Bargaining and Coalition Formation Dr James Tremewan (james.tremewan@univie.ac.at) Early experiments on non-cooperative bargaining models
Early experiments on non-cooperative bargaining models Early experiments on non-cooperative bargaining models • An Experimental Analysis of Ultimatum Bargaining, G¨ uth et al (1982). • Looks at one round of bargaining and finds significant deviations from predictions. • Testing Noncooperative Bargaining Theory: A Preliminary Study, Binmore et al (1985). • Looks at two rounds of bargaining and fnds results more supportive of predictions. • A Further Test of Noncooperative Bargaining Theory: Comment, Neelin et al (1988). • Looks at two, three, and five rounds of bargaining, and... • An Experimental Study of Sequential bargaining, Ochs and Roth (1989) • Investigates the experiments of the previous three papers more methodically and gains new insights. 2/19
G¨ uth et al (1982) An Experimental Analysis of Ultimatum Bargaining, G¨ uth et al (1982) • Paper includes the first published experiment on the Ultimatum Game (also another bargaining experiment). • Subjects were graduate students of economics (apparently unlikely to be familiar with game theory). • Each played one UG as either proposer or receiver for somewhere between four and 10 DM. • Same subjects repeat the experiment one week later with (possibly) different amount and partner, but same role. 3/19
G¨ uth et al (1982) Results • Each experiment consisted of 21 games. • Average demands: 65% (naive); 69% (experienced). • 50/50 splits: 7 (naive); 3 (experienced). • Rejections: 2 (naive); 6 (experienced). • Demands were far from SPE prediction (100%). • Rejections occured, counter to SPE prediction. • With experience, demands moved closer to SPE, but number of rejections increased. 4/19
Binmore et al (1985) Testing Noncooperative Bargaining Theory: A Preliminary Study, Binmore et al (1985) • Two stage game: • Stage 1: Cake is worth 100 pence. Player 1 makes proposal ( X ). Player 2 accepts (1 receives X , 2 receives 1 − X ) or rejects (game continues). • Stage 2: Cake is worth 25 pence. Player 2 makes proposal ( X ′ ). Player 1 accepts (1 receives X ′ , 2 receives 1 − X ′ ) or rejects (both receive zero). • SPE outcome: Player 1 demands 74-76 pence and Player 2 accepts. • After initial play (Game A), subjects exchange roles (Game B). 5/19
Binmore et al (1985) Results: Opening Demands 6/19
Neelin et al (1988) A Further Test of Noncooperative Bargaining Theory: Comment, Neelin et al (1988) • Subjects (intermediate micro-theory students) with played either two, three, or five round alternating offer bargaining games. • Two rounds: Cake worth $5, then $1.25. • Three rounds: Cake worth $5, then $2.50, then $1.25. • Five rounds: Cake worth $5, then $1.70, then $0.58, then $0.20, then $0.07. • Amounts chosen such that SPE predicts demand of $3.75 in first round in each game. 7/19
Neelin et al (1988) Results: First round 8/19
Neelin et al (1988) Results • Two round game replicates Game B of previous study: modal demand as per SPE. • Three round game: modal demand equal split... fairness? • Five round game: modal demand leaves $1.70 for Player 2... =size of round two cake. • Common thread: In each case Player 1 leaves Player 2 the maximum they could receive if the game continues to the next round. 9/19
Interlude Interlude: the story so far • Subjects do exploit their strategic advantage to some extent in longer games, when they understand it (e.g. after experience, or micro-theory class). • In two round games, SPE was a good predictor. • In longer games, less so. Subjects seem to only look one round ahead, but still reason to some extent a la game theory. • In the UG, initial demands are typically a long way from SPE prediction, and substantial offers are rejected. Why the difference? Possibilities: • In equilibrium, Receiver is indifferent between accepting and rejecting. For small offers they may reject at almost no cost. Proposer must guess minimum to make rejection costly. • In SPE, initial offer in UG is blatently unfair... in longer games, less so. • In longer games there is a new focal point (the size of cake in second round); in UG only zero offer or 50-50 split. 10/19
Interlude Interlude: the story so far • Difficult to draw conclusions from comparing these studies: • Different subject pools (level of knowledge of game theory etc.). • Different experimental instructions: • Binmore et al: ”YOU WILL BE DOING US A FAVOUR IF YOU SIMPLY SET OUT TO MAXIMIZE YOUR WINNINGS.” • Neelin et al: ”You will be discussing the theory this experiment is designed to test in class.” • Different degrees of anonymity: • G¨ uth et al: subjects could see each other and used pencil and paper. • Binmore et al: subjects interacted via computer. • Multiple things changed at once, so cannot isolate cause of effects, e.g. Neelin et al changes number of rounds and discount factors simultaneously. 11/19
