- Email: [email protected]
Risk taking in selection contests
Games and Economic Behavior 42 (2003) 172–179 www.elsevier.com/locate/geb
Note
Risk taking in selection contests Hans K. Hvide a,∗ and Eirik G. Kristiansen b a Department of Finance, Norwegian School of Economics and Business, Hellev. 30, 5045 Bergen, Norway b Department of Economics, Norwegian School of Economics and Business and Norges Bank
(The Central Bank of Norway), Hellev. 30, 5045 Bergen, Norway Received 5 October 1999
Abstract We study contests where the strategic variable is the degree of risk taking rather than the amount of effort. The selection efficiency of such contests is examined. We show that the selection efficiency of a contest may be improved by limiting the competition in two ways; by having a small number of contestants, and by restricting contestant quality. 2002 Elsevier Science (USA). All rights reserved. JEL classification: C44; D29; D83; J41 Keywords: Contest; Risk taking; Selection; Tournament
1. Introduction In a contest or a tournament, rewards are based on the relative performance of the contestants. Contests serve two different purposes. First, tournaments can provide incentives to work hard, as in (Lazear and Rosen, 1981). Second, tournaments can serve as a selection mechanism. In this paper we focus on the selection role of contests, in the case where risk taking is the strategic variable of the contestants. Employees involved in a promotion process or tenure process, for example, may choose tasks that differ in risk profile to show off their abilities. We investigate the selection efficiency of contests in which the contestants optimize their choice of risk, given the risk taking of others. Who will come out on top, bad types or * Corresponding author.
E-mail addresses: [email protected] (H.K. Hvide), [email protected] (E.G. Kristiansen). 0899-8256/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved. doi:10.1016/S0899-8256(02)00538-9
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good types? In what way will the selection efficiency depend on, for example, the quality of the contestant pool? Answering such questions can be important, e.g., to understand the efficiency of promotion processes in firms.1 We confine ourselves to a simple static game where risk taking is the contestants’ only choice variable, and focus on the selection efficiency of contests as a function of the number of contestants and of the quality of the contestant pool. Two natural conjectures are that the selection efficiency improves with the quality of the contestant pool, and that the selection efficiency improves with the number of contestants. Tougher competition makes tougher winners. We show that neither conjecture necessarily holds true. The model we work with has two types of agents, a low type and a high type, each type with two possible pure strategies, safe and risky. The risky strategy induces a (not necessarily mean preserving) spread in the probability distribution of individual output compared to the safe strategy. For a given risk level, the high type’s output dominates the low type’s output. We focus on what seems to be the most natural measure of selection efficiency of a contest; the probability of a high type agent winning it (Π ). The main result is that Π may decrease with a pool of agents of higher quality, i.e., an increase in the share of high ability agents in the pool. To see the underlying intuition, notice that increasing the quality of the pool has two effects. The first is the statistical effect: a higher quality of the pool increases Π , holding the strategies of the types fixed. The second effect is the equilibrium effect: increasing the quality of the pool shifts the equilibrium of the game to one with increased risk taking. The latter effect may decrease Π . Thus we show that the statistical effect’s positive influence on Π may be dominated by the equilibrium effect’s negative influence on Π . Although it has often been argued that contests serve both motivation and selection functions (see, e.g., (Lazear and Rosen, 1981; Schlicht, 1988)), the tournament literature has mostly focused on the case with homogenous agents, where selection problems in the sense discussed here do not arise. Papers that do consider the case with heterogeneous agents restrict the discussion to how a tournament reward structure may motivate agents to work hard. One exception is Rosen (1986, Section V), which considers both the motivation function and the selection function of contests, in a setting where effort rather than risk taking is the choice variable. However, Rosen confines attention to the case where there is purely public information about types. Another exception is Meyer (1991), who considers the selection efficiency in a tournament where the agents act non-strategically.2 The efficiency of various selection procedures is a main topic in the statistical decision theory (see, e.g., (Gibbons et al., 1977)). By focusing on selection efficiency as the measure 1 Another example is the selection of fund managers in financial markets, where empirical studies show that investors tend to select fund managers with the highest rate of return previous year. Furthermore, these studies, e.g., Chevalier and Ellison (1997), show that competition for prospective investments has impact on fund managers’ risk taking. Kristiansen (2001) and Goriaev et al. (2001) model risk taking and selection in fund manager markets. 2 Bhattacharya and Guasch (1988) considers a different selection problem, in that the aim in their paper is to induce a high level of effort, by making each contest consist of homogenous contestants, rather than choosing the most able agents. Hvide (2002) and Prat and Palomino (2001) consider the interaction of effort and risk taking in a tournament.
