- Email: [email protected]
Level-k reasoning in contests
Economics Letters 108 (2010) 149–152
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
Level-k reasoning in contests Mark Bernard ⁎ Department of Economics, Stockholm School of Economics, P.O. Box 6501, 113 83 Stockholm, Sweden
a r t i c l e
i n f o
Article history: Received 14 August 2009 Received in revised form 29 March 2010 Accepted 8 April 2010 Available online 18 April 2010
a b s t r a c t We introduce level-k reasoning to contest theory, compare its predictions to those of Nash Equilibrium and relate to the experimental evidence. © 2010 Elsevier B.V. All rights reserved.
Keywords: Contests Cognitive hierarchy JEL classification: D03 D74 P16
1. Introduction
2. The general setup
Contest theory has mostly used Nash equilibrium to solve models and derive predictions.1 The current paper instead introduces level-k reasoning to contest theory. Variants of the level-k framework have been studied and used by Camerer et al. (2004), Costa-Gomes et al. (2001), Costa-Gomes and Crawford (2006), Nagel (1995) and Stahl and Wilson (1994, 1995). Players are taken to be rational, but do not necessarily hold consistent beliefs about other players. They are assumed to think that they are “a little smarter” than their opponents. We investigate (a) if and how level-k predictions differ from those of Nash equilibrium in standard Tullock contests, and (b) how well the level-k approach explains experimental evidence from contests. As for (a), we find that relative to Nash equilibrium, level-k thinkers exert less effort for any finite level k, but there is monotone convergence. This immediately affects our answer to (b) since most of the experimental literature finds spending to be higher than predicted. However, we cannot definitively refute the level-k approach on these grounds as most studies implicitly assume subjects to maximize expected material payoffs, which is questionable. We discuss this briefly. The rest of the paper is organized as follows: Section 2 presents the competing theories in a general setup. Section 3 covers the theoretical predictions of level-k reasoning as juxtaposed to Nash equilibrium for two-player Tullock contests. Section 4 relates to the empirical evidence, and Section 5 concludes.
Suppose that N ≥ 2 risk-neutral players compete for a prize of value y, which we normalize to 1. Player i 2 {1, 2, ..., N} can influence his probability of winning the prize by exerting an effort xi ≥ 0. Let the subscript −i denote all players other than i. A contest success function (CSF) is a probability measure p(x) = (p1(x1, x− 1), ..., pN (xN, x− N)) that maps players' efforts into winning probabilities. We assume anonymity throughout, i.e. permutations of effort profiles lead to corresponding permutations of the components of p. This allows us to drop subscripts. Skaperdas (1996) has axiomatized a class of CSF as follows:
⁎ Tel.:+46 8 736 9255; fax: + 46 8 31 32 07. E-mail address: [email protected]. 1 Konrad (2008) provides an overview of the field. 0165-1765/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2010.04.022
pðxi ; x−i Þ =
f ðxi Þ ; ∑Nj= 1 f xj
where f is a positive and strictly increasing function. The cost of effort is taken to be linear, with unit marginal cost, so that player i's expected utility becomes i
U ðxi ; x−i Þ = pðxi ; x−i Þ−xi ;
2.1. Nash equilibrium The strategy profile x⁎ = {x1⁎, x2⁎, ..., xN⁎} is a Nash equilibrium if for all i, i
i
þ
U ðxi ; x−i Þ ≥ U ðxi ; x−i Þ ∀xi ∈ R 0 :
150
M. Bernard / Economics Letters 108 (2010) 149–152
2.2. The level-k model The level-k model assigns different cognitive levels to players, upon which the latter's actions depend. Let Tk denote a player of level k. Usually, T0 is assumed to randomize uniformly over a meaningful subset of the strategy space. In our model, it seems natural to choose [0, 1] as the relevant interval, since there is no point in expending more than there is to win. For k ≥ 1, Tk is assumed to believe that he faces only Tk− 1 opponents who act independently of each other, and to play a best response given this belief 2. More specifically, if i is T1, he solves
max
xi ∈½0;1
∫ ⋯N−1 ∫pðxi ; x−i Þdx1 ::: dxi−1 dxi ½0;1
+ 1 :::dxN −xi
:
The problem being symmetric, we drop player subscripts and henceforth use level superscripts instead (so xk is Tk's action). Let x1 be the maximizer to the above problem and assume that it exists and is unique. Then T2 solves 2 1 1 1 2 −x g: max fp x ; x ; x ; :::; x
Fig. 1. Level-1 best response as a function of r.
