Bargaining and Coalition Formation Dr James Tremewan (james.tremewan@univie.ac.at) Cooperative models of bargaining
Introduction An preliminary clarification: what do people bargain over? • Key questions: • Do people bargain over money or ”utility”? • Can we make interpersonal comparisons of utility? 2/38
Introduction Are interpersonal comparisons of utility possible? 3/38
Introduction Tiresias (Mark Rothko) 4/38
Introduction What is a utility function? A very brief outline. 1 • We assume that people have preferences over outcomes (and lotteries over outcomes), and that these preferences guide decision making. • Someone who prefers apples over oranges will choose an apple rather than an orange. • If preferences satisfy certain conditions (e.g. transitivity) they can be represented by a ”nice” utility function that allow us to use the tools of Expected Utility Theory (known as a ”von Neuman-Morgenstern” utility function). • Non-uniqueness: If a set of preferences can be represented by a utility function U ( x ), it can also be represeted by utility functions aU ( x ) + b where a ≥ 0. 1 For more details see, for example, Mas-Colell et al (1995), chapters 1,3, and 6. 5/38
Introduction Are interpersonal comparisons of utility possible? • Zeus and Hera had an argument over whether men or women enjoyed sex more. The only way they found to resolve this question was to ask Tiresias, who was originally a man but had been turned into a woman by Hera, and was later changed back into a man. • The choice of specific utility function for each person is arbitrary, and the utility associated with any particular outcome could be assigned any value for each person. • Furthermore, identical preference orderings result in identical choices and can be represented by the same utility function, but we can say nothing about the intensity of changes in outcomes. • Two people have identical preferences orderings and prefer apples to oranges. They will make the same decisions in all circumstances, but when when forced to take an orange rather than an apple, one may suffer dreadfully while the other is only mildly put out. 6/38
Introduction Interpersonal comparisons: Summary • When considering bargaining outcomes we will often be interested in ideas of equality. In both theory and empirical data we must always be aware of whether we are talking about equality of outcomes (e.g. money) or utility. • Two people simultaneously pick up a 10 note from the footpath. Should they share it 50-50? • What if one is homeless and the other a millionaire? • Our theoretical framework does not allow interpersonal comparisons of utility, but one can often compare outcomes. • Possibilities for interpersonal comparisons of utility: • Subjective reports of happiness or satisfaction (but does one person’s ”very happy” describe the same ”reality” as another’s?) • Neurological or physiological measurement? 7/38
The Axiomatic Approach The Bargaining Problem • Two bargainers (players), i ∈ { 1 , 2 } . • Set of possible agreements A , and disagreement event D . • Each player has ”well behaved” preference ordering over A ∪ { D } such that we can assign each a vNM expected utility function u i . • Let S ⊂ R 2 be the set of all utility pairs that can be outcomes of agreements, and d i = u i ( D ). • Nash (1950) defines a bargaining problem as the pair � S , d � where • d ∈ S • there exists s ∈ S such that s i > d i for i = 1 , 2 (i.e. both players can benifit from bargaining). • S is compact (closed and bounded) and convex (allows us to solve maximization problems on set). 2 • Note that bargaining occurs purely over utilities . 2 note that these assumptions can be justified by allowing ”probabilistic agreements.” 8/38
The Axiomatic Approach The Bargaining Problem ✻ ✂ ❅ ✂ ❅ ❅ ✂ S ❇ ✂ u 2 ❇ ❇ ❇ d ◗ r ◗ ✁ ◗ ✁ ◗ ✁ ✲ u 1 9/38
The Axiomatic Approach Bargaining Solutions • Let B be the set of all bargaining problems � S , d � . • A bargaining solution is a functon f : B → R 2 that assigns to each bargaining problem � S , d � ∈ B a unique element of S • Instead of explicitly modelling process, Nash’s approach was to identify some characteristics ”reasonable” solutions should have ( axioms ) and define a solution as an outcome that satisfied those characteristics. • The Nash bargaining solution is the unique element of S that satisfies a set of four particular axioms. 10/38
