Productive and unproductive competition: a unified framework

Abstract

Conventional theories of competition classify contests as being either “productive”, when the competitive efforts generate a surplus for society, or “unproductive”, when competition generates no social surplus and merely distributes already existing resources. These two discrete categories of competition create a division of real-world situations into analytical categories that fails to recognize the entire spectrum of competitive activities. Taking the existing models of productive and unproductive competition as benchmark idealizations, this paper revisits the relationship between the privately and socially optimal levels of competition in the full range of intermediate cases, as well as in the extremum cases of destructive and super-productive competition.

Appendix

Proof of Proposition 1 and of Corollary 2

The privately optimal investments in effort $e_i^*, \, i \in \{1,2\}$ are given by:

$$\frac{\left( z + h(e_j) \right) \, h^{'}(e_i^{*}) \, V}{\left( z + h(e_i^*) + h(e_j) \right) ^2} = 1 \quad i,j=1,2; \, i \ne j$$

(6)

The second order sufficiency conditions for $e_i^*$ to be an interior Nash equilibrium, i.e.,

$$\frac{\left( z + h(e_j)\right) \, h{''}(e_i^*) \, V }{\left( z + h(e_i^*) + h(e_j) \right) ^2} - \frac{2 \left( z + h(e_j) \right) \, \left( h{'}(e_i^*) \right) ^2 \, V }{\left( z + h(e_i^*) + h(e_j) \right) ^3}$$

(7)

for $i,j=1,2; \, i \ne j$, are satisfied since $h{'}(e_i) > 0$ and $h{''}(e_i) < 0$, $\forall i$. In equilibrium, $e_1^* = e_2^* = e^{*}$, where $e^{*}$ is given by:

$$\frac{\left( z + h(e^*) \right) \, h'(e^*) \, V}{\left( z + 2 \, h(e^*) \right) ^2} = 1$$

(8)

To ensure that $e^*$ is an equilibrium effort per player with payoff function $R_i$, we have to show that $e^*$ is in fact a global optimum for $R_i$ given the other player is choosing $e^*$. But this follows straightforwardly from the concavity of $R_i$ in $e_i$, as shown in (7). Thus, the sufficiency condition for $e^{*}$ to be an interior Nash equilibrium is clearly satisfied and this implies a unique pure strategy equilibrium. The socially optimal investments in effort $e_i^{**}, \, i \in \{1,2\},$ are computed as the solutions of the following FOCs:

$$\frac{z \, h{'}(e_i^{**}) \, T}{\left( z + h(e_i^{**}) + h(e_j) \right) ^2} = 1 \quad i,j=1,2; \, i \ne j$$

(9)

The second order sufficiency conditions for $e_i^{**}$ to be an interior Nash equilibrium, i.e.,

$$\frac{z \, h{''}(e_i^{**}) \, T}{\left( z + h(e_i^{**}) + h(e_j) \right) ^2} - \frac{2 \, z \, \left( h{'}(e_i^{**}) \right) ^2 \, T}{\left( z + h(e_i^{**}) + h(e_j) \right) ^3}$$

(10)

for $i,j=1,2; \, i \ne j$, are satisfied since $h{'}(e_i) > 0$ and $h{''}(e_i) < 0$, $\forall i$. In equilibrium, $e_1^{**} = e_2^{**} = e^{**}$, where $e^{**}$ is given by:

$$\frac{z \, h{'}(e^{**}) \, T}{\left( z + 2 \, h(e^{**}) \right) ^2} = 1$$

(11)

The second order sufficiency condition for $e^{**}$ to be an interior Nash equilibrium is clearly satisfied. The first order conditions (8) and (11) lead to

$$\left( \frac{z + 2 h(e^*)}{z + 2 h(e^{**})} \right) ^2 \, \frac{h'(e^{**})}{h'(e^{*})} = \left( 1 + \frac{h(e^{*})}{z} \right) \, \frac{V}{T}$$

(12)

The LHS in (12) is larger than 1 if and only if $e^{*} \ge e^{**}$. Conversely, the RHS in (12) is always larger than 1 when $V \ge T$. When $V < T$, the RHS in (12) is larger than 1 if and only if $w \le \underline{w}$, where:

$$\underline{w}=1+ \frac{h(e^*)}{z}$$

(13)

Thus, $w \le \underline{w}$ is a necessary and sufficient condition for $e^{*} \ge e^{**}$ when $V < T$. $\square$

