Paper.dvi

The Contest Winner: Gifted or Venturesome? Otto von Guericke Universit¨at Magdeburg and CESifo We examine the chance of winning a contest when participants dif- fer in both their talent and their attitude towards risk. For the case ofCARA preferences, we show that the agent’s probability of winning isincreasing (decreasing) in the own (opponent’s) skill level but decreas-ing (increasing) in the own (opponent’s) degree of risk aversion. Weconclude that the chance of winning may be higher for a low-skilledagent with a low degree of risk aversion than for a high-skilled agentwith a high degree of risk-aversion. In many circumstances, especiallyin selection contests, such an outcome is undesirable.
Keywords: Selection Contest, Asymmetric Players, Risk Aver- 80333 Munich, Germany, Tel.: +49–89–289–25707, Fax: +49–89–289–25701, E-mail:[email protected].
May the best man win! Following this dictum, in many areas of life con-tests are used with the objective of selecting the most able. In business,entrepreneurs conduct job interviews or assessment centers in order to hireor promote the agent with the highest skills. In politics, elections are runwhere the candidates try to convince the voters of their abilities for holdingoffice. In sports, athletes compete against each other in order to single outthe strongest one.
But is it really the most gifted one who has the best chance of winning such a selection contest? Is the winner of the famous Tour de France reallythe most talented cyclist or simply the guy who fears least the consequencesof doping? Is the president-elect indeed best suited for holding office orjust the one who had the guts to invest more money into the campaign?Seemingly, an agent’s success during a contest does not depend on her skillsonly but also on her attitude towards risk.
Despite this observation, the economic literature on asymmetric contests has mainly focussed on models where agents differ in just one dimension. Inparticular, the influence of skill heterogeneity and asymmetric risk aversionon the probability of winning a contest have not yet been analyzed simultane-ously but only apart. On the one hand, Baik (1994) shows that, with respectto skills, the more talented agent has, ceteris paribus, the higher probabilityof winning the contest. On the other hand, authors like Skaperdas and Gan(1995) or Cornes and Hartley (2003) demonstrate for the case of constantabsolute risk aversion that the less risk averse agent invests, ceteris paribus,more and has a higher probability of winning.
Given these results, now consider a contest between agent A, who is high skilled but highly risk averse, and agent B, who is low skilled but barely riskaverse. Two natural questions arise: i) Who has the higher probability ofwinning the contest? ii) Who spends more effort? To study this topic more closely, we engage a simple model of a two-person contest where agents differ in both their skill levels and their degree of con-stant absolute risk aversion.1 We show that the agent’s probability of winningis increasing (decreasing) in the own (opponent’s) skill level but decreasing(increasing) in the own (opponent’s) degree of risk aversion. Accordingly,the above questions cannot be answered generally without ambiguity. Inparticular, there will be cases in which the less able/risk-averse agent spendsmore effort and has a higher probability of winning the contest than the more 1 Note the difference to the model of Hvide (2002), where both players have the same attitude towards risk ex-ante but where risk taking is a strategic variable that is endoge-nously determined in equilibrium of the contest.
Usually, in selection contests, such an outcome is undesirable.2 The de- signer of the contest should therefore take measures in order to reduce theriskiness of the contest if she does not want to discriminate against risk averseagents. However, this is not a trivial task. As we will show, for example, it isa priori not clear whether the contest prize should be increased or decreasedin order to improve the situation.
The remainder of this paper is organized as follows: The formal model is set up in Section 2 and used to derive the main results in Section 3. Section4 illustrates the results and the tradeoff between talent and attitude towardsrisk by means of some numerical examples. Finally, possible extensions andscope for further research are discussed in Section 5.
In this section, we first introduce the basic assumptions we use in our analysisof the contest game. We then consider the individual maximization problemsand derive a general condition characterizing the equilibrium of this game.
There are N agents participating in a winner-take-all contest competing forsome rent R > 0. Each agent i ∈ {1, . . . , N } has an initial wealth endowmentIi and can spend some resources xi ∈ [0, Ii] in order to improve her probabilityof winning pi. This probability is determined by the following contest successfunction (CSF): is an increasing concave function of xi satisfying fi(0) = 0. For the sake of concreteness and ease of calculation we assumeN = 2 and where θi > 0 is a parameter expressing agent i’s skill level. We also referto θi as agent i’s talent for the task required within the contest. Note thatequation (2) states, reasonably enough, a complementarity between talentand effort, which is standard in the related literature (e.g. Baik, 1994).
