This paper suggests a potential rationale for the recent empirical finding that overconfident agents tend to self-select into more competitive environments (e.g. Dohmen and Falk, forthcoming). In particular, it shows that moderate overconfidence in a contest can improve the agent's performance relative to an unbiased opponent and can even lead to an advantage for the overconfident agent in absolute
terms

Ludwig, Sandra

Wichardt, Philipp C.

Wickhorst, Hanke

English

Open Access LMU

Sandra Ludwig und Philipp C. Wichardt und Hanke Wickhorst:Overconfidence Can Improve an Agent’s Relativeand Absolute Performance in ContestsMunich Discussion Paper No. 2010-35Department of EconomicsUniversity of MunichVolkswirtschaftliche FakultätLudwig-Maximilians-Universität MünchenOnline at http://epub.ub.uni-muenchen.de/11885/Overconfidence Can Improve an Agent’s Relative andAbsolute Performance in Contests*Sandra Ludwig†Dept. of Economics, LMU MunichPhilipp C. Wichardt‡Dept. of Economics, University of BonnHanke Wickhorst§Mu¨nster School of Business and EconomicsThis Version: July 5, 2010Abstract. This paper suggests a potential rationale for the recent empirical find-ing that overconfident agents tend to self-select into more competitive environments(e.g. Dohmen and Falk, forthcoming). In particular, it shows that moderate over-confidence in a contest can improve the agent’s performance relative to an unbiasedopponent and can even lead to an advantage for the overconfident agent in absoluteterms.Keywords: Overconfidence, Contests.JEL classification: D21, D44, D82.∗Acknowledgments: We would like to thank Alexander Dilger, several participants of the ESAEuropean Meeting 2009 and an anonymous reviewer for helpful comments and suggestions. Fi-nancial support from the German Research Foundation (DFG), SFB/TR 15 at the University ofMunich, is gratefully acknowledged.†e-mail: sandra.ludwig@lrz.uni-muenchen.de‡e-mail: philipp.wichardt@uni-bonn.de.§Postal address for correspondence: Institute for Economic Education, University of Mu¨nster,Scharnhorststr. 100, D-48151 Mu¨nster, Germany; tel.: +49 (0)251 83 25339, fax: +49 (0)251 8328429, e-mail: hanke.wickhorst@uni-muenster.de.1 IntroductionRecent empirical evidence shows that overconfident agents tend to self-select intomore competitive environments than unbiased agents (e.g. Dohmen and Falk, forth-coming; Bartling et al., 2009). At first sight, this may seem puzzling as overcon-fidence in contests is commonly found to reduce individual welfare due to sub-optimally high effort choices, which – if at all – are beneficial for the principal(cf. Santos-Pinto, 2010; Ando, 2004).1However, an effect that seems to have gone unnoticed in the literature on over-confidence in contests is that the high effort of the overconfident agent may also leadto a comparative payoff-advantage of the biased agent due to an increased probabil-ity of success. In fact, although both Ando (2004) and Santos-Pinto (2010) touchon individual welfare effects of overconfidence in contests, neither of them considersrelative payoff effects in their analysis (and we are not aware of any other paperthat does so). Yet, as shown below, the induced increase in the probability of suc-cess of an overconfident agent may not only reverse the relative performance of theagents. It may even overcompensate the biased agent for his additional effort and,thereby, increase his payoff above the rational benchmark (if the bias is sufficientlysmall). Thus, overconfident agents may actually be correct in believing that theyhave a comparative advantage in more competitive environments such as economiccontests.In the sequel, we demonstrate these positive effects of overconfidence in a simplemodel of imperfectly discriminating contests in the tradition of Tullock (1980).2 Model and ResultsThe Model. Consider a standard two agent Tullock contest with linear effort costswhere agents compete for a winner-price wH (the loser gets wL, with ∆w = wH −wL).2 In order to simplify the exposition, we restrict attention to a contest successfunction with a discriminatory power of 1. While not affecting the general thrust ofthe argument, the assumption, for example, guarantees the existence of equilibrium.Moreover, assume that one agent (agent 1) is