Transcript Ultimatum Game - Indiana University

Ultimatum Game
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Two players bargain (anonymously) to divide a
fixed amount between them.
P1 (proposer) offers a division of the “pie”
P2 (responder) decides whether to accept it
If accepted both player gets their agreed upon
shares
If rejected players receive nothing.
What do game theorists say?
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Ariel Rubenstein (1982)
 showed that there exist a unique subgame perfect Nash
equilibrium solution to this problem
 D= ( -  , )
So the rational solution was predicting that proposer
should offer the smallest possible share and responder
would accept it.
Experimental data is inconsistent !
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Güth, Schmittberger, Schwarze (1983)
They did the first experimental study on this game.
 The mean offer was 37% of the “pie”
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Since then several other studies has been conducted to
examine this gap between experiment and theory.
Almost all show that humans disregard the rational
solution in favor of some notion of fairness*.
The average offers are in the region of 40-50% of the pie
 About half of the responders reject offers below 30%
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Analyzed data by Spiegel et al. (1994)
Güth et el. Experiment
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A sample of 42 economics students was divided by two.
By random one group was assigned to the role of player 1.
The other took role of player 2
P1’s had to divide a pie C which was varied between DM4
and DM10
A week later the subjects were invited to play the game
again
In the first experiment the mean offer was .37C
In the replication after a week, the offer were somewhat
less generous,but still considerably greater than epsilon.
Mean offer was .32 C
Experiment 1
Experiment 2
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When a responder rejects a positive offer, he signals that his
utility function has non-monetary argument.
When an allocator makes high offer it is either
A taste for fairness
 Fear of rejection
 Both
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Further experiments reveal that both explanations have
some validity
Kahneman,Knetch,Thaler (1986b)
investigated two questions
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Will proposers be fair even if their offers can not
be rejected.
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Subjects had to divide $20 either by 18 and 2 or
equal splits.
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Of the 161 subjects, 122 (76%) divided it evenly
Will subjects sacrifice money to punish a
proposer who behaved unfairly to someone else
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The answer was yes by 74%
Details of second experiment.
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Same subjects were told they would be matched
with two of the previous proposers
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One of those who took $18 for himslef (U)
One of those who took $10 and split it evenly(E)
They could either get $6 and pay $6 to U
Or they could get $5 and pay $5 to E
74% decided to take the smaller reward.
Some background
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Replicator dynamics, is a system of deterministic
difference or differential equations in bilogical
models.
Neutrally stable strategy
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Does not require a higher payoff to win
Mutant can coexist(after it appears) with a neutrally
stable strategy in the system
It can not replace a neutrally strategy.
Assumptions of the model
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Pie is set to 1
Players are equally likely to be in either of the two roles
When acting as proposer, the player offers the amount p
When acting as responder, the player rejects any offer less
than q
share kept by proposer should not be smaller than his
demanding offer q as responder so 1- p>= q
Expected payoff for a player using S1=(p1,q1)
against a player using S2 = (p2,q2)
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1- p1 + p2
1 - p1
P2
0
p1>=q2 & P2 >= q1
p1>=q2 & p2 < q1
p1< q2 & p2>= q1
p1 < q2 & p2 < q1
In the mini game with only two possible
offers l, h : 0< l < h < 1/2
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Assigning four strategies G1 to G4 to
G1= (l,l) : reasonable
G2 =(h,l)
G3 = (h,h) : fair
G4 = (l,h) : gready or..
…
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Replicator equation is used to describe the change
in frequnecies x1, x2, x3
It resembels a population dynamics where
successful strategies spread either by cultural
imitation or biological reproduction.
Their claim is that :
reason dominates fairness
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Reasonable strategy G1 will
eventually reach fixation
Mixed population of G1 and
G3 players will converge to
pure G1 or G3
Mixed population of G1 and
G2 players will always tend to
pure G1
Mixed population of G2 and
G3 players are neutrally stable
Role of information: accepting low offer
affects reputation
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If we assume the average offer of an h-proposer to
an l-responder is lowered by an amount a
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in a mixture of h-proposers, G2 and G3, G3 dominates.
Depending on the initial condition, either the
reasonable strategy G1, or fair strategy G3 reaches
fixation
In the extreme case, when we h-proposer have full
information about responder, G3 reaches fixation
where as mixture of G1 and G2 are neutrally stable.
Having some information this time
fairness dominates
Full game : continuum of all strategies
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In a population of n players
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Individuals leave a number of offspring proportional
to their total payoff
Offspring adopt the strategy of their parents plus or
minus some small random error
Evolutionary dynamics leads to a state where all
players adopt strategies that are close to the
rational strategy
How about some Information ?
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If the proposer can sometime
obtain information
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like what offers have been
accepted by the responder in
the past,
Then the process would lead
again to the evolution of
fairness
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If a large fraction w of
players is informed about any
one accepted offer
Conclusion?!
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This agrees with findings on the emergence of the
cooperation and bargaining behavior.