Ultimatum Game <ul><li>Two players bargain (anonymously) to divide a fixed amount between them. </li></ul><ul><li>P1  (pro...
What do game theorists say? <ul><li>Ariel Rubenstein (1982)  </li></ul><ul><ul><li>showed that there exist a unique subgam...
Experimental data is inconsistent ! <ul><li>Güth, Schmittberger, Schwarze (1983) </li></ul><ul><ul><li>They did the first ...
Analyzed data by Spiegel et al. (1994)
Güth et el. Experiment <ul><li>A sample of 42 economics students was divided by two.  </li></ul><ul><li>By random one grou...
Experiment 1
Experiment 2
<ul><li>When a responder rejects a positive offer, he signals that his utility function has non-monetary argument. </li></...
Kahneman,Knetch,Thaler (1986b) investigated two questions <ul><li>Will proposers be fair even if their offers can not be r...
Details of second experiment. <ul><li>Same subjects were told they would be matched with two of the previous proposers </l...
Some background <ul><li>Replicator dynamics, is a system of deterministic difference or differential equations in bilogica...
Assumptions of the model <ul><li>Pie is set to 1 </li></ul><ul><li>Players are equally likely to be in either of the two r...
Expected payoff for a player using S1=(p1,q1) against a player using S2 = (p2,q2) <ul><li>1- p1 + p2  p1>=q2 & P2 >= q1 </...
In the mini game with only two possible offers  l, h : 0< l < h < 1/2 <ul><li>Assigning four strategies G1 to G4 to </li><...
… <ul><li>Replicator equation is used to describe the change in frequnecies x1, x2, x3 </li></ul><ul><li>It resembels a po...
Their claim is that :  reason dominates fairness <ul><li>Reasonable strategy G1 will eventually reach fixation </li></ul><...
Role of information: accepting low offer affects reputation <ul><li>If we assume the average offer of an h-proposer to an ...
Having some information this time fairness dominates
Full game : continuum of all strategies <ul><li>In a population of n players  </li></ul><ul><ul><li>Individuals leave a nu...
How about some Information ? <ul><li>If the proposer can sometime obtain information </li></ul><ul><ul><li>like what offer...
Conclusion?! <ul><li>This agrees with findings on the emergence of the cooperation and bargaining behavior.  </li></ul>
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Ultimatum

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Ultimatum

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

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