overconfident while the other (agent 2)is rational. In particular, to capture agent 1’s overly optimistic view on his abilities,1See, e.g., Yates (1990) for some background on the discussion of overconfidence in psychology.2Assuming effort costs to be linear essentially simplifies the subsequent exposition but is notcrucial for the qualitative results to be derived.2assume that he has a biased perception of his effort cost, i.e. c˜1 = c1 − b where c1 isagent 1’s true cost of effort and 0 < b < c1. The rational agent 2, by contrast, hasa correct view about his effort cost, i.e. c˜2 = c2 > 0. The resulting maximisationproblem of agent i, then, is given by:maxeieiei + e−i∆w + wL − c˜iei , (1)which gives rise to concave reaction functions with positive (negative) first deriva-tives for small (large) values of e−i. For the case of e1 = e2 = 0, assume that eachagent wins the contest with probability 0.5. Again, the assumption is not restrictiveas in equilibrium both agents will exert strictly positive effort.Finally, assume that both agents are informed about their own perceived effortcost c˜i (but not about a potential own bias) and the perceived effort cost of theiropponent c˜−i; for the rational agent this is equivalent to assuming that he knowshis true effort cost as c˜1 = c1. In effect, the assumption ensures that both agentsbest respond to their opponent’s action so that attention is restricted entirely tothe effects of overconfidence while informational issues are set aside.3 A standardargument, then, shows that the corresponding Nash equilibrium effort levels are:e∗1 =∆wc2(c˜1 + c2)2=∆wc2(c1 + c2 − b)2 (2)ande∗2 =∆wc˜1(c˜1 + c2)2=∆w(c1 − b)(c1 + c2 − b)2 . (3)Thus, if both agents are rational, i.e. b = 0, equilibrium effort levels are:eBM1 =∆wc2(c1 + c2)2and eBM2 =∆wc1(c1 + c2)2, (4)which we will consider as the benchmark for our analysis.Aggregate Effects. To begin with, note that irrespective of agent 2’s effort cost, agent1’s effort increases in his bias because∂e∗1∂b= 2∆wc2(c1+c2−b)3 > 0 (as b < c1). The effort of3From an applied point of view, assuming agents to best respond to each other’s actions seemsreasonable as an approximation, for example, in settings where some ex ante learning about theopponent’s reaction function is possible, e.g. because information about past behaviour is readilyavailable. Such learning need not affect the biased self-perception of overconfident agents, though,as for example Pulford and Colman (1997) show that for such agents feedback about eventualoutcomes essentially leaves their biases unaffected.3the rational agent 2, by contrast, decreases in agent 1’s overconfidence if agent 1’sperceived effort cost is smaller than agent 2’s true cost, i.e. if c˜1 < c2 it holds that∂e∗2∂b= ∆w(c1−c2−b)(c1+c2−b)3 < 0 (as b < c1); it increases otherwise. Similar to previous results(e.g. Santos-Pinto, 2010), combining these two effects yields that overconfidence isbeneficial for the principal (who wants to maximise aggregate efforts) as the sum ofthe efforts in the case with overconfidence (equations (2) and (3)) is greater than thesum of the efforts in the rational benchmark (equations (4)): e∗1 + e∗2 > eBM1 + eBM2 .4Individual Effects. In order to assess the relative individual payoff effects, considerthe agents’ expected equilibrium payoffs. These are given by:U∗i =e∗ie∗i + e∗−i∆w + wL − cie∗i , (5)where the true effort cost has to be taken into account. Accordingly, the payoffdifference between the overconfident agent 1 and the rational agent 2 is:∆U∗ := U∗1 − U∗2 =e∗1 − e∗2e∗1 + e∗2∆w − c1e∗1 + c2e∗2. (6)Inserting equations (2) and (3) into (6) gives:∆U∗ =∆w(c˜1 + c2)2[c2(c2 + c˜1− c1)− c˜12] = ∆w(c1 + c2 − b)2 [c2(c2− b)− (c1− b)2]. (7)Calculating the first derivative with respect to b shows that becoming moreoverconfident increases agent 1’s relative performance compared to agent 2 if thebias is moderate, b < c1+c23:5∂∆U∗∂b=∆w(c1 + c2 − b)3 [(c1 + c2 − b)(−c2 + 2(c1 − b)) + 2(c2(c2 − b)− (c1 − b)2)]=∆wc2(c1 + c2 − b)3 (c1 + c2 − 3b). (8)Furthermore, it follows from (7) that∆U∗ ≥ 0, ifc˜1 < c1 ≤ c2,c˜1 < c2 < c1 and c2 ≥ 45c1 and b ∈ [b1, b2], (9)4Similarly, underconfidence (i.e. b < 0 and agent 1 overestimates his effort cost) is detrimentalfor the principal.5The derivative is also positive