Coalitions in Negotiation
work in progress
Sylvie Thoron
GREQAM
2006/2007
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Introduction
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Introduction - Motivation
• The economic theory focuses on individual agents.
• Different kind of individual behavior: maximizing,
with bounded rationality, forward looking (perfect
rationality), backward looking (learning)…
• Without interactions between different agents
(general equilibrium).
• Interacting agents (game theory)
• In all these cases, decisions are taken by individual
agents.
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Introduction - Motivations
• However, group of agents or coalitions, as well as
individual agents, are active elements of real
economic systems.
• Examples can be found at each level:
– Consumers form associations to protect their interests,
workers form trade unions…
– Politicians form coalitions to win elections
– Firms are coalitions themselves, interact with other
coalitions, form coalitions (mergers, cartels…)
– Jurisdictions
– Nations are coalitions, form coalitions (trade
agreements, free trade area, environmental
agreements…)
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Introduction - Questions
Framework: Game theory, cooperative and noncooperative.
– Why coalitions form? (return to scales, production of
public goods, power, …)
– Which coalitions will form?  stability of coalitions
and coalition structures
– How do coalitions form? Endogenous formation of
coalitions
– How do coalitions interact between each other?
(behavior among coalitions)
– How are payoffs shared among the members of a given
coalition (behavior within coalitions)
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How to define a game?
• Cooperative game theory
– coalitional function games
• Non-cooperative game theory
– Normal form games
– Extensive form games
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Introduction - Applications
–
–
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Focus on negotiation: formation of
agreements.
Industrial Organization (mergers, cartels and
collusion)
International Economics (international trade
agreements, customs unions, free trade areas)
Public goods (environmental agreements)
Political Science (measure of voting power)
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First Week:
Cooperative Approach
Coalitional Form Games
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Section 1 Value
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Section 1 Value
1.1Definitions
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Coalitional Functions
• We denote a (TU) game by (N,v), in which N is a set of
players and v a coalitional function.
•
Definition: a coalitional function (characteristic function
or partition function) is a mapping which associates to
any coalition S a payoff:
S  v(S)  R+,
which will be called the worth, or value of coalition S.
• The incremental value of player i to coalition S, i S is:
v(S) - v(S\{i}).
• By extension, we define the incremental value of
coalition S to coalition T, if S is included in T
v(T) - v(T\S).
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Superadditivity
• A game is superadditive if and only if the sum of
the worth of two disjoint coalitions cannot be
larger than the worth of the union of both
coalitions.
S,T,ST ,vST v(S)v(T)
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Cooperative solutions
• We have defined coalitional function games.
• What are the solutions of these games?
• By solution we mean sharing rule.
• An imputation is a payoff vector (x1,…,xn) such that:
xi > v(i), i N (individually rational) and S xi = v(N) (efficient)
• Which criteria? => axiomatic approach
• But what do we have to share? The worth of the grand coalition? The
worth of smaller coalitions?
• We will analyze one solution concept:
– Shapley value
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Shapley Value
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Shapley value
• We consider a characteristic function game (N,v).
• A value associates to every game v and every
player a real number.
i N,v R, i  N
• Positive interpretation of the Shapley value:
Question: What would be the expected outcome of
a bargaining to share v(N)?
• Normative interpretation of the Shapley value:
Question: What would be a « fair » division of the
worth of the grand coalition v(N)?
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A heuristic approach to the
Shapley value
• Partners have to negotiate about the sharing of v(N).
• Players agree on three points:
– Each partner must be remunerated at the level of her
contribution (her incremental value).
– Problem: this incremental value depends on the group
the player joins. We need to calculate a kind of average
of the different incremental values.
– Each one of the n! orders has the same probability to
appear.
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The Shapley value
Heuristic Description
• Players have to meet in a bargaining room to share what they
can obtain all together: v(N).
• They arrive sequentially and the order in which they do so is
determined by chance, with all arrival orders equally probable.
• Each player, when she enters the room, demands and is
promised the amount which her participation contributes to the
value of the coalition already in the room.