T2 knows this and best-responds to x1. And so on. Notice that the first-order condition to Eq. (2) reads
x ∈½0;1 2
Assume existence and uniqueness again and iterate on k to find the best replies for all cognitive levels3. We shall now focus on a widely used class of CSF, the so-called Tullock CSF, and restrict attention to the case of two players.
r−1 r r x1 x0 ∫ r r 2 dx0 = 1: x1 + x0 0 1
For Tk, k ≥ 2, the FOC (given uniqueness of xk− 1) reads
3. Tullock contests
r−1 r r xk xk−1 k r r 2 = 1: x + xk−1
3.1. Nash equilibrium Let us assume that p is of the Tullock (1980) form, i.e.
pðx1 ; x2 Þ =
8 > < > :
ðx1 Þr ðx1 Þ + ðx2 Þr
if x1 + x2 N 0;
1= 2
otherwise;
r
r : 4
ð4Þ
The integrals in Eqs. (2) and (3) are generally not analytically solvable. Therefore, we decided to rely on numerical calculations. Using MATLAB, it seems safe to state
for some r N 0. For r ≤ 2, as e.g. Pérez-Castrillo and Verdier (1992) show, there exists a unique symmetric Nash equilibrium in pure strategies which is given by x⁎ =
ð3Þ
ð1Þ
For r ≤ 1, the symmetric equilibrium is also the unique Nash equilibrium. For r N 2, there are no pure strategy equilibria, but mixed ones4. We restrict the analysis to r ≤ 2.
Conjecture 1. x1 b x⁎ for all r 2 (0, 2]. Fig. 1 shows the calculated level-1 best replies as a function of r5. The straight line plots Nash equilibrium efforts, the strictly concave function below represents level-1 best responses. From this we can immediately infer the following core result, which is illustrated in Fig. 2: Proposition 1. Assuming that conjecture 1 is true, the sequence k ⁎ ⁎ {xk}∞ k= 1 converges monotonically to x . In particular, x ≤ x for any k 2 Z++.
3.2. The level-k model Proof. This follows immediately from proposition 2 in Pérez-Castrillo and Verdier (1992). 5
Since T0 randomizes uniformly over [0, 1], T1 solves r 1 x 1 r dx0 −x : max ∫ 1 r x + x0 x1 ∈½0;1 0 1
ð2Þ
2 Camerer et al. (2004) assume instead a prior Poisson distribution over all types, which Tk then truncates at k − 1. 3 If Tk's maximizer is not unique, he is assumed to randomize uniformly over the set argmax. Tk + 1 takes this into account. 4 See Baye, Kovenok, and de Vries (1994).
By strict concavity of p, for r ≤ 1, our result holds even for the more general case where Tk best replies to a distribution over {T0, T1, ..., Tk− 1}, as in Camerer et al. (2004). Interestingly, thus, level-k reasoning results in systematic underspending relative to Nash equilibrium. This poses a challenge to the model, as we shall see in our review of the experimental evidence. Before that, we focus attention on the widely used parameter choice r = 1, for which the model can be solved analytically. 5
Codes are provided upon request.
M. Bernard / Economics Letters 108 (2010) 149–152
151
Fig. 2. Best replies of levels 1 through 4 (colored red; straight blue line indicates Nash equilibria).