if b > c1 + c2 but this can never hold as b < c1 and ci > 0.4where b1 =2c1−c2−√c2(5c2−4c1)2and b2 =2c1−c2+√c2(5c2−4c1)2.6 Hence, a necessarycondition for U∗2 > U∗1 to hold, i.e. for ∆U∗ < 0, is that c2 < c1. In particular,U∗2 > U∗1 holds irrespective of the size of b if agent 2 is sufficiently more skilled thanagent 1, i.e. if c2 <45c1, otherwise, i.e. if45c1 ≤ c2 < c1, U∗2 > U∗1 holds only ifb 6∈ [b1, b2]. Put differently, if agent 1 is at least as skilled as agent 2, ∆U∗ is positiveirrespective of the level of overconfidence; see Figure 1 for an illustration.0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,4500,050,10,150,20,250,3bPayoffsFigure 1: Expected equilibrium payoffs as a function of b for parameters: c1 = c2 =0.5; ∆w = 1 and b ∈ [0, 0.45]. The continuous line shows an agent’s expected payoffin equilibrium if both agents are rational. The other lines refer to the expectedpayoffs of the overconfident agent (dotted line) and the rational agent (dashed line)in the contest with one rational and one overconfident agent.Moreover, the comparative advantage of agent 1 can persist even if agent 2 isthe more skilled agent (cf. equation (9)). Accordingly, overconfidence can lead toa situation where the biased agent 1 has a greater chance of winning the contestdespite being the less able one. In this case, the induced increase in agent 1’s effortleads to an increase in the winning probability that outweighs agent 1’s higher cost;6When c˜1 < c2 < c1, we solve c2(c2 − b)− (c1 − b)2 ≥ 0 for b (cf. equation (7)). The inequalityis quadratic in b, and b1 and b2 are the two existing zeros for c2 ≥ 45c1. Both b1 and b2 are inthe feasible range, i.e. 0 < b1 < b2 < c1. Moreover, the inequality holds (implying that ∆U∗ ispositive) if b ∈ [b1, b2]; otherwise the inequality is violated.5thus agent 1 is better off than his opponent, although he exerts the higher effort andhas the higher effort cost per unit of effort.7Finally, a comparison of agent 1’s payoff for the case of b > 0 with that of thebenchmark scenario, b = 0, shows that being overconfident may indeed even improveagent 1’s absolute payoff asU∗1 − UBM1 > 0 ifc22 − c21c2> b, (10)i.e. if agent 1 is the more skilled agent (c2 > c1) and the bias is sufficiently small.To see that (10) holds, note that:U∗1 − UBM1 =∆wc2(c1 + c2 − b)−∆wc1c2(c1 + c2 − b)2 +∆wc1c2 −∆wc2(c1 + c2)(c1 + c2)2=(∆wc22 −∆wc2b)(c1 + c2)2 −∆wc22(c1 + c2 − b)2(c1 + c2)2(c1 + c2 − b)2=b∆wc2(c1 + c2)2(c1 + c2 − b)2 [c22 − c21 − c2b]which is positive ifc22−c21c2> b.By contrast, compared to the benchmark situation with b = 0, the rationalagent 2 is always worse off when paired with an overconfident agent 1 (b > 0), i.e.U∗2 − UBM2 < 0, (11)becauseU∗2 − UBM2 =(∆wc1 −∆b)(c1 + c2 − b)−∆wc2(c1 − b)(c1 + c2 − b)2+∆wc1c2 −∆wc1(c1 + c2)(c1 + c2)2=∆wc2b(c1 + c2)2(c1 + c2 − b)2 [2c1b + c2b− 2c1c2 − 2c21]is always smaller than zero as b < c1.Proposition 1 below summarises the main points of the above analysis.7If agent 1 is instead underconfident, i.e. b < 0, all effects are reversed: Becoming less under-confident increases agent 1’s effort and his performance relative to agent 2, and ∆U∗ > 0 is onlypossible if agent 1 is the more able agent.6Proposition 1 For the above described Tullock-contest between an overconfidentagent 1 with a strictly positive bias (b > 0) and a rational agent 2, it holds that:1. The principal is strictly better off than in the case without overconfidence (b =0) as e∗1 + e∗2 > eBM1 + eBM2 .2. Agent 1 overexerts effort with respect to his ability, i.e.∂e∗1∂b> 0.3. Agent 2 reduces his effort compared to the rational benchmark (b = 0), i.e.∂e∗2∂b< 0, if c1 − c2 < b.4. Agent 2 is always worse off if agent 1 is overconfident than if he is not, i.e.U∗2 < UBM2 .5. For small biases (b < c1+c23), agent 1’s relative performance as measured bythe difference in equilibrium payoffs ∆U∗ = U∗1 − U∗2 is increasing in b.6. Agent 1’s equilibrium payoff is larger than that of agent 2, i.e. ∆U∗ > 0, ifc1 ≤ c2, or if c1 > c2 ≥ 45c1 and b ∈ [b1, b2].7. Agent 1 has an absolute payoff advantage from being overconfident if his truecost of effort is smaller than that of agent 2 and if his bias is small, i.e.U∗1 > UBM1 for c1 < c2 and 0 < b <c22−c21c2.Extension to two overconfident agents. As a last step of