Developing Principles in Bargaining

Motivation Consider a purely distributive bargaining situation where impasse is costly to both sides How should we determine who gets what? Are there principles both sides can agree to that can help lead to agreement. How does the possibility of outside resolution (arbitration) affect outcomes?

Overview Principles of “good” bargaining agreements Bargaining “solutions” from these principles Arbitration Mediation

Utility Bargaining typically involves many facets In labor negotiations: Wage levels Benefits Employment security Contract duration Workplace conditions Assessing these requires a comparison of the cost to the one side versus the benefit to another of various concessions

Utility again Therefore, it helps to think of a possible agreement as providing some utility to each side Utility will not generally line up with dollar costs. Example: It costs management $2 million/year to improve benefits The value to employees of these benefits is estimated to be $3 million/year. Thus, even though benefits are a concession, there is an integrative aspect to the negotiation.

A Bargaining Problem A bargaining problem is a question of how to allocate utilities (which we’ll value in $) among two or more parties. U is a utility possibility set u * is a status quo point A bargaining solution assigns utility outcomes (u 1, u 2 ) for every set U and status quo point u *.

Scale-free Solutions It’s hard to measure utility (and there’s a lying is a possibility). What we know about measuring utility is that it is like a temperature scale Which is hotter—32 deg F or 0 deg C? Both the same, just a change in the unit of measure Since we can only measure utility this way, we would not like our “solution” to be “scale free”

Principle 1:Scale Free-ness A bargaining solution is scale free if for every  i > 0,  i, when u 1,u 2 is the solution to (U,u * ), then  i u i +  i is the solution to (U’, u * ’) U’ = (  1 u +  1,  2 u +  2 ) and u * ’ = (  1 u * +  1,  2 u * +  2 ) Note: This lets us transform the problem such that u * is always at the origin. Principle 1: A bargaining solution should be scale free

Comments Scale free measure means that if we’re concluding an international agreement, our principles for arriving at a solution should not depend on the currency in which we are negotiating.

Principle 2: Symmetry A second principle we might agree to is that if our bargaining situations are exactly alike, then an agreement should split things equally as well. A bargaining problem is symmetric if u 1 * =u 2 * and when (u 1, u 2 )  U then (u 2, u 1 )  U. Principle 2: If a bargaining problem is symmetric then its solution is symmetric, i.e. u 1 = u 2.

Comments There are strong psychological foundations for symmetry. In a variety of experiments people exhibit “inequality aversion” Equal treatment is considered often essential to any system of justice

Principle 3: No money left on the table We might desire to impose bargaining solutions where all gains from negotiation are exhausted. That is, we cannot give one party more utility without taking utility away from the first party. Principle 3: If u  U and u’  U and u i ’ > u i for i=1,2 then u is NOT a bargaining solution.

Comments Money left on the table and perceived fairness might be in conflict. Consider the following situation: Choose between two allocations: 1. You get $1,000 and partner gets $1, You get $1,000 and partner gets $100,000  Many people prefer 1 to 2.

The Road Not Taken Suppose that we are originally going to split $100. If we fail to agree, we each get nothing. We agree to a split. Now suppose that we are to split $100, but that the set of feasible agreements requires that my rival gets at least $45 of the $100. Does this matter to the bargaining outcome?

Principle 4: Alternatives not chosen don’t matter This principle states that in the above situation, we should still agree to a split. If we remove alternatives that we did not choose in our bargaining solution, we might desire that our solution remain the same. Principle 4: Suppose (U, u * ) and (U’, u *’ ) are bargaining problems with U  U’. Then if the optimal bargain in U’ is (u 1 ’,u 2 ’)  U, then the optimal bargaining outcome in U is (u 1 ’,u 2 ’).

Some Bargaining Solutions Philosophers and others have proposed a variety of bargaining solutions for “just” allocations in distributive bargaining problems. What principles do these solutions satisfy?

Egalitarian Solutions Choose an outcome giving equal utility to each side and lying on the utility frontier. Since the solution lies on the utility frontier, there’s no money left on the table. If we delete options from the negotiation, it doesn’t change the outcome so Principle 4 holds. If everyone is symmetric, this specifies a symmetric outcome, so Principle 2 holds.

A Problem Suppose that we are going to divide $100. Everyone is symmetric, so we divide Player 1 protests and argues that he values each dollar twice as much as player 2. The bargaining solution gives $33 to 1 and the rest to 2. This does not satisfy scale free-ness! It’s also a dumb strategy for player 1.

Utilitarian Solution Choose an outcome maximizing the sum of the utilities. Since the solution lies on the utility frontier, there’s no money left on the table. If we delete options from the negotiation, it doesn’t change the outcome so Principle 4 holds.

A Problem Two players split $1 U 1 = 2x U 2 = x Solution: Everything to Player 1 Transform U 2 = 3x New solution: Everything to Player 2

Nash Solution Choose an allocation that maximizes the product of the utilities. This satisfies all of the principles. In fact, it is the only bargaining solution satisfying all the principles.

Example Two players split $1 U 1 = 2x U 2 = x Nash solution max 2x(1 – x) 2 – 4x = 0 x = ½

Transform the Problem U 1 = 2x U 2 = 3x max2x(3(1 – x)) 6 – 12x = 0 x = ½

Utility Frontier u1 u2

Outside Options u1 u2 2’s outside option 1’s outside option

Bargaining Solution u1 u2 2’s outside option 1’s outside option Bargaining Solution

Improved Outside Option u1 u2 2’s outside option 1’s old outside option 1’s new outside option Old solution

New Solution u1 u2 2’s outside option 1’s old outside option 1’s new outside option Old solution New solution

Comments 1’s improved outside option netted some additional surplus in the bargaining But it was less than 1 for 1

Bargaining Solution u1 u2 2’s outside option 1’s outside option Bargaining Solution

Destroying Possibilities u1 u2 Bargaining Solution 1 arranges it so that 2 cannot get More than this amount

Comments Notice that this tactic by 1 does nothing to change the bargaining solution. By irrelevance of options not taken

Arbitration Arbitration affects the outside options of each bargainer In the case of final offer arbitration, the arbiter is required to choose between the two final positions in the negotiation before impasse was reached But now your negotiating position affects your outside option. Does this help or harm negotiated outcomes?