turning up the heat - philipp stracke(x)= 1 p 1 x2;x 2[0;1] expected contestant effort = m(xe)= z 1...
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Turning Up the HeatThe Demoralization Effect of Competition in Contests
Dawei Fang1 Thomas Noe2 Philipp Strack3
1Department of Economics, Gothenburg
2Said Business School, Oxford
3Department of Economics, UC Berkeley
1
Motivation
Motivation
• Contests are ubiquitous• sales contests• promotion contests• college admissions• exam grades• R&D contests
• But many of these competitions are “soft”.• Salary differences among managers receiving different performance
ratings are small (Medoff and Abraham, 1980)• Corporate sales contests often set multiple prizes (Backes-Gellner and
Pull, 2013)• Some firms employ large promotion rates (Ariga et al., 1999)• Some schools use easy grading curves: “As, Bs, and tuition fees”
2
Basic insight
• The classic tradeoff: increasing competitiveness• increases efficiency, but also• increases inequality
• This paper shows: In a very standard framework for contestcompetitions, all-pay auctions, that there is no tradeoff.
• When the marginal cost of effort is increasing, making contest morecompetitive leads to contestant demoralization, and thereby• reduces the cost efficiency of individual contestant effort• leads to second order stochastically dominated effort distributions• increases risk-taking under conventional statistical measures if cost
convexity effect is non-decreasing in the level of effort.
3
Is demoralization just a theoretical oddmint?
• Demoralization is consistent with property interpreted data from fieldand lab experiments• We develop simple statistical measures to identify demoralization effects
in contests• Using these measures, we identify demoralization effects in laboratory and
field experiments• Develop predictions for the interaction between competitiveness and
reward levels required for valid empirical tests of contest models
• But why are some contests so competitive?• Derive general conditions under which, competitive contest designs are
optimal despite demoralization
4
Testable implications: A sample
• Less competitive bonus schemes lead to higher worker output
• Small classes lead to higher student effort
• Competitive prizes improve the outcome in R&D contests when costconvexity is limited
5
The model
The model
6
The all-pay contest
• n≥ 2 homogeneous agents compete in an all-pay contest.
• Each agent i simultaneously chooses an effort (outcome) xi and incurs aneffort cost c(xi).
• c is differentiable, strictly increasing, and weakly convex, with c(0) = 0.
• There is an ordered vector of prizes v ∈ Rn+: v1 ≤ ·· · ≤ vn and
0 = v1 < vn.
• Agent with kth highest effort wins prize vk. Ties are broken randomly.
• Agents are risk neutral.
• An agent’s payoff equals his prize less his effort cost.
• Solution concept: symmetric Nash equilibrium
8
Equilibrium
• The effort distribution in any symmetric equilibrium must satisfy (Barutand Kovenock, 1998):• No point mass• No rents• Interval support [0, x̄], where c(x̄) = vn
LEMMA 1. EQUILIBRIUM
There exists a unique symmetric equilibrium. In this equilibrium, each agent’seffort distribution Fv is given implicitly by
exp. contest reward︷ ︸︸ ︷n
∑i=1
vi
(n−1i−1
)Fv(x)i−1 (1−Fv(x))
n−i =
effort cost︷︸︸︷c(x) , x ∈ [0, x̄].
10
What is increased competition?
• Three notions of a more competitive environment• Greater prize inequality: Rewards to super-stars increase relative to
rewards to stars• Increase contest size: More students take exam under the same grading
curve• Entry by new contestants without an increase in contest rewards: New
firms enter a stagnant market.
• Basic result: Under all three notions of increased competitiveness:
Increased competitiveness =⇒ Demoralization
12
Competition: inequality
Prize inequality effects:Inequity and inefficiency
13
The tale of two contests
Contest Type Performance Prizes
Worst Median Best
Last-place elimination (E) 0 1 1Winner take all (wta) 0 0 2
15
Equilibrium in Contest E
• Three contestants: A, B, and C
• Each contestant chooses a random performance level with distribution
• Effort cost: c(x) = x2
• Equilibrium in Contest E
FE(x) = 1−√
1− x2,x ∈ [0,1]
• Expected contestant effort =
µ(XE) =∫ 1
0x fE(x)dx =
∫ 1
0(1−FE(x))dx =
π
4≈ 0.785
16
Why is this an equilibrium?