the analysis, we brieflyconsider the case of two overconfident agents with different biases but identical trueeffort cost c. For the sake of argument, we assume that agent 1’s bias is larger thanthe bias of agent 2, i.e. 0 < b2 < b1 < c.8 In this case, assuming as before that bothagents know their own perceived cost c˜i and the perceived cost of their opponentc˜−i and solving the corresponding maximisation problem described in equation (1),we obtain the following equilibrium effort levels:eˆ1 =∆w(c− b2)(2c− b1 − b2)2 and eˆ2 =∆w(c− b1)(2c− b1 − b2)2 . (12)Moreover, inserting equilibrium efforts into the agents’ utility functions (cf. equation(5)), the expression for ∆U becomes:∆Uˆ := Uˆ1 − Uˆ2 = ∆w(2c− b1 − b2)2︸ ︷︷ ︸>0[b22 − b21 + c(b1 − b2)]. (13)8If b1 = b2, both agents trivially have the same expected payoffs as the maximisation problemis symmetric.7And, as ∆w > 0, the condition for the more biased agent 1 to be better off, ∆Uˆ > 0,can be simplified to(b1 − b2)(c− (b1 + b2)) > 0 . (14)Thus, the more biased agent 1 is better off (worse off) than agent 2 whenever thesum of the biases is smaller (larger) than c.Moreover, regarding comparative effects, if c = b1 + b2 (so that ∆Uˆ = 0) butb2 < b1,9 the more biased agent 1, taking b2 as given, would gain a comparativeadvantage from having a lower bias. In particular, if b1 was lowered while keeping b2fixed, the situation would change to one with c > b1 + b2 so that ∆Uˆ would becomenegative provided that b1 remains larger than b2. Similarly, taking b1 as given, theless biased agent 2 would gain a comparative advantage from an increase in his biasso that c < b1 + b2 while keeping b2 < b1. Accordingly, if also biases were a matterof choice, the only stable situation would be the one with c = b1 + b2 and b1 = b2: ifbi was increased (decreased) in such a case, the situation would change to one withbi > b−i and c < bi + b−i (bi < b−i and c > bi + b−i) so that agent i would be worseoff compared to his opponent. Put differently, while both agents always have thesame expected payoffs if b1 = b2, it is only when b1 = b2 and also b1 + b2 = c thatno agent would generate a comparative advantage from having a different level ofoverconfidence.Proposition 2 For the above described Tullock-contest between two overconfidentagents with identical true effort cost c and biases 0 < bi < c, i = 1, 2, it holds that:1. The agents expected payoffs are identical whenever b1 = b2 or b1 + b2 = c.2. If b1 + b2 < c and b1 6= b2, the expected payoff is larger for the agent with thelarger bias.3. If b1 + b2 > c and b1 6= b2, the expected payoff is larger for the agent with thesmaller bias.4. If b1 + b2 = c and b1 = b2 (and only then), no agent would gain a comparativeadvantage over his opponent from a change in his bias.Finally, it deserves a mention that the continuity of the agents’ reaction andutility functions ensures that the main results of the analysis — (a) that overcon-fidence may outweigh a higher cost of effort (cf. Proposition 1.6) and (b) that the9See Footnote 8.8relative advantage of being more (or less) overconfident in the case with two biasedagents and equal true effort cost depends on the sum of the biases (cf. Proposition 2)— also transfer to the more general case of two overconfident agents with differenteffort cost (at least for some range of parameters).ReferencesAndo, M. (2004): Overconfidence in Economic Contests, Working Paper, NihonUniversity.Bartling, B., E. Fehr, M.A. Marchal, and D. Schunk (2009), “Egalitarianism andCompetitiveness,” American Economic Review (Papers and Proceedings), 99,93-98.Dohmen, T., and A. Falk (forthcoming), “Performance Pay and Multi-dimensionalSorting: Productivity, Preferences and Gender,” American Economic Review.Santos-Pinto, L. (2010): Positive Self-Image in Tournaments, International Eco-nomic Review, 51, 475-496.Pulford, B. and A. Colman (1997). “Overconfidence: Feedback and Item DifficultyEffects,” Personality and Individual Differences, 23, 125-133.Tullock, G. (1980): “Efficient Rent-Seeking,” pp. 97-112 in: Buchanan, J.M. Tolli-son, R.D. und Tullock, G. (Ed.), Toward a Theory of the Rent-Seeking Society,College Station: Texas A&M University Press.Yates, J.F. (1990): Judgement and Decision Making, Englewood Cliffs, NJ: Pren-tice Hall.9