3.3. Illustration of the case r = 1 With r = 1, level-1 expected utility can be expressed as h i1 1 1 1 0 1 1 0 1 1 1 U x ; x = x ln x + x −x = x ln 1 + 1 −x : 0 x
follows immediately that for any strictly interior x1, xk converges to x⁎ monotonically. We have thus proved Proposition 2. Suppose r = 1. Then the sequence {xk}∞ k= 1 converges monotonically to x⁎. In particular, xk b x⁎ for any k 2 Z++. 4. Experimental evidence
The level-1 FOC is then 1 1 + 1: ln 1 + 1 = x 1 + x1 Both sides of the equation are strictly decreasing, strictly convex functions. As x1 approaches 0, the LHS approaches + ∞, while the RHS approaches 2. As x1 approaches 1, the LHS approaches ln2 and the RHS 3 approaches N ln2. Thus, there exists a unique x̂1 2 (0,1) satisfying the 2 FOC, which is 0.1885 b 0.25 = x⁎. To see that xk converges to the Nash equilibrium effort as k approaches infinity, notice that from Eq. (4) we have, for k ≥ 2, k
x =
There exists a growing literature that investigates behavior in contests. The standard assumption is that subjects' expected material payoffs are a fair enough representation of their preferences over outcomes in the experiment. Using Weibull's (2004) terminology, the game is identified with the game protocol. Given this assumption, most studies find evidence of significant overspending relative to Nash equilibrium, even after allowing for learning6. Level-k theory apparently fares worse than Nash equilibrium7. One way to address this issue could be to manipulate level-0 behavior. Crawford and Iriberri (2008) very elegantly discuss and work with different assumptions on T0 in their application of level-k
pffiffiffiffiffiffiffiffiffiffi k−1 k−1 : xk−1 −x =F x
Obviously, the strictly concave function F has a unique stable fixed 1 4
point at x⁎ = , the Nash equilibrium, at which the slope of F is 0. It
6 See Önculer and Croson (2005) for a survey, and Abbink et al. (2009) for a recent example. 7 Typically r = 1 is used, so by proposition 2 the problem obtains even barring numerical methods.
152
M. Bernard / Economics Letters 108 (2010) 149–152
reasoning to auctions. For instance, they successfully explain overspending patterns in first-price auctions. This strongly suggests applying such modifications here, too. For instance, concentrating probability mass around the maximizer of the deterministic bestreply function could stimulate level-1 spending. However, in case r = 1 it is impossible to get x1 N x⁎ since the maximizer of the bestreply function coincides with Nash equilibrium. Indeed, proposition 2 holds for any specification of level-0 behavior with bounded support. Thus, unfortunately, this approach is not promising in the present case. A fundamental question in the interpretation of experimental evidence is whether players' preferences are adequately captured. The claim that level-k theory fares worse at explaining the data holds if subjects maximize expected material payoffs. Assuming instead that subjects also anticipate some (dis-)utility from winning (losing) per se can reconcile both level-k theory and Nash equilibrium with the data.8 5. Conclusion This paper introduced level-k reasoning to contest theory and assumed players to think that they are “a little smarter” than their opponents. Our purpose was to find out (a) if and how level-k predictions differ from those of Nash equilibrium in standard Tullock contests, and (b) how well the level-k approach explains experimental evidence from contests. As for (a), we found that relative to Nash equilibrium, level-k thinkers exert less effort for any finite level-k, but there is monotone convergence. Interestingly, most of the experimental literature on contests finds spending/effort to be higher than predicted. As for (b), one might thus infer that level-k theory loses the race to standard Nash equilibrium. However, such a claim may be premature if players care for more than the expected monetary reward. A natural extension of this paper would be to compute the levelk replies for other classes of contest success functions, for instance the logistic one as discussed by e.g. Skaperdas (1996), to see whether the underspending result is robust to the choice of conflict technology.
8 For instance, Holt and Sherman (1994) and Filiz-Ozbay and Ozbay (2007) attempt to explain overbidding in first-price auctions using such assumptions.
Acknowledgments I am indebted to Magnus Johannesson, Erik Mohlin, Karl Wärneryd and an anonymous referee for their valuable comments.