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More Definitions
• Consider a game (U,v) (U is the « universe » of
players) and consider N a carrier of v:
SU, v(S)  v(SN)
• i is a dummy player if
SU, v(Si)  v(S)
• If v and w are two characteristic functions, v + w
is a characteristic function such that:
(v + w)(S)= v(S) + w(S)
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Axiomatization
• Axiom 1 Anonymity (Symmetry): the value is invariant to
any permutation p (a mapping of U onto itself)
pi (pv) = i (v)
What each partner can obtain or contribute should not depend on her
name.
• Axiom 2 Efficiency: For each carrier N,
SiN i (v) = v(N)
Each null player gets zero.
• Axiom 3 Additivity
i (v + w) = i (v) + i (w)
Two negotiations about different objects are independent: What a partner
can obtain as a result of two negotiations is just the sum of what she
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can get in each one.
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Theorem
A unique value function exists which satisfies
Axioms 1-3, for games with finite carriers;
this is the Shapley Value.
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i  N,
The Shapley value
i  N , v  
n  s !s  1!S  N ,iS vS   vS \ i 
n!
i: a partner
N: the set of partners
V: the characteristic function
n: the size of N
S: the size of coalition S
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Proof
• The Shapley value satisfies the three axioms.
• What about the unicity? Insight into the proof
• For any coalition S C N, we can define a unanimity game V by:
VS (T) = 1 if S C T
VS (T) = 0 otherwise
• For each unanimity game VS , there is a unique value which satisfies
the three axioms:
i (VS) = 0 if i  N\S
i (VS) = 1/|S| if i  S
• Property of unanimity games: The class of unanimity games forms a
basis for the class of all the (characteristic function) games. In other
words, any game is a linear combination of unanimity games.
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Applications
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Application of the Shapley value 1
Airport Cost Game
• Example: An airport new landing runway needs to be
constructed.
• It will be used by three planes of different size.
• A bigger plane needs a longer runway.
• How to share the cost of constructing the runway among the
three planes?
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Airport Cost Game
How to share the cost of constructing the runway among
the three planes?
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Airport Cost Game
How to share the cost of constructing the runway
among the three planes?
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Airport Cost Game
How to share the cost of constructing the runway among
the three planes?
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Airport Cost Game
How to share the cost of constructing the runway among
the three planes?
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Application of the Shapley value 1:
Airport game
Littlechild and Owen (1973)
• How to allocate the cost of constructing or
maintaining a public facility among users?
• How to share the cost of a capacity?
• n planes of different size: N={1,…,n}
• Construction of a runway
• m types of plane: N = N1 U N2 U … U Nm
• Ci construction necessary for type i:
C1<…<Cm
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• For a coalition SCN, the runway necessary has to
be long enough for the largest plane of the
coalition.
• ==> cost sharing game:
C(S) = Cj(S) With j(S) = Max {j | S Nj = O}
• Application of the Shapley value ==> allocation of
costs:
Fj = C1/n + (C2-C1)/(n-n1) +…+ (Cj-Cj-1)/(n-nj-1)
• Application to the Birmingham Airport
(investment and policy pricing in 1968-1969)
==> The real fees appear to be similar to the Shapley
value.
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Application of the Shapley Value 2:
Measuring voting power
• How to measure the power of different voters in a given
procedure?
• The measurement of voting power is done “à priori”, before
preferences of voters on the alternatives are known.
• However, ex post, the probability to win will depend on the
interest groups.
•  In a simple game (or voting game), players form
coalitions with the only objective to win a vote
V is a simple game if and only if:
V(S) = 0 or 1 for each S
If V(S) = 1, S is a winning coalition
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Each voting procedure can be represented by a simple game
Examples: If N is the set of voters, which coalitions are winning in the
different procedures?
• The simple majority procedure can be represented by the following
simple game: simple majority game
•V(S) = 1 for each S such that s > n/2
•V(S) = 0 otherwise
• The dictatorship procedure: unanimity game
VK(S) = 1 for each S such that K S
= 0 otherwise
• Weighted majority game [M;w1,…,wn]
• M is the minimal number of votes to win (the majority)
• n the number of voters
• wi the weight of voter i, i.e. the number of votes
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Example of a weighted majority game: the procedure for
electing a president in the US
Two stages
1. First Election of the Great Electors in each state  electoral
college
2. Second stage: the electoral college elects the president by
simple majority rule. The number of great electors for each
state depends on the population of the state.