2 In the context of sales contests, Bono (2008) characterizes a situation where it might be desirable to promote less risk averse managers, since they exert, ceteris paribus, moreeffort.
Introducing risk aversion into the analysis of contests we follow the ap- proach proposed by Skaperdas and Gan (1995) and Cornes and Hartley(2003), respectively and assume that the preferences of agent i can be ex-pressed by the following utility function which exhibits constant absolute riskaversion (CARA): where αi is agent i’s constant degree of absolute risk aversion. We includethe limit case of a risk-neutral player i with ui(Wi) = Wi into the analysis,and refer to this situation as one in which αi = 0.
Individual objectives and equilibrium conditions The contest is organized as a Cournot-Game. The players simultaneouslychoose their effort levels xi in order to maximize their expected utility Euifrom consumption Wi, which equals Ii − xi + R if agent i wins the contestand Ii − xi otherwise. Hence, for i, j ∈ {1, 2}, i = j, Eui = piui(Ii − xi + R) + (1 − pi)ui(Ii − xi) It is easily verified that β is an increasing function of α and δ is a decreasingfunction of α (Skaperdas and Gan, 1995, supplementary appendix to Propo-sition 2). The first order condition (FOC) for an interior solution of agenti’s maximization problem yields Equation (8) implicitly defines the reaction function of agent i, i.e. her opti-mal effort xi as a function of the opponent’s effort xj. Under the assumptions made, the rent seeking game has a unique Nash equilibrium in pure strate-gies, as shown by Cornes and Hartley (2003, Propostioin 3.3) and Yamazaki(2008), respectively. In order to compute this equilibrium under differentparameter constellations, we divide the FOC of agent 1 by the FOC of agent2, and note that p′1 = x2 . Denoting by q := p2 = θ2x2 the competitive balance Equation (9) can be transformed into a quadratic equation for the compet-itive balance in equilibrium; as θ2δ(α1) > 0, only the positive root yields a A value q < 1 indicates that agent 1’s probability of winning exceeds the oneof agent 2, i.e. p1 > p2, and vice versa for q > 1.
In this section, we compare the outcome of the contest for different parameterconstellations and derive some insightful comparative static results. We startwith the case of symmetric, i.e. identical, players as a benchmark. We thenanalyze successively how the outcome changes if we introduce heterogeneitybetween the agents with respect to their skills only, their risk attitude only,and both skill and risk aversion.
In this subsection, we derive the equilibrium of the contest as well as itscomparative statics properties for the case of identical players.
Proposition 1 Suppose θ1 = θ2 = θ > 0 and α1 = α2 = α ≥ 0. (a) The equilibrium is symmetric with equal winning probabilities p∗1 = p∗2 = 1 (i.e. q = 1) and effort levels, which do not depend on the skill level 3 Some authors use the difference in winning probabilities as an alternative measure of ‘competitive balance’ or ‘closeness’ of the contest (see e.g. Runkel, 2006a,b).
(b) The effort levels are increasing in the prize: ∂x∗sym > 0. (c) The effort levels are decreasing in the degree of risk aversion: ∂x∗sym < 0. The proof can be found in Appendix A. Parts (a) and (b) of Proposition1 are straightforward generalizations of the respective results in the case ofrisk-neutral players (Baik, 1994) and, as such, very intuitive: The intensityof competition among equal competitors does not depend on the (skill) levelthe contest takes place at, but is positively related to the rent offered.
Part (c) of Proposition 1 resolves the general ambiguity result of Konrad and Schlesinger (1997) for the case of preferences with CARA. Note thata higher degree of risk aversion has two opposing effects on the individualinvestment decision. On the one hand, there is the so called gambling effect:Since participation in the contest comes along with an uncertain payment, itmay be regarded as a lottery, which the agents invest the less into the morerisk averse they are. On the other hand, there is a so called effect of self-protection: By spending more effort, the players can reduce their probabilityof loosing the contest. Therefore, the more risk averse they are the morethey invest. Under our assumptions, the gambling effect outweighs the effectof self protection.
