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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.

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motivation
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
Overview
  • Principles of “good” bargaining agreements
  • Bargaining “solutions” from these principles
  • Arbitration
  • Mediation
utility
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
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
  • 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 (u1, u2) for every set U and status quo point u*.
scale free solutions
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
Principle 1:Scale Free-ness
  • A bargaining solution is scale free if for every i > 0, i, when u1,u2 is the solution to (U,u*), then iui + i is the solution to (U’, u*’)
    • U’ = (1u + 1, 2u + 2) and u*’ = (1u* + 1, 2u* + 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
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
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 u1*=u2* and when (u1, u2)  U then (u2, u1)  U.
  • Principle 2: If a bargaining problem is symmetric then its solution is symmetric, i.e. u1 = u2.
comments1
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
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 ui’> ui for i=1,2 then u is NOT a bargaining solution.
comments2
Comments
  • Money left on the table and perceived fairness might be in conflict.
  • Consider the following situation:
    • Choose between two allocations:
      • You get $1,000 and partner gets $1,000
      • You get $1,000 and partner gets $100,000
    • Many people prefer 1 to 2.
the road not taken
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 50-50 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
Principle 4: Alternatives not chosen don’t matter
  • This principle states that in the above situation, we should still agree to a 50-50 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 (u1’,u2’)U, then the optimal bargaining outcome in U is (u1’,u2’).
some bargaining solutions
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
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
A Problem
  • Suppose that we are going to divide $100.
  • Everyone is symmetric, so we divide 50-50.
  • 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
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 problem1
A Problem
  • Two players split $1
  • U1 = 2x
  • U2 = x
  • Solution: Everything to Player 1
  • Transform U2 = 3x
  • New solution: Everything to Player 2
nash solution
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
Example
  • Two players split $1
  • U1 = 2x
  • U2 = x
  • Nash solution
    • max 2x(1 – x)
    • 2 – 4x = 0
    • x = ½
transform the problem
Transform the Problem
  • U1 = 2x
  • U2 = 3x
    • max2x(3(1 – x))
    • 6 – 12x = 0
    • x = ½
outside options
Outside Options

u2

1’s outside option

2’s outside option

u1

bargaining solution
Bargaining Solution

u2

1’s outside option

Bargaining

Solution

2’s outside option

u1

improved outside option
Improved Outside Option

u2

1’s old outside option

1’s new outside option

Old solution

2’s outside option

u1

new solution
New Solution

u2

1’s old outside option

1’s new outside option

Old solution

New solution

2’s outside option

u1

comments3
Comments
  • 1’s improved outside option netted some additional surplus in the bargaining
  • But it was less than 1 for 1
bargaining solution1
Bargaining Solution

u2

1’s outside option

Bargaining

Solution

2’s outside option

u1

destroying possibilities
Destroying Possibilities

u2

1 arranges it so that 2 cannot get

More than this amount

Bargaining

Solution

u1

comments4
Comments
  • Notice that this tactic by 1 does nothing to change the bargaining solution.
    • By irrelevance of options not taken
arbitration
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?