Bargaining and Coalition Formation Dr James Tremewan - - PowerPoint PPT Presentation

Bargaining and Coalition Formation

Dr James Tremewan (james.tremewan@univie.ac.at) Cooperative models of bargaining

Introduction

An preliminary clarification: what do people bargain over?

• Key questions:
• Do people bargain over money or ”utility”?
• Can we make interpersonal comparisons of utility?

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Introduction

Are interpersonal comparisons of utility possible?

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Introduction

Tiresias (Mark Rothko)

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Introduction

What is a utility function? A very brief outline.1

• We assume that people have preferences over outcomes (and

lotteries over outcomes), and that these preferences guide decision making.

• Someone who prefers apples over oranges will choose an apple

rather than an orange.

• If preferences satisfy certain conditions (e.g. transitivity) they

can be represented by a ”nice” utility function that allow us to use the tools of Expected Utility Theory (known as a ”von Neuman-Morgenstern” utility function).

• Non-uniqueness: If a set of preferences can be represented by a

utility function U (x), it can also be represeted by utility functions aU (x) + b where a ≥ 0.

1For more details see, for example, Mas-Colell et al (1995), chapters

1,3, and 6.

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Introduction

Are interpersonal comparisons of utility possible?

• Zeus and Hera had an argument over whether men or women

enjoyed sex more. The only way they found to resolve this question was to ask Tiresias, who was originally a man but had been turned into a woman by Hera, and was later changed back into a man.

• The choice of specific utility function for each person is arbitrary,

and the utility associated with any particular outcome could be assigned any value for each person.

• Furthermore, identical preference orderings result in identical

choices and can be represented by the same utility function, but we can say nothing about the intensity of changes in outcomes.

• Two people have identical preferences orderings and prefer

apples to oranges. They will make the same decisions in all circumstances, but when when forced to take an orange rather than an apple, one may suffer dreadfully while the other is only mildly put out.

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Introduction

Interpersonal comparisons: Summary

• When considering bargaining outcomes we will often be

interested in ideas of equality. In both theory and empirical data we must always be aware of whether we are talking about equality of outcomes (e.g. money) or utility.

• Two people simultaneously pick up a 10 note from the
• footpath. Should they share it 50-50?
• What if one is homeless and the other a millionaire?
• Our theoretical framework does not allow interpersonal

comparisons of utility, but one can often compare outcomes.

• Possibilities for interpersonal comparisons of utility:
• Subjective reports of happiness or satisfaction (but does one

person’s ”very happy” describe the same ”reality” as another’s?)

• Neurological or physiological measurement?

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The Axiomatic Approach

The Bargaining Problem

• Two bargainers (players), i ∈ {1, 2}.
• Set of possible agreements A, and disagreement event D.
• Each player has ”well behaved” preference ordering over A ∪ {D}

such that we can assign each a vNM expected utility function ui.

• Let S ⊂ R2 be the set of all utility pairs that can be outcomes
• f agreements, and di = ui (D).
• Nash (1950) defines a bargaining problem as the pair S, d

where

• d ∈ S
• there exists s ∈ S such that si > di for i = 1, 2 (i.e. both

players can benifit from bargaining).

• S is compact (closed and bounded) and convex (allows us to

solve maximization problems on set).2

• Note that bargaining occurs purely over utilities.

2note that these assumptions can be justified by allowing ”probabilistic

agreements.”

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The Axiomatic Approach

The Bargaining Problem

✻ ✲

u1 u2

✂ ✂ ✂ ✂ ❅ ❅ ❅ ❇ ❇ ❇ ❇ ✁ ✁ ✁ ◗ ◗ ◗ ◗

S d

r

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The Axiomatic Approach

Bargaining Solutions

• Let B be the set of all bargaining problems S, d.
• A bargaining solution is a functon f : B → R2 that assigns to

each bargaining problem S, d ∈ B a unique element of S

• Instead of explicitly modelling process, Nash’s approach was to

identify some characteristics ”reasonable” solutions should have (axioms) and define a solution as an outcome that satisfied those characteristics.

• The Nash bargaining solution is the unique element of S that

satisfies a set of four particular axioms.

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The Axiomatic Approach

Nash’s Axioms: Invariance to Equivalent Utility Representations (INV)

• As previously pointed out, there are many different utility

functions that can represent the same preference order over

• utcomes.
• Loosely speaking, INV states that the choice of utility functions

should not affect the outcome represented by the solution.

• Formally: Suppose that the bargaining problem S′, d′ is
• btained from S, d by the transformations si → αisi + βi,

where αi > 0. Then fi (S′, d′) = αifi (S, d) + βi.

