Transcript Repeated contests with fatigue Dmitry Ryvkin Florida State University Economic Science Assocoation Meeting

Repeated contests with fatigue
Dmitry Ryvkin
Florida State University
Economic Science Assocoation Meeting
Rome 2007
Contests
Players compete for prizes by expending resources:
- rent-seeking (Lockard and Tullock, 2001)
- labor market contracts (Prendergast 1999, Lazear 1999)
- R&D competition (Taylor 1995)
- elections (Klumpp and Polborn 2005)
- sports (Szymanski 2003)
Key idea: submission of the highest effort does not guarantee a win
Static model: contest success function (CSF)
pi  pi (e1 , e2 ,..., eN )
Questions: choice of effort, efficiency of contracts, dissipation of rent, etc.
Dynamic contests
Competition may occur in several stages (rounds)
- with elimination (Rosen 1986)
- without elimination (Harris and Vickers 1985, 1987; Konrad and Kovenock 2005, 2006)
Examples:
- patent races
- “up-or-out” rules
- sports
Questions: efficiency, design
Repeated contests
- Competition occurs repeatedly (either continuously or in discrete “stages”)
- The winner is the player who first reaches a target
Continuous races (Harris and Vickers):
B
A
Best-of-(2n-1) contests (Ferrall and Smith 1999; Konrad and Kovenock 2005, 2006):
(0,0)
(1,0)
(0,1)
(2,0)
(0,2)
(1,1)
(2,1)
(1,2)
Repeated contests
Theory: complex equilibria even for simplest stage games (Konrad and Kovenock, Harris
and Vickers)
Empirical results: “burning out”
a) no strategic choices of effort at a given stage
b) no dependence of effort on the standing in the series
This work:
- Best-of-(2n-1) contests with fatigue
- An experimental study
Goals:
- See if there is strategic behavior within a given stage, and if players change their
behavior depending on their standing in the series
The model
Best-of-(2n-1) contests with fatigue
Players
two identical risk-neutral players repeatedly making binary choices of effort
Stage game
players 1, 2 choose effort levels x, y {0,1} (0=“low”, 1=“high”)
probability for player 1 to win a stage contest:
Advantage parameter
Fatigue
 1 / 2,

pxy  (1  a) / 2,
(1  a) / 2,

x y
x  1, y  0
x  0, y  1
Fatigue parameter
net fatigue of player 1 at stage t:
t 1
Ft  f  ( xs  y s )
s 1
winning probability for player 1 at stage t:
p xt yt | xt 1 yt 1 ,..., x1 y1  p xt yt  Ft
The player who is the first to win n stages wins the whole match and gets a payoff of 1.
Equilibria
No fatigue (f=0)
a finitely repeated game with a dominant strategy
low (0)
high (1)
low (0)
1/2, 1/2
(1-a)/2, (1+a)/2
high (1)
(1+a)/2, (1-a)/2
1/2, 1/2
Prediction: burnout
Fatigue (f >0)
- the stage game still has a dominant strategy (high effort)
- payoffs acquire history dependence
Prediction: strategic choices of effort (low effort is optimal sometimes)
Experimental setup
This is a model-induced experiment
Instructions basically explain the model using references to sports as examples
Subjects: undergraduate students from Florida State University
Interface: separated computer terminals, zTree (Fischbacher)
Treatments
low advantage/high fatigue
(4 matches)
high advantage/low fatigue
(4 matches)
1 (n=4)
a = 0.2, f = 0.12
a = 0.6, f = 0.06
2 (n=2)
a = 0.2, f = 0.4
a = 0.6, f = 0.2
3 (n=6)
a = 0.2, f = 0.08
a = 0.6, f = 0.04
4 (n=2)
a = 0.2, f = 0
a = 0.6, f = 0
Random re-matching after each match; 32 matches total (192 time periods)
Treatment 4 is the no-fatigue treatment
Experimental hypotheses
[The hypotheses are based in the model]
1. Burnout without fatigue (basic rationality, clarity of instructions)
2. More likely low effort in longer matches
3. More likely low effort for higher f
4. More likely high effort for higher a
5. More likely low effort for higher net fatigue F
Key question: when subjects are aware of the presence of fatigue, will
they nevertheless burn out or choose low effort at least sometimes?
Results I
Summary statistics: % of high and low effort
In treatments 1-3 % of high effort is different between low advantage/high fatigue
and high advantage/low fatigue cases at less than 1% significance level. In
treatment 4 the difference is rejected at 5%.
Regression analysis
The model
Results
*
All estimates except (*) are significant at less than 1% level
Conclusions
1. Subjects do optimize effort when they are aware of fatigue,
at least in the artificial setting
2. Subjects’ decisions depend on their standing in the series
3. Extensions:
a) compare the behavior with actual equilibrium predictions;
b) experiments with real tasks