Traditionally, the literature conducts the comparative statics with respect to the dissipation rate, which is defined as the fraction ρ of the rent that is ‘wasted’ in form of aggregate effort, i.e. ρ := Corollary 1 For θ1 = θ2 = θ > 0 and α1 = α2 = α > 0, the equilibriumrent dissipation rate equals and is decreasing in both the prize R and the degree of risk aversion α. The proof can be found in Appendix A. As shown by Hillman and Samet(1987) in a more general framework, less than the full rent will be dissipatedif agents are risk averse. The comparative statics of Corollary 1 are in linewith the simulations run by Hillman and Katz (1984) for the case of logarith-mic utilities and with the results in Long and Vousden (1987) for contests with rents that are divisible among agents.4 Put differently, for the caseof preferences with CARA, the intuition holds that less of the rent will bewasted if the agents are more risk averse or if the stakes are higher.
In this subsection, we derive the equilibrium of the contest as well as itscomparative statics properties for the case of players with the same degreeof risk aversion but different skill levels.
Proposition 2 Suppose, without loss of generality, θ1 > θ2 > 0 and α1 =α2 = α ≥ 0. i.e. the player with the higher skill level has the better probability ofwinning, p∗1 > p∗2. (b) The higher the agent’s skill level, the better her probability of winning, i.e. ∂q < 0 < ∂q . (i) Under risk neutrality (α = 0), competitive balance is independent of the prize, i.e. ∂q = 0. (ii) In the case of risk aversion (α > 0), the higher the prize, the better the winning probability of the more talented agent, i.e. ∂q < 0. (d) The higher the degree of risk aversion, the better the winning probability of the more talented agent, i.e. ∂q < 0. The proof can be found in Appendix A. Parts (a) and (b) of Proposition2 are straightforward generalizations of the respective results in the caseof risk-neutral players (Baik, 1994) and, as such, very intuitive: The moretalented agent has the better chance of winning and this probability is thehigher the larger the gap in skills is. Part (c) of Proposition 2 shows that the 4 They contrast, though, to the diametric result in Fabella (1992). However, Konrad and Schlesinger (1997, footnote 11) report that his “result is not correct as the papercontains several serious errors”.
neutrality result for risk neutral agents (Runkel, 2006a, Proposition 1 (b))does not hold for risk averse players: A higher prize increases the ‘riskiness’of the contest which is worse for the less skilled agent being more likely toloose. A similar intuition also drives the result in Part (d) of Proposition 2.
Can there also be said something about the equilibrium effort levels? If the agents are risk neutral, they will exert the same effort level x∗1 = x∗2 = in equilibrium, which will be maximal for equal skills θ 1994). Under risk aversion, however, the equilibrium effort levels will differif and only if agents differ in skills. To see this, consider the relative effortξ := x∗2 = θ1 q in equilibrium; then the following statements hold.
Corollary 2 Suppose, without loss of generality, θ1 > θ2 > 0 and α1 = α2 =α > 0. (a) The higher the skill level of agent i the higher her relative equilibrium effort, i.e. ∂ξ < 0 < ∂ξ . (b) In equilibrium, the agent with the higher skill level exerts more effort, The proof can be found in Appendix A. The results confirm the comple-mentary character of effort and skill and back up the much cited anecdotalevidence for talent to come along with diligence.
In this subsection, we derive the equilibrium of the contest as well as itscomparative statics properties for the case of players with the same skilllevel but different degrees of risk aversion.
Proposition 3 Suppose, without loss of generality, θ1 = θ2 = θ > 0 andα1 > α2 ≥ 0. (b) For small differences in risk aversion, the player with the higher degree of risk aversion exerts less effort and has the smaller probability ofwinning, i.e. α1 = α2 + ε, with ε > 0 sufficiently small, implies x∗1 < x∗2and p∗1 < p∗2. (c) The higher the prize, the smaller the winning probability of the more risk averse agent, i.e. ∂q > 0. The proof can be found in Appendix A. Part (b) of Proposition 3 reproducesthe respective results in Skaperdas and Gan (1995, Proposition 2b), Cornesand Hartley (2003, Proposition 3.4), and Bono (2008, Proposition 1) for thespecific framework at hand. The intuition here is similar to the correspondingresult in the symmetric equilibrium. Since, under the assumptions made,the gambling effect outweighs the effect of self-protection, the less risk averseagent will spend more effort and thus have a higher chance of winning thecontest. An analogous reasoning explains the result of Part (c). An increasingprize raises the ‘riskiness’ of the contest which is worse for the more risk averseagent.
