Learning Dynamics for
Mechanism Design
An Experimental Comparison of Public
Goods Mechanisms
P.J. Healy
[email protected]
California Institute of Technology
The Repeated Public Goods
Implementation Problem
• Example: Condo Association “special assessment”
– Fixed set of agents regularly choosing public good levels.
– Goal is to maximize efficiency across all periods
– What mechanism should be used?
• Questions:
– Are the “one-shot” mechanisms the best solution to the
repeated problem?
– Can one simple learning model approximate behavior in a
variety of games with different equilibrium properties?
– Which existing mechanisms are most efficient in the
dynamic setting?
Previous Experiments on
Public Goods Mechanisms I
• Dominant Strategy (VCG) mechanism experiments
– Attiyeh, Franciosi and Isaac ’00
– Kawagoe and Mori ’01 & ’99 pilot
– Cason, Saijo, Sjostrom, & Yamato ’03
– Convergence to strict dominant strategies
– Weakly dominated strategies are observed
Previous Experiments on
Public Goods Mechanisms II
• Nash Equilibrium mechanisms
– Voluntary Contribution experiments
– Chen & Plott ’96
– Chen & Tang ’98
– Convergence iff supermodularity (stable equil.)
• Results consistent with best response behavior
A Simple Learning Model
• k-period Best Response model
– Agents best respond to pure strat. beliefs
– Belief = unweighted average of the others’
strategies in the previous k periods
• Needs convex strategy space
– Rational behavior, inconsistent beliefs
– Pure strategies only
A Simple Learning Model: Predictions
– Strictly dominated strategies: never played
– Weakly dominated strategies: possible
– Always converges in supermodular games
– Stable/convergence => Nash equilibrium
– Can be very unstable (cycles w/ equilibrium)
A New Set of Experiments
• New experiments over 5 public goods mechanisms
–
–
–
–
–
Voluntary Contribution
Proportional Tax
Groves-Ledyard
Walker
Continuous VCG (“cVCG”) with 2 parameters
• Identical environment (endow., prefs., tech.)
• 4 sessions each with 5 players for 50 periods
• Computer Interface
– History window & “What-If Scenario Analyzer”
The Environment
• Agents: i  N
N 5
• Private Good: xi Public Good: y
Endowments: (i ,0)
2
• Preferences: ui ( xi , y )  bi y  ai y  xi
i  (ai , bi )
• Technology: y  x / 
• Mechanisms: mi  M i
y(m)  f (m1 , m2 ,, mn )
ti (m, y)  gi (m1 , mn , y(m))
xi  i  ti (m, y)
The Mechanisms
y   mi
• Voluntary Contribution
y   mi
• Proportional Tax
• Groves-Ledyard
• Walker
• VCG
ti 
iN
y   mi
iN
y   mi
ti 
iN
M i  R 2
iN
y
n
ti 
ti   mi
y
n
 y   n 1
 
n 2 n
mi  i 2   2i 

 mi 1  mi 1   y
mi  (aˆi , bˆi )
y  yNPO (aˆ, bˆ)
zi  y NPO\{i} (aˆ , bˆ)

 

