hbox {with probability}\,1-p_i (x_i ,x_j ) \\ \end{array} }} \right. \end{aligned}$$ (16)

Hence, (16) can be described as \(\Gamma (i,j,\left\{ {W,0,b-1,b-1,b,b} \right\} )\), where \(b\in \left( {0,1} \right) \), and the best response function is

$$\begin{aligned} x_i =-x_j +\sqrt{Wx_j /(1-b)} \end{aligned}$$ (17)

When \(a=b\) the best response functions (15) and (17) and the equilibrium effort expenditures in the two contests are exactly the same. This result indicates that some contests with endogenous prizes, as in Chung (1996), are strategically equivalent to contests with externalities, as in Lee and Kang (1998). Also note that, although both contests are strategically equivalent, they are not payoff equivalent. In particular, the contest defined by (16) results in higher expected payoff than the contest defined by (14), providing a clear Pareto ranking between the two contests. Hence, a benevolent contest designer, such as the government trying to maximize the total social welfare, may opt to choose a contest that elicits the same level of expenditures and, at the same time, results in Pareto improvement for both contestants.

Next, we consider a “limited liability” contest introduced by Skaperdas and Gan (1995), where the loser’s payoff is independent of the efforts expended.Footnote 10 The authors motivate this example by stating that contestants may be entrepreneurs who borrow money to spend on research and development and thus are not legally responsible in case of a loss. The loser of such a contest is unable to repay the loan and goes bankrupt. In such a case, \(W>0\), \(\alpha _1 =-1\), and the other parameters in \(\Omega \) are zero. The payoff is:

$$\begin{aligned} \pi _i (x_i ,x_j )=\left\{ {{\begin{array}{ll} W-x_i \,&{}\quad \hbox {with probability}\,p_i (x_i ,x_j )\\ 0 \,&{}\quad \hbox {with probability}\,1-p_i (x_i ,x_j ) \\ \end{array} }} \right. \end{aligned}$$ (18)

The best response function for player \(i\) is:

$$\begin{aligned} x_i =-x_j +\sqrt{x_j^2 +Wx_j } \end{aligned}$$ (19)

For a contest to be strategically equivalent to \(\Gamma (i,j,\left\{ {W,0,-1,0,0,0} \right\} )\) the required restrictions from (10) are \(\beta _2 -\beta _1 -\alpha _2 =0\), \(\alpha _1 =-1\), and \(L=0\). When we impose further restrictions \(\alpha _2 =-1\), \(\beta _2 =-1\), and \(\beta _1 =0\) we obtain a contest with the following payoff function:

$$\begin{aligned} \pi _i (x_i ,x_j )=\left\{ {{\begin{array}{ll} W-x_i &{}\quad \hbox {with probability}p_i (x_i ,x_j ) \\ -x_i -x_j &{}\quad \hbox {with probability}1-p_i (x_i ,x_j ) \\ \end{array} }} \right. \end{aligned}$$ (20)

This contest can be interpreted as a “full liability” contest, since the loser has to pay in full the expenditures of both players. Note that although (18) is strategically equivalent to (20), the “full liability” contest is (by definition) more risky than the “limited liability” contest. In (18) players do not have to worry about what happens in the case of a loss, since they are not legally responsible. In contrast, the loser in (20) has to pay the expenditures of both players. Therefore, equivalence between (18) and (20) holds only under the assumption of risk neutrality. Moreover, it is easy to verify from (9) that contests (18) and (20) are not payoff equivalent. The equilibrium payoff in the “full liability” contest is \(E^{*}\left( \pi \right) =0\) and in the “limited liability” contest it is \(E^{*}\left( \pi \right) =W/3\). This is another specific example in which a contest designer can step in, if he is interested in overall payoff. Since the two contests are effort equivalent, but the limited liability contest provides the players with a higher payoff, and presumably will need less monitoring than the full liability contest, a contest designer may be interested to implement a limited liability contest instead of a full liability contest.

