ackground notes on bargaining Cooperative bargaining - bargaining theory related to game theory - nice book is Muthoo (1999) argaining theory with applications; - also, Dixit and Skeath's game theory text has a nice introduction. - Nash worked on both cooperative and non-cooperative games; - Cooperative game? binding agreements can be written (enforcement) - Cooperative bargaining? focus on outcomes, not process argaining in a household framework: - 2 adults, A, - agreement drawn from some set P - frequently, in household models, expenditure on private vs public goods (3-good model) Page 1 of 7
- disagreement outcomes d = { d, S d } Econ 400 (divorce, non-cooperative outcome within marriage) - preferences over outcomes: represented by cardinal utility functions u A, u - utility payoff set U={ u ( a), u ( a) a P} - pairs of utilities assigned to each possible outcome in P; utility of disagreement point is u = i u 1 ( d I ), i = A, - impt: care only about outcomes, not bargaining process itself - As if bargaining over utility levels Ass'ms: 1. U is closed, bounded, convex; 2. u U 3. u = ( u, u ) U such that u > u, i = A, A argaining problem: (U,u ) i i Page 2 of 7
argaining solution: a rule applied to all bargaining problems, to pick a unique point from U as the outcome: 2 s:( U, u) R Nash bargaining solution 4 requirements for reasonable solution - together imply a unique solution 1. efficiency: must be no feasible bargain which makes at least one party better off, other no worse off; 2. linear invariance: solution should be invariant to linear transformations of utility functions (cardinal utility, remember); Page 3 of 7
3. symmetry: background: bargaining game is symmetric if i) u = u and ii) U, the set of (utility) outcomes, is symmetric about the 45 degree line; alternatively, ( uu, ) Uif and only if ( u, u) U 1 2 2 1 axiom: if bargaining game is symmetric, solution outcome must be symmetric: s ( U, u) = s ( U, u). 4. Independence of irrelevant alternatives (IIA) (always tricky) -about psychology and the negotiation process. - Consider two bargaining games ( Uu, ) and ( U*, u, ) with U* U. If suu (, ) U*, then su ( *, u) suu (, ) =. So existence of utility pairs which are not part of the solution will not affect the solution. Reasonable? Page 4 of 7
u, u Nash's theorem? The unique solution meeting these requirements is the utility pair ( u, u ) which solves a max ( u u )( u u ) s.t. u u, i= A, and ( u, u ) U A i i Characterization of Nash solution? Each player obtains disagreement utility + half of the surplus generated by the cooperation (symmetry). Pictures? - for symmetry, bargaining set is symmetric about 45 degree line; line from disagreement point to tangency of level curve of objective function with (concave)pareto frontier bisects tangent (extended to axes). If have transferable utility, so that utility can be transferred between players on a one-to-one basis using money, then the standard bargaining solution is particularly simple: Page 5 of 7
1. max joint utility - utility frontier has linear segment; 2. calculate individual utilities - split the surplus equally; 3. calculate money transfers to accomplish achieve this split. Other solutions? 1. asymmetric Nash: objective function is ( u ) A ( ) A u π A u u π exponents reflect bargaining power; outcome depends on relative bargaining power, with more "powerful" agent obtaining higher share of surplus. - outcome has A A A u u π u u π = Page 6 of 7
2. Kalai-Smorodinsky solution - doesn't satisfy IIA: consider outcome if one agent can make a take-it-or-leave-it offer. Outcome would be on Pareto frontier, at point where proposer has all surplus and other has disagreement outcome. K-S solution chooses point on Pareto frontier giving all agents utilities proportional to the expected utilities that would obtain if each agent equally likely to be proposer of TOL offer. (ref: MWG, p. 844) So what? Assigns utilities - via consumption bundles - to household members based on characteristics of those members - especially, disagreement outcomes. Nash solution pins down unique efficient point. Alternative approach: just rely on efficiency: assume household reaches a point on the Pareto frontier. Page 7 of 7