Proof of Proposition 3 and of Corollary 4

Player i’s probability of winning the contest is increasing in his expenditure at a decreasing rate:

$$\frac{\partial p_{i,s}}{\partial e_i} = \frac{z + e_{j,s} (1 - \theta ^2)}{(z + e_{i,s} + e_{j,s} + \theta ( e_{i,s} + e_{j,s}) )^2} > 0$$

(14)

$$\frac{\partial ^2 p_{i,s}}{\partial e_i^2} = - \frac{2 (1+\theta ) (z + e (1 - \theta ^2))}{z + e_{i,s} + e_{j,s} + \theta ( e_{i,s} + e_{j,s}) )^3} < 0$$

(15)

The marginal impact of an increase of one player’s effort on the other player’s probability of winning the contest is defined as:

$$\frac{\partial p_{i,s}}{\partial e_j} = - \frac{e_{i,s} (1- \theta ^2) - \theta z}{(z + e_{i,s} + e_{j,s} + \theta ( e_{i,s} + e_{j,s}) )^2}$$

(16)

$$\frac{\partial ^2 p_{i,s}}{\partial e_j^2} = \frac{2 (1 + \theta ) (e_{i,s} (1 - \theta ^2) - \theta z)}{(z + e_{i,s} + e_{j,s} + \theta ( e_{i,s} + e_{j,s}) )^3}$$

(17)

Player i’s probability of winning the contest is decreasing in the other player’s expenditure at an increasing rate. This implies $e_{i,s} (1- \theta ^2) - \theta z > 0$.

The privately optimal investments in effort $e_{i,s}^{*}, \, i \in \{1,2\},$ are given by:

$$\frac{\left( z + e_{j,s} \left( 1-\theta ^2\right) \right) \, V}{\left( z+ (e_{i,s}^* + e_{j,s}) \, (1+ \theta )\right) ^2} = 1 \quad i,j=1,2; \, i \ne j$$

(18)

with $\theta \in (0,1]$. The second order sufficiency conditions for $e_{i,s}^{*}$ to be an interior Nash equilibrium, i.e.,

$$- \frac{2 (1 + \theta ) (z + e_{j,s} (1 - \theta ^2)) \, V}{\left( z + (e_{i,s}^* + e_{j,s}) \, (1+ \theta )\right) ^3}$$

(19)

are always satisfied $\forall i,j=1,2, \, i \ne j$. In equilibrium, $e_{1,s}^{*} = e_{2,s}^{*} = e_{s}^{*}$, where $e_{s}^{*}$ is given by:

$$\frac{\left( z + e_{s}^* \left( 1-\theta ^2\right) \right) \, V}{\left( z+ 2 \, e_{s}^* (1+\theta ) \right) ^2} = 1$$

(20)

The second order sufficiency condition for $e_{s}^{*}$ to be an interior Nash equilibrium is clearly satisfied. By deriving $e_{s}^*$ from (20) and $e^{*}$ from the linear specification of (8), it follows that $e_{s}^* < e^{*}$ for $\theta \in (0,1]$. The socially optimal investments in effort $e^{**}_{i,s}, \, i \in \{1,2\},$ are computed as the solutions of the following FOCs:

$$\frac{z (1 + \theta ) \, T}{\left( z+ (e_{i,s}^{**} + e_{j,s}) \, (1 + \theta )) \right) ^2} = 1 \quad i,j=1,2; \, i \ne j$$

(21)

The second order sufficiency conditions for $e_{i,s}^{**}$ to be an interior Nash equilibrium, i.e.,

$$- \frac{2 z (1 + \theta )^2 \, T}{\left( z+ (e_{i,s}^{**} + e_{j,s}) \, (1 + \theta )) \right) ^3}$$

(22)

are always satisfied for $i,j=1,2, \, i \ne j$. In equilibrium, $e_{1,s}^{**} = e_{2,s}^{**} = e_{s}^{**}$, where $e_{s}^{**}$ is given by:

$$\frac{z (1 + \theta ) \, T}{\left( z+ 2 \, e_{s}^{**} (1+\theta ) \right) ^2} = 1$$

(23)

The second order sufficiency condition for $e_{s}^{*}$ to be an interior Nash equilibrium is clearly satisfied.

