modeling bargaining behavior in ultimatum games
DESCRIPTION
Modeling Bargaining Behavior in Ultimatum Games. Haijin Lin and Shyam Sunder Japan Association for Evolutionary Economics Ochanomizu, Tokyo, Japan March 25-26, 2000. Ultimatum Game. Two players divide a given amount (the pie =1) between them - PowerPoint PPT PresentationTRANSCRIPT
Modeling Bargaining Behavior in Ultimatum Games
Haijin Lin and Shyam Sunder
Japan Association for Evolutionary EconomicsOchanomizu, Tokyo, Japan
March 25-26, 2000
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
2
Ultimatum Game
Two players divide a given amount (the pie =1) between them
Player 1 proposes a division (d for self, 1-d) for player 2
Player 2 may either accept (in which case the proposal is implemented) or reject (in which case both players receive 0)
Game ends
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
3
Subgame Perfect Equilibrium Assume that payoff from the game is the
only argument in player 2s preferences Preferences are increasing in personal
payoffs Player 2 should accept any positive amount In subgame perfect equilibrium, player 1
demands all but the smallest fraction of the pie, and player 2 accepts this fraction
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
4
Laboratory Results Few observations correspond to the subgame
perfect equilibrium Modal Player 1 proposal is 50/50 split In most observations, Player 1s demands lie
in 50 to 70 percent range Player 2 rejects a significant number of
proposals that offer them positive amounts, even significantly positive amounts
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
5
Why This Discrepancy Between Data and Theory?
Guth, Schmittberger and Schwarze (1982) Average demand of player 1 = 67% Rejection rate by player 2 = 20% Explanations: subjects often rely on what
they consider a fair or justified result. Ultimatum difficult to exploit because Player
2 willing to punish Player 1 who asks for too much
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
6
Why This Discrepancy Between Data and Theory?
Binmore, Shaked, and Sutton (1985): 1-stage and 2-stage game with .25 discount
In 1-stage game, modal demand=75 %(vs.1) In 2-stage game, modal demand in round 1
=50% (vs.75%), rejection rate =15% Differences consistent with subgame perfect
Eq. predictions for the two games Useful predictive role for game theory
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
7
Why This Discrepancy Between Data and Theory?
Guth and Tietz (1988): two 2-stage games, with discounts of 90 and 10 percent
Discount 0.1, demand 76 to 67% (vs. 90%) Discount 0.9, demand 70 to 59 % (vs.10%) Game theoretic solution seems to have
predictive power only when its solutions are in socially acceptable range
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
8
Why This Discrepancy Between Data and Theory?
Neelin, Sonnenschein, and Spiegel (1988): two-, three- and five-period games
Data for all three games close to perfect eq. predictions of two-period games
Reject both Stahl/Rubinstein theory as well as the equal split model
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
9
Why This Discrepancy Between Data and Theory?
Ochs and Roth (1989): eight games, 2- and 3-period, different discount rates for players
Different aspects of the data are consistent with conclusions of different prior studies
Frequent disadvantageous counter proposals Perceptions of fairness may be important Anticipating fairness preferences of others
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
10
Why This Discrepancy Between Data and Theory?
Thaler (1988): utility includes arguments other than money
Guth and Tietz (1990): players shift between strategic and equitable thinking hierarchically (consider one at a time)
Kennan and Wilson (1993): games of incomplete informationModel uncertainty about otherspreferences
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
11
Why This Discrepancy Between Data and Theory?
Bolton (1991): utility function with two arguments, income and share of the pie
Forsythe, Horowitz, Savin, Sefton (1994): dictator game modal offer is subgame perf.; reject fairness as dominant factor
Kahneman, Knetch, Thaler (1986): subjects willing to sacrifice personally to punish-reward past behavior seen to be unfair-fair
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
12
What do we do? Reanalyze prior data to develop a model Abandon extreme assumption about Player
1s belief about Player 2s strategy Use data to estimate reasonable models
(static and dynamic) of Player 2s strategy and Player 1s beliefs
Compare predictions of the models on remaining data
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
13
To Those Who Share Their Data: Thank You!
Slembeck (1999): 19 fixed pairs play 20 consecutive rounds of single-stage ultimatum game
Guth et al. (1982) Guth and Tietz (1988) Ochs and Roth (1989) Neelin et al. (1988)
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
14
Player 2s Decisions
Do the chances of acceptance of a proposal by Player 2 depend on the size of Player 1s demand?