References Abbink, K., Brandts, J., Herrmann, B., Orzen, H., 2009. Inter-Group Conflict and IntraGroup Punishment in an Experimental Contest Game. American Economic Review, forthcoming. Baye, M.R., Kovenock, D., de Vries, C.G., 1994. The solution to the Tullock Rent-Seeking Game when R N 2: mixed-strategy equilibria and mean dissipation rates. Public Choice 81 (3), 363–380. Camerer, C.F., Ho, T.H., Chong, J.K., 2004. A cognitive hierarchy model of games. Quarterly Journal of Economics 119 (3), 861–898. Costa-Gomes, M., Crawford, V.P., Broseta, B., 2001. Cognition and behavior in normalform games: an experimental study. Econometrica 69 (5), 1193–1235. Costa-Gomes, M., Crawford, V.P., 2006. Cognition and behavior in two-person guessing games: an experimental study. American Economic Review 96 (5), 1737–1768. Crawford, V.P., Iriberri, N., 2008. Level-k auctions: can a non-equilibrium model of strategic thinking explain the winner's curse and overbidding in private-value auctions? Econometrica 75 (6), 1721–1770. Filiz-Ozbay, E., Ozbay, E.Y., 2007. Auctions with anticipated regret: theory and evidence. American Economic Review 97 (4), 1407–1418. Holt, C.A., Sherman, R., 1994. The loser's curse. American Economic Review 84 (3), 642–652. Nagel, R., 1995. Unraveling in guessing games: an experimental study. American Economic Review 85 (5), 1313–1326. Konrad, K.A., 2008. Strategy in Contests: An Introduction. Oxford University Press, New York. Önculer, A., Croson, R.T.A., 2005. Rent seeking for a risky rent: a model and experimental investigation. Journal of Theoretical Politics 17, 403–429. Perez-Castrillo, J.D., Verdier, T., 1992. A general analysis of rent-seeking games. Public Choice 73 (3), 335–350. Skaperdas, S., 1996. Contest success functions. Economic Theory 7, 283–290. Stahl, D., Wilson, P.W., 1994. Experimental evidence on players' models of other players. Journal of Economic Behavior & Organization 25 (3), 309–327. Stahl, D., Wilson, P.W., 1995. On players' models of other players: theory and experimental evidence. Games and Economic Behavior 10, 218–254. Tullock, G., 1980. Efficient rent-seeking. In: Buchanan, J.M., Tollison, R.D., Tullock, G. (Eds.), Toward a theory of the rent-seeking society. Texas A & M University Press, Austin. Weibull, J.W., 2004. Testing game theory. In: Huck, Steffen (Ed.), Advances in Understanding Strategic Behaviour: Game Theory, Experiments and Bounded Rationality. Essay in Honour of Werner Güth. Palgrave MacMillan, New York.
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
Level-k reasoning in contests Mark Bernard ⁎ Department of Economics, Stockholm School of Economics, P.O. Box 6501, 113 83 Stockholm, Sweden
a r t i c l e
i n f o
Article history: Received 14 August 2009 Received in revised form 29 March 2010 Accepted 8 April 2010 Available online 18 April 2010
a b s t r a c t We introduce level-k reasoning to contest theory, compare its predictions to those of Nash Equilibrium and relate to the experimental evidence. © 2010 Elsevier B.V. All rights reserved.
Keywords: Contests Cognitive hierarchy JEL classification: D03 D74 P16
1. Introduction
2. The general setup
Contest theory has mostly used Nash equilibrium to solve models and derive predictions.1 The current paper instead introduces level-k reasoning to contest theory. Variants of the level-k framework have been studied and used by Camerer et al. (2004), Costa-Gomes et al. (2001), Costa-Gomes and Crawford (2006), Nagel (1995) and Stahl and Wilson (1994, 1995). Players are taken to be rational, but do not necessarily hold consistent beliefs about other players. They are assumed to think that they are “a little smarter” than their opponents. We investigate (a) if and how level-k predictions differ from those of Nash equilibrium in standard Tullock contests, and (b) how well the level-k approach explains experimental evidence from contests. As for (a), we find that relative to Nash equilibrium, level-k thinkers exert less effort for any finite level k, but there is monotone convergence. This immediately affects our answer to (b) since most of the experimental literature finds spending to be higher than predicted. However, we cannot definitively refute the level-k approach on these grounds as most studies implicitly assume subjects to maximize expected material payoffs, which is questionable. We discuss this briefly. The rest of the paper is organized as follows: Section 2 presents the competing theories in a general setup. Section 3 covers the theoretical predictions of level-k reasoning as juxtaposed to Nash equilibrium for two-player Tullock contests. Section 4 relates to the empirical evidence, and Section 5 concludes.
