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
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In labor negotiations:
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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
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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.
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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
 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

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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 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
 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 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.
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 ui’ > ui for
i=1,2 then u is NOT a bargaining solution.
Comments
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Money left on the table and perceived
fairness might be in conflict.
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Consider the following situation:
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Choose between two allocations:
1.
You get $1,000 and partner gets $1,000
2.
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.
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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
 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
 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.
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Since the solution lies on the utility frontier,
there’s no money left on the table.
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If we delete options from the negotiation, it
doesn’t change the outcome so Principle 4
holds.
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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 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!
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It’s also a dumb strategy for player 1.
Utilitarian Solution
 Choose an outcome maximizing the sum of
the utilities.
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Since the solution lies on the utility frontier,
there’s no money left on the table.
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If we delete options from the negotiation, it
doesn’t change the outcome so Principle 4
holds.
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
 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
 U1 = 2x
 U2 = x
 Nash solution
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max 2x(1 – x)
2 – 4x = 0
x=½
Transform the Problem
 U1 = 2x
 U2 = 3x
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max2x(3(1 – x))
6 – 12x = 0
x=½
Utility Frontier
u2
u1
Outside Options
u2
1’s outside option
2’s outside option
u1
Bargaining Solution
u2
1’s outside option
Bargaining
Solution
2’s outside option
u1
Improved Outside Option
u2
1’s old outside option
1’s new outside option
Old solution
2’s outside option
u1
New Solution
u2
1’s old outside option
1’s new outside option
Old solution
New solution
2’s outside option
u1
Comments
 1’s improved outside option netted some
additional surplus in the bargaining
 But it was less than 1 for 1
Bargaining Solution
u2
1’s outside option
Bargaining
Solution
2’s outside option
u1
Destroying Possibilities
u2
1 arranges it so that 2 cannot get
Bargaining
More than this amount
Solution
u1
Comments
 Notice that this tactic by 1 does nothing to
change the bargaining solution.
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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?