Ecient Rent-Seeking,&quot;

Egalitarianism and Competitiveness,&quot; American Economic Review (Papers and

Judgement and Decision Making,

Overcon Feedback and Item Diculty Eects,&quot;

Overcon in Economic Contests, Working Paper,

Positive Self-Image in Tournaments,

Overconfidence Can Improve an Agent's Relative and Absolute Performance in Contests

Universität München: Elektronischen Publikationen

Ludwig, Sandra; Wichardt, Philipp C. und Wickhorst,
Hanke:
Overconfidence Can Improve an Agent’s Relative
and Absolute Performance in Contests
Munich Discussion Paper No. 2010-35
Department of Economics
University of Munich
Volkswirtschaftliche Fakultät
Ludwig-Maximilians-Universität München
Online at https://doi.org/10.5282/ubm/epub.11885
Overconfidence Can Improve an Agent’s Relative and
Absolute Performance in Contests*
Sandra Ludwig†
Dept. of Economics, LMU Munich
Philipp C. Wichardt‡
Dept. of Economics, University of Bonn
Hanke Wickhorst§
Münster School of Business and Economics
This Version: July 5, 2010
Abstract. This paper suggests a potential rationale for the recent empirical find-
ing that overconfident agents tend to self-select into more competitive environments
(e.g. Dohmen and Falk, forthcoming). In particular, it shows that moderate over-
confidence in a contest can improve the agent’s performance relative to an unbiased
opponent and can even lead to an advantage for the overconfident agent in absolute
terms.
Keywords: Overconfidence, Contests.
JEL classification: D21, D44, D82.
∗Acknowledgments: We would like to thank Alexander Dilger, several participants of the ESA
European Meeting 2009 and an anonymous reviewer for helpful comments and suggestions. Fi-
nancial support from the German Research Foundation (DFG), SFB/TR 15 at the University of
Munich, is gratefully acknowledged.
†e-mail: sandra.ludwig@lrz.uni-muenchen.de
‡e-mail: philipp.wichardt@uni-bonn.de.
§Postal address for correspondence: Institute for Economic Education, University of Münster,
Scharnhorststr. 100, D-48151 Münster, Germany; tel.: +49 (0)251 83 25339, fax: +49 (0)251 83
28429, e-mail: hanke.wickhorst@uni-muenster.de.
1 Introduction
Recent empirical evidence shows that overconfident agents tend to self-select into
more competitive environments than unbiased agents (e.g. Dohmen and Falk, forth-
coming; Bartling et al., 2009). At first sight, this may seem puzzling as overcon-
fidence in contests is commonly found to reduce individual welfare due to sub-
optimally high effort choices, which – if at all – are beneficial for the principal
(cf. Santos-Pinto, 2010; Ando, 2004).1
However, an effect that seems to have gone unnoticed in the literature on over-
confidence in contests is that the high effort of the overconfident agent may also lead
to a comparative payoff-advantage of the biased agent due to an increased probabil-
ity of success. In fact, although both Ando (2004) and Santos-Pinto (2010) touch
on individual welfare effects of overconfidence in contests, neither of them considers
relative payoff effects in their analysis (and we are not aware of any other paper
that does so). Yet, as shown below, the induced increase in the probability of suc-
cess of an overconfident agent may not only reverse the relative performance of the
agents. It may even overcompensate the biased agent for his additional effort and,
thereby, increase his payoff above the rational benchmark (if the bias is sufficiently
small). Thus, overconfident agents may actually be correct in believing that they
have a comparative advantage in more competitive environments such as economic
contests.
In the sequel, we demonstrate these positive effects of overconfidence in a simple
model of imperfectly discriminating contests in the tradition of Tullock (1980).
2 Model and Results
The Model. Consider a standard two agent Tullock contest with linear effort costs
where agents compete for a winner-price wH (the loser gets wL, with ∆w = wH −
wL).
2 In order to simplify the exposition, we restrict attention to a contest success
function with a discriminatory power of 1. While not affecting the general thrust of
the argument, the assumption, for example, guarantees the existence of equilibrium.
