Contests, private provision of public goods and evolutionary stability

Economics Letters 138 (2016) 34–37

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Contests, private provision of public goods and evolutionary stability Andreas Wagener ∗ University of Hannover, School of Economics and Management, Koenigsworther Platz 1, 30167 Hannover, Germany

highlights • The private provision of public goods can be incentivized by a contest. • Evolutionary stability of public goods game with contest is studied. • If evolutionary stability and Nash equilibrium coincide, the outcome is efficient.

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Article history: Received 2 September 2015 Received in revised form 2 November 2015 Accepted 13 November 2015 Available online 2 December 2015

abstract We study evolutionary stability for public goods games incentivized by a contest. In a quasi-linear setting, we derive conditions such that evolutionary stability, Nash equilibrium and efficient solution coincide. © 2015 Elsevier B.V. All rights reserved.

JEL classification: C72 D74 H41 Keywords: Public goods games Contests Finite-player ESS

1. Introduction Contests and lotteries can be helpful devices to incentivize the private provision of public goods (Morgan, 2000; Kolmar and Wagener, 2012). Adding to a situation where a public good is provided through private contributions a contest that rewards higher contributions by improving the contributor’s chances of winning a rent or prize can alleviate the under-provision problem for public goods. Under the assumption of Nash play, a suitable combination may even implement an efficient level of the public good (Kolmar and Wagener, 2012). We study how contests and the private provision of public goods work together in an (direct) evolutionary framework. We analyze finite-player evolutionarily stable strategies (ESS) of a quasilinear private provision game for a public good (henceforth: public goods game) that is combined with a contest where individual success increases in one’s contributions to the public good. We relate

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http://dx.doi.org/10.1016/j.econlet.2015.11.018 0165-1765/© 2015 Elsevier B.V. All rights reserved.

the ESS to the efficient solution and to the Nash equilibrium of that game and study conditions such that the outcomes coincide. Our results are as follows:

• The finite-player ESS in the provision-with-contest game is identical to the ESS of the contest alone and, thus, depends on the prize and the contest success function only (Result 1). The ESS predicts the outcome when players care about their relative, rather than about absolute, payoffs. As the public goods game with contest is a generalized aggregative game, its ESS is globally stable and stochastically stable. I.e., knowing the ESS shortcuts the full study of specific dynamic processes of learning, imitation, reproduction etc. • The motive of spite inherent in evolutionary play exacerbates both the under-provision in the public goods game and the over-investment problem in the contest. By balancing these opposing trends, a suitable combination of contest and public goods game, characterized in Result 2, implements the efficient level of the public good in evolutionary play. • With Tullock contests, the conditions on prize and contest success function such that the efficient level of the public good is provided turn out to be same for Nash and for evolutionary

A. Wagener / Economics Letters 138 (2016) 34–37

play. They establish a trade-off between the value of the prize and contest decisiveness (Result 3). • Generally, if Nash equilibrium and ESS coincide in a public goods game with contest, they both implement the efficient level of the public good (Result 4). The necessary condition trades off the value of the prize and the negative spillovers in the contest (Result 5). 2. Public goods game with a contest Notation. We consider symmetric normal form games with n ≥ 2 identical players, indexed by i. Each player’s strategy set is given by a closed interval [0, m] for some m > 0. Vectors in Rn are denoted in bold-face: x = (x1 , . . . , xn ). We write 1 = (1, . . . , 1). Symmetric vectors are recognizable by superscripts or other adornments. E.g., for scalar xE we denote by xE = xE · 1. We write x−i for the vector of all xj except for xi . Goods. Endowed with an amount m > 0 of a dual-use good, each individual decides how much of it she spends on contributing to a public good (xi ∈ [0, m]) or to consume directly (m − xi ). The provision level, g, of the public n good equals the sum of individual contributions: g = g (x) = i=1 xi . There is no way to provide the public good other than by players’ contributions xi . In particular, the contest designer (see below) is not able to add to, or subtract from, the public good. Contest. To incentivize contributions to the public good, a contest is installed. With probability pi it pays a prize of value z > 0 to individual i. The prize is financed by equal lump-sum taxes z /n and measured in terms of private consumption. Winning probabilities pi are determined by a contest success function (CSF) that depends on own contributions, xi , and an aggregate, h(x), of all contributions: pi = p(xi , h(x)).

