The Encyclopedia of Economic Principals

Chapter 13

Bargaining, Coalitions, and Cooperative Value

Outside options, patience and marginal contributions decide who gets what.

Four of the chapter's worked examples, made interactive: outside options in Nash bargaining, patience in alternating offers, the Shapley value of a venture, and when a stable cost-sharing deal exists.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Outside options move the bargain

How does a better outside option change the payment in a licensing deal?

The Nash solution splits the surplus over the two outside options equally. Raising one side's option moves the payment toward it by half the increase.

Equation, written in LaTeX: N(x)=x(120-x).

Equation, written in LaTeX: N'(x)=120-2x=0,

Equation, written in LaTeX: N_1(x)=(x-30)(120-x),

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The deal creates 120 thousand of surplus. x is the payment to the data supplier, who keeps x; the software firm keeps 120 - x. Each side's outside option is what it gets if talks fail.

Predict first. If both sides gain an outside option of 30, where does the payment land?

Your prediction

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Figure: Outside options move the bargain. Nash product curve over the payment, peaking at x = 60.0 with outside options 0 and 0.
Supplier outside option: 0, Firm outside option: 0
Constructed example: the chapter's hypothetical license (options 0 and 30 for the supplier); supplier options of 15 and 45 and a firm option of 30 are added for comparison.

Calculated values

Payment x
60.0
Supplier gain over its option
60.0
Firm gain over its option
60.0
Nash product
3,600.00

Maximize (x - 0)(120 - x - 0). The derivative is (120 - x - 0) - (x - 0) = 0, so x = (120 + 0 - 0) / 2 = 60.0. The supplier gains 60.0 - 0 = 60.0 and the firm 120 - 60.0 - 0 = 60.0: the surplus over the two options, 120, is split equally.

Worked steps

  1. x = (120 + 0 - 0) / 2 = 60.0
  2. Supplier gain = 60.0 - 0 = 60.0
  3. Firm gain = 120 - 60.0 - 0 = 60.0
  4. Product = 60.0 x 60.0 = 3,600.00

Use the idea

Before bargaining, improve your credible alternative; it moves the split even when the deal's value is unchanged.

Where the conclusion applies

Linear utilities, a divisible payment and credible outside options.

Check your understanding: With a supplier option of 45 and a firm option of 0, what are the payment and each party's gain?
Maximize (x - 45)(120 - x): x = 82.5; gains 37.5 each.

Chapter 13 source: section "Nash bargaining solution".

Demonstration 2 of 4

Patience is bargaining power

How do the two sides' patience divide a surplus when they alternate offers?

Each side's share depends on how costly waiting is for the other. A more patient firm has a more valuable counteroffer, so the union must offer it more.

Equation, written in LaTeX: x=\frac{1-0.90}{1-(0.90)(0.90)}=\frac{0.10}{0.19}\approx0.526316.

Equation, written in LaTeX: x=\frac{1-0.80}{1-(0.90)(0.80)}=\frac{0.20}{0.28}\approx0.714286.

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A 100,000 surplus. The union proposes first; a rejected offer means a counteroffer the next day. Discount factors are the value of a dollar tomorrow to each side. x is the union's share.

Predict first. If the firm becomes more patient (0.95), does the union's share rise or fall?

Your prediction

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Figure: Patience is bargaining power. Split bar of the surplus and a grid of union shares; with factors 0.9 and 0.9 the union keeps $52,631.58.
Union discount factor: 0.9, Firm discount factor: 0.9
Constructed example: the chapter's hypothetical union and firm (0.90 and 0.90, firm 0.80); factors of 0.80 for the union and 0.95 for either side are added for comparison.

Calculated values

Union share
0.526316
Union keeps
$52,631.58
Firm gets
$47,368.42
Firm's value of rejecting
$47,368.42

x = (1 - 0.9) / (1 - 0.9 x 0.9) = 0.10 / 0.1900 = 0.526316. The union keeps $52,631.58 and offers $47,368.42. If the firm rejects, it proposes tomorrow and keeps its proposer share, worth 0.9 x $52,631.58 = $47,368.42 today, the same as the offer, so it accepts at once.

Worked steps

  1. du x df = 0.9 x 0.9 = 0.8100
  2. x = 0.10 / 0.1900 = 0.526316
  3. Union = 100,000 x 0.526316 = $52,631.58
  4. Firm = 100,000 - $52,631.58 = $47,368.42

Use the idea

In negotiations, the side that can wait longer at lower cost gets the larger share.