Suppose that N ≥ 2 risk-neutral players compete for a prize of value y, which we normalize to 1. Player i 2 {1, 2, ..., N} can influence his probability of winning the prize by exerting an effort xi ≥ 0. Let the subscript −i denote all players other than i. A contest success function (CSF) is a probability measure p(x) = (p1(x1, x− 1), ..., pN (xN, x− N)) that maps players' efforts into winning probabilities. We assume anonymity throughout, i.e. permutations of effort profiles lead to corresponding permutations of the components of p. This allows us to drop subscripts. Skaperdas (1996) has axiomatized a class of CSF as follows:
⁎ Tel.:+46 8 736 9255; fax: + 46 8 31 32 07. E-mail address: [email protected]. 1 Konrad (2008) provides an overview of the field. 0165-1765/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2010.04.022
pðxi ; x−i Þ =
f ðxi Þ ; ∑Nj= 1 f xj
where f is a positive and strictly increasing function. The cost of effort is taken to be linear, with unit marginal cost, so that player i's expected utility becomes i
U ðxi ; x−i Þ = pðxi ; x−i Þ−xi ;
2.1. Nash equilibrium The strategy profile x⁎ = {x1⁎, x2⁎, ..., xN⁎} is a Nash equilibrium if for all i, i
i
þ
U ðxi ; x−i Þ ≥ U ðxi ; x−i Þ ∀xi ∈ R 0 :
150
M. Bernard / Economics Letters 108 (2010) 149–152
2.2. The level-k model The level-k model assigns different cognitive levels to players, upon which the latter's actions depend. Let Tk denote a player of level k. Usually, T0 is assumed to randomize uniformly over a meaningful subset of the strategy space. In our model, it seems natural to choose [0, 1] as the relevant interval, since there is no point in expending more than there is to win. For k ≥ 1, Tk is assumed to believe that he faces only Tk− 1 opponents who act independently of each other, and to play a best response given this belief 2. More specifically, if i is T1, he solves
max
xi ∈½0;1
∫ ⋯N−1 ∫pðxi ; x−i Þdx1 ::: dxi−1 dxi ½0;1
+ 1 :::dxN −xi
:
The problem being symmetric, we drop player subscripts and henceforth use level superscripts instead (so xk is Tk's action). Let x1 be the maximizer to the above problem and assume that it exists and is unique. Then T2 solves 2 1 1 1 2 −x g: max fp x ; x ; x ; :::; x
Fig. 1. Level-1 best response as a function of r.
T2 knows this and best-responds to x1. And so on. Notice that the first-order condition to Eq. (2) reads
x ∈½0;1 2
Assume existence and uniqueness again and iterate on k to find the best replies for all cognitive levels3. We shall now focus on a widely used class of CSF, the so-called Tullock CSF, and restrict attention to the case of two players.
r−1 r r x1 x0 ∫ r r 2 dx0 = 1: x1 + x0 0 1
For Tk, k ≥ 2, the FOC (given uniqueness of xk− 1) reads
3. Tullock contests
r−1 r r xk xk−1 k r r 2 = 1: x + xk−1
3.1. Nash equilibrium Let us assume that p is of the Tullock (1980) form, i.e.
pðx1 ; x2 Þ =
8 > < > :
ðx1 Þr ðx1 Þ + ðx2 Þr
if x1 + x2 N 0;
1= 2
otherwise;
r
r : 4
ð4Þ
The integrals in Eqs. (2) and (3) are generally not analytically solvable. Therefore, we decided to rely on numerical calculations. Using MATLAB, it seems safe to state
for some r N 0. For r ≤ 2, as e.g. Pérez-Castrillo and Verdier (1992) show, there exists a unique symmetric Nash equilibrium in pure strategies which is given by x⁎ =
ð3Þ
ð1Þ
For r ≤ 1, the symmetric equilibrium is also the unique Nash equilibrium. For r N 2, there are no pure strategy equilibria, but mixed ones4. We restrict the analysis to r ≤ 2.
Conjecture 1. x1 b x⁎ for all r 2 (0, 2]. Fig. 1 shows the calculated level-1 best replies as a function of r5. The straight line plots Nash equilibrium efforts, the strictly concave function below represents level-1 best responses. From this we can immediately infer the following core result, which is illustrated in Fig. 2: Proposition 1. Assuming that conjecture 1 is true, the sequence k ⁎ ⁎ {xk}∞ k= 1 converges monotonically to x . In particular, x ≤ x for any k 2 Z++.