The Axiomatic Approach Nash’s Axioms: Invariance to Equivalent Utility Representations (INV) • As previously pointed out, there are many different utility functions that can represent the same preference order over outcomes. • Loosely speaking, INV states that the choice of utility functions should not affect the outcome represented by the solution. • Formally: Suppose that the bargaining problem � S ′ , d ′ � is obtained from � S , d � by the transformations s i → α i s i + β i , where α i > 0. Then f i ( S ′ , d ′ ) = α i f i ( S , d ) + β i . 11/38
The Axiomatic Approach Nash’s Axioms: Symmetry (SYM) • It is assumed that any asymmetry in the players bargaining ability is captured by S and d . • It therefore seems reasonable that two players in the same positions should experience the same outcome. • A bargaining problem � S , d � is defined to be symmetric if d 1 = d 2 and ( s 1 , s 2 ) ∈ S if and only if ( s 2 , s 1 ) ∈ S . • If the bargaining problem � S , d � is symmetric, then f 1 ( S , d ) = f 2 ( S , d ) • Note that this has nothing to do with ”fairness”, just that relabelling should not not alter the strategic situation. 12/38
The Axiomatic Approach Nash’s Axioms: Independence of Irrelevant Alternatives (IIA) • Suppose for a given set of alternatives in a bargaining problem a particular outcome is chosen as the bargaining solution. Now if we define a new problem by removing one or more of the alternatives which were not the bargaining solution of the original problem , then the new solution will be the same as the old one. • Formally: If � S , d � and � T , d � are bargaining problems with S ⊂ T and f ( T , d ) ∈ S , then f ( S , d ) = f ( T , d ). 13/38
The Axiomatic Approach Nash’s Axioms: Pareto Efficiency (PAR) • Players should not agree on a particular outcome if one of them can be made better off without harming the other. • Formally: Suppose � S , d � is a bargaining problem, s ∈ S , t ∈ S , and t i > s i for i = 1 , 2. Then f ( S , d ) � = s . 14/38
The Axiomatic Approach The Nash bargaining solution • Remarkably, the preceding four axioms identify a unique solution for any bargaining problem. • Theorem: There is a unique bargaining solution f N : B → R 2 satisfying the axions INV, SYM, IIA, and PAR. It is given by f N ( S , d ) = arg max ( s 1 − d 1 ) ( s 2 − d 2 ) . ( d 1 , d 2 ) ≤ ( s 1 , s 2 ) ∈ S • Proof: See Osborne and Rubinstein pgs 13 & 14. Go through this at home (some of the simpler parts of the proof may be in the test, but you will not be expected to be able to reproduce it all.) 15/38
The Axiomatic Approach Application: Dividing a Dollar: The role of disagreement points • Two individuals can divide a dollar in any way they wish. If they fail to agree, they receive d i for i = 1 , 2. They may discard some of the money. Players are expected value maximisers, i.e. u i = x where x is their share of the money. • A = { ( a 1 , a 2 ) ∈ R 2 : a 1 + a 2 ≤ 1 and a i ≥ 0 for i = 1 , 2 } (= S ) • D = ( d 1 , d 2 ) • PAR implies that no money is wasted: if player 1 receives x 1 then player 2 receives x 2 = 1 − x 1 . • f N ( S , d ) = arg max ( x 1 − d 1 ) (1 − x 1 − d 2 ). 16/38
The Axiomatic Approach Application: Dividing a Dollar: The role of disagreement points • Solution: x 1 = 1+ d 1 − d 2 , x 2 = 1 − d 1 + d 2 2 2 • d 1 = d 2 ⇒ each player receives half (implied directly by SYM). • A player’s share is increasing in their own disagreement payoff (outside option) and decreasing in the other player’s disagreement payoff. • Player’s have an incentive to overstate their outside option: calls into question perfect information about other’s utility from outcomes. 17/38
The Axiomatic Approach Application: Dividing a Dollar: The role of risk-aversion • Two individuals can divide a dollar in any way they wish. If they fail to agree, they both get nothing. They may discard some of the money. Players care only about the amount they get and prefer more rather than less. • A = { ( a 1 , a 2 ) ∈ R 2 : a 1 + a 2 ≤ 1 and a i ≥ 0 for i = 1 , 2 } • D = (0 , 0) • Assume the players’ preferences can be represented by the utility functions u i = x r i where r 1 ≥ r 2 , i.e. player 2 is more risk averse than player 1. 18/38
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