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of the success of a contest, instead of, e.g., aggregate output, our work is in that sense closer to statistical decision theory than to the tournament literature. We should therefore note that the strategic element makes the noise in the selection process we study endogenous, while the noise in the selection processes studied by statistical decision theory is exogenous. Thus, the statistical decision theory literature only considers statistical effects, while we consider the interaction between statistical and equilibrium effects.3
2. The model We assume that the principal can only observe the rank of the outputs of the agents, and awards a prize to the agent with the highest rank. There are n risk-neutral agents competing for the prize, whose value is normalized to 1. The individual output space Z consists of four elements; Z := {z1 , z2 , z3 , z4 }, where z1 < z2 < z3 < z4 (tied winners have an equal chance of obtaining the prize). There are two types of agents, low (l) and high (h), with θ denoting the share of the h type in the pool from which the n agents are drawn. Both types are assumed to have an opportunity cost of participation equal to zero, and hence the group of contestants is a true random sample from the pool. Agents of each type have two pure strategies, safe (s) and risky (r). If an l type agent chooses s, her output is z2 with certainty. If an h agent chooses s, her output is z3 with certainty. If an l type agent chooses r, her output is z1 with probability 1 − x, and z4 with probability x. If an h type agent plays r, her output is z1 with probability 1 − y, and z4 with probability y, where y > x. Outputs are assumed to be statistically independent. As can easily be seen, the case when the distribution of output under r is a mean preserving spread of the distribution of output under s is a special case. We do not exclude mixed strategies, and thus the (mixed) strategy space has the usual continuity properties. We assume that there are no costs associated with risk taking, and hence the expected utility for an agent equals her win probability.4
3. Equilibrium analysis We consider the incomplete information game Γ (n, θ ), where an agent knows n and θ and her own type, but does not know the type of the other contestants. A strategy is a mapping from the type space T , where T := {l, h}, to the action space C, where C := {s, r}. We denote the set of symmetric pure strategies by S, where S := {(s, s), (s, r), 3 Dekel and Scotchmer (1999) find an evolutionary pressure towards risk-loving preferences provided that the breeding pattern is determined by a contest. The focus of Dekel and Scotchmer (1999) is very different from ours (there is, e.g., no discussion of selection efficiency), but the models applied are similar. In a related setting, Warneryd (2002) shows that types with different risk attitudes may coexist in a stable equilibrium. 4 Notice that the discrete output space restricts the possible risk taking, in that risk can only be increased by putting more probability weight on the endpoints z1 and z4 , something that would not be the case with a continuous output space. In the working paper version of the paper, Hvide and Kristiansen (2000), we consider the case where output is normally distributed, and where agents can choose the level of variance of their output. The results there are in line with the present results.
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(r, s), (r, r)}, with the l type’s action written first. We confine attention to symmetric Bayes–Nash equilibria (BNE), i.e., strategy tuples where all agents maximize their probability of winning given the strategy of the other agents, and where all agents of the same type play the same strategy. The key endogenous variable is the probability of an h type agent winning the prize in a BNE, denoted by Π(Γ ). 3.1. Quality of contestant pool To analyze the effect of increasing the quality of the contestant pool, we consider the case n = 2. Straightforward calculations reveal that there are unique equilibria, and moreover that all four elements of S can be equilibrium strategies depending on the values of the parameters (θ, x, y).5 Proposition 1. Contestant quality. (i) In a low-quality pool, a marginal increase in contestants’ average quality increases selection efficiency. (ii) In a medium-quality, pool a marginal increase in contestants’ average quality may decrease selection efficiency. (iii) In a high-quality pool, a marginal increase in contestants’ average quality increases selection efficiency. Proof. See the Appendix. ✷ Figure 1 illustrates the results in Proposition 1. Figure 1 depicts a typical example of Π as a function of θ , given by the bold line. For θ < θ 0 , the equilibrium is (s, s), and small increases in θ only induces a statistical effect on Π , implying that Π increases with θ . For a θ > θ 1 , the equilibrium is (r, s), and the same argument applies. For θ 0 θ < θ 1 , however, Π can decrease with θ . The movement A → C is the total effect on Π from increasing θ from θ 0 to θ 1 . The total effect can be decomposed into the statistical effect A → B, which is positive, and the equilibrium effect B → C, which is negative. 3.2. Number of contestants We now consider the effect on Π from increasing n. Example 1. Number of contestants. Π may decrease when the number of contestants increases from 2 to 3. 5 Recall that x (y) is the probability of an l (h) agent obtaining Z if she plays r. With both x and y large, y (r, r) is the equilibrium, which is natural. In the case where both x and y are small, (s, s) is the equilibrium. That seems counterintuitive since in that equilibrium an l agent loses with certainty if the other agent is an h type. The intuition behind the (s, s) equilibrium is that the probability of a l type winning against an h type (by playing r) is sufficiently small for the l type to rather care about her best chance of winning were she to play against another l type agent.