Reanalyze the data for the first and the last 10 rounds separately (no important change)
Result: Relative frequency of acceptance by Player 2 declines as Player 1 demands more
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
15
Figure 1: Frequency of Acceptance in Slembeck (1999) data(No. of observations at the top of each bar)
0
0.2
0.4
0.6
0.8
1
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
Fraction Demanded by Player 1
Fre
qu
ency
of
Acc
epta
nce
by
Pla
yer
2
5 4 3
7 171
88
33
16 1835
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
16
Figure 2: Frequency of acceptance in data from other prior studies(No. of the observations at the top of each bar)
Panel C: Neelin et al (1988)
0
0.2
0.4
0.6
0.8
1
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
Fraction Demanded by P layer 1
1
1 2
1 4
Panel A: Guth (1982)
0
0.2
0.4
0.6
0.8
1
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
Fraction Demanded by P layer 1
10
88
8
5
3
Panel E: 3rd round data in O chs/Roth(1989)
0
0.2
0.4
0.6
0.8
1
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
Fraction Demanded by P layer 1
4
2
1 1
Panel D: 2nd round data in O chs/Roth(1989)
0
0.2
0.4
0.6
0.8
1
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
Fraction Demanded by P layer 1
4
9
3
5
2 2
Panel F: Summary of data from five panels
0
0.2
0.4
0.6
0.8
1
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
Fraction Demanded by P layer 1
121
2415
189
11
Panel B: Guth/Tietz(1988)
0
0.2
0.4
0.6
0.8
1
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
Fraction Demanded by P layer 1
1
5
4
3
2 1
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
17
Player 1s Beliefs About Player 2s Decision Rule
Traditional Assumption Player 2 will accept all proposals d1 < 1
with probability 1 Player 2 will reject proposals d1 = 1
Yields subgame perfect equilibrium
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
18
Figure 3: Assumptions about Player 2s Behavior (A)
1
1
Fraction Demanded by Player 1Fre
quen
cy o
f A
ccep
tanc
e
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
19
Player 1s Beliefs About Player 2s Decision Rule
Alternative assumption: probability of acceptance declines in a straight line from d1=0 to d1=1.
With this belief, Player 1 maximizes his expected payoff if he demands a 50/50 split
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
20
Figure 4: Assumptions about Player 2s Behavior (B)
1
1
Fraction Demanded by Player 1
Fre
quen
cy o
f A
ccep
tanc
e
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
21
Player 1s Beliefs About Player 2s Decision Rule
More generally, define a class of decreasing functions of d1 including the above two as special cases
We pick rectangular hyperbolas that pass through points (1,0) and (0,1)
This is a single parameter (a) family of functions so we can easily estimate the parameter from the data
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
22
Specification of Probability of Acceptance Function
2
211
)2
211(2
2
2
2 a
ad
af
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
23
Figure 5: One Family of Probability Functions with Parameter a
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Fraction Demanded by P layer 1
a=0 a=0.3 a=0.5 a=0.8a=1 a=5 a=10 a=100
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
24
Estimation of Probability Function from Data
One half of Slembeck data (odd numbered rounds from odd numbered pairs, even numbered rounds from even pairs), 190
Least squares estimate: min Max. Likelihood Estimate:
max
n
iii adDS
1
2)),((
n
i
Di
Di
ii adadL1
1)),(1(),(
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
25
Figure 6: Static Model Estimated by Using Slembeck Data (1999)(LSE(a) = 1.57; MLE(a) = 1.46)
0
0.2
0.4
0.6
0.8
1
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
Fraction Demanded by Player 1
Freq
uenc
y of
Acc
epta
nce
by
Play
er 2
f(true) f(LSE) f(MLE)
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
26
Panel B: Relative Frequency and Estimated Model for Second Half of the Slembeck Data(LSE (a) = 1.42, MLE (a) = 1.34)
0
0.2
0.4
0.6
0.8
1
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
Fraction Demanded by P layer 1
f(true) f(LSE) f(M LE)
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
27
Statistically
Cannot reject the null hypothesis that the parameter estimates of a from the two halves of the Slembecks data are equal (Chows test)
Estimates from the whole sample:LSE(a) = 1.49MLE(a) = 1.4
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
28
Optimal Demand of Player 1 Conditional on a given value of parameter
a, we can derive the expected value maximizing demand of Player 1:
See Figure 7 For a = 1.4, d* = 0.62; a = 1.49, d* = 0.61
2
224222*
211
21211
a
aaaaaad
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
29
Figure 7: Dependence of Optimal Demand d on Parameter a
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Value of P arameter a
d (9A) d (9B)
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
30
How Do Player 1s Decisions Correspond to the Model
Two continuous lines in Fig. 8 are the expected value of demand d (MLE, LSE)
Vertical bars are the actual frequency of demand (modal d in range 0.4 < d 0.5)
Modal value less than the optimal point estimates (0.62, 0.61)
Crude inverted U correspondence of demand frequency to expected value
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
31
Figure 8: Expected Value and Actual Frequency of Demand (d) in Slembeck (1999) Data
(Frequency of Acceptance at the top of each bar)(LSE (a) = 1.49, MLE (a) = 1.40)
0
30
60
90
120
150
180
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95
Fraction Demanded by Player 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Frequency of Demand Rewards(LSE) Rewards(MLE)
1 1 1 0.857
0.836
0.614
0.333
0.1875 0.167
0.086
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
32
Static Vs. Dynamic Model Static model organizes data (compare to
perfect and fairness predictions) Assumed parameter a unchanged over
rounds of subject experienceacross pairs of playersbetween paired player 2 and belief of player 1
Can we do better by allowing learning and estimating separately for each pair?