• Consider contestant A:
• Contestant A wins a prize of 1 if and only if she is not the worstperformer.
• Contestant A is the worst performer if both B and C outperform her.
• So if contestant A chooses effort x, the probability that A is the worstperformer is
P{XEB > x & XE
C > x}= P{XEB > x}P{XE
C > x}= (1−FE(x))2
• So, the expected contest reward to A from effort x equals
prize︷︸︸︷1 ×
Prob. Winning︷ ︸︸ ︷(1− (1−FE(x))2)
17
Expected contest rewards
Exp. Contest Reward
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
x -Effort
18
Reward = cost: Equilibrium
Exp. Contest Reward
Cost
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
x -Effort
19
Switch from E to WTA
• Suppose the contest design is changed to WTA
• and contestants B and C follow their E-contest strategies
• What is the expected contest reward for A?• Contestant A wins with effort x in the WTA contest if and only if A tops
both B and C:• Probability:
P{XEB < x & XE
C < x}= P{XEB < x}P{XE
C < x}= (FE(x))2
• Winner’s prize = 2• Expected contest reward:
Prize︷︸︸︷2 ×
Prob. of Winning︷ ︸︸ ︷(FE(x))2
20
Reward 6= Cost: Disequilibrium
Exp. Contest Reward
Cost
0.0 0.2 0.4 0.6 0.8 1.00.0
0.5
1.0
1.5
2.0
x -Effort
21
A’s response
• A maximizes the difference between expected contest rewards and effortcosts• So A’s best response, holding the responses of B and C fixed, is to
increase effort to x = 1• So, it seems that switching to WTA increases A’s effort• But, unfortunately for A, B and C will also change their strategies• They would also want to choose effort level x = 1.• But if A, B and C choose effort level x = 1. A, B, and C will tie and split
equally the winners’ prize of 2.• Their contest reward equals
contest reward =13×2 < c(x = 1) = 1 = effort cost.
so, not an equilibrium
22
Equilibrium in WTA contest
• Under WTA, the highest effort level must exceed the highest effort levelin E. Why?• The highest effort level earns the winner’s prize with probability 1• The winners prize is 2• In equilibrium, cost equals expected reward.• So, at the highest level of effort under WTA, x̄wta,
c(x̄wta) = 2 =⇒ (x̄wta)2 = 2 =⇒ x̄wta =√
2
• Thus, effort spreads out
• Spreading out effort is inefficient
23
Effort and efficiency
• Cost efficiency and effort:
Cost efficiency =Effort
Effort Cost=
xc(x)
=1x↓ in x
Effort Cost×Cost efficiency = Effort
• In both the WTA contest and E contest, expected effort costs equalexpected contest rewards
• Expected contest rewards are the same in both contests, so expectedeffort costs are the same.
• In the WTA, effort costs are spent less efficiently, reaching for occasionalvery high performance levels
24
Equilibrium in the WTA
• Each contest chooses a random performance level with distribution
Fwta(x) =x√2, x ∈ [0,
√2]
• Expected contestant effort =
µ(Xwta) =∫ √2
0(1−Fwta(x))dx =
1√2≈ 0.707
25
Reward = cost: Equilibrium
Exp. Contest Reward
Cost
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0
0.5
1.0
1.5
2.0
x -Effort
26
Effort in the E contest SSD dominates effort in the WTA contest
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0
0.2
0.4
0.6
0.8
1.0
x -Effort
1 - F
1 - FE
1 - Fwta
27
Pop quiz: Field Experiment 1 (Lim et al., 2009)
• 15 salespeople compete
• In Treatment 1, the winner gets 300$
v = (0,0,0,0,0,0,0,0,0,0,0,0,0,0,300)
• In Treatment 2, the 5 best performing salespeople get 60$
v = (0,0,0,0,0,0,0,0,0,0,60,60,60,60,60)
• Question: which reward scheme better motivates the sales staff?