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Fig. 1. Improved contestant quality and selection efficience.
Proof. See the Appendix. ✷ Example 1 shows that adding a contestant, starting from a small n, may harm the selection efficiency, Π .6 When the number of agents is already large, then adding a player presumably has no equilibrium effect since both types play risky already. Intuitively, Π may decrease for increases from a small n, but must increase for increases starting from a large n. Example 2 shows that, starting with two contestants, the (positive) statistical effect from adding an infinite number of contestants may be dominated by the (negative) equilibrium effect. The example builds on a useful result from Dekel and Scotchmer (1999). Example 2. Π may be larger for 2 contestants than for an infinite number of contestants. Proof. Proposition 3 in (Dekel and Scotchmer, 1999) shows that there exists a finite n, denoted n*, such that for all n larger than n∗ , (r, r) is the unique equilibrium. It follows that (r, r) is the only equilibrium for an infinite number of contestants. Consequently, with an infinite number of contestants, the winner has output equal to z4 , with probability 1. By the law of large numbers, the share of h agents that achieve z4 is just y, and the share of l agents that achieve z4 is equal to x. Thus Π(∞) = θy/(θy + (1 − θ )x). Now consider θ = 12 , x = 15 , y = 14 . With those parameter values (see the Appendix for full derivation) we have Π(∞) = 59 < 34 = Π(2). ✷ To summarize, our main results are that Π can be non-monotone in n and in θ , due to the negative equilibrium effect of increases in n or θ on Π . These results were shown for n = 2. Since the non-monotonicity in θ is the most surprising result, in the working paper version of the paper we generalize it to hold for all n and θ . In other words, for all (n, θ ) 6 Note also that if a switch from a safe to a risky strategy yields a sufficiently large reduction in expected
output, an increase in the number of contestants (which induce more risk taking) may reduce expected aggregated output.
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there exists (x, y) such that a (small) increase in θ induces a decrease in Π . In the working paper, we also discuss whether a principal can solve the selection problem by modifying the contest, by constructing several different contests, or by increasing the number of prizes. The central message from that discussion is that these instruments can partially mitigate— but not solve—the selection problem in the context of the current model.
Acknowledgments We thank two referees, Tore Ellingsen, Terje Lensberg, Jim March, Tore Nilssen, Andrea Prat, seminar participants at EARIE 99, University of Bergen, and at the Norwegian School of Economics and Business. Hvide is grateful to the Maritime Program at NHH for financial support, and to the Eitan Berglas School of Economics in Tel Aviv for its kind hospitality. Appendix. Proofs We first derive BNE for n = 2. We use the following convention: Ui (j, k) denotes the win probability of an agent of type i when agents of her own type (including herself) play strategy j and agents of the other type play strategy k. For example, UH (s, r) denotes the win probability of an h agent when all h agents (including herself) play s, and all l agents play r. The individual payoffs in the symmetric tuples (when all agents of the same type choose the same strategies) are: 1 1 + (1 − θ )(y − x) , 2 1 UH (s, r) = 1 − θ − x + θ x, 2 1 UH (r, s) = θ + (1 − θ )y, 2 1 UH (s, s) = 1 − θ, 2 UH (r, r) =
1 (1 + θ x − θy), 2 1 UL (s, r) = (1 + θ ) − θy, 2 1 UL (r, s) = (1 − θ ) + xθ, 2 1 UL (s, s) = (1 − θ ). 2
UL (r, r) =
For individual deviations, we use the following convention: Ui (j, k) denotes the win probability of an agent of type i when she plays strategy −j , other agents of her own type play strategy j , and agents of the other type play strategy k. Since the payoff from letting −j be a mixed strategy is a convex combination of playing s and playing r, we only need to consider pure strategy deviations. For example, UH (s, r) denotes the win probability of an h agent playing r, when all other h agents play s, and all l agents play r. The individual payoffs from individual deviation are: UH (r, r) = θ (1 − y) + (1 − θ )(1 − x), 1 1 UH (s, r) = θy + (1 − θ ) xy + y(1 − x) + (1 − x)(1 − y) , 2 2 UH (r, s) = θ (1 − y) + (1 − θ ), UH (s, s) = y, UL (r, r) = θ (1 − y) + (1 − θ )(1 − x), 1 1 UL (s, r) = θ xy + x(1 − y) + (1 − x)(1 − y) + (1 − θ )x, 2 2 UL (r, s) = (1 − θ )(1 − x), UL (s, s) = x.