Moreover, assume that one agent (agent 1) is overconfident while the other (agent 2)
is rational. In particular, to capture agent 1’s overly optimistic view on his abilities,
1See, e.g., Yates (1990) for some background on the discussion of overconfidence in psychology.
2Assuming effort costs to be linear essentially simplifies the subsequent exposition but is not
crucial for the qualitative results to be derived.
2
assume that he has a biased perception of his effort cost, i.e. c̃1 = c1 − b where c1 is
agent 1’s true cost of effort and 0 < b < c1. The rational agent 2, by contrast, has
a correct view about his effort cost, i.e. c̃2 = c2 > 0. The resulting maximisation
problem of agent i, then, is given by:
max
ei
ei
ei + e−i
∆w + wL − c̃iei , (1)
which gives rise to concave reaction functions with positive (negative) first deriva-
tives for small (large) values of e−i. For the case of e1 = e2 = 0, assume that each
agent wins the contest with probability 0.5. Again, the assumption is not restrictive
as in equilibrium both agents will exert strictly positive effort.
Finally, assume that both agents are informed about their own perceived effort
cost c̃i (but not about a potential own bias) and the perceived effort cost of their
opponent c̃−i; for the rational agent this is equivalent to assuming that he knows
his true effort cost as c̃1 = c1. In effect, the assumption ensures that both agents
best respond to their opponent’s action so that attention is restricted entirely to
the effects of overconfidence while informational issues are set aside.3 A standard
argument, then, shows that the corresponding Nash equilibrium effort levels are:
e∗1 =
∆wc2
(c̃1 + c2)2
=
∆wc2
(c1 + c2 − b)2
(2)
and
e∗2 =
∆wc̃1
(c̃1 + c2)2
=
∆w(c1 − b)
(c1 + c2 − b)2
. (3)
Thus, if both agents are rational, i.e. b = 0, equilibrium effort levels are:
eBM1 =
∆wc2
(c1 + c2)2
and eBM2 =
∆wc1
(c1 + c2)2
, (4)
which we will consider as the benchmark for our analysis.
Aggregate Effects. To begin with, note that irrespective of agent 2’s effort cost, agent
1’s effort increases in his bias because
∂e∗
1
∂b
= 2∆wc2
(c1+c2−b)3
> 0 (as b < c1). The effort of
3From an applied point of view, assuming agents to best respond to each other’s actions seems
reasonable as an approximation, for example, in settings where some ex ante learning about the
opponent’s reaction function is possible, e.g. because information about past behaviour is readily
available. Such learning need not affect the biased self-perception of overconfident agents, though,
as for example Pulford and Colman (1997) show that for such agents feedback about eventual
outcomes essentially leaves their biases unaffected.
3
the rational agent 2, by contrast, decreases in agent 1’s overconfidence if agent 1’s
perceived effort cost is smaller than agent 2’s true cost, i.e. if c̃1 < c2 it holds that
∂e∗
2
∂b
= ∆w(c1−c2−b)
(c1+c2−b)3
< 0 (as b < c1); it increases otherwise. Similar to previous results
(e.g. Santos-Pinto, 2010), combining these two effects yields that overconfidence is
beneficial for the principal (who wants to maximise aggregate efforts) as the sum of
the efforts in the case with overconfidence (equations (2) and (3)) is greater than the
sum of the efforts in the rational benchmark (equations (4)): e∗1 + e
∗
2 > e
BM
1 + e
BM
2 .
4
Individual Effects. In order to assess the relative individual payoff effects, consider
the agents’ expected equilibrium payoffs. These are given by:
U∗i =
e∗i
e∗i + e
∗
−i
∆w + wL − cie∗i , (5)
where the true effort cost has to be taken into account. Accordingly, the payoff
difference between the overconfident agent 1 and the rational agent 2 is:
∆U∗ := U∗1 − U∗2 =
e∗1 − e∗2
e∗1 + e
∗
2
∆w − c1e∗1 + c2e∗2. (6)
Inserting equations (2) and (3) into (6) gives:
∆U∗ =
∆w
(c̃1 + c2)2
[c2(c2 + c̃1 − c1)− c̃12] =
∆w
(c1 + c2 − b)2
[c2(c2 − b)− (c1 − b)2]. (7)
Calculating the first derivative with respect to b shows that becoming more
overconfident increases agent 1’s relative performance compared to agent 2 if the
bias is moderate, b < c1+c2
3
:5
∂∆U∗
∂b
=
∆w
(c1 + c2 − b)3
[(c1 + c2 − b)(−c2 + 2(c1 − b)) + 2(c2(c2 − b) − (c1 − b)2)]
=
∆wc2
(c1 + c2 − b)3
(c1 + c2 − 3b). (8)
Furthermore, it follows from (7) that
∆U∗ ≥ 0, if