•
Assumption: each great elector votes for the candidate
preferred by the majority of her state. Therefore, the different
great electors of a same state vote in the same way.
•
 The result can be different from a direct majority
procedure: a narrow majority in a densely populated state,
like California, can affect an election’s outcome more than
wide majorities in several small states.
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•Banzahf index (1965): for each player, we count the number of
swings:
S is a swing for player i if and only if:
iS, VS   1 and VS \
i  0
•i is a pivot player for coalition S.
bi = number of swing for i / number of possible coalitions to
which i could belong
•Shapley-Shubik index (1954) (application of the Shapley value
to simple games): the order in which a voter joins a coalition is
relevant.
Fi = (S S,swing for i (s-1)!(n-s)!)/n!
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Example: comparison of SS-index and Banzahf index
[3; 2, 1, 1]
A,B, C
1. Shapley-Shubik index
There are 3! Orders in which A, B and C can declare
their support for a bill. In each order we look for pivot voter
(swing voter).
2. Banzhaf index
We do not take into account the order in which the voters
join a coalition. We only consider winning coalitions and for
each of them we look for swing voter.
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Section 2 Shapley Value with
Coalition Structure
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Coalition Structure Value
• Players belong to coalitions in a structure:
B = (B1 ,…, Bm )
• Assume that this coalition structure is binding:
players who do not belong to the same coalition
cannot cooperate.
• A coalition structure value associates to every
game v, every coalition structure and every player
a real number: i (v,B).
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Coalition Structure Value
• Aumann & Drèze IJGT (1974)
• A Shapley value defined for a given coalition
structure
• Each coalition forms and gets its worth.
• How do the members of each coalition share this
worth?
• Can we say that the computation of the Shapley
value in a game with fixed coalition structure boils
down to the computation of the SV for each of the
elements of the coalition structure?
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• Axiom 1: Relative efficiency
(v,B)(Bk) = v(Bk)
• Axiom 2: Symmetry (anonimity)
• Axiom 3: Additivity
i (v + w, B)(N) = i (v, B) + i (w, B)
• Axiom 4: Null-player condition
i (v, p) = 0 if i is a dummy
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Restriction property:
• For each SCN, denote by v|S the game on S
defined for all TCS by (v|S)(T) = v(T).
• Theorem: There is a unique coalition structure
value which satisfies Axioms 1-4, it is given for all
Bk and all i Bk by:
i (v,B) = i (v|Bk)
• The restriction of the value is the value of the
restriction of the game.
• Intuitive when there are no externalities.
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Coalition Structure Value
• Owen (1977), Hart and Kurz (1983)
• The coalition structure is binding: players
who do not belong to the same coalition
cannot cooperate.
• Coalitions do not form to get their worth.
• Coalitions form to be in a better position to
bargain whith the others on how to devide
the worth of the grand coalition v(N).
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•
•
•
•
The set of possible random orders is restricted
by the coalition structure.
Orders consistent with the coalition structure are
retained: coalitions of the structure are blocks in
the consistent orders.
The different consistent orders appear with the
same probability.
Given a game V and a CS p, we say that the
game among coalitions is inessential if:


 
V k 1,...,mBk  k 1,...,mV Bk
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Axiomatic Approach
• Axiom 1: Efficiency (N is a carrier)
(v, B)(N) = S i (v, B) = v(N)
• Axiom 2: Anonymity (symmetry)
• Axiom 3: Additivity
i (v + w, B) = i (v, B) + i (w, B)
• Axiom 4: Inessential Game
For each coalition of the structure
(v, B)(Bk) = v(Bk)
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Theorem
• The unique coalition structure value satisfying
Axioms 1-4 is given by:

  
 i v, B   E v Pi  i  v Pi
• Where the expectation E is over all random orders on
a carrier N of v that are consistent with B and Pi
denotes the random set of predecessors of i.