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First consider equilibrium (r, r). Notice that the payoff from individual deviation is the same for an h agent and an l agent, and moreover that UH (r, r) > UL (r, r). Thus we only have to check a deviation from an l agent. An l agent follows the supposed equilibrium strategy if 12 (1 − θy + θ x) > θ (1 − y) + (1 − θ )(1 − x), which implies that y > (1 + θ x − 2x)/θ . Now consider equilibrium (s, s). An l agent follows the supposed equilibrium strategy if x < 12 (1 − θ ). The condition for an h agent is y < 1 − 12 θ . Now consider equilibrium (r, s). An lagent follows the supposed equilibrium strategy if 12 (1 − θ ) + xθ > (1 − θ )(1 − x), which implies that x > 12 (1 − θ ). The condition for the h type is 12 (1 + (1 − θ )(y − x)) > θy + (1 − θ )( 12 xy + y(1 − x) + 12 (1 − x)(1 − y)), which implies that y < 12 . Finally, consider equilibrium (s, r). An l agent sticks if 12 (1 + θ ) − θy > θ ( 12 xy + x(1 − y) + 12 (1 − x)(1 − y)) + (1 − θ )x, which implies that x < (1 − θy)/(2 − θ ). The condition for the h type is 12 θ + (1 − θ )y > θ (1 − y) + (1 − θ ), which implies that y > 1 − 12 θ . Uniqueness of BNE, given (x, y, θ ), follows directly from the argument. Proof of Proposition 1. We first prove (i) and (iii) and then prove (ii). (i) From the equilibrium structure derived above, we have that for θ < min[1 − 2x, 2 − 2y], (s, s) is a unique equilibrium strategy. As can easily be verified, Π(s, s) = 1 − (1 − θ )2 , increases with θ . (iii) From above, we have that for θ > (1 − 2x)/(y − x), (r, r) is a unique equilibrium. As can be easily verified, Π(r, r) = θ 2 + 2(1 − θ )θ (y(1 − x) + 12 xy + 12 (1 − x)(1 − y)), which increases in θ . (ii) We show that if min[1 − 2x, 2 − 2y] < θ < (1 − 2x)/(y − x), Π can decrease in θ . Consider θ = 1 − 2x < 2 − 2y. In this case, (s, s) is the equilibrium. A small increase in θ induces the equilibrium to switch to (s, r), and Π decreases, as can easily be verified. ✷ Proof of Example 1. Note that if n = 2, θ = 12 , x = 15 , y = 14 , then (s, s) is the unique BNE. That gives Π(n = 2, θ = 12 ) = θ 2 + 2θ (1 − θ ) = 34 = 150/200. Now increase n to 3. In that case, (s, s) is no longer a BNE since 1 1 1 1 2 = UL (s, s) = 1− < UL (s, s) = . 3 2 12 5 However, (r, s) is indeed the BNE since (a) UL (r, s) = 67/300 > UL (r, s) = 48/300. While on the other hand, (b) UH (s, r) = 532/1200 > UH (s, r) = 319/1200. Thus Π decreases: 1 97 150 = θ 3 + 2θ 2 (1 − θ )(1 − x) + 2θ (1 − θ )2 (1 − x)2 = Π n = 3, θ = < . 2 200 200
✷
References Bhattacharya, S., Guasch, J.L., 1988. Heterogeneity, tournaments, and hierarchies. J. Polit. Economy 96, 867– 881. Chevalier, J., Ellison, G., 1997. Risk taking by mutual funds as a response to incentives. J. Polit. Economy 105, 1167–1200. Dekel, E., Scotchmer, S., 1999. On the evolution of attitudes towards risk in winner-take-all games. J. Econ. Theory 87, 125–143. Gibbons, J.D., Olkin, I., Sobel, M., 1977. Selecting and Ordering Populations: A New Statistical Methodology. Wiley, New York. Goriaev, A., Palomino, F., Prat, A., 2001. Dynamic incentives in the money management tournament, Working paper, London School of Economics. Hvide, H.K., 2002. Tournament rewards and risk taking. J. Lab. Econ. 20, 877–898. Hvide, H.K., Kristiansen, E.G., 2000. Risk taking in selection contests. Working paper, Norwegian School of Economics and Business Administration.