c̃1 < c1 ≤ c2,
c̃1 < c2 < c1 and c2 ≥ 45c1 and b ∈ [b1, b2],
(9)
4Similarly, underconfidence (i.e. b < 0 and agent 1 overestimates his effort cost) is detrimental
for the principal.
5The derivative is also positive if b > c1 + c2 but this can never hold as b < c1 and ci > 0.
4
where b1 =
2c1−c2−
√
c2(5c2−4c1)
2
and b2 =
2c1−c2+
√
c2(5c2−4c1)
2
.6 Hence, a necessary
condition for U∗2 > U
∗
1 to hold, i.e. for ∆U
∗ < 0, is that c2 < c1. In particular,
U∗2 > U
∗
1 holds irrespective of the size of b if agent 2 is sufficiently more skilled than
agent 1, i.e. if c2 <
4
5
c1, otherwise, i.e. if
4
5
c1 ≤ c2 < c1, U∗2 > U∗1 holds only if
b 6∈ [b1, b2]. Put differently, if agent 1 is at least as skilled as agent 2, ∆U∗ is positive
irrespective of the level of overconfidence; see Figure 1 for an illustration.
0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45
0
0,05
0,1
0,15
0,2
0,25
0,3
b
P
a
y
o
ff
s
Figure 1: Expected equilibrium payoffs as a function of b for parameters: c1 = c2 =
0.5; ∆w = 1 and b ∈ [0, 0.45]. The continuous line shows an agent’s expected payoff
in equilibrium if both agents are rational. The other lines refer to the expected
payoffs of the overconfident agent (dotted line) and the rational agent (dashed line)
in the contest with one rational and one overconfident agent.
Moreover, the comparative advantage of agent 1 can persist even if agent 2 is
the more skilled agent (cf. equation (9)). Accordingly, overconfidence can lead to
a situation where the biased agent 1 has a greater chance of winning the contest
despite being the less able one. In this case, the induced increase in agent 1’s effort
leads to an increase in the winning probability that outweighs agent 1’s higher cost;
6When c̃1 < c2 < c1, we solve c2(c2 − b) − (c1 − b)2 ≥ 0 for b (cf. equation (7)). The inequality
is quadratic in b, and b1 and b2 are the two existing zeros for c2 ≥ 45c1. Both b1 and b2 are in
the feasible range, i.e. 0 < b1 < b2 < c1. Moreover, the inequality holds (implying that ∆U
∗ is
positive) if b ∈ [b1, b2]; otherwise the inequality is violated.
5
thus agent 1 is better off than his opponent, although he exerts the higher effort and
has the higher effort cost per unit of effort.7
Finally, a comparison of agent 1’s payoff for the case of b > 0 with that of the
benchmark scenario, b = 0, shows that being overconfident may indeed even improve
agent 1’s absolute payoff as
U∗1 − UBM1 > 0 if
c22 − c21
c2
> b, (10)
i.e. if agent 1 is the more skilled agent (c2 > c1) and the bias is sufficiently small.
To see that (10) holds, note that:
U∗1 − UBM1 =
∆wc2(c1 + c2 − b) − ∆wc1c2
(c1 + c2 − b)2
+
∆wc1c2 − ∆wc2(c1 + c2)
(c1 + c2)2
=
(∆wc22 − ∆wc2b)(c1 + c2)2 − ∆wc22(c1 + c2 − b)2
(c1 + c2)2(c1 + c2 − b)2
=
b∆wc2
(c1 + c2)2(c1 + c2 − b)2
[c22 − c21 − c2b]
which is positive if
c2
2
−c2
1
c2
> b.
By contrast, compared to the benchmark situation with b = 0, the rational
agent 2 is always worse off when paired with an overconfident agent 1 (b > 0), i.e.
U∗2 − UBM2 < 0, (11)
because
U∗2 − UBM2 =
(∆wc1 − ∆b)(c1 + c2 − b) − ∆wc2(c1 − b)
(c1 + c2 − b)2
+
∆wc1c2 − ∆wc1(c1 + c2)
(c1 + c2)2
=
∆wc2b
(c1 + c2)2(c1 + c2 − b)2
[2c1b + c2b − 2c1c2 − 2c21]
is always smaller than zero as b < c1.
Proposition 1 below summarises the main points of the above analysis.
7If agent 1 is instead underconfident, i.e. b < 0, all effects are reversed: Becoming less under-
confident increases agent 1’s effort and his performance relative to agent 2, and ∆U∗ > 0 is only
possible if agent 1 is the more able agent.
6
Proposition 1 For the above described Tullock-contest between an overconfident
agent 1 with a strictly positive bias (b > 0) and a rational agent 2, it holds that:
1. The principal is strictly better off than in the case without overconfidence (b =
0) as e∗1 + e
∗
2 > e
BM
1 + e
BM
2 .
2. Agent 1 overexerts effort with respect to his ability, i.e.
∂e∗
1
∂b
> 0.
3. Agent 2 reduces his effort compared to the rational benchmark (b = 0), i.e.
∂e∗
2
∂b
< 0, if c1 − c2 < b.
4. Agent 2 is always worse off if agent 1 is overconfident than if he is not, i.e.
U∗2 < U
BM
2 .
5. For small biases (b < c1+c2
3
), agent 1’s relative performance as measured by
the difference in equilibrium payoffs ∆U∗ = U∗1 − U∗2 is increasing in b.
6. Agent 1’s equilibrium payoff is larger than that of agent 2, i.e. ∆U∗ > 0, if
c1 ≤ c2, or if c1 > c2 ≥ 45c1 and b ∈ [b1, b2].
7. Agent 1 has an absolute payoff advantage from being overconfident if his true
cost of effort is smaller than that of agent 2 and if his bias is small, i.e.
U∗1 > U
BM
1 for c1 < c2 and 0 < b <
c2
2
−c2
1
c2
.