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
33
Specifying the Dynamic Model jth player 2 adjusts her threshold acceptance value yjt by first
order adaptive processyjt = jyjt-1 + (1-j)dit
Represent jth player 2s decision in period t by Djt
Djt= 0 if dit > yit, and Djt = 1 if dit yit
ith player 1submits his demand dit= xit, leaving no money on the table
ith player 1 adjusts xit based on player 2s decision
xit+1 = xitiDjt i
(1-Djt)
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
34
Estimating the Dynamic Model
Use the data for each pair to estimate three parameters (yj0, j, i )
Use Excel Solver to obtain two sets of estimates by minimizing
(1) the sum of squared errors between actual and estimated player 1s decision: t (dit - d`it)2
(2) the sum of squared errors between actual and estimated player 2s decision: t (Dit - D`it)2
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
35
Table 1: Parameter Estimates from Dynamic Model(Pairs 1-10)
Parameter Estimates from Player 1 Decision OptimizationPair Y0 Alpha Beta Sum of Squared Errors R2
1 0.5 - - 0 -2 0.5 - - 0 -3 0.66 0.84 1 0.04 04 0.44 1 1.18 0.06 0.835 0.44 0.96 1.07 0.021 0.9146 0.42 0.97 1.04 0.068 0.8427 0.5 0.97 1.07 0.048 0.8978 0.63 0.83 1.06 0.006 0.9179 0.41 0.83 1.12 0.0257 0
10 0.55 0.85 1 0.018 0.38
Average/Overall 0.505 0.91 1.07 0.0287 0.598
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
36
Table 1: Parameter Estimates from Dynamic Model(Pairs 1-10)
Parameter Estimates from Player 2 Decision Optimization
Pair Y0 Alpha Beta Sum of Squared Errors1 0.5 - - 02 0.5 - - 03 0.55 0.84 1 24 0.5 0.9 1.08 75 0.47 0.95 1.08 96 0.47 0.95 1.08 67 0.38 0.96 1.07 58 0.56 0.77 1.02 89 0.5 0.64 1.09 9
10 0.55 0.85 1 4Average/Overall 0.5 0.86 1.05 5
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
37
Dynamic Parameter Estimates (Player 1 Decision Optimization) Player 2s initial threshold: 0.41-0.66 (0.51)
Player 2s adaptive parameter:0.81-1 (0.91)
Player 1s adaptive parameter: 1-1.12 (1.07)
Variation explained on average: 60 percent
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
38
Figure 9: Dynamic Model Parameter Estimates from Player 1 Decision Optimization
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 2 3 4 5 6 7 8 9 10
Pair Number
Est
imat
ed P
aram
eter
s an
d S
qu
ared
E
rro
r
Y(0)
Alpha
Beta
Sq.Error
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
39
Dynamic Parameter Estimates (Player 2 Decision Optimization) Player 2s initial threshold: 0.38-0.56 (0.5) Player 2s adaptive parameter:0.64-0.96
(0.86) Player 1s adaptive parameter: 1-1.09 (1.05) Errors in predicting Player 2 decision on
average: 25 percent
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
40
Dynamic Model Parameter Estimates from Player 2 Decision Optimization
0
0.2
0.4
0.6
0.8
1
1.2
1 2 3 4 5 6 7 8 9 10
Pair Number
Est
imat
ed P
aram
eter
s an
d S
qu
ared
E
rro
r (x
0.1) Y(0)
Alpha
Beta
Sq.Error
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
41
Model Comparisons
Compare various aspects of data against perfect equilibrium, fairness, global static, and dynamic modelsTime series of Player 1 decisionsCumulative frequency of Player 1 decisionsDistribution of rewards between Players 1 and 2Efficiency/Acceptance rate
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
42
Model Comparison: Time Series of Player 1 Decisions
Panels of Fig. 10 show actual and model (static and dynamic) time series of di for pairs 3-8 (pairs 1 and 2 used 50/50 splits)
Static model captures the mean of the process across all pairs
Dynamic model tracks time series better by using up more degrees of freedom (yet to deal with the dependency problem)
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
43
Figure 10: Time Series Charts of Actual and Dynamic Model Demands of Player 1
Pair 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12 14 16 18 20
Round Number
Re
lati
ve
De
ma
nd
of
Pla
ye
r 1
Actual Demand Dyn. Model Demand Static Model
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
44
Figure 10: Time Series Charts of Actual and Dynamic Model Demands of Player 1
Pair 4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12 14 16 18 20