Ochs and Roth (1989) Ochs and Roth (1989) • Ameliorates some of the concerns in the previous slide. • Introduces a new factor: • In some treatments the two subjects have different discount factors. • Achieved by altering exchange rates: subjects always bargain over 100 tokens, but exchange rate varies between subjects and between rounds. • Subjects undergraduate economics students; played 10 rounds of stranger-matching in one role only; mediated by computer (each role in seperate room); one round paid at random. • 4 x 2 design: • Different discount rates: ( δ 1 , δ 2 ) ∈ { (0 . 4 , 0 . 4) , (0 . 6 , 0 . 4) , (0 . 6 , 0 . 6) , (0 . 4 , 0 . 6) } . • Two or three period games. 12/19
Ochs and Roth (1989) 13/19
Ochs and Roth (1989) Results: SPE as prediction • Poor point prediction of first period offer: average within 2 standard deviations of SPE only in Cell 1. • Also fails to account for qualitative differences: • In 2-period game δ 1 should not affect offer, but significant difference between 1-2 and 3-4 (”more patient” P1 offers less). • Holding discount factors constant, initial offer should be lower in 3-period than 2-period game: true in only two of four comparisons. • SPE predicts a qualitative difference in means in 25 comparisons: direction correct in 17 out of 25 (coin toss correct 12.5 times). • SPE predicts zero rejections, but 125/760 first period offers rejected (10/76 in final round). 14/19
Ochs and Roth (1989) Observed (unpredicted) regularities • Consistent first-mover advantage in all cells. • Discount factor of player 1 had influence even in 2-period games. • Substantial fraction of offers rejected. • Observed mean agreements deviated from SPE in direction of equal division. • A substantial fraction of rejected offers were followed by counterproposals where the new proposer got less than they had just rejected - strong evidence that utility is determined by more than simply own earnings. 15/19
Ochs and Roth (1989) Explaining the regularities • Authors discuss ”mimimum acceptance thresholds” (in absolute terms): can explain first three regularities, but not the last two. • Authors claim that disutility of ”unfair” offers, where unfairness is deviation from 50-50 split as the most promising explanation of their data (explains all five regularities). • Note that this does not mean that ”fair” offers imply that proposer is fair, just uncertainty over others’ fairness concerns. • Evidence of heterogenous fairness concerns as private info: • 16/76 P1s always offered 50% in first period. • 28/76 P1s started high in first round and reduced offers in later rounds if accepted. • 14/76 P1s started low in first round offered more in later rounds if rejected. • Some genuine nice guys, and others searching for the biggest cut they can get. 16/19
Discussion General Discussion of Experimental Evidence • Experimental results appear to find significant deviations from game theoretic predictions. • Offers are too fair. • Not enough efficiency (some offers are rejected). • Failure to perform backward induction more than one round. • But so far the theory has been about perfect information games; have the experiments involved perfect information? • These experiments involved perfect information about payoffs... but not about preferences! • It is preferences over payoffs that determine actions! • Way forward: • Build model of fairness concerns, and allow for imperfect information about other’s preferences. • Models of bounded rationality (not for this course). • Game Theory is dead! Long live Game Theory! 17/19
Mid-term Exam Mid-term Exam • Part 1: Maths • Find the Nash Bargaining solution given a particular set of utilities or profits. (25%) • Solve a slight variation on the alternating bargaining model you have seen in slides. (25%) • Part 2: Short essays. Four of the following six topics will be in the exam. Write a short essay on two. (25% each) • Explain the differences between the axiomatic and non-cooperative approaches to modeling bargaining. What are the advantages of each? • What are some testable predictions of the Nash Bargaining Solution? To what extent are these predictions supported by experimental or empirical evidence? • How can infinitely repeated alternating bargaining games be implemented in the laboratory? How well does the experimental evidence support the predictions? *(Not in slides) 18/19
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