Extension to two overconfident agents. As a last step of the analysis, we briefly
consider the case of two overconfident agents with different biases but identical true
effort cost c. For the sake of argument, we assume that agent 1’s bias is larger than
the bias of agent 2, i.e. 0 < b2 < b1 < c.
8 In this case, assuming as before that both
agents know their own perceived cost c̃i and the perceived cost of their opponent
c̃−i and solving the corresponding maximisation problem described in equation (1),
we obtain the following equilibrium effort levels:
ê1 =
∆w(c − b2)
(2c − b1 − b2)2
and ê2 =
∆w(c − b1)
(2c − b1 − b2)2
. (12)
Moreover, inserting equilibrium efforts into the agents’ utility functions (cf. equation
(5)), the expression for ∆U becomes:
∆Û := Û1 − Û2 =
∆w
(2c − b1 − b2)2
︸ ︷︷ ︸
>0
[b22 − b21 + c(b1 − b2)]. (13)
8If b1 = b2, both agents trivially have the same expected payoffs as the maximisation problem
is symmetric.
7
And, as ∆w > 0, the condition for the more biased agent 1 to be better off, ∆Û > 0,
can be simplified to
(b1 − b2)(c − (b1 + b2)) > 0 . (14)
Thus, the more biased agent 1 is better off (worse off) than agent 2 whenever the
sum of the biases is smaller (larger) than c.
Moreover, regarding comparative effects, if c = b1 + b2 (so that ∆Û = 0) but
b2 < b1,
9 the more biased agent 1, taking b2 as given, would gain a comparative
advantage from having a lower bias. In particular, if b1 was lowered while keeping b2
fixed, the situation would change to one with c > b1 + b2 so that ∆Û would become
negative provided that b1 remains larger than b2. Similarly, taking b1 as given, the
less biased agent 2 would gain a comparative advantage from an increase in his bias
so that c < b1 + b2 while keeping b2 < b1. Accordingly, if also biases were a matter
of choice, the only stable situation would be the one with c = b1 + b2 and b1 = b2: if
bi was increased (decreased) in such a case, the situation would change to one with
bi > b−i and c < bi + b−i (bi < b−i and c > bi + b−i) so that agent i would be worse
off compared to his opponent. Put differently, while both agents always have the
same expected payoffs if b1 = b2, it is only when b1 = b2 and also b1 + b2 = c that
no agent would generate a comparative advantage from having a different level of
overconfidence.
Proposition 2 For the above described Tullock-contest between two overconfident
agents with identical true effort cost c and biases 0 < bi < c, i = 1, 2, it holds that:
1. The agents expected payoffs are identical whenever b1 = b2 or b1 + b2 = c.
2. If b1 + b2 < c and b1 6= b2, the expected payoff is larger for the agent with the
larger bias.
3. If b1 + b2 > c and b1 6= b2, the expected payoff is larger for the agent with the
smaller bias.
4. If b1 + b2 = c and b1 = b2 (and only then), no agent would gain a comparative
advantage over his opponent from a change in his bias.
Finally, it deserves a mention that the continuity of the agents’ reaction and
utility functions ensures that the main results of the analysis — (a) that overcon-
fidence may outweigh a higher cost of effort (cf. Proposition 1.6) and (b) that the
9See Footnote 8.
8
relative advantage of being more (or less) overconfident in the case with two biased
agents and equal true effort cost depends on the sum of the biases (cf. Proposition 2)
— also transfer to the more general case of two overconfident agents with different
effort cost (at least for some range of parameters).
References
Ando, M. (2004): Overconfidence in Economic Contests, Working Paper, Nihon
University.
Bartling, B., E. Fehr, M.A. Marchal, and D. Schunk (2009), “Egalitarianism and
Competitiveness,” American Economic Review (Papers and Proceedings), 99,
93-98.
Dohmen, T., and A. Falk (forthcoming), “Performance Pay and Multi-dimensional
Sorting: Productivity, Preferences and Gender,” American Economic Review.
Santos-Pinto, L. (2010): Positive Self-Image in Tournaments, International Eco-
nomic Review, 51, 475-496.
Pulford, B. and A. Colman (1997). “Overconfidence: Feedback and Item Difficulty
Effects,” Personality and Individual Differences, 23, 125-133.
Tullock, G. (1980): “Efficient Rent-Seeking,” pp. 97-112 in: Buchanan, J.M. Tolli-
son, R.D. und Tullock, G. (Ed.), Toward a Theory of the Rent-Seeking Society,
College Station: Texas A&M University Press.
Yates, J.F. (1990): Judgement and Decision Making, Englewood Cliffs, NJ: Pren-
tice Hall.
9


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Overconfidence Can Improve an Agent's Relative and Absolute Performance in Contests

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