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The Axiomatic Approach

Nash’s Axioms: Symmetry (SYM)

• It is assumed that any asymmetry in the players bargaining

ability is captured by S and d.

• It therefore seems reasonable that two players in the same

positions should experience the same outcome.

• A bargaining problem S, d is defined to be symmetric if

d1 = d2 and (s1, s2) ∈ S if and only if (s2, s1) ∈ S.

• If the bargaining problem S, d is symmetric, then

f1 (S, d) = f2 (S, d)

• Note that this has nothing to do with ”fairness”, just that

relabelling should not not alter the strategic situation.

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The Axiomatic Approach

Nash’s Axioms: Independence of Irrelevant Alternatives (IIA)

• Suppose for a given set of alternatives in a bargaining problem a

particular outcome is chosen as the bargaining solution. Now if we define a new problem by removing one or more of the alternatives which were not the bargaining solution of the original problem, then the new solution will be the same as the old one.

• Formally: If S, d and T, d are bargaining problems with

S ⊂ T and f (T, d) ∈ S, then f (S, d) = f (T, d).

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The Axiomatic Approach

Nash’s Axioms: Pareto Efficiency (PAR)

• Players should not agree on a particular outcome if one of them

can be made better off without harming the other.

• Formally: Suppose S, d is a bargaining problem, s ∈ S, t ∈ S,

and ti > si for i = 1, 2. Then f (S, d) = s.

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The Axiomatic Approach

The Nash bargaining solution

• Remarkably, the preceding four axioms identify a unique solution

for any bargaining problem.

• Theorem: There is a unique bargaining solution f N : B → R2

satisfying the axions INV, SYM, IIA, and PAR. It is given by f N (S, d) = arg max

(d1,d2)≤(s1,s2)∈S

(s1 − d1) (s2 − d2) .

• Proof: See Osborne and Rubinstein pgs 13 & 14. Go through

this at home (some of the simpler parts of the proof may be in the test, but you will not be expected to be able to reproduce it all.)

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The Axiomatic Approach

Application: Dividing a Dollar: The role of disagreement points

• Two individuals can divide a dollar in any way they wish. If they

fail to agree, they receive di for i = 1, 2. They may discard some

• f the money. Players are expected value maximisers, i.e. ui = x

where x is their share of the money.

• A = {(a1, a2) ∈ R2 : a1 + a2 ≤ 1 and ai ≥ 0 for i = 1, 2} (= S)
• D = (d1, d2)
• PAR implies that no money is wasted: if player 1 receives x1

then player 2 receives x2 = 1 − x1.

• f N (S, d) = arg max (x1 − d1) (1 − x1 − d2).

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The Axiomatic Approach

Application: Dividing a Dollar: The role of disagreement points

• Solution: x1 = 1+d1−d2

2

, x2 = 1−d1+d2

2

• d1 = d2 ⇒ each player receives half (implied directly by SYM).
• A player’s share is increasing in their own disagreement payoff

(outside option) and decreasing in the other player’s disagreement payoff.

• Player’s have an incentive to overstate their outside option: calls

into question perfect information about other’s utility from

• utcomes.

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The Axiomatic Approach

Application: Dividing a Dollar: The role of risk-aversion

• Two individuals can divide a dollar in any way they wish. If they

fail to agree, they both get nothing. They may discard some of the money. Players care only about the amount they get and prefer more rather than less.

• A = {(a1, a2) ∈ R2 : a1 + a2 ≤ 1 and ai ≥ 0 for i = 1, 2}
• D = (0, 0)
• Assume the players’ preferences can be represented by the utility

functions ui = xri where r1 ≥ r2, i.e. player 2 is more risk averse than player 1.

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The Axiomatic Approach

Application: Dividing a Dollar: The role of risk-aversion

• S = {(s1, s2) ∈ R2 : (s1, s2) = (ar1

1 , ar2 2 ) for some (a1, a2) ∈ A}

✻ ✲

u1 u2 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S d = (0, 0)

✉

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The Axiomatic Approach

Application: Dividing a Dollar: The role of risk-aversion

• PAR implies that no money is wasted: if player 1 receives x1

then player 2 receives x2 = 1 − x1.

• f N (S, d) = arg max (xr1) (1 − x)r2.
• Solution: x1 =

r1 r1+r2, x2 = r2 r1+r2

• A player’s share is decreasing in their degree of risk aversion, and

increasing in the other’s risk aversion (here big ri ⇒ less risk aversion).