(1)

The CSF has standard properties: it  is a probability on the set of n agents (0 ≤ p(x, h(x)) ≤ 1 and i=1 p(xi , h(x)) = 1 for all n x ∈ R+ ). It strictly increases in x and strictly decreases in h; the aggregate h(x) strictly increases in all its arguments. For simplicity, p and h are twice differentiable. Moreover, h(x) is symmetric, i.e., h(π(x)) = h(x) for all permutations π of x. The latter assumption implies that, for all x and whenever x′i = x′j , p(x′i , h(x)) = p(x′j , h(x))

and

∂ p(x′j , h(x)) ∂ p(x′i , h(x)) = . ∂ xi ∂ xj

dpi dxi

=

d dxi

p(xi , h(x)) =

∂ p(xi , h(x)) ∂ p(xi , h(x)) ∂ h(x) + , ∂ xi ∂h ∂ xi

can be decomposed into two parts. The (positive) ‘‘aggregate∂ p(xi ,h(x)) taking’’ effect measures how an increase in one’s con∂ xi tribution raises the odds, given that the aggregate, h(x), of all contributions remains unchanged. The (negative) ‘‘aggregate∂ p(xi ,h(x)) ∂ h(x) changing’’ effect measures how increasing xi de∂h ∂ xi creases i’s winning chances via a raise in aggregate contributions. Example: Tullock CSF. A prominent class of CSFs that satisfies our assumptions are Tullock CSFs: p(xi , h(x)) = xri /h(x) with h(x) :=

n 

xrj .

Preferences. Individuals have quasi-linear preferences over the (expected) consumption of the private good, m − z /n − xi + pi z, and the public good: ui = m − z /n − xi + pi z + v(g )

= m − z /n − xi + p(xi , h(xi , x−i )) · z + v(g (xi , x−i )) = u(xi , x−i ). Function v is strictly increasing and strictly concave: v ′ (g ) > 0 > v ′′ (g ). We assume v ′ (0) > 1 > nv ′ (nm − z ) for all z we consider. These inequalities preclude that all resources should optimally be devoted to, respectively, the private consumption or the public good. Writing ui as U (xi , x) = m − z /n − xi + p(xi , h(x))z + v(g (x)),

(3)

shows that the public goods game with a contest is a generalized aggregative game: for each player, payoffs depend on the own action, xi , and on symmetric aggregates, h and g, of all strategies, x. 3. Efficiency and Nash equilibrium Efficiency. By the separability and the strict concavity of v , the efficient1 level of the public good, g ∗ , is uniquely given by the Samuelson condition: nv ′ (g ∗ ) = 1

g ∗ = v ′−1 (1/n).

or

(4)

Denote by x := g /n the contribution that, if made by everybody, efficiently provides the public good: given available resources,  x∗ = x∗ 1 is the symmetric solution to the problem maxx j uj (x). It satisfies ∗

n  ∂ uj (x∗ ) j =1

∂ xi

∗

= 0 for all i.

(5)

Nash equilibrium. A symmetric Nash equilibrium is a contribution level, xN , such that u(xN ) ≥ u(xi , xN−i ) for all xi ∈ [0, m − z /n]. In the public goods game with contest, it satisfies

− 1 + v ′ (nxN ) + z ·

d dxi

p(xNi , h(xN )) = 0.

(6)

Efficiency and Nash play. Without contest, the provision of the public good is inefficiently low: v ′ (nxN ) > v ′ (nx∗ ) for p = 0 or z = 0. Adding a contest may remedy this. Combining (4) and (6), efficiency requires that the CSF locally satisfies z·

From (1), the marginal effect of own contributions on an individual’s winning probability,

35

dp(x∗i , h(x∗ )) dxi

=

n−1 n

.

(7)

(Kolmar and Wagener, 2012, Eq. (13)). Condition (7) indicates that a contest designer can trade off a sharper decisiveness (greater sensitivity of pi with respect to xi ) for a larger prize. 4. Evolutionary stability A strategy xE is called an evolutionarily stable strategy (ESS) if u(xE , x, xE , . . . , xE ) ≥ u(x, xE , . . . , xE )





n −2







n−1



for all x ≥ 0. An ESS is a strategy that, when played by all players, cannot be invaded by single mutations: any deviating player earns a lower payoff than the non-deviating players (relative payoff comparison).