28
Prize inequality
DEFINITION OF PRIZE INEQUALITY
Let w ∈Pn and v ∈Pn be two prize vectors of equal length and sum ofprizes, ∑
ni=1 wi = ∑
ni=1 vi. w is more unequal than v if the Lorenz curve of w is
constantly lower than the Lorenz curve of v, i.e.,
k
∑i=1
wi ≤k
∑i=1
vi for all k ∈ {1, . . . ,n}.
30
Changes in prize inequality
A prize vector becomes more equal when
• Pigou-Dalton transfer: prizes are shifted from better to worse performingagents, e.g.,
(0,1,3)→ (0,2,2)
• Averaging: replace a subset of prizes by the subset average, e.g.,
(0,1,2,3,4)→ (0,2,2,2,4)
• Interpretation: more equal prizes reduce competition
31
THEOREM 1. DEMORALIZATION EFFECT OF PRIZE INEQUALITY
Suppose w ∈Pn is more unequal than v ∈Pn, then
• µ(Xw)≤ µ(Xv) and
• Xw �SSD Xv.
32
Example: Motivating effort through bonuses
2 4 6 8 10 12 14 16 18
0.10
0.12
0.14
0.16
0.18
0.20
0.22
Number of Prizes k
Av
era
ge
Ou
tpu
t → less competitive →
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Output
CD
F
FIGURE 1: On the left: the average output in a situation with n = 20 agents,c(x) = x2, a total budget of one and k identical nonzero prizesv = (0, . . . ,0,1/k, . . . ,1/k). On the right: the CDF for 4 nonzero prizes (red) and 8nonzero prizes (blue).
33
Field experiment 1 revisited
(v11,v12,v13,v14,v15) Average Sales Std. Deviation
Treatment 1 (0,0,0,0,300) 897.07 $ 778.31 $Treatment 2 (60,60,60,60,60) 1391.00 $ 1096.97 $
TABLE 1: Average sales in a field experiment performed by Lim et al (2009) withn = 15 salespeople in each contest where the lowest 10 prizes equaled zerov1 = v2 = . . .= v10 = 0. The difference in average sales is statistically significant.
Return to performance riskiness
34
Another field experiment: Field Experiment 2 (Lim et al., 2009)
(v10,v11,v12,v13,v14,v15) Average Sales Std. Deviation
Treatment 3 (0,40,40,40,40,40) 358.33 $ 181.98 $Treatment 4 (7,13,23,36,51,70) 333.33 $ 211.05 $
TABLE 2: Average sales in another field experiment performed by Lim et al (2009)with n = 15 salespeople in each contest where the lowest 9 prizes equaled zerov1 = v2 = . . .= v9 = 0 and the prize vector in Treatment 4 marginally failed to bemore unequal than the prize vector in Treatment 3. The difference in average sales isnot statistically significant.
35
Competition: Crowding
Contest size effects:Bigger is not better
36
This contest is getting pretty crowded
• Consider the simple 2-contestant contest with prize vector (0,1), andeffort cost, c(x) = x2
• Equilibrium effort distribution: F2(x) = x2.• Expected effort: 2/3
• Scale up the contest ten-fold, increasing the number of prizes andnumber of contestants
• Results in a 20-contestant contest with prize vector
(
×10︷ ︸︸ ︷0,0, . . . ,0,
×10︷ ︸︸ ︷1,1, . . . ,1)
• prize inequality is unaffected by the 10 fold scaling• the size of rewards is not affected• the reward per contestant is not affected
• So why should scaling matter?
38
Effect of scaling on incentives
• Under the 2-contestant equilibrium strategy, F2, scaling makes thedistribution of the sample median more peeked around the median.
• Beating sample median effort results in winning the prize
• So, the ten-fold scaling greatly increases the marginal rewards for effortlevels close to the median of F2
• Holding the strategies of the other contestants fixed, each contestantwould increase expected effort to x = 0.825 which “almost” ensureswinning the prize.