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Kristiansen, E.G., 2001. Agency, performance, and selection: The market for fund managers. Working paper, Norwegian School of Economics and Business Administration. Lazear, E., Rosen, S., 1981. Rank-order tournaments as optimum labor contracts. J. Polit. Economy 89, 841–864. Meyer, M.A., 1991. Learning from coarse information: Biased contests and career profiles. Rev. Econ. Stud. 58, 15–41. Prat, A., Palomino, F., 2001. Risk taking and optimal contracts for money managers. Working paper, London School of Economics. Rosen, S., 1986. Prizes and incentives in elimination tournaments. Amer. Econ. Rev. 76, 701–715. Schlicht, E., 1988. Promotions, elections, and other contests: A comment. J. Inst. Theoretical Econ. 144, 94–99. Warneryd, K., 2002. Rent, risk, and replication: Preference adaptation in winner-take-all markets. Games Econ. Behav., forthcoming.
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Risk taking in selection contests Hans K. Hvide a,∗ and Eirik G. Kristiansen b a Department of Finance, Norwegian School of Economics and Business, Hellev. 30, 5045 Bergen, Norway b Department of Economics, Norwegian School of Economics and Business and Norges Bank
(The Central Bank of Norway), Hellev. 30, 5045 Bergen, Norway Received 5 October 1999
Abstract We study contests where the strategic variable is the degree of risk taking rather than the amount of effort. The selection efficiency of such contests is examined. We show that the selection efficiency of a contest may be improved by limiting the competition in two ways; by having a small number of contestants, and by restricting contestant quality. 2002 Elsevier Science (USA). All rights reserved. JEL classification: C44; D29; D83; J41 Keywords: Contest; Risk taking; Selection; Tournament
1. Introduction In a contest or a tournament, rewards are based on the relative performance of the contestants. Contests serve two different purposes. First, tournaments can provide incentives to work hard, as in (Lazear and Rosen, 1981). Second, tournaments can serve as a selection mechanism. In this paper we focus on the selection role of contests, in the case where risk taking is the strategic variable of the contestants. Employees involved in a promotion process or tenure process, for example, may choose tasks that differ in risk profile to show off their abilities. We investigate the selection efficiency of contests in which the contestants optimize their choice of risk, given the risk taking of others. Who will come out on top, bad types or * Corresponding author.
E-mail addresses: [email protected] (H.K. Hvide), [email protected] (E.G. Kristiansen). 0899-8256/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved. doi:10.1016/S0899-8256(02)00538-9
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good types? In what way will the selection efficiency depend on, for example, the quality of the contestant pool? Answering such questions can be important, e.g., to understand the efficiency of promotion processes in firms.1 We confine ourselves to a simple static game where risk taking is the contestants’ only choice variable, and focus on the selection efficiency of contests as a function of the number of contestants and of the quality of the contestant pool. Two natural conjectures are that the selection efficiency improves with the quality of the contestant pool, and that the selection efficiency improves with the number of contestants. Tougher competition makes tougher winners. We show that neither conjecture necessarily holds true. The model we work with has two types of agents, a low type and a high type, each type with two possible pure strategies, safe and risky. The risky strategy induces a (not necessarily mean preserving) spread in the probability distribution of individual output compared to the safe strategy. For a given risk level, the high type’s output dominates the low type’s output. We focus on what seems to be the most natural measure of selection efficiency of a contest; the probability of a high type agent winning it (Π ). The main result is that Π may decrease with a pool of agents of higher quality, i.e., an increase in the share of high ability agents in the pool. To see the underlying intuition, notice that increasing the quality of the pool has two effects. The first is the statistical effect: a higher quality of the pool increases Π , holding the strategies of the types fixed. The second effect is the equilibrium effect: increasing the quality of the pool shifts the equilibrium of the game to one with increased risk taking. The latter effect may decrease Π . Thus we show that the statistical effect’s positive influence on Π may be dominated by the equilibrium effect’s negative influence on Π . Although it has often been argued that contests serve both motivation and selection functions (see, e.g., (Lazear and Rosen, 1981; Schlicht, 1988)), the tournament literature has mostly focused on the case with homogenous agents, where selection problems in the sense discussed here do not arise. Papers that do consider the case with heterogeneous agents restrict the discussion to how a tournament reward structure may motivate agents to work hard. One exception is Rosen (1986, Section V), which considers both the motivation function and the selection function of contests, in a setting where effort rather than risk taking is the choice variable. However, Rosen confines attention to the case where there is purely public information about types. Another exception is Meyer (1991), who considers the selection efficiency in a tournament where the agents act non-strategically.2 The efficiency of various selection procedures is a main topic in the statistical decision theory (see, e.g., (Gibbons et al., 1977)). By focusing on selection efficiency as the measure 1 Another example is the selection of fund managers in financial markets, where empirical studies show that investors tend to select fund managers with the highest rate of return previous year. Furthermore, these studies, e.g., Chevalier and Ellison (1997), show that competition for prospective investments has impact on fund managers’ risk taking. Kristiansen (2001) and Goriaev et al. (2001) model risk taking and selection in fund manager markets. 2 Bhattacharya and Guasch (1988) considers a different selection problem, in that the aim in their paper is to induce a high level of effort, by making each contest consist of homogenous contestants, rather than choosing the most able agents. Hvide (2002) and Prat and Palomino (2001) consider the interaction of effort and risk taking in a tournament.