In this subsection, we argue that the agent with the higher skill level mightnevertheless be very likely to lose the contest if, at the same time, she ex-hibits a higher degree of risk aversion. Such an outcome often is undesirable,especially in the case of a selection contest. A selection contest aims at find-ing out who is most productive in fulfilling a certain task and, hence, wantsto rank the agents according to their skills rather than attitude towards risk.
Examples range from job interviews to tournaments in sports with its dictum:May the best man win! Part (a) of Proposition 2 shows that for a given equal degree of risk aver- sion agent 1, with the higher skill level, has a higher chance of winning. Part(b) of Proposition 3 states that, starting from the symmetric equilibrium,agent 1 has a smaller winning probability if his degree of risk aversion in-creases marginally. Therefore, by the continuity of the relevant functions,the implicit functions theorem implies the following Corollary 3 There exists a set of parameter values with θ1 > θ2 and α1 > α2such that q > 1, i.e. p∗1 < p∗2. Corollary 3 characterizes a situation where the presence of heterogeneity withrespect to the agents’ attitude towards risk induces some kind of failure ofthe contest: The probability of selecting the agent with the lower skills ishigher than the probability of selecting the agent with the higher skills. Putdifferently, the venturesome has a good chance of beating the gifted.
Moreover, note that a higher prize on the one hand increases the chance of winning for the agent with the higher skill level (Proposition 2(c)), but onthe other hand decreases the chance of winning for the agent with the higher degree of risk aversion (Proposition 3(c)). Consequently, from the viewpointof contest design, it is a priori not clear how the prize of the contest should bechosen in order not to end up in such an undesirable situation characterizedby Corollary 3. The numerical examples presented in the next section willclarify these remarks.
In this section we illustrate our findings by means of some numerical exam-ples.
First, we provide an example for a situation in which the chance of winningis higher for a low-skilled agent with a low degree of risk aversion (the ven-turesome) than for a high-skilled agent with a high degree of risk-aversion(the gifted).
Example 1 Suppose θ1 = 10, θ2 = 9, α1 = 1, α2 = 1. Accordingly, player 1 (the gifted) is more talented but also more risk averse than player 2 (theventuresome). (a) For R = ln(4) we compute q = 1, i.e. the gifted and the venturesome have the same probability of winning the contest. (b) For R = ln(9) we compute q = 1161+21 > 1, i.e. the gifted has a lower probability of winning the contest than the venturesome. Furthermore, the example in the previous subsection shows that an increasingcontest prize R may increase the winning probability of the venturesome. Inthis subsection, however, we provide an example for a situation, in which theopposite is true.
Example 2 Suppose θ1 = 2, θ2 = 1, α1 = 1, α2 = 1. Again, player 1 (the gifted) is more talented but also more risk averse than player 2 (theventuresome). (a) For R = ln(4) we compute q = 1, i.e. the gifted has a higher probability of winning the contest than the venturesome. (b) For R = ln(9) we compute q = 1 < 1 , i.e. the winning probability of the talented increases even further as the contest prize R increases. We have examined the chance of winning a contest when participants differ inboth their talent and their attitude towards risk. For the case of CARA pref-erences, we have shown that the agent’s probability of winning is increasing(decreasing) in the own (opponent’s) skill level but decreasing (increasing)in the own (opponent’s) degree of risk aversion. Hence there are situationsin which the chance of winning is higher for a low-skilled agent with a lowdegree of risk aversion than for a high-skilled agent with a high degree ofrisk-aversion.
Since such an outcome is undesirable in many circumstances, the contest should be designed in order to reduce the riskiness of the game. In ourmodel, the only parameter available for contest design is the contest prize.
We have shown, by the means of a numerical example, that the optimal prizedepends, however, on the parameters of the model. To extend the analysis inthis direction, one could think of a framework in which the contest designerhas more instruments at hand. Imagine, for example, a situation where thedesigner can influence the agents’ effort costs (which are normalized to 1 inour model).
Many scandals in business, politics, and sports lead to the impression that the agent’s success during a selection contest is not always based on her supe-rior skills but the result of her cheating. Sportsmen dope, politicians betray,managers bribe. In real life, besides plain effort, cheating is an illegal butoften applied possibility for the agent to enhance the probability of winning.