39
Reward 6= effort: Disequilibrium
Exp. Contest Reward
Cost
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
x -Effort
40
Equilibrium in the scaled contest
• But, if all contestants shifted to x = 0.825, all contestants would earn lessthan their cost of effort.
• Equilibrium in the scaled contest:• F20(x) = I−1
x2 (10,10), (I−1 is Inverse Beta Distribution.)• Effort shifts away from the median toward extreme effort levels• Expected effort≈ 0.575
41
Effort in scaled contest is SSD dominated
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
x -Effort
1 - F
1 - F2
1 - F20
42
Pop quiz: Effect changing class size
• Consider a simple grading scheme that gives a Pass to better 50% ofstudents and a Fail to the rest.
• Suppose you have one class of 20, and one of 40 students.
• Question: which class of students exert higher effort on average?
43
Increasing contest size
DEFINITION OF CONTEST SCALING
Prize vector w ∈Pn·s is a scaling of v ∈Pn if
w = (v1,v1, . . . ,v1︸ ︷︷ ︸s times
,v2,v2, . . . ,v2︸ ︷︷ ︸s times
, . . .)
Fixing average reward and prize inequality, contest size is increased when
• contest scaling: e.g., increasing # agents when grading on a curve isadopted
• consolidation: consolidating identical contests into a single large contest
45
THEOREM 2. DEMORALIZATION EFFECT OF CONTEST SIZE
If w ∈Pn·s is a scaling of v, then
• µ(Xw)≤ µ(Xv) and
• Xw �SSD Xv.
46
Example continued
5 10 15 20Contest Scale
0.58
0.60
0.62
0.64
0.66
Average Effort
FIGURE 2: Average effort in an s-scaling the 2-contestant contest example,1≤ s≤ 20. 50% of the contestants receive a prize of 1, the rest receive a prize of 0.
47
Competition: Entry
Entry:Mixing level and size effects
Making demoralized contestants
48
Entry
• Generally, the effect of mixed prize level and contest size changes isindeterminate
• The exception: Adding entrants without adding rewards• Size effect: more contestants• Prize level effect: Lower prize rewards per capita.
DEFINITION OF ENTRY
Prize vector w ∈Pn+1 is an entrant transformation of v ∈Pn if
w = (0,v1,v2, . . . ,vn).
49
Entry discourages individual, but increases total effort
THEOREM 3. ENTRANT EFFECT ON EFFORT
If w ∈Pn+1 is an entrant transformation of v, then individual effort is smaller
µ(Xw)< µ(Xv) and Xw �FSD Xv
while total effort is larger
E
[n+1
∑i=1
Xwi
]≥ E
[ n
∑i=1
Xvi
]
50
Risk taking
Competition and risk-taking:Variances, like first impressions,
can be deceiving
51
The competitiveness/riskiness relation
• If we define risk in the Rothchild and Stiglitz (1971) sense of “increasingrisk,” i.e., SSD, then increasing competitiveness always increasesriskiness.
• If riskiness and risk-taking are defined by conventional statisticalmeasures, e.g., variance, standard deviation, coefficient of variation, willdemoralization increase risk-taking?
52
Sufficient condition
PROPOSITION 1. INCREASING VARIANCE, STANDARD DEVIATION, AND
COEFFICIENT OF VARIATION
• If c is subquadratic (i.e., less convex than the square function x ↪→ x2),increasing prize inequality or contest scale increases variance, standarddeviation, and coefficient of variation of contest outcomes.
• So the competitiveness/risk-taking relation holds under conventionalmeasures of riskiness if the cost function is either quadratic or linear.
• Useful for lab experiments.
53
When cost is superquadratic, variance and standard deviation maynot increase with competitiveness
• Suppose c(x) = x3. Consider the two prize vectors:
v = (0, . . . ,0︸ ︷︷ ︸101
,0.01, . . . ,0.01︸ ︷︷ ︸100
) w = (0, . . . ,0︸ ︷︷ ︸200
,1)
• Numerical calculations of contest statistics:
Prize vector µ(X) σ(X) σ(X)/µ(X)v 0.116 0.101 0.869w 0.015 0.085 5.752
• Driving force: (a) Variance and standard deviation are not scale-invariantand (b) demoralization reduces the scale of effort.