(2)

j =1

Parameter r > 0 measures the decisiveness of the contest. For r = 1, the contest is a lottery. To avoid technical problems we assume r ≤ n/(n − 1) whenever we discuss Tullock contests.

1 The efficient level solves max

g ,(ci )i=1,...,n

 i

(ci + v(g ))

s.t. g +



ci ≤ mn − z ,

i

where the right-hand side denotes available resources after financing the prize.

36

A. Wagener / Economics Letters 138 (2016) 34–37

6. Tullock contests

Result 1. An ESS in the public goods game with contest satisfies z·

∂ p(xEi , h(xE )) = 1. ∂ xi

(8)

Proof. Following Tanaka (2000), denote the payoff difference between an individual that deviates from a common strategy and an individual that sticks with it by

ψ(x, x′ ) := u(x1 [x, x′ ]) − u(x2 [x, x′ ]), where, for any x, x′ , we write x1 [x, x′ ] and x2 [x, x′ ] for x1 = (x; x′ , . . . , x′ ) and x2 = (x′ ; x, x′ , . . . , x′ ). Strategy xE is an ESS if and only if it solves maxx≥0 ψ(x, x′ ) for x′ = xE , i.e., iff xE = arg maxx≥0 ψ(x, xE ). Since components of x1 and x2 sum up to the same amount g (x1 ) = g (x2 ), we get

ψ(x, x′ ) = x′ − x + z · p(x, h(x1 )) − p(x′ , h(x2 )) . 





d dx

p(x, h(x1 )) −

d dx

p(x′ , h(x2 ))

Result 3. In a public goods game with Tullock contest, all contests with rz = g ∗

(11) N

∗

E

∗

N

E

lead to x = x and to x = x (and, consequently, to x = x ). Proof. With (2) we get for x′ = x′ · 1 that dp(x′ , h(x′ )) dx

(9)

=

(n − 1)r n2 x ′

and

∂ p(x′ , h(x′ )) r = ′. ∂x nx

Plugging the first expression into (7) and the second into (10) reveals that (11) must hold in both cases. 

Calculate:

∂ ψ(x, x′ ) = −1 + z · ∂x

Generically, neither in public goods games nor in contests do the ESS and the Nash equilibrium coincide. Public goods games with Tullock contests, however, exemplify that, in a combination of the two mechanisms, Nash equilibrium and ESS can be identical. Moreover, this even leads to efficiency2 :



With Tullock contests, the conditions such that the public goods game implements the efficient amount of the public good are identical for Nash equilibrium and for ESS. Suitable contests are characterized by a trade-off between decisiveness and prize.

∂ p(x, h(x1 )) ∂ p(x, h(x1 )) ∂ h(x1 ) + ∂x ∂h ∂x  ∂ p(x′ , h(x2 )) ∂ h(x2 ) . − ∂h ∂x 

= −1 + z ·

7. Equivalence of ESS and Nash equilibrium

A necessary condition for ESS is ∂∂x ψ(xE , xE ) = 0. Evaluate this at x = x′ = xE (such that x1 = x2 = xE ) and use the symmetry of h to get (8).  From (9), the ESS of the public goods game with contest is identical to the ESS of the contest alone: the public good is irrelevant for individuals’ relative performance vis-à-vis each other. For Tullock contests, (8) confirms that the ESS is given by xE = rz /n (Hehenkamp et al., 2004). Public goods games and contests are, singly and combined, generalized aggregative games. Hence, their ESS is at the same time globally stable (i.e., robust against arbitrary numbers of identical mutations) and leads to stochastically stable states in imitation dynamics (Alós-Ferrer and Ania, 2005). Condition (8) also holds for these solution concepts.

The observation that the equivalence of Nash equilibrium and ESS implies efficiency is not confined to Tullock CSFs: Result 4. In all public goods games with contest, xN = xE implies g = g ∗. Proof. For symmetric games with uniform preferences, differentiable payoffs, and interval strategy sets, xE = xN necessitates ∂ uj (xN ) ∂ xi

= 0 for all i ̸= j (Hehenkamp et al., 2010, Corollary 2).3 Since

a Nash equilibrium itself requires holds. 