3.2. The level-k model Proof. This follows immediately from proposition 2 in Pérez-Castrillo and Verdier (1992). 5
Since T0 randomizes uniformly over [0, 1], T1 solves r 1 x 1 r dx0 −x : max ∫ 1 r x + x0 x1 ∈½0;1 0 1
ð2Þ
2 Camerer et al. (2004) assume instead a prior Poisson distribution over all types, which Tk then truncates at k − 1. 3 If Tk's maximizer is not unique, he is assumed to randomize uniformly over the set argmax. Tk + 1 takes this into account. 4 See Baye, Kovenok, and de Vries (1994).
By strict concavity of p, for r ≤ 1, our result holds even for the more general case where Tk best replies to a distribution over {T0, T1, ..., Tk− 1}, as in Camerer et al. (2004). Interestingly, thus, level-k reasoning results in systematic underspending relative to Nash equilibrium. This poses a challenge to the model, as we shall see in our review of the experimental evidence. Before that, we focus attention on the widely used parameter choice r = 1, for which the model can be solved analytically. 5
Codes are provided upon request.
M. Bernard / Economics Letters 108 (2010) 149–152
151
Fig. 2. Best replies of levels 1 through 4 (colored red; straight blue line indicates Nash equilibria).
3.3. Illustration of the case r = 1 With r = 1, level-1 expected utility can be expressed as h i1 1 1 1 0 1 1 0 1 1 1 U x ; x = x ln x + x −x = x ln 1 + 1 −x : 0 x
follows immediately that for any strictly interior x1, xk converges to x⁎ monotonically. We have thus proved Proposition 2. Suppose r = 1. Then the sequence {xk}∞ k= 1 converges monotonically to x⁎. In particular, xk b x⁎ for any k 2 Z++. 4. Experimental evidence
The level-1 FOC is then 1 1 + 1: ln 1 + 1 = x 1 + x1 Both sides of the equation are strictly decreasing, strictly convex functions. As x1 approaches 0, the LHS approaches + ∞, while the RHS approaches 2. As x1 approaches 1, the LHS approaches ln2 and the RHS 3 approaches N ln2. Thus, there exists a unique x̂1 2 (0,1) satisfying the 2 FOC, which is 0.1885 b 0.25 = x⁎. To see that xk converges to the Nash equilibrium effort as k approaches infinity, notice that from Eq. (4) we have, for k ≥ 2, k
x =
There exists a growing literature that investigates behavior in contests. The standard assumption is that subjects' expected material payoffs are a fair enough representation of their preferences over outcomes in the experiment. Using Weibull's (2004) terminology, the game is identified with the game protocol. Given this assumption, most studies find evidence of significant overspending relative to Nash equilibrium, even after allowing for learning6. Level-k theory apparently fares worse than Nash equilibrium7. One way to address this issue could be to manipulate level-0 behavior. Crawford and Iriberri (2008) very elegantly discuss and work with different assumptions on T0 in their application of level-k
pffiffiffiffiffiffiffiffiffiffi k−1 k−1 : xk−1 −x =F x
Obviously, the strictly concave function F has a unique stable fixed 1 4
point at x⁎ = , the Nash equilibrium, at which the slope of F is 0. It
6 See Önculer and Croson (2005) for a survey, and Abbink et al. (2009) for a recent example. 7 Typically r = 1 is used, so by proposition 2 the problem obtains even barring numerical methods.