• Note that this has nothing to do with the fact that

u1 (x) ≥ u2 (x) ∀x. It is easy to see the solution to the maximisation problem is unchanged if u2 = 100xr2 (as implied by INV).

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The Axiomatic Approach

Alternative bargaining solutions

• Drop the symmetry axiom. For each α ∈ (0, 1) we can define a

solution by arg max

(d1,d2)≤(s1,s2)∈S

(s1 − d1)α (s2 − d2)(1−α) . The variable α is often interpreted as the relative bargaining power of player 1.

• Replacing IIA with a ”monotonicity” axiom gives the

Kalai-Smorodinksy solution: has some attractive features.

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The Axiomatic Approach

Experimental tests of the Nash bargaining solution

• Two-Person Bargaining: An Experimental Test of the Nash

Axioms, Nydegger and Owen (1974)

• Tests each of Nash’s four axioms.
• How sensitive are bargaining outcomes to changes in

disagreement payoffs?, Anbarci and Feltovich (2011)

• Tests the prediction that players with higher disagreement

payoffs gain a larger share.

• Risk Aversion in Bargaining: An Experimental Study, Murnighan

et al (1988).

• Tests the predictions about the effect of player’s risk aversion
• n outcomes.

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Nydegger and Owen (1974)

Two-Person Bargaining: An Experimental Test of the Nash Axioms, Nydegger and Owen (1974)

• An early experiment. Experimental methodology not well

developed, and computers unavailable.

• All bargaining face-to-face across table. All rules were common
• knowledge. $1 show-up fee.
• Treatment 1: Bargaining over $1. In case of disagreement, the

dollar is lost.

• Treatment 2: As Treatment 1, but player 2 could receive no

more than 60 cents (to test IIA).

• Treatment 3: Bargaining over 60 poker chips. Player 1 could

cash them in for 2 cents/chip, Player 2 for 1 cent/chip (to test INV).

• Subjects: 20 male undergraduate students per treatment.

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Nydegger and Owen (1974)

Results

• Treatment 1: All 10 pairs split money equally (consistent with

SYM, PAR).

• Treatment 2: All 10 pairs split money equally (consistent with

IIA).

• Treatment 3: All 10 pairs divided the chips to equalize monetary

payoffs (contradicting INV which predicts there should be no difference from Treatment 1).

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Nydegger and Owen (1974)

Shortcomings

• Assumes EV maximization, whereas people tend to be

risk-averse.

• Lack of anonymity.
• Weak tests of the theory:
• Many possible explanations for equal split in symmetric game

(weak test of SYM).

• With equal split so salient (no other reasonable outcome)

disagreement unlikely (weak test of PAR).

• Only one of many ways of constraining the set of bargaining
• utcomes (weak test of INV).

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Anbarci and Feltovich (2011)

How sensitive are bargaining outcomes to changes in disagreement payoffs?, Anbarci and Feltovich (2011)

• Subjects bargain over a fixed sum (using either Nash Demand

Game or Unstructured Bargaining Game).

• Disagreement payoffs are randomly determined: 5-25% of cake

(unfavoured player) or 25-45% of cake (favoured player).

• (Assuming risk-neutrality) NBS predicts
• δx1

δd1

• =
• δx2

δd2

• = 1

2 and

• δx1

δd2

• =
• δx2

δd1

• = −1

2 and

• δx1

δd1

• +
• δx2

δd2

• = 1

(see previous set of slides)

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Anbarci and Feltovich (2011) 27/38

Anbarci and Feltovich (2011) 28/38

Anbarci and Feltovich (2011) 29/38

Anbarci and Feltovich (2011)

Conclusions

• Effects are in the correct direction, but too small (around 0.25,

and significantly less than 0.5).

• Can this be explained by risk-aversion? No. Authors show that

with risk-aversion, NBS ⇒

• δx1

δd1

• +
• δx2

δd2

• > 1
• However, authors show that with a utility function including

fairness concerns, NBS ⇒

• δx1

δd1

• +
• δx2

δd2

• < 1

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Murnighan et al (1988)

Risk Aversion in Bargaining: An Experimental Study, Murnighan et al (1988)

• As shown in the previous set of slides, in a simple divide the

dollar game with zero disagreement payoffs, NBS predicts the less risk-averse player will gain more.

• This paper identifies two bargaining games, one where increased

risk-aversion should lead to lower shares and one higher shares.

• Risk preferences of subjects are elicited, and high risk-aversion

subjects bargain with low risk-aversion subjects.

• Some support is found for the risk-aversion hypothesis, but not

so strong.