The first order conditions (20) and (23) lead to

$$\left( \frac{z + 2 \, e_{s}^{*} (1+ \theta )}{z + 2\, e_{s}^{**} (1+\theta )} \right) ^2 = \left( 1 + \frac{e^*_{s} (1- \theta ^2) - \theta z}{z (1+ \theta )} \right) \, \frac{V}{T}$$

(24)

The LHS in (24) is larger than 1 if and only if $e_{s}^{*} \ge e_{s}^{**}$. Conversely, the RHS in (24) is larger than 1 if and only if $w \le \underline{w}_{s}$, where:

$$\underline{w}_s=1+ \frac{e_{s}^* (1- \theta ^2) - \theta z}{z (1+\theta )}$$

(25)

Thus, $w \le \underline{w}_s$ is a necessary and sufficient condition for $e_{s}^{*} \ge e_{s}^{**}$. Since $e_{s}^* < e^*$, when $e^{*}_{s} (1 - \theta ^2) - \theta z > 0$, $\underline{w}> \underline{w}_s > 1$. When $e^{*}_{s} (1 - \theta ^2) - \theta z = 0$, $\underline{w} > \underline{w}_s = 1$; when $e^{*}_{s} (1 - \theta ^2) - \theta z < 0$, $0< \underline{w}_s< 1 < \underline{w}$. $\square$

Proof of Proposition 5 and Corollary 6

Player i’s probability of winning the contest is increasing in his expenditure at a decreasing rate:

$$\frac{\partial p_{i,sm}}{\partial e_{i,sm}} = \frac{z + (e_{j,sm} + e_{k,sm}) (1 - \theta ) (1 + 2 \theta )}{(z + (e_{i,sm}+ e_{j,sm} + e_{k,sm}) (1 + 2 \theta ) )^2} > 0$$

(26)

$$\frac{\partial ^2 p_{i,sm}}{\partial e_{i,sm}^2} = - \frac{6 z + 2 (e_{j,sm} + e_{k,sm}) (1 - \theta ) (1 + 2 \theta )^2}{(z + (e_{i,sm}+ e_{j,sm} + e_{k,sm}) (1 + 2 \theta ) )^3} < 0$$

(27)

with $\theta \in (0,1]$. The marginal impact of an increase of one player’s effort on the other players probability of winning the contest is defined as follows:

$$\frac{\partial p_{i,sm}}{\partial e_{j,sm}} = - \frac{e_{i,sm} (1 - \theta ) (1 + 2 \theta ) - \theta z}{(z + (e_{i,sm}+ e_{j,sm} + e_{k,sm}) (1 + 2 \theta ) )^2}$$

(28)

$$\frac{\partial ^2 p_{i,sm}}{\partial e_{j,sm}^2} = \frac{2 (1 + 2 \theta ) (e_{i,sm} (1 - \theta ) (1 + 2 \theta ) - \theta z)}{(z + (e_{i,sm}+ e_{j,sm} + e_{k,sm}) (1 + 2 \theta ) )^3}$$

(29)

with $i,j,k = 1,2,3, \, i\ne j \ne k$. Player i’s probability of winning the contest is decreasing in the other player’s expenditures at an increasing rate. This implies $e_{i,sm} (1 - \theta ) (1 + 2 \theta ) - \theta z> 0$.

The privately optimal investments in effort $e_{i,sm}^{*}, \, i \in \{1,2,3\}$ are given by:

$$\frac{(z + (e_{j,sm} + e_{k,sm}) (1 - \theta ) (1 + 2 \theta )) \, V}{(z + (e^{*}_{i,sm}+ e_{j,sm} + e_{k,sm}) (1 + 2 \theta ) )^2} = 1 \quad i,j,k = 1,2,3; \, i\ne j \ne k$$

(30)

The second order sufficiency conditions for $e_{i,sm}^{*}$ to be an interior Nash equilibrium, i.e.,

$$- \frac{( 6z + 2 (e_{j,sm} + e_{k,sm}) (1 - \theta ) (1 + 2 \theta )^2 ) \, V}{(z + (e^{*}_{i,sm}+ e_{j,sm} + e_{k,sm}) (1 + 2 \theta ))^3}$$

(31)

are always satisfied $\forall i,j,k \in \{1,2,3\}$. In equilibrium, $e_{1,sm}^{*} = e_{2,sm}^{*} = e_{3,sm}^{*} = e_{sm}^{*}$, where $e_{sm}^{*}$ is given by:

$$\frac{(z + 2 e^{*}_{sm} (1-\theta ) (1 + 2 \theta )) V}{(z + 3 e^{*}_{sm} (1 + 2 \theta ))^2} = 1$$

(32)