Where the conclusion applies

Complete information, a fixed surplus, alternating offers without a deadline, and immediate agreement in equilibrium.

Check your understanding: With a union factor of 0.90 and a firm factor of 0.95, what does the union keep?
x = 0.05 / (1 - 0.855) = 0.05 / 0.145 = 0.3448; the union keeps $34,482.76.

Chapter 13 source: section "Rubinstein alternating-offers result".

Demonstration 3 of 4

Splitting a venture by marginal contribution

How should a three-firm venture's value be shared by what each member adds?

Averaging marginal contributions over all arrival orders rewards a firm for the value it adds, not for its standalone worth. A firm that adds nothing anywhere gets nothing.

Equation, written in LaTeX: v(AB)=60, v(AC)=20, v(BC)=30, v(ABC)=120.

Equation, written in LaTeX: (\phi_A,\phi_B,\phi_C)=(\frac{130}{3},\frac{145}{3},\frac{85}{3})\approx(43.33,48.33,28.33).

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v(S) is what coalition S can earn, in thousands; single firms earn 0. A firm's Shapley value is its average marginal contribution over the six orders in which the three could arrive.

Predict first. If v(ABC) rises by 30, how is the extra split?

Your prediction

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Figure: Splitting a venture by marginal contribution. Bars of Shapley values: A 43.33, B 48.33, C 28.33, summing to 120.
Grand coalition value v(ABC): 120, Coalition technology: Base
Constructed example: the chapter's hypothetical venture (v(ABC) = 120 and the dummy case); grand values of 100 and 150 are added for comparison.

Calculated values

A
43.33
B
48.33
C
28.33
Sum
120.00
Totals over six orders (A, B, C)
260, 290, 170

Over the six arrival orders A adds 0, 0, 60, 90, 20, 90 (total 260), B adds 60, 100, 0, 0, 100, 30 (total 290) and C adds 60, 20, 60, 30, 0, 0 (total 170). Dividing by six gives 43.33, 48.33 and 28.33; the exact shares sum to (260 + 290 + 170) / 6 = 120.

Worked steps

  1. A: 260 / 6 = 43.33
  2. B: 290 / 6 = 48.33
  3. C: 170 / 6 = 28.33
  4. Check: (260 + 290 + 170) / 6 = 120

Use the idea

Use the Shapley value to divide joint gains when contributions depend on who else is in.

Where the conclusion applies

Transferable value and a characteristic function known to all.

Check your understanding: With v(ABC) = 150, what are the Shapley values?
Each gains 30 x 2 / 6 = 10: (53.33, 58.33, 38.33).

Chapter 13 source: section "Shapley value".

Demonstration 4 of 4

When does a stable cost-sharing deal exist?

Can three hospitals share savings so that no pair would rather go it alone?

Bondareva-Shapley: the core is nonempty exactly when no balanced collection of coalitions claims more than the grand coalition can deliver.

Equation, written in LaTeX: \frac12(60)+\frac12(60)+\frac12(60)=90\leq100.

Equation, written in LaTeX: 2(x_A+x_B+x_C)\geq210,

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Single hospitals save 0, each pair saves the pair value and all three save 100. The core is the set of splits of 100 that give every pair at least its own saving.

Predict first. What is the largest pair saving that leaves the core nonempty?

Your prediction

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Figure: When does a stable cost-sharing deal exist? Bars of the weighted pair claim 90.00 against the grand coalition's 100.
Saving for each pair: 60
Constructed example: the chapter's hypothetical hospitals (pairs 60 and 70); pair savings of 50 and 200/3 are added for comparison.

Calculated values

Weighted claim
90.00
Grand coalition
100
Core
nonempty
Equal split
33.33
Each pair gets under equal split
66.67

With weight 1/2 on each pair the claim is 0.5 x 60 x 3 = 90.00 against 100. The claim is within 100, so the core is nonempty: the equal split 33.33 each gives every pair 66.67, more than 60.

Worked steps

  1. Claim = 0.5 x 60 + 0.5 x 60 + 0.5 x 60 = 90.00
  2. Balanced: 90.00 <= 100
  3. Core nonempty

Use the idea

Before promising every subgroup more than it could get alone, add up what the subgroups could claim together.

Where the conclusion applies

Transferable savings, symmetric hospitals and binding agreements.

Check your understanding: With a pair saving of 66.67, is the core empty?
No: the claim 1.5 x 200/3 = 100 = v(ABC), so the core is the single point (33.33, 33.33, 33.33).

Chapter 13 source: section "Bondareva-Shapley theorem".