Modeling Bargaining Behavior in Ultimatum Games

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

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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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)


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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


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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


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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

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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)

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1

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95


10

88

8

5

3

Panel E: 3rd round data in O chs/Roth(1989)

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0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95


4

2

1 1

Panel D: 2nd round data in O chs/Roth(1989)

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0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95


4

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2 2

Panel F: Summary of data from five panels

0

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0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95


121

2415

189

11

Panel B: Guth/Tietz(1988)

0

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1

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95


1

5

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3

2 1


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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


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Figure 3: Assumptions about Player 2s Behavior (A)

1

1

Fraction Demanded by Player 1Fre

quen

cy o

f A

ccep

tanc

e


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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


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Figure 4: Assumptions about Player 2s Behavior (B)

1

1


Fre

quen

cy o

f A

ccep

tanc

e


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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


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Specification of Probability of Acceptance Function

2

211

)2

211(2

2

2

2 a

ad

af


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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


a=0 a=0.3 a=0.5 a=0.8a=1 a=5 a=10 a=100


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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(),(


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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


Freq

uenc

y of

Acc

epta

nce

by

Play

er 2

f(true) f(LSE) f(MLE)


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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

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0.8

1

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95


f(true) f(LSE) f(M LE)


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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


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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


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Figure 7: Dependence of Optimal Demand d on Parameter a

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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)


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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


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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


0

0.05

0.1

0.15

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0.25

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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


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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?


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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)


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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


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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


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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


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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


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Figure 9: Dynamic Model Parameter Estimates from Player 1 Decision Optimization

0

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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


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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


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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


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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


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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)


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Figure 10: Time Series Charts of Actual and Dynamic Model Demands of Player 1

Pair 3

0

0.1

0.2

0.3

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0.5

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0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 20

Round Number

Re

lati

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De

ma

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of

Pla

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r 1

Actual Demand Dyn. Model Demand Static Model


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Pair 4

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Round Number

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Pair 5

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Round Number

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Pair 6

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0 2 4 6 8 10 12 14 16 18 20

Round Number

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Pair 7

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Pair 8

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Round Number

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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)


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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


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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)


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Figure 11: Model Comparison: Distribution of Rewards Between the Two Players

0

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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


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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)


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Figure 12: Efficiency and Probability of Acceptance by Player 2

0

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Subgame Perf. Fairness Static Model andActual

Dynamic Model

Models

Eff

icie

ncy

/Acc

epta

nce

Pro

bab

ilit

y


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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?


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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


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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

Modeling Bargaining Behavior in Ultimatum Games

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