54
Sufficient condition for inequality to increase risk under scale-invariantrisk measures
DEFINITION OF value-preserving convexificationw ∈Pn is a value-preserving convexification of v ∈Pn if ∑
ni=1 wi = ∑
ni=1 vi
and
(wj+1−wj)(vi+1− vi)≥ (wi+1−wi)(vj+1− vj), 1≤ i < j≤ n−1
PROPOSITION 2.Suppose c is geometrically convex, i.e.,
x ↪→ c′(x)c(x)/x
is nondecreasing.
If w�VPC v, then Xw/µ(Xw) is a mean-preserving spread of Xv/µ(Xv).
55
• The geometric convexity condition is much weaker than the subquadraticcondition: simply requires that the effect of convexity not decrease aseffort increases.
• Almost all “standard” convex cost functions are geometrically convex,e.g., power law, exponential, and polynomials with positive coefficients.
• The VPC condition simply requires that the inequality is increasedtransfers that disproportionately increase the largest prizes. Excludesinequality increasing transfers from the bottom to the middle of the prizeschedule.
• Field Experiment 1 (Lim et al., 2009) involves a VPC. Although standarddeviation is lower in WTA, coefficient of variation is higher:(778.31/897.07≈ 0.87 > 0.79≈ 1096.97/1391).
Results in Lim et al.
56
Sufficient condition for contest scaling to increase riskiness underscale-invariant measures
• Simple contests are contests with only two distinct prize values (e.g.,promotion contests, college admissions, winner-take-all contests).
PROPOSITION 3.Suppose c is geometrically convex. If v is a simple contest and w is a scalingof v, then Xw/µ(Xw) is a mean-preserving spread of Xv/µ(Xv).
57
Importance of controlling for total prize
PROPOSITION 3. REWARD LEVEL EFFECT ON PERFORMANCE
RISKINESS
If w = λv, λ > 1, performance is less risky (according to all scale-invariantmeasures of riskiness) under w than under v if c is geometrically convex. If cis a power law (which is geometrically linear), then performance riskinessremains the same.
• Controlling for reward-level effects essential for specify valid empiricaltests of the contest model
• Proposition 3 can be used to generate variance bounds on the effect ofreward level changes on effort variance.
58
Best shot
Competition in best shot contests:If you don’t care about averageeffort, you might not care about
demoralization
59
Designer welfare in development contests
• Let Xmax denote the best outcome.
• In development contests, the contest designer maximizes
E[u(Xmax)], where u is increasing and convex.
• E.g., u(Xmax) = Xmax or u(Xmax) = max[Xmax−K,0], where K dependson development costs, market conditions, etc.
• u is convex in Xmax if development timing is flexible (Dixit and Pindyck,1994).
60
Results
PROPOSITION 4. THE EFFECT OF COMPETITION ON BEST SHOT
If c is logconcave (i.e., less convex than the exponential function), increasingcompetitiveness (prize inequality or contest consolidation) increases designerwelfare in development contests.
• Standard convex cost functions, such as power law, exponential, andquadratic, are both logconcave and geometrically convex.
• Intuition: Increasing competitiveness produces two counteracting effects• positive side: an increase in upside performance dispersion• negative side: a decrease in average performance
61
• Proposition 4 is consistent with the stylized fact that most developmentcontests have a winner-take-all prize schedule.
• If effort costs are very convex, it can be beneficial to use multiple prizesin development contests.
62
Conclusion
Conclusion
63
Conclusion
• We find a demoralization effect of competition under an all-pay contestframework.
• Increasing competition reduces average effort and increases performanceriskiness in the sense of SSD.
• Demoralization leads to qualifications for increased competition toincrease outcome riskiness under conventional measures of riskiness.
• Demoralization also leads to qualifications for increased competition tobenefit designer welfare in development contests.
64
Thank you for your time!
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