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Application
Kauppi and Widgren
Economic Policy (2004)
• In the European Union, most measures are adopted
under qualified majority voting rules.
• Consequence: A country member of the EU can end
up enforcing laws that its government opposed.
• This occurs more often in countries which our not
powerful in the EU decision making process.
•  The importance of national voting power in the
European Union.
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Application
Kauppi and Widgren
Economic Policy (2004)
• How can we measure the countries’ voting power?
• We can apply the power indices we know, SS and B,
to the EU rules.
• How can we verify the accuracy of the voting power
indices?
• We cannot measure directly the power of each
country.
• We can measure the EU budget allocation across
members: one manifestation of power.
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Application
Kauppi and Widgren
Economic Policy (2004)
• Assumption: the EU allocation across members is one
manifestation of power.
• Implicit assumption: each country is purely selfish
and « budjet maximizer ».
• Alternative assumption: distribution of EU spending
among members is based on «needs» rather than
power.
– Commun Agricultural Policy (CAP)
– Structural & Cohesion Funds  funds allocated to the
poorest countries in the EU.
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Application
Kauppi and Widgren
Economic Policy (2004)
• How to test whether the « power » view or the « needs » view
provide a better explanation of the EU budget allocation.
• First regression: the budget share can be explained by
– SS power index
– The member’s share of EU agricultural output
– The member’s per capita income relative to the EU-wide per capita
income.
Sit = a + b1Pit + b2 Ait + b3yit + uit
We can interpret b1 (SSI) and b2 (AGRI) parameters estimation as the share of the budget
allocation which is explained by the SSI versus AGRI variables.
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Application
Kauppi and Widgren
Economic Policy (2004)
– Second regression: using a modified SS index
– To take into account coalitions
– The key problem is to identify the most relevant
groupings
– By considering variations of the SS index, we
improve the fit necessarily.
– But they choose the best grouping: FranceGermany statistically.
– They grouped together the countries which are
more likely to vote together.
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Second Week:
Strategic Approach
Normal Form Games
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A very general framework
• We want to use a strategic approach to analyse the
stability of coalitions
• The set of players (individual, firms, nations…):
N = {1,…,n}
• 1) First step: the game of coalition formation
in which each player chooses the coalition she wants
to belong to.
• 2) Second step: the game between coalitions that
determines the payments.
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First step: Endogenous Coalition
Structures
• Normal form games of coalition formation
(N, Pi, Si)r
– Si is a set of strategies (wishes) to form
coalitions
– r is a rule mapping a coalition structure B to
each strategy profile  (vector of strategies)
r

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B
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Equilibria for
Normal Form Games
• Definition: For a given normal form game (N, Pi,
Si),  is a Nash equilibrium if and only if no
player has any incentive to deviate unilaterally
Pi() >= Pi(’, ) i N, ’ Si
• Definition: For a given normal form game (N, Pi,
Si),  is a strong Nash equilibrium if and only if
no coalition M has any incentive to deviate
unilaterally
M N, i M, and ’ Si
Pi() < = Pi(’, )
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Exemple: Open membership
games
• D’Aspremont et al CJE (1983), Thoron CJE
(98)
• N = {1,…,n} a set of symmetric players
• Si = {C,R} = {1,0}
• r:   k = S i=1,…,n i
• Only one coalition is formed: the cartel, the
agreement.
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Application 1: Stable Cartels
D’Aspremont et al. CJE (1983) Motivation
• Why do cartels form?
• Firms have an incentive to form a cartel to
decrease competition (collusion).
• The cartel members decrease their production in
order to increase the price.
• Why are cartels “unstable”? Stigler (1968)
• The non-members benefit from the high price
without bearing the cost of the cut producing:
there are positive externalities
• They have an incentive to free ride
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• First step: open membership game
• N is symmetrical  The size k is the only relevant
characteristic. A cartel of size k is formed.
• Second step: oligopoly game
•  PC(k) utility of each cartel member
Each cartel member maximizes the sum of its
partners’ payoffs.