While it is intuitive that the availability of a cheating technology increasesthe riskiness of the contest, without a formal analysis it is not clear how theavailability of such a technology influences the relative winning probabilitiesof the gifted and the venturesome.
Since the case of risk neutral agents is discussed extensively in the literature(see e.g. Baik, 1994), we concentrate on risk averse agents.
(a) For θ1 = θ2 = θ > 0 and α1 = α2 = α > 0 equation (10) immediately implies q = 1. Hence, the equilibrium is symmetric with equal winningprobabilities p∗1 = p∗2 = 1 and effort levels x∗ be easily computed from the FOC (8).
The equilibrium rent dissipation rate can be computed immediately from theequilibrium effort levels in Proposition 1. Denoting A := αR we calculate where the last inequality is verified, again, using the identity eX = for any real X. From this inequality it is also apparent that ρsym < 1.
Again, we concentrate on risk averse agents.
(a) For θ1 > θ2 > 0 and α1 = α2 = α > 0, the value of q can be immediately computed from equation (10). Moreover, θ1 − θ2 > 0 and, hence, (b) We have to show that ∂q < 0 < ∂q . One easily verifies that either which is obviously true as the root-term exceeds e2αR θ1−θ2 .
(d) and (c) (ii) Again denoting A := αR, we have to show that ∂q < 0.
Using θ1 − θ2 > 0, one easily verifies that this is equivalent to Suppose θ1 > θ2 > 0 and α1 = α2 = α > 0.
(a) We have to show that ∂ξ < 0 < ∂ξ . One easily verifies that either = 1 and applying part (a) we conclude ξ < 1, i.e. x∗1 > x∗2, for all θ1 > θ2.
Suppose θ1 = θ2 = θ > 0 and α1 ≥ α2 > 0.
(a) The value of q can be immediately computed from equation (10).
which is positive, since δ(α) > 0 and The last inequality can be verified using the identity eX = Applying the result (11) and noting that q|α ε > 0 such that α1 = α2 + ε implies q > 1 and thus p∗1 < p∗2 for all 0 < ε < ¯ε. However, as θ1 = θ2, p∗1 < p∗2 is possible ifand only if x∗1 < x∗2.
(c) The higher the prize, the smaller the winning probability of the more risk averse agent, i.e. ∂q > 0: So far, we have only examples (e.g. for 2, we compute q|R=ln(4) < q|R=ln(9)), but no Baik, K. H. (1994). Effort levels in contests with two asymmetric players.
Southern Economic Journal, 61:367–378.
Bono, J. W. (2008). Sales contests, promotion decisions and heterogeneous risk. Managerial and Decisions Economics, forthcoming.
Cornes, R. and Hartley, R. (2003). Risk aversion, hetergeneity and contests.
Fabella, R. V. (1992). A note on the Tullock dissipation rate. Journal of Economic Behavior and Organization, 19:245–249.
Hillman, A. L. and Katz, E. (1984). Risk-averse rent seekers and the social cost of monopoly power. The Economic Journal, 94:104–110.
Hillman, A. L. and Samet, D. (1987). Dissipation of contestable rents by small numbers of contenders. Public Choice, 54:63–82.
Hvide, H. K. (2002). Tournament rewards and risk taking. Journal of Labor Konrad, K. and Schlesinger, H. (1997). Risk aversion in rent-seeking and rent-augmenting games. The Economic Journal, 107:1671–1683.
Long, N. V. and Vousden, N. (1987). Risk-averse rent seeking with shared rents. The Economic Journal, 97:971–985.
Runkel, M. (2006a). Optimal contest design, closeness and the contest success function. Public Choice, 129:217–231.
Runkel, M. (2006b). Total effort, competitive balance and the optimal contest success function. European Journal of Political Economy, 22:1009–1013.
Skaperdas, S. and Gan, L. (1995). Risk aversion in contests. The Economic Yamazaki, T. (2008). On the existence and uniqueness of pure-strategy Nash equilibrium in risk-averse rent-seeking games. Working Paper Series No.56,Faculty of Economics, Niigata University.

Source: http://www.socialpolitik.ovgu.de/sozialpolitik_media/papers/Sahm_Marco_uid644_pid579.pdf