• Authors hypothesize that bigger stakes may increase effect, and

also that any risk-aversion effect is dominated by ”focal-point” effect.

• Some evidence that risk-aversion weakens bargaining position

also found in Dickinson, Theory and Decision (2009).

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Conclusion

Nash bargaining solution: pros and cons

• Pros:
• It is general, in the sense that it does not relate only to a

particular bargaining process. Can be widely applied.

• Captures some key features of bargaining, such as importance
• f disagreement payoffs and risk preferences.
• Easily calculated, so widely used as a component in bigger

models without adding much complexity.

• Cons:
• People do appear to make inter-personal comparisons of utility,

which violates INV.

• Does not account for focal points which may exist outside the

formal strategic structure of the game.

• In some cases precise features of the bargaining process may be

important (possibly violating IIA).

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Bargaining in Marriage

Bargaining in Marriage: early models of family expenditure

• Consumption is often determined by family units rather than

individuals: macroeconomics requires a model of household decision-making.

• Early approaches considered family decisions to be made by a

single utility-maximising agent subject to a family budget constraint:

• The traditional neo-classical approach treats households as

rational individuals.

• In Becker (1974,1981), decisions are made by a single

”altruistic” family member who internalizes the effect of decisions on other family members.

• These “unitary models” imply that the source of income and
• utside options are irrelevant to consumption decisions.

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Bargaining in Marriage

Empirical evidence against “unitary model”

• Higher relative income of wife ⇒ greater household expenditure
• n restaurant meals, child care, and women’s clothing and less
• n alcohol and tobacco (Phipps and Burton, 1992).
• However higher wages for wife ⇒ lower relative price of child

care, restaurant meals, etc. so may not imply bargaining.

• Increase in female unearned income improves child health
• utcomes in developing countries more than unearned male

income (e.g. Thomas, 1990; Haddad and Hoddinott, 1995).

• No price effect, but may be correlation between e.g. ownership
• f assets and past (and hence current) labour supply.
• Best evidence: Change in UK child benefit in 1970s meant

exogenous increase in income of women relative to men which led to:

• Increase in expenditure on women’s and children’s clothing

relative to men’s (Lundberg, Pollack and Wales, 1997).

• Increase in expenditure on cigarettes relative to pipe tobacco

and cigars (Ward-Batts, 2002).

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Bargaining in Marriage

Nash Bargaining and Marriage (McElroy, 1990)

• Need to define disagreement payoffs and bargaining set.
• Disagreement payoff could be utility of staying single (if

comprehensive prenuptial agreement is possible), utility upon divorce, or utility in ”non-cooperative marriage.”

• Unmarried utility functions: Um

0 (x0, x1, x3), Uf 0(x0, x2, x4).

• x1 (x2) is a good consumed by m (f ).
• x3 (x4) is leisure time of m (f ).
• x0 is a private good that would be a household good if married.
• x = (x0, x1, x2, x3, x4)′ have prices p = (p0, p1, p2, p3, p4)′.
• Maximise utility s.t. (p0x0 + p1x1 + p3x3 = Im + p3T (for m).
• Im is non-wage income.
• T is time endowment
• Disagreement payoff: V m

0 (p0, p1, p3, Im; αm).

• αm is a vector of “extrahousehold environmental parameters.”

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Bargaining in Marriage

• Extrahousehold Environmental Parameters (EEPs): Variables

that shift the maximum value of utility attainable by the individual outside the marriage, i.e. the disagreement payoff.

• Examples:
• Gender ratio (affects the probability of finding a good

alternative partner).

• Parents’ wealth: in rural India divorce not possible but women

may be able to return to parents.

• Tax regime and government transfers.
• Social norms: societal attitude towards divorce.
• Utilities in marriage: Um(x), Uf (x).
• Nash bargaining solution: x which maximises

(Um(x) − V m

0 (p0, p1, p3, Im; αm))

• Uf (x) − V f

0 (p0, p2, p4, If ; αf )

• s.t. p′x = (p3 + p4)T + Im + If

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Bargaining in Marriage

Some applications

• Subsidy for single mothers’ children:
• Unitary model impliess this would be a disincentive to marry.
• However bargaining model implies an increase in utility for

married women as well, so no disincentive.

• Policy implications for equal pay, divorce law etc.
• Tax law: should rates be based on joint or seperate incomes?
• Strategic (and perhaps inneficient) investment in (e.g.)

premarital education to improve bargaining power.

• Effect of size of dowry on daughter’s welfare (Suen et al, 2003)

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Bargaining in Marriage 38/38