•  PF(k) utility of each non-member
Each frange member maximizes its own payoff.
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A cartel of size k is stable iff it satisfies:
• Internal stability
PC(k) > PF(k-1)
No member has any incentive to leave the cartel
• External stability
PC(k+1) < PF(k).
No non member has any incentive to join the cartel.
• A strategy profile  is a Nash equilibrium of the
open membership game r iff
– k = S i=1,…,n i
– k satisfies the internal and external stabilities
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Theorem: stable cartels exist
• By definition, the cartel of size k = 1 is internally
stable
• By definition, k = n is externally stable
• If k is not externally stable, then k + 1 is internally
stable:
PF(k) < Pc(k + 1)
• Let us start with k = 1 and let us increase the size
untill the cartel is stable.
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Application 1: Stable cartel in a
Cournot oligopoly
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• Two properties of positive externalities:
• (P1) PF.k PFk1,k1,...,n1
• (P2) PF1PC1 and
PFk PCk ,k2,...,n1
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Application 2: IEA (International
Environmental Agreement)
• Yi (1997), Ray and Vohra (2001), Thoron (2004)
• Contribution to a public good
• N = {1,…,n}
• Each player is endowed with one unit of private good.
• If player i provides xi units of public good for a cost in private goods
c(x) = cx2, c > 0
• Amount of public good: X = S i = 1,…,n xi
• Benefit from consuming the public good: g(X)
• Player i’s net payoff is: Pi(x1,…,xn) = g(X) – c(xi)
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Application 2: IEA (International
Environmental Agreement)
• First step: open membership game to reach an agreement
N = {1,…,n} identical countries, k countries reach an agreement
to contribute to a public good (control of polution)
• Second step: Contribution to a public good game
• Each signatory i chooses to provide xi units of public good to
maximize the sum of its partners’ utilities
• Each non signatory i chooses to provide xi units of public good
to maximize its own utility
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Application 2: IEA (International
Environmental Agreement)
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Interpretation of the Internal and external
stability in this framework:
• When a country withdraws, the remaining signatories reduce
their abatement levels, and hence punish the country.
• When the agreement is internally stable this is because this
punishment is higher than the cost saving from withdrawing.
• When a country joins the IEA, the other signatories increase
their abatement levels, and hence reward the country for
acceding the agreement
• When the agreement in externally stability, this is because this
reward is not enough to compensate the increase in cost of the
potential new signatory.
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Address Games
• We do not want to restrict the number of
coalitions to be formed
• Address game:
• Set of strategies: a set of addresses
Si = (a1,…,al) with l > n.
• The new rule r: On each « location », an
alliance is formed by the inhabitants.
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Application 1
• Yi and Shin IJIO (2000): “Endogenous
Formation of Research Coalition with
Spillovers »
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Exclusive Membership
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Normal Form Games of Coalition
Formation
• Exclusive membership games Hart and Kurz (83)
Each player chooses the coalition she wants to belong to.
The strategy is a wish.
iN,SiiSi /SiN and iSi
D rule: a coalition is formed by all the players who have
announced the same wish, whether or not this wish can be
realized.
G rule: a coalition is formed if and only if all the members
have announced the same coalition.
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D rule: a coalition is formed by all the players
who have announced the same wish, whether
or not this wish can be realized.
BD T N /i, jT if and only if SiS j
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G rule: a coalition is formed if and only if all
the members have announced the same
coalition.


BG  Ti /iN
Ti S i if Si S j jS i
Ti 
i otherwise
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Hart and Kurz (1983)
• First step: G or D game of coalition formation
==> the outcome is a coalition structure B
• Second step: Determination of the Coalition
Structure Value HK(v, B)
==> payoffs
• A coalition structure is stable if it is generated by a
strong Nash equilibrium
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Hart and Kurz (1983)
• If the payoffs are determined by HK (v,p),
Hart and Kurz (83) prove that there exist
games in which there is no stable coalition
structure.
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Third Week
Strategic Approach
Extensive Form Games
Networks and Coalitions
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