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of the success of a contest, instead of, e.g., aggregate output, our work is in that sense closer to statistical decision theory than to the tournament literature. We should therefore note that the strategic element makes the noise in the selection process we study endogenous, while the noise in the selection processes studied by statistical decision theory is exogenous. Thus, the statistical decision theory literature only considers statistical effects, while we consider the interaction between statistical and equilibrium effects.3
2. The model We assume that the principal can only observe the rank of the outputs of the agents, and awards a prize to the agent with the highest rank. There are n risk-neutral agents competing for the prize, whose value is normalized to 1. The individual output space Z consists of four elements; Z := {z1 , z2 , z3 , z4 }, where z1 < z2 < z3 < z4 (tied winners have an equal chance of obtaining the prize). There are two types of agents, low (l) and high (h), with θ denoting the share of the h type in the pool from which the n agents are drawn. Both types are assumed to have an opportunity cost of participation equal to zero, and hence the group of contestants is a true random sample from the pool. Agents of each type have two pure strategies, safe (s) and risky (r). If an l type agent chooses s, her output is z2 with certainty. If an h agent chooses s, her output is z3 with certainty. If an l type agent chooses r, her output is z1 with probability 1 − x, and z4 with probability x. If an h type agent plays r, her output is z1 with probability 1 − y, and z4 with probability y, where y > x. Outputs are assumed to be statistically independent. As can easily be seen, the case when the distribution of output under r is a mean preserving spread of the distribution of output under s is a special case. We do not exclude mixed strategies, and thus the (mixed) strategy space has the usual continuity properties. We assume that there are no costs associated with risk taking, and hence the expected utility for an agent equals her win probability.4
3. Equilibrium analysis We consider the incomplete information game Γ (n, θ ), where an agent knows n and θ and her own type, but does not know the type of the other contestants. A strategy is a mapping from the type space T , where T := {l, h}, to the action space C, where C := {s, r}. We denote the set of symmetric pure strategies by S, where S := {(s, s), (s, r), 3 Dekel and Scotchmer (1999) find an evolutionary pressure towards risk-loving preferences provided that the breeding pattern is determined by a contest. The focus of Dekel and Scotchmer (1999) is very different from ours (there is, e.g., no discussion of selection efficiency), but the models applied are similar. In a related setting, Warneryd (2002) shows that types with different risk attitudes may coexist in a stable equilibrium. 4 Notice that the discrete output space restricts the possible risk taking, in that risk can only be increased by putting more probability weight on the endpoints z1 and z4 , something that would not be the case with a continuous output space. In the working paper version of the paper, Hvide and Kristiansen (2000), we consider the case where output is normally distributed, and where agents can choose the level of variance of their output. The results there are in line with the present results.
