Ultimatum Game

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? Ariel Rubenstein (1982)

showed that there exist a unique subgame perfect Nash equilibrium solution to this problem D= (π - ε , ε)

So the rational solution was predicting that proposer should offer the smallest possible share and responder would accept it.

Experimental data is inconsistent !

Güth, Schmittberger, Schwarze (1983) They did the first experimental study on this game. The mean offer was 37% of the “pie”

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%

Analyzed data by Spiegel et al. (1994)

Güth et el. Experiment 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

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

Further experiments reveal that both explanations have some validity

Kahneman,Knetch,Thaler (1986b)investigated two questions Will proposers be fair even if their offers can not

be rejected. Subjects had to divide $20 either by 18 and 2 or

equal splits. Of the 161 subjects, 122 (76%) divided it evenly

Will subjects sacrifice money to punish a proposer who behaved unfairly to someone else The answer was yes by 74%

Details of second experiment.

Same subjects were told they would be matched with two of the previous proposers 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

Replicator dynamics, is a system of deterministic difference or differential equations in bilogical models.

Neutrally stable strategy 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 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)

1- p1 + p2 p1>=q2 & P2 >= q1 1 - p1 p1>=q2 & p2 < q1 P2 p1< q2 & p2>= q1 0 p1 < q2 & p2 < q1

In the mini game with only two possible offers l, h : 0< l < h < 1/2 Assigning four strategies G1 to G4 to G1= (l,l) : reasonable G2 =(h,l) G3 = (h,h) : fair G4 = (l,h) : gready or..

…

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

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

If we assume the average offer of an h-proposer to an l-responder is lowered by an amount a 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

In a population of n players 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 ?

If the proposer can sometime obtain information like what offers have been

accepted by the responder in the past,

Then the process would lead again to the evolution of fairness If a large fraction w of players

is informed about any one accepted offer

Conclusion?!

This agrees with findings on the emergence of the cooperation and bargaining behavior.