Alexeev and Leitzel (1996) study a “rent-shrinking” contest \(\Gamma (i,j,\left\{ W,0,-1,-1,\right. \left. -1,0 \right\} )\), where the winning prize value decreases by the total effort expenditures. From (10), a strategically equivalent contest would require \(\beta _2 -\beta _1 -\alpha _2 =2\), \(\alpha _1 =-1\) and \(L=0\). A “lazy winner” contest \(\Gamma (i,j,\left\{ {W,0,-1,-2,0,0} \right\} )\) of Chowdhury and Sheremeta (2011a), in which the marginal cost of winning (\(\alpha _1 =-1)\) is lower than the marginal cost of losing (\(\alpha _2 =-2)\), definitely satisfies these restrictions. Moreover, the two contests are also payoff equivalent. The equivalence between the “rent-shrinking” and “lazy winner” contests enables the designer to achieve the same equilibrium rent dissipation using two alternative contests. Nevertheless, the “lazy winner” contest is, arguably, easier to implement and it is less susceptible to the collusion problem mentioned in Alexeev and Leitzel (1996).

In many cases, a contest designer can use different policy tools to implement a certain contest. Using the same procedure as before it can be shown that under certain restrictions, contests with endogenous valuations (Amegashie 1999), contests with differential cost structure (Chowdhury and Sheremeta 2011a), and contests with taxes (Glazer and Konrad 1999), are strategically equivalent. Specifically, Glazer and Konrad (1999) study a contest \(\Gamma (i,j,\left\{ {\left( {1-t} \right) w,0,-\left( {1-t} \right) ,-1,0,0} \right\} )\), in which a part of the rent seeker’s non-negative profit is taxed with tax rate \(t\in (0,1)\). Amegashie (1999) studies a contest \(\Gamma (i,j,\left\{ {W,0,-(1-m),-1,0,0} \right\} )\), in which the winner’s prize value is a linear function of own effort spent. Chowdhury and Sheremeta (2011a)) study the “lazy winner” contest \(\Gamma (i,j,\left\{ {W,0,\alpha _1 ,\alpha _2 ,0,0} \right\} )\), in which the marginal cost of winning is lower than the marginal cost of losing, i.e., \(|\alpha _1 |<|\alpha _2 |\).

Using condition (7), when \(\left( {1-t} \right) w=W\), \(\alpha _1 -\alpha _2 =t=m\), and \(\alpha _1 =\left( {t-1} \right) =(m-1)\) then the three contests are strategically and effort equivalent. The first condition means that the take-home winning payoff in the contest with tax should be the same as the basic winning prize of the other two contests. The second condition implies that the tax rate in the tax contest should be same as the endogenous valuation margin in the endogenous valuation contest. Moreover, the difference between winning and losing marginal cost, which has the same effect in the lazy winner contest, should also be the same. Finally, and most intuitively, the marginal cost of winning in the lazy winner contest is same as the “effective” marginal costs of winning in the other two contests.

The equivalence between these three seemingly unrelated contests conveys an important message. It shows that the designer can either use policy tools, such as taxes, or contests with alternative cost structure to achieve the same objective, and it may be easier to implement the policy from one domain to another. Moreover, the three contests do not necessarily generate the same equilibrium payoffs. The equilibrium payoff (under the restriction of strategic equivalence) in Glazer and Konrad (1999) is \(E^{*}\left( \pi \right) =\left( {1-t} \right) ^{2}\hbox {W}/(4-3t)\), in Amegashie (1999) it is \(E^{*}\left( \pi \right) =(1-t)W/(4-3t)\), and in Chowdhury and Sheremeta (2011a) it is \(E^{*}\left( \pi \right) =(1-t)\hbox {W}/(2-3t)\). Hence, a contest designer, such as a government trying to maximize the social welfare, can achieve a Pareto improvement by choosing a specific contest structure that generates the highest payoffs for players yet results in the same equilibrium efforts. In specific, implementing a contest designed with endogenous valuation might be preferable to the designer than implementing a contest with taxation.