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(r, s), (r, r)}, with the l type’s action written first. We confine attention to symmetric Bayes–Nash equilibria (BNE), i.e., strategy tuples where all agents maximize their probability of winning given the strategy of the other agents, and where all agents of the same type play the same strategy. The key endogenous variable is the probability of an h type agent winning the prize in a BNE, denoted by Π(Γ ). 3.1. Quality of contestant pool To analyze the effect of increasing the quality of the contestant pool, we consider the case n = 2. Straightforward calculations reveal that there are unique equilibria, and moreover that all four elements of S can be equilibrium strategies depending on the values of the parameters (θ, x, y).5 Proposition 1. Contestant quality. (i) In a low-quality pool, a marginal increase in contestants’ average quality increases selection efficiency. (ii) In a medium-quality, pool a marginal increase in contestants’ average quality may decrease selection efficiency. (iii) In a high-quality pool, a marginal increase in contestants’ average quality increases selection efficiency. Proof. See the Appendix. ✷ Figure 1 illustrates the results in Proposition 1. Figure 1 depicts a typical example of Π as a function of θ , given by the bold line. For θ < θ 0 , the equilibrium is (s, s), and small increases in θ only induces a statistical effect on Π , implying that Π increases with θ . For a θ > θ 1 , the equilibrium is (r, s), and the same argument applies. For θ 0 θ < θ 1 , however, Π can decrease with θ . The movement A → C is the total effect on Π from increasing θ from θ 0 to θ 1 . The total effect can be decomposed into the statistical effect A → B, which is positive, and the equilibrium effect B → C, which is negative. 3.2. Number of contestants We now consider the effect on Π from increasing n. Example 1. Number of contestants. Π may decrease when the number of contestants increases from 2 to 3. 5 Recall that x (y) is the probability of an l (h) agent obtaining Z if she plays r. With both x and y large, y (r, r) is the equilibrium, which is natural. In the case where both x and y are small, (s, s) is the equilibrium. That seems counterintuitive since in that equilibrium an l agent loses with certainty if the other agent is an h type. The intuition behind the (s, s) equilibrium is that the probability of a l type winning against an h type (by playing r) is sufficiently small for the l type to rather care about her best chance of winning were she to play against another l type agent.
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Fig. 1. Improved contestant quality and selection efficience.
Proof. See the Appendix. ✷ Example 1 shows that adding a contestant, starting from a small n, may harm the selection efficiency, Π .6 When the number of agents is already large, then adding a player presumably has no equilibrium effect since both types play risky already. Intuitively, Π may decrease for increases from a small n, but must increase for increases starting from a large n. Example 2 shows that, starting with two contestants, the (positive) statistical effect from adding an infinite number of contestants may be dominated by the (negative) equilibrium effect. The example builds on a useful result from Dekel and Scotchmer (1999). Example 2. Π may be larger for 2 contestants than for an infinite number of contestants. Proof. Proposition 3 in (Dekel and Scotchmer, 1999) shows that there exists a finite n, denoted n*, such that for all n larger than n∗ , (r, r) is the unique equilibrium. It follows that (r, r) is the only equilibrium for an infinite number of contestants. Consequently, with an infinite number of contestants, the winner has output equal to z4 , with probability 1. By the law of large numbers, the share of h agents that achieve z4 is just y, and the share of l agents that achieve z4 is equal to x. Thus Π(∞) = θy/(θy + (1 − θ )x). Now consider θ = 12 , x = 15 , y = 14 . With those parameter values (see the Appendix for full derivation) we have Π(∞) = 59 < 34 = Π(2). ✷ To summarize, our main results are that Π can be non-monotone in n and in θ , due to the negative equilibrium effect of increases in n or θ on Π . These results were shown for n = 2. Since the non-monotonicity in θ is the most surprising result, in the working paper version of the paper we generalize it to hold for all n and θ . In other words, for all (n, θ ) 6 Note also that if a switch from a safe to a risky strategy yields a sufficiently large reduction in expected
output, an increase in the number of contestants (which induce more risk taking) may reduce expected aggregated output.
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there exists (x, y) such that a (small) increase in θ induces a decrease in Π . In the working paper, we also discuss whether a principal can solve the selection problem by modifying the contest, by constructing several different contests, or by increasing the number of prizes. The central message from that discussion is that these instruments can partially mitigate— but not solve—the selection problem in the context of the current model.