152
M. Bernard / Economics Letters 108 (2010) 149–152
reasoning to auctions. For instance, they successfully explain overspending patterns in first-price auctions. This strongly suggests applying such modifications here, too. For instance, concentrating probability mass around the maximizer of the deterministic bestreply function could stimulate level-1 spending. However, in case r = 1 it is impossible to get x1 N x⁎ since the maximizer of the bestreply function coincides with Nash equilibrium. Indeed, proposition 2 holds for any specification of level-0 behavior with bounded support. Thus, unfortunately, this approach is not promising in the present case. A fundamental question in the interpretation of experimental evidence is whether players' preferences are adequately captured. The claim that level-k theory fares worse at explaining the data holds if subjects maximize expected material payoffs. Assuming instead that subjects also anticipate some (dis-)utility from winning (losing) per se can reconcile both level-k theory and Nash equilibrium with the data.8 5. Conclusion This paper introduced level-k reasoning to contest theory and assumed players to think that they are “a little smarter” than their opponents. Our purpose was to find out (a) if and how level-k predictions differ from those of Nash equilibrium in standard Tullock contests, and (b) how well the level-k approach explains experimental evidence from contests. As for (a), we found that relative to Nash equilibrium, level-k thinkers exert less effort for any finite level-k, but there is monotone convergence. Interestingly, most of the experimental literature on contests finds spending/effort to be higher than predicted. As for (b), one might thus infer that level-k theory loses the race to standard Nash equilibrium. However, such a claim may be premature if players care for more than the expected monetary reward. A natural extension of this paper would be to compute the levelk replies for other classes of contest success functions, for instance the logistic one as discussed by e.g. Skaperdas (1996), to see whether the underspending result is robust to the choice of conflict technology.
8 For instance, Holt and Sherman (1994) and Filiz-Ozbay and Ozbay (2007) attempt to explain overbidding in first-price auctions using such assumptions.
Acknowledgments I am indebted to Magnus Johannesson, Erik Mohlin, Karl Wärneryd and an anonymous referee for their valuable comments.
References Abbink, K., Brandts, J., Herrmann, B., Orzen, H., 2009. Inter-Group Conflict and IntraGroup Punishment in an Experimental Contest Game. American Economic Review, forthcoming. Baye, M.R., Kovenock, D., de Vries, C.G., 1994. The solution to the Tullock Rent-Seeking Game when R N 2: mixed-strategy equilibria and mean dissipation rates. Public Choice 81 (3), 363–380. Camerer, C.F., Ho, T.H., Chong, J.K., 2004. A cognitive hierarchy model of games. Quarterly Journal of Economics 119 (3), 861–898. Costa-Gomes, M., Crawford, V.P., Broseta, B., 2001. Cognition and behavior in normalform games: an experimental study. Econometrica 69 (5), 1193–1235. Costa-Gomes, M., Crawford, V.P., 2006. Cognition and behavior in two-person guessing games: an experimental study. American Economic Review 96 (5), 1737–1768. Crawford, V.P., Iriberri, N., 2008. Level-k auctions: can a non-equilibrium model of strategic thinking explain the winner's curse and overbidding in private-value auctions? Econometrica 75 (6), 1721–1770. Filiz-Ozbay, E., Ozbay, E.Y., 2007. Auctions with anticipated regret: theory and evidence. American Economic Review 97 (4), 1407–1418. Holt, C.A., Sherman, R., 1994. The loser's curse. American Economic Review 84 (3), 642–652. Nagel, R., 1995. Unraveling in guessing games: an experimental study. American Economic Review 85 (5), 1313–1326. Konrad, K.A., 2008. Strategy in Contests: An Introduction. Oxford University Press, New York. Önculer, A., Croson, R.T.A., 2005. Rent seeking for a risky rent: a model and experimental investigation. Journal of Theoretical Politics 17, 403–429. Perez-Castrillo, J.D., Verdier, T., 1992. A general analysis of rent-seeking games. Public Choice 73 (3), 335–350. Skaperdas, S., 1996. Contest success functions. Economic Theory 7, 283–290. Stahl, D., Wilson, P.W., 1994. Experimental evidence on players' models of other players. Journal of Economic Behavior & Organization 25 (3), 309–327. Stahl, D., Wilson, P.W., 1995. On players' models of other players: theory and experimental evidence. Games and Economic Behavior 10, 218–254. Tullock, G., 1980. Efficient rent-seeking. In: Buchanan, J.M., Tollison, R.D., Tullock, G. (Eds.), Toward a theory of the rent-seeking society. Texas A & M University Press, Austin. Weibull, J.W., 2004. Testing game theory. In: Huck, Steffen (Ed.), Advances in Understanding Strategic Behaviour: Game Theory, Experiments and Bounded Rationality. Essay in Honour of Werner Güth. Palgrave MacMillan, New York.