bargaining and coalition...

Bargaining and Coalition Formation

Dr James Tremewan ([email protected])

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)4/38

Introduction

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

• We assume that people have preferences over outcomes (andlotteries over outcomes), and that these preferences guidedecision making.• Someone who prefers apples over oranges will choose an apple

rather than an orange.

• If preferences satisfy certain conditions (e.g. transitivity) theycan be represented by a ”nice” utility function that allow us touse the tools of Expected Utility Theory (known as a ”vonNeuman-Morgenstern” utility function).

• Non-uniqueness: If a set of preferences can be represented by autility function U (x), it can also be represeted by utilityfunctions aU (x) + b where a ≥ 0.

1For more details see, for example, Mas-Colell et al (1995), chapters1,3, 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 thisquestion was to ask Tiresias, who was originally a man but hadbeen turned into a woman by Hera, and was later changed backinto a man.

• The choice of specific utility function for each person is arbitrary,and the utility associated with any particular outcome could beassigned any value for each person.

• Furthermore, identical preference orderings result in identicalchoices and can be represented by the same utility function, butwe 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 allcircumstances, but when when forced to take an orange ratherthan an apple, one may suffer dreadfully while the other is onlymildly put out.

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Introduction

Interpersonal comparisons: Summary

• When considering bargaining outcomes we will often beinterested in ideas of equality. In both theory and empirical datawe must always be aware of whether we are talking aboutequality 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 interpersonalcomparisons 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 outcomesof 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.”8/38


The Bargaining Problem

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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 toidentify some characteristics ”reasonable” solutions should have(axioms) and define a solution as an outcome that satisfiedthose characteristics.

• The Nash bargaining solution is the unique element of S thatsatisfies a set of four particular axioms.

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Nash’s Axioms: Invariance to Equivalent Utility

Representations (INV)

• As previously pointed out, there are many different utilityfunctions that can represent the same preference order overoutcomes.

• Loosely speaking, INV states that the choice of utility functionsshould not affect the outcome represented by the solution.

• Formally: Suppose that the bargaining problem 〈S ′, d ′〉 isobtained from 〈S , d〉 by the transformations si → αisi + βi ,where αi > 0. Then fi (S ′, d ′) = αi fi (S , d) + βi .

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Nash’s Axioms: Symmetry (SYM)

• It is assumed that any asymmetry in the players bargainingability is captured by S and d .

• It therefore seems reasonable that two players in the samepositions should experience the same outcome.

• A bargaining problem 〈S , d〉 is defined to be symmetric ifd1 = d2 and (s1, s2) ∈ S if and only if (s2, s1) ∈ S .

• If the bargaining problem 〈S , d〉 is symmetric, thenf1 (S , d) = f2 (S , d)

• Note that this has nothing to do with ”fairness”, just thatrelabelling should not not alter the strategic situation.

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Nash’s Axioms: Independence of Irrelevant

Alternatives (IIA)

• Suppose for a given set of alternatives in a bargaining problem aparticular outcome is chosen as the bargaining solution. Now ifwe define a new problem by removing one or more of thealternatives which were not the bargaining solution of the originalproblem, then the new solution will be the same as the old one.

• Formally: If 〈S , d〉 and 〈T , d〉 are bargaining problems withS ⊂ T and f (T , d) ∈ S , then f (S , d) = f (T , d).

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Nash’s Axioms: Pareto Efficiency (PAR)

• Players should not agree on a particular outcome if one of themcan 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) 6= s.

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The Nash bargaining solution

• Remarkably, the preceding four axioms identify a unique solutionfor 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 throughthis at home (some of the simpler parts of the proof may be inthe test, but you will not be expected to be able to reproduce itall.)

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Application: Dividing a Dollar: The role of

disagreement points

• Two individuals can divide a dollar in any way they wish. If theyfail to agree, they receive di for i = 1, 2. They may discard someof the money. Players are expected value maximisers, i.e. ui = xwhere 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 x1then player 2 receives x2 = 1− x1.

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

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disagreement points

• Solution: x1 = 1+d1−d22

, x2 = 1−d1+d22

• 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’sdisagreement payoff.

• Player’s have an incentive to overstate their outside option: callsinto question perfect information about other’s utility fromoutcomes.

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risk-aversion

• Two individuals can divide a dollar in any way they wish. If theyfail to agree, they both get nothing. They may discard some ofthe money. Players care only about the amount they get andprefer 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 utilityfunctions ui = x ri where r1 ≥ r2, i.e. player 2 is more risk aversethan player 1.

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risk-aversion

• S = {(s1, s2) ∈ R2 : (s1, s2) = (ar11 , ar22 ) for some (a1, a2) ∈ A}

6

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....................

.....................