Acknowledgments We thank two referees, Tore Ellingsen, Terje Lensberg, Jim March, Tore Nilssen, Andrea Prat, seminar participants at EARIE 99, University of Bergen, and at the Norwegian School of Economics and Business. Hvide is grateful to the Maritime Program at NHH for financial support, and to the Eitan Berglas School of Economics in Tel Aviv for its kind hospitality. Appendix. Proofs We first derive BNE for n = 2. We use the following convention: Ui (j, k) denotes the win probability of an agent of type i when agents of her own type (including herself) play strategy j and agents of the other type play strategy k. For example, UH (s, r) denotes the win probability of an h agent when all h agents (including herself) play s, and all l agents play r. The individual payoffs in the symmetric tuples (when all agents of the same type choose the same strategies) are: 1 1 + (1 − θ )(y − x) , 2 1 UH (s, r) = 1 − θ − x + θ x, 2 1 UH (r, s) = θ + (1 − θ )y, 2 1 UH (s, s) = 1 − θ, 2 UH (r, r) =
1 (1 + θ x − θy), 2 1 UL (s, r) = (1 + θ ) − θy, 2 1 UL (r, s) = (1 − θ ) + xθ, 2 1 UL (s, s) = (1 − θ ). 2
UL (r, r) =
For individual deviations, we use the following convention: Ui (j, k) denotes the win probability of an agent of type i when she plays strategy −j , other agents of her own type play strategy j , and agents of the other type play strategy k. Since the payoff from letting −j be a mixed strategy is a convex combination of playing s and playing r, we only need to consider pure strategy deviations. For example, UH (s, r) denotes the win probability of an h agent playing r, when all other h agents play s, and all l agents play r. The individual payoffs from individual deviation are: UH (r, r) = θ (1 − y) + (1 − θ )(1 − x), 1 1 UH (s, r) = θy + (1 − θ ) xy + y(1 − x) + (1 − x)(1 − y) , 2 2 UH (r, s) = θ (1 − y) + (1 − θ ), UH (s, s) = y, UL (r, r) = θ (1 − y) + (1 − θ )(1 − x), 1 1 UL (s, r) = θ xy + x(1 − y) + (1 − x)(1 − y) + (1 − θ )x, 2 2 UL (r, s) = (1 − θ )(1 − x), UL (s, s) = x.
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First consider equilibrium (r, r). Notice that the payoff from individual deviation is the same for an h agent and an l agent, and moreover that UH (r, r) > UL (r, r). Thus we only have to check a deviation from an l agent. An l agent follows the supposed equilibrium strategy if 12 (1 − θy + θ x) > θ (1 − y) + (1 − θ )(1 − x), which implies that y > (1 + θ x − 2x)/θ . Now consider equilibrium (s, s). An l agent follows the supposed equilibrium strategy if x < 12 (1 − θ ). The condition for an h agent is y < 1 − 12 θ . Now consider equilibrium (r, s). An lagent follows the supposed equilibrium strategy if 12 (1 − θ ) + xθ > (1 − θ )(1 − x), which implies that x > 12 (1 − θ ). The condition for the h type is 12 (1 + (1 − θ )(y − x)) > θy + (1 − θ )( 12 xy + y(1 − x) + 12 (1 − x)(1 − y)), which implies that y < 12 . Finally, consider equilibrium (s, r). An l agent sticks if 12 (1 + θ ) − θy > θ ( 12 xy + x(1 − y) + 12 (1 − x)(1 − y)) + (1 − θ )x, which implies that x < (1 − θy)/(2 − θ ). The condition for the h type is 12 θ + (1 − θ )y > θ (1 − y) + (1 − θ ), which implies that y > 1 − 12 θ . Uniqueness of BNE, given (x, y, θ ), follows directly from the argument. Proof of Proposition 1. We first prove (i) and (iii) and then prove (ii). (i) From the equilibrium structure derived above, we have that for θ < min[1 − 2x, 2 − 2y], (s, s) is a unique equilibrium strategy. As can easily be verified, Π(s, s) = 1 − (1 − θ )2 , increases with θ . (iii) From above, we have that for θ > (1 − 2x)/(y − x), (r, r) is a unique equilibrium. As can be easily verified, Π(r, r) = θ 2 + 2(1 − θ )θ (y(1 − x) + 12 xy + 12 (1 − x)(1 − y)), which increases in θ . (ii) We show that if min[1 − 2x, 2 − 2y] < θ < (1 − 2x)/(y − x), Π can decrease in θ . Consider θ = 1 − 2x < 2 − 2y. In this case, (s, s) is the equilibrium. A small increase in θ induces the equilibrium to switch to (s, r), and Π decreases, as can easily be verified. ✷ Proof of Example 1. Note that if n = 2, θ = 12 , x = 15 , y = 14 , then (s, s) is the unique BNE. That gives Π(n = 2, θ = 12 ) = θ 2 + 2θ (1 − θ ) = 34 = 150/200. Now increase n to 3. In that case, (s, s) is no longer a BNE since 1 1 1 1 2 = UL (s, s) = 1− < UL (s, s) = . 3 2 12 5 However, (r, s) is indeed the BNE since (a) UL (r, s) = 67/300 > UL (r, s) = 48/300. While on the other hand, (b) UH (s, r) = 532/1200 > UH (s, r) = 319/1200. Thus Π decreases: 1 97 150 = θ 3 + 2θ 2 (1 − θ )(1 − x) + 2θ (1 − θ )2 (1 − x)2 = Π n = 3, θ = < . 2 200 200
✷
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