∂ ui (xN ) ∂ xi

= 0 for all i, condition (5)

Result 4 works more generally: in all games, if xN = xE the necessary condition for x = xN = xE to be efficient is satisfied. Finally, we derive a necessary condition for xN = xE in the current framework. For symmetric profiles x′ = x′ · 1, define

∂ p(x′ , h(x′ )) ∂ h(x′ ) · ∂h ∂x

5. ESS and efficiency

A(x′ ) :=

The ESS in a public goods game without contest is xE = 0 (Possajennikov, 2003), implying provision failure. Conversely, the ESS in contests alone goes along with considerable rent dissipation (Hehenkamp et al., 2004). Combining both mechanisms balances these opposing trends and even implements the efficient level of the public good in evolutionary play. From (8) we get

as the marginal reduction of an individual’s winning probability if somebody else slightly changes her contribution; A captures the contest externalities.

Result 2. For an ESS in a public goods game with contest to implement the efficient level of the public good, CSF and prize must satisfy

∂ p(x∗i , h(x∗ )) z· = 1. ∂ xi

(10)

Similar as in (7), contest design can substitute value of prize for contest decisiveness: for given aggregate contributions, the more strongly the likelihood to win the prize increases in individual contributions, the smaller the prize needed to induce efficiency. While similar in structure with (7), (10) only refers to the aggregate-taking effect in the CSF; this reflects that aggregatechanging effects are irrelevant in ESS.

Result 5. If a public goods game with contest leads to xE = xN = x∗ then prize z and CSF p(x, h(x)) satisfy z · A(x∗ ) = −1/n.

(12)

Proof. Given x, define B(x) :=

n ∂ p(xi , h(x)) ∂ p(xi , h(x))  ∂ h(x) + · . ∂ xi ∂h ∂ xj j =1

2 For the Nash equilibrium, this has been shown in Kolmar and Wagener (2012, Ex. 1). 3 This can also be seen directly: For xE = xN plug (8) into (6) to obtain that

v ′ (nxN ) + z ∂ p(x ∂,hh(x N

∂

uj (xN )

∂ xi

= 0.

N ))

∂ h( x N ) ∂ xi

= 0 must hold which is precisely the condition

A. Wagener / Economics Letters 138 (2016) 34–37

Evaluate this at a symmetric x′ = x′ · 1 and use the symmetry of h to obtain: B(x′ · 1) =

∂ p(x′ , h(x′ )) ∂ h(x′ ) ∂ p(x′ , h(x′ )) +n· · =: b(x′ ). ∂x ∂h ∂x

Hence, dp(x′ , h(x′ )) dx

= b(x′ ) − (n − 1)A(x′ ) and

∂ p(x′ , h(x′ )) = b(x′ ) − nA(x′ ). ∂x Conditions (7) and (10), thus, can be rewritten, respectively, as z b(xN ) − (n − 1)A(xN ) =





n−1 n

and

 z b(xE ) − nA(xE ) = 1. 

Equalities x′ = xE = xN = x∗ then necessitate (12).



37

For the Tullock CSF, A(x′ ) = −r /(n2 x′r ). Using this in (12) confirms (11). As in our previous results, (12) exhibits a trade-off between the value of the prize, z, and the sensitivity of the CSF. References Alós-Ferrer, C., Ania, A.B., 2005. The evolutionary stability of perfectly competitive behavior. Econom. Theory 26, 497–516. Hehenkamp, B., Leininger, W., Possajennikov, A., 2004. Evolutionary equilibrium in Tullock contests: Spite and overdissipation. Eur. J. Polit. Econ. 20, 1045–1057. Hehenkamp, B., Possajennikov, A., Guse, T., 2010. On the equivalence of Nash and evolutionary equilibrium in finite populations. J. Econ. Behav. Organ. 73, 254–258. Kolmar, M., Wagener, A., 2012. Contests and the private provision of public goods. South. Econ. J. 79, 161–179. Morgan, J., 2000. Financing public goods by means of lotteries. Rev. Econom. Stud. 67, 761–784. Possajennikov, A., 2003. Evolutionary foundations of aggregate-taking behavior. Econom. Theory 21, 921–928. Tanaka, Y., 2000. Stochastically stable states in an oligopoly with differentiated goods. J. Math. Econom. 34, 235–253.