Bargaining and Coalition Formation

Transcription

1 1 These slides are based largely on chapter 2 of Osborne and Rubenstein (1990), Bargaining and Markets Bargaining and Coalition Formation Dr James Tremewan (james.tremewan@univie.ac.at) 1

2 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. 2/17

3 3/17 The Bargaining Problem S u 2 d u 1

4 4/17 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.

5 5/17 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.

6 6/17 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.

7 7/17 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 (S, d) S, then f (S, d) = f (T, d).

8 8/17 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.

9 9/17 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.)

10 10/17 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 ).

11 11/17 Application: Dividing a Dollar: The role of disagreement points Solution: x 1 = 1+d 1 d 2, x 2 2 = 1 d 1+d 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.

12 12/17 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.

13 13/17 Application: Dividing a Dollar: The role of risk-aversion S = {(s 1, s 2 ) R 2 : (s 1, s 2 ) = (a r 1 1, a r 2 2 ) for some (a 1, a 2 ) A} u 2 d = (0, 0) S u

14 14/17 Application: Dividing a Dollar: The role of risk-aversion 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 r 1 ) (1 x) r 2. Solution: x 1 = r 1 r 1 +r 2, x 2 = r 2 r 1 +r 2 A player s share is decreasing in their degree of risk aversion, and increasing in the other s risk aversion (here big r i less risk aversion). Note that this has nothing to do with the fact that u 1 (x) u 2 (x) x. It is easy to see the solution to the maximisation problem is unchanged if u 2 = 100x r 2 (as implied by INV).

15 15/17 Application: Negotiating wages Negotiation between firm and union over wages and employment. The union represents L workers who can obtain a wage of w 0 outside the firm. Firm s production function: f (l) where l is the number of workers employed. Assume f is strictly concave, f (0) = 0, and f (l) > l(w 0 ) for some l. Price of output =1. Agreement is a wage-employment pair (w, l). Utility of union is total earnings of members: lw + (L l)w 0. Utility of firm is profit: f (l) lw. Disagreement payoffs: Union: Lw 0, Firm: 0.

16 16/17 Application: Negotiating wages Surplus to be divided is firm s profit. PAR l chosen to maximise f (l) lw 0 l. Diff between firm s utility and disagreement payoff: f (l ) l w Diff between union s utility and disagreement payoff: l (w w 0 ) Wage is determined by: arg max w w 0 (f (l ) l w) l (w w 0 ) Solution: w = (w 0 + f (l )/l )/2 The negotiated wage is the average of the outside wage and the average product of labour.

17 17/17 Alternative bargaining solutions Drop the symmetry axiom. For each α (0, 1) we can define a solution by arg max (s 1 d 1 ) α (s 2 d 2 ) (1 α). (d 1,d 2 ) (s 1,s 2 ) S The variable α is often interpreted as the relative bargaining power of player 1. Replacing IIA with a monotonicity axiom gives the Kalai-Smorodinksy solution: has some attractive features.