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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. .
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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= ( - , ) 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%
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
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.
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.