Round Number
Re
lati
ve
De
ma
nd
of
Pla
ye
r 1
Actual Demand Dyn. Model Demand Static Model
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
45
Figure 10: Time Series Charts of Actual and Dynamic Model Demands of Player 1
Pair 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12 14 16 18 20
Round Number
Re
lati
ve
De
ma
nd
of
Pla
ye
r 1
Actual Demand Dyn. Model Demand Static Model
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
46
Figure 10: Time Series Charts of Actual and Dynamic Model Demands of Player 1
Pair 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12 14 16 18 20
Round Number
Re
lati
ve
De
ma
nd
of
Pla
ye
r 1
Actual Demand Dyn. Model Demand Static Model
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
47
Figure 10: Time Series Charts of Actual and Dynamic Model Demands of Player 1
Pair 7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12 14 16 18 20
Round Number
Re
lati
ve
De
ma
nd
of
Pla
ye
r 1
Actual Demand Dyn. Model Demand Static Model
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
48
Figure 10: Time Series Charts of Actual and Dynamic Model Demands of Player 1
Pair 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12 14 16 18 20
Round Number
Re
lati
ve
De
ma
nd
of
Pla
ye
r 1
Actual Demand Dyn. Model Demand Static Model
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
49
Model Comparison: Cumulative Frequency of Player 1 Decisions Figure 8A compares the cumulative
frequency of Player 1s decisions against perfect eq., fairness benchmarks, and the global static model
Fairness corresponds to the mode Global static captures the average (Add dynamic model)
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
50
Fig. 8A: Cumulative Frequency of Player 1 Rel. Demand: Subgame Perfect., Static Model, and Actual
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Player 1 Relative Demand
Cu
mu
lati
ve F
req
uen
cy
Optimal Static Model
Subgame Perfect
Actual
Fairness
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
51
Model Comparison: Distribution of Rewards Between Players
Figure 11 maps distribution of profits of players 1 and 2 in two dimensions
Actual data is closer to global static model
(add dynamic model)
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
52
Figure 11: Model Comparison: Distribution of Rewards Between the Two Players
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Reward to Player 1
Re
wa
rd t
o P
lay
er
2
Subgame Perf.
Fairness
Static Model
Actual
Dynamic Model
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
53
Model Comparison: Efficiency or Probability of Acceptance
Perfect equilibrium and fairness benchmarks have 100 percent rate of acceptance and therefore 100 percent efficiency
Through estimation criterion, global static model efficiency = actual acceptance rate (by definition)
(Add dynamic model efficiency)
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
54
Figure 12: Efficiency and Probability of Acceptance by Player 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Subgame Perf. Fairness Static Model andActual
Dynamic Model
Models
Eff
icie
ncy
/Acc
epta
nce
Pro
bab
ilit
y
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
55
Concluding Remarks Many arguments made for abandoning
perfect equilibrium model in favor of fairness, altruism, etc.
Instead, we could abandon assumption that Player 2 will accept any positive amountsEconomy of assumptionsGain in data organizing power
But why do Players 2 reject small amounts?
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
56
Why Reject Small Amounts?
Externalities in lab environment How isolated are subject beliefs,
expectations and behavior between lab and outside?
Experimental economics depends on an assumption of continuity across lab walls
If lab game is a piece of the larger game of life, rejection may be individually rational
04/22/23 Lin and Sunder, Modeling Bargaining Behavior in Ultimatum
Games
57
Experiments and Theory
Use laboratory data to infer subject beliefs Estimated beliefs can be plugged into the
models to compare against new data Mutual contributions between theory and
experiments Hope for convergence of models and data