S

d = (0, 0)u

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risk-aversion

• PAR implies that no money is wasted: if player 1 receives x1then player 2 receives x2 = 1− x1.

• f N (S , d) = arg max (x r1) (1− x)r2 .

• Solution: x1 = r1r1+r2

, x2 = r2r1+r2

• A player’s share is decreasing in their degree of risk aversion, andincreasing in the other’s risk aversion (here big ri ⇒ less riskaversion).

• Note that this has nothing to do with the fact thatu1 (x) ≥ u2 (x)∀x . It is easy to see the solution to themaximisation problem is unchanged if u2 = 100x r2 (as implied byINV).

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Alternative bargaining solutions

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

arg max(d1,d2)≤(s1,s2)∈S

(s1 − d1)α (s2 − d2)(1−α) .

The variable α is often interpreted as the relative bargainingpower of player 1.

• Replacing IIA with a ”monotonicity” axiom gives theKalai-Smorodinksy solution: has some attractive features.

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Experimental tests of the Nash bargaining solution

• Two-Person Bargaining: An Experimental Test of the NashAxioms, Nydegger and Owen (1974)• Tests each of Nash’s four axioms.

• How sensitive are bargaining outcomes to changes indisagreement 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, Murnighanet al (1988).• Tests the predictions about the effect of player’s risk aversion

on 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 welldeveloped, and computers unavailable.

• All bargaining face-to-face across table. All rules were commonknowledge. $1 show-up fee.

• Treatment 1: Bargaining over $1. In case of disagreement, thedollar is lost.

• Treatment 2: As Treatment 1, but player 2 could receive nomore than 60 cents (to test IIA).

• Treatment 3: Bargaining over 60 poker chips. Player 1 couldcash them in for 2 cents/chip, Player 2 for 1 cent/chip (to testINV).

• Subjects: 20 male undergraduate students per treatment.

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Results

• Treatment 1: All 10 pairs split money equally (consistent withSYM, PAR).

• Treatment 2: All 10 pairs split money equally (consistent withIIA).

• Treatment 3: All 10 pairs divided the chips to equalize monetarypayoffs (contradicting INV which predicts there should be nodifference from Treatment 1).

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Shortcomings

• Assumes EV maximization, whereas people tend to berisk-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

outcomes (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∣∣∣∣δx1δd1

∣∣∣∣ =

∣∣∣∣δx2δd2

∣∣∣∣ =1

2and


∣∣∣∣ =


∣∣∣∣ = −1

2

and ∣∣∣∣δx1δd1

∣∣∣∣ +


∣∣∣∣ = 1

(see previous set of slides)26/38


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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 thatwith risk-aversion, NBS ⇒∣∣∣∣δx1δd1

∣∣∣∣ +


∣∣∣∣ > 1

• However, authors show that with a utility function includingfairness concerns, NBS ⇒∣∣∣∣δx1δd1

∣∣∣∣ +


∣∣∣∣ < 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 theless risk-averse player will gain more.

• This paper identifies two bargaining games, one where increasedrisk-aversion should lead to lower shares and one higher shares.

• Risk preferences of subjects are elicited, and high risk-aversionsubjects bargain with low risk-aversion subjects.

• Some support is found for the risk-aversion hypothesis, but notso strong.

• Authors hypothesize that bigger stakes may increase effect, andalso that any risk-aversion effect is dominated by ”focal-point”effect.

• Some evidence that risk-aversion weakens bargaining positionalso 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

of 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 thanindividuals: macroeconomics requires a model of householddecision-making.

• Early approaches considered family decisions to be made by asingle utility-maximising agent subject to a family budgetconstraint:• 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 ofdecisions on other family members.

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

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Empirical evidence against “unitary model”• Higher relative income of wife ⇒ greater household expenditure

on restaurant meals, child care, and women’s clothing and lesson 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

outcomes in developing countries more than unearned maleincome (e.g. Thomas, 1990; Haddad and Hoddinott, 1995).• No price effect, but may be correlation between e.g. ownership

of 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 whichled 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).34/38


Nash Bargaining and Marriage (McElroy, 1990)

• Need to define disagreement payoffs and bargaining set.

• Disagreement payoff could be utility of staying single (ifcomprehensive prenuptial agreement is possible), utility upondivorce, or utility in ”non-cooperative marriage.”• Unmarried utility functions: Um

0 (x0, x1, x3), U f0 (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: Vm0 (p0, p1, p3, Im;αm).

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

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• Extrahousehold Environmental Parameters (EEPs): Variablesthat shift the maximum value of utility attainable by theindividual 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), U f (x).

• Nash bargaining solution: x which maximises

(Um(x)− Vm0 (p0, p1, p3, Im;αm))

(U f (x)− V f

0 (p0, p2, p4, If ;αf ))

s.t. p′x = (p3 + p4)T + Im + If

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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 formarried 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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