n 1
2
2



ti   y   (b j y  a j y )     (b j zi  a j zi ) 
zi 
n
j i

  j i

Experimental Results I: Choosing k
• Which value of k minimizes the M.A.D. across all
mechanisms, sessions, players and periods?
Model
k=1
k=2
k=3
k=4
k=5
k=6
k=7
k=8
k=9
k=10
2-50
1.407
-
3-50
1.394
1.240
-
4-50
1.284
1.135
1.097
-
5-50
1.151
0.991
0.963
0.952
-
• k=5 is the most accurate
6-50
1.104
0.967
0.940
0.932
0.924
-
7-50
1.088
0.949
0.925
0.915
0.9114
0.9106
-
8-50
1.072
0.932
0.904
0.898
0.895
0.897
0.899
-
9-50
1.054
0.922
0.888
0.877
0.876
0.881
0.884
0.884
-
10-50
1.054
0.913
0.883
0.866
0.860
0.868
0.873
0.874
0.879
-
11-50
1.049
0.910
0.875
0.861
0.853
0.854
0.863
0.864
0.870
0.875
Walker Session 2 Player 1
15
Message
10
5
0
-5
-10
0
10
20
30
40
50
40
50
Period
Walker Session 2 Player 2
15
Message
10
5
0
-5
-10
0
10
20
30
Period
Walker Session 2 Player 3
15
Message
10
5
0
-5
-10
0
10
20
30
40
50
40
50
Period
Walker Session 2 Player 4
15
Message
10
5
0
-5
-10
0
10
20
30
Period
Walker Session 2 Player 5
15
Message
10
5
0
-5
-10
0
10
20
30
40
50
40
50
Period
Groves-Ledyard Session 1 Player 1
6
Message
4
2
0
-2
-4
0
10
20
30
Period
Experimental Results: 5-B.R. vs. Equilibrium
• Null Hypothesis: E[| m  BR |]  E[| m  EQ |]
t
i
t
i
t
i
t
i
• Non-stationarity => period-by-period tests
• Non-normality of errors => non-parametric tests
– Permutation test with 2,000 sample permutations
• Problem: If EQit  BRit then the test has little power
• Solution:
– Estimate test power as a function of ( EQit  BRit ) / 
– Perform the test on the data only where power is sufficiently large.
0.9
0.95
0.8
0.94
0.7
0.93
0.6
0.92
0.5
0.91
0.4
0.89
0.3
0.86
0.2
0.8
0.1
0.67
0
0
0.5
1
1.5
2
2.5
( -  )/
a
b
3
x
3.5
4
4.5
5
0
0
0.95
0
1
Prob. H False Given Reject H
Estimated Test Power
Simulated Test Power
5-period B.R. vs. Equilibrium
• Voluntary Contribution (strict dom. strats): EQit  BRit
• Groves-Ledyard (stable Nash equil): EQit  BRit
• Walker (unstable Nash equil): 73/81 tests reject H0
– No apparent pattern of results across time
• Proportional Tax: 16/19 tests reject H0
Interesting properties of the
2-parameter cVCG mechanism
• Best response line in 2-dimensional strategy space
Best Response in the cVCG mechanism
Convert data to polar coordinates 

•
i
• Dom. Strat. = origin, B.R. line = 0-degree line
s
, ri
s
Experimental Results III: Efficiency
• Outcomes are closest to Pareto optimal in cVCG
– cVCG > GL ≥ PT > VC > WK (same for efficiency)
– Sensitivity to parameter selection
• Variance of outcomes:
– cVCG is lowest, followed by Groves-Ledyard
– Walker has highest
• Walker mechanism performs very poorly
– Efficiency below the endowment
– Individual rationality violated 42% of last 10 periods
Discussion & Conclusions
• Data are consistent with the learning model.
– Repercussions for theoretical research
• Should worry about dynamics
– k-period best response studied here, but other learning
models may apply
• Example: Instability of the Walker mechanism
• cVCG mechanism can perform efficiently
• Open questions:
– cVCG behavior with stronger conflict between
incentives and efficiency
– Sensitivity of results to parameter changes
– Effect of “What-If Scenario Analyzer” tool
Efficiency Confidence Intervals - All 50 Periods
Efficiency
1
No Pub Good
0.5
Walker
VC
PT
Mechanism
GL
cVCG
Av e rage Public Good Le v e ls
9
Pers 1-50
Public Good Level
8
Pareto Optimal
Pers 41-50
7
6
5
4
3
2
1
0
VC
PT
GL
WK
Mechanism
VCG
VCG*
Standard Deviation of PG Levels
7
Periods 1-50
Standard Deviation
6
Periods 41-50
5
4
3
2
1
0
VC
PT
GL
WK
Mechanism
VCG
VCG*
Voluntary Contribution Mechanism
Results