The second order sufficiency condition for $e_{sm}^{*}$ to be an interior Nash equilibrium is clearly satisfied. By deriving $e^{*}_{s}$ from (20) and $e^{*}_{sm}$ from (32), it can be shown that $e^{*}_{s} > e^{*}_{sm}$, for all $V,z > 0$ such that $e^{*}_{s}, e^{*}_{sm} > 0$. In the absence of spillover (i.e., $\theta = 0$), (32) becomes:

$$\frac{(z + 2 e^{*}_{m} ) V}{(z + 3 e^{*}_{m})^2} = 1$$

(33)

By deriving $e^{*}$ from the linear specification of (8) and $e^{*}_{m}$ from (33), it follows that $e_m^{*} < e^{*}$, and that $3 e^{*}_{m} > 2 e^{*}$, $\forall V, \, z >0$. The socially optimal investments in effort $e^{**}_{i,sm}, \, i \in \{1,2,3\}$, are computed as the solutions of the following FOCs:

$$\frac{z (1 + 2 \theta ) T}{(z + (e^{**}_{i,sm}+ e_{j,sm} + e_{k,sm}) (1 + 2 \theta ) )^2} = 1 \quad i,j,k = 1,2,3; \, i\ne j \ne k$$

(34)

The second order sufficiency conditions for $e^{**}_{i,sm}$ to be an interior Nash equilibrium, i.e.,

$$- \frac{2 z (1 + 2 \theta )^2 T}{(z + (e^{**}_{i,sm}+ e_{j,sm} + e_{k,sm}) (1 + 2 \theta ) )^3}$$

(35)

are always satisfied. In equilibrium, $e^{**}_{1,sm} = e^{**}_{2,sm} = e^{**}_{3,sm} = e^{**}_{sm}$, where $e^{**}_{sm}$ is given by:

$$\frac{z (1 + 2 \theta ) T}{(z + 3 e^{**}_{sm} (1 + 2 \theta ) )^2} = 1$$

(36)

The second order sufficiency condition for $e^{**}_{sm}$ to be an interior Nash equilibrium is clearly satisfied. In the absence of spillover (i.e., $\theta = 0$), (36) becomes:

$$\frac{z \, T}{(z + 3 e^{**}_{m})^2} = 1$$

(37)

The first order conditions (32) and (36) lead to

$$\left( \frac{z + 3 e_{sm}^{*} (1 + 2 \theta )}{z + 3 e_{sm}^{**} (1 + 2 \theta )}\right) ^2 = \left( 1 + \frac{2 (e^{*}_{sm} (1 - \theta ) (1 + 2 \theta ) - \theta z)}{z (1 + 2 \theta )} \right) \frac{V}{T}$$

(38)

The LHS in (38) is larger than 1 if and only if $e_{sm}^{*} \ge e_{sm}^{**}$. Conversely, the RHS in (38) is always equal or larger than 1 if an only if $w \le \underline{w}_{sm}$, where:

$$\underline{w}_{sm} = 1 + \frac{2 (e^{*}_{sm} (1 - \theta ) (1 + 2 \theta ) - \theta z)}{z (1 + 2 \theta )}$$

(39)

Thus, $w \le \underline{w}_{sm}$ is a necessary and sufficient condition for $e_{sm}^{*} \ge e_{sm}^{**}$. Given $e^{*}_{s} > e^{*}_{sm}$, by comparing (25) and (39) it follows that $\underline{w}_{s} < \underline{w}_{sm}$ when $2 e^{*}_{sm} > e^{*}_s$. In the absence of spillover (i.e., $\theta = 0$), (38) becomes:

$$\frac{(z + 3 e^{*}_{m})^2}{(z + 3 e^{**}_{m})^2} = \left( 1 + \frac{2 e^{*}_{m}}{z}\right) \, \frac{V}{T}$$

(40)

The LHS in (40) is larger than (equal to) 1 if and only if $e_{m}^{*} > e_{m}^{**}$ ($e_{m}^{*} = e_{m}^{**}$). Conversely, the RHS in (40) is always equal or larger than 1 when $V \ge T$. When $V < T$, the RHS in (40) is equal or larger than 1 if an only if $w \le \underline{w}_{m}$, where:

$$\underline{w}_{m} = 1 + \frac{2 e^{*}_{m}}{z}$$

(41)

Given $e^{*}_{m} < e^{*}$ and $3 e^{*}_m > 2 e^{*}$, by comparing (13) and (41), it can be easily verified that $\underline{w} < \underline{w}_{m}$. $\square$