networks and coalitions masterae2 - m2 1 coalitions in negotiation work in progress sylvie thoron...

Networks and Coalitions MasterAE2 - M2

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Coalitions in Negotiationwork 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 - QuestionsFramework: Game theory, cooperative and non-

cooperative. – 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

– 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.

)()(,,, TvSvTSvTSTS


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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 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.

• 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)?

NivNi ,R ,


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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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• Consider a game (U,v) (U is the « universe » of players) and consider N a carrier of v:

• i is a dummy player if

• If v and w are two characteristic functions, v + w is a characteristic function such that:

(v + w)(S)= v(S) + w(S)

)( )( , NSvSvUS

)( )( , SviSvUS

More Definitions


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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, i 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 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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The Shapley value

!

\!1!,

N,i

,i n

iSvSvssnvN SiNS

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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• 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\Si (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.

Proof


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Applications


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Application of the Shapley value 1Airport 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 GameHow to share the cost of constructing the runway among

the three planes?


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Airport Cost GameHow to share the cost of constructing the runway

among the three planes?


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the three planes?


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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: j = 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 SIf V(S) = 1, S is a winning coalition


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Each voting procedure can be represented by a simple gameExamples: 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 is a pivot player for coalition S.

i = 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.

i = (S,swing for i (s-1)!(n-s)!)/n!

0 \V and 1 V , iSSSi


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Example: comparison of SS-index and Banzahf index[3; 2, 1, 1] A,B, C

1. Shapley-Shubik indexThere 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 indexWe 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: = (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,)(Bk) = v(Bk)

• Axiom 2: Symmetry (anonimity)

• Axiom 3: Additivityi(v + w, )(N) = i(v, ) + i(w, )

• Axiom 4: Null-player conditioni(v, ) = 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 Bk by:

i(v,) = 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 , we say that the game among coalitions is inessential if:

mk kkmk BVBV ,...,1,...,1


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Axiomatic Approach• Axiom 1: Efficiency (N is a carrier)(v, )(N) = i(v, ) = v(N)

• Axiom 2: Anonymity (symmetry)

• Axiom 3: Additivityi(v + w, ) = i(v, ) + i(w, )

• Axiom 4: Inessential GameFor each coalition of the structure

(v, )(Bk) = v(Bk)


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Theorem• The unique coalition structure value satisfying

Axioms 1-4 is given by:

• 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.

iii PviPvEBv ,


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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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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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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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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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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: France-

Germany 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, i, i)– i is a set of strategies (wishes) to form

coalitions– is a rule mapping a coalition structure B to

each strategy profile (vector of strategies)


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Equilibria for Normal Form Games

• Definition: For a given normal form game (N, i, i), is a Nash equilibrium if and only if no player has any incentive to deviate unilaterally

i() >= i(’, ) i N, ’ i

• Definition: For a given normal form game (N, i, i), is a strong Nash equilibrium if and only if no coalition M has any incentive to deviate unilaterally

M N, i M, and ’ i

i() < = i(’, )


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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

• i = {C,R} = {1,0} • : k = i=1,…,n i

• Only one coalition is formed: the cartel, the agreement.


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Application 1: Stable CartelsD’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 C(k) utility of each cartel memberEach cartel member maximizes the sum of its partners’

payoffs.

F(k) utility of each non-memberEach frange member maximizes its own payoff.


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A cartel of size k is stable iff it satisfies:• Internal stability

C(k) > F(k-1) No member has any incentive to leave the cartel

• External stabilityC(k+1) < F(k).

No non member has any incentive to join the cartel.

• A strategy profile is a Nash equilibrium of the open membership game iff – k = 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:

F(k) < c(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) .

• (P2) and

1,...,1,1 nkkk FF

11 CF

1,...,2, nkkk CF


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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 = i = 1,…,n xi

• Benefit from consuming the public good: g(X)

• Player i’s net payoff is: i(x1,…,xn) = g(X) – c(xi)


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• First step: open membership game to reach an agreementN = {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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• 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.

Interpretation of the Internal and external stability in this framework:


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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

i = (a1,…,al) with l > n.

• The new rule : 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.

rule: a coalition is formed by all the players who have announced the same wish, whether or not this wish can be realized.

rule: a coalition is formed if and only if all the members have announced the same coalition.

iiiii SiNSSNi and /,


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rule: a coalition is formed by all the players who have announced the same wish, whether or not this wish can be realized.

ji SSTjiNTB ifonly and if ,/


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rule: a coalition is formed if and only if all the members have announced the same coalition.

NiTB i /

iji

ii SjSSST if

otherwise iT i


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Hart and Kurz (1983)

• First step: or game of coalition formation

==> the outcome is a coalition structure B• Second step: Determination of the Coalition

Structure Value (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 (v,), 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 masterae2 - m2 1 coalitions in negotiation work in progress sylvie thoron...

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