2. Networks and Coalitions MasterAE2 - M2 1 Coalitions in Negotiation work in progress Sylvie Thoron GREQAM 2006/2007

3. Networks and Coalitions MasterAE2 - M2 2 Introduction

4. Networks and Coalitions MasterAE2 - M2 3 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.

5. Networks and Coalitions MasterAE2 - M2 4 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…)

6. Networks and Coalitions MasterAE2 - M2 5 Introduction - Questions Framework: 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)

7. Networks and Coalitions MasterAE2 - M2 6 How to define a game? Cooperative game theory –coalitional function games Non-cooperative game theory –Normal form games –Extensive form games

8. Networks and Coalitions MasterAE2 - M2 7 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)

9. Networks and Coalitions MasterAE2 - M2 8 First Week: Cooperative Approach Coalitional Form Games

10. Networks and Coalitions MasterAE2 - M2 9 Section 1 Value

11. Networks and Coalitions MasterAE2 - M2 10 Section 1 Value 1.1Definitions

12. Networks and Coalitions MasterAE2 - M2 11 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).

13. Networks and Coalitions MasterAE2 - M2 12 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.

14. Networks and Coalitions MasterAE2 - M2 13 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 (x 1,…,x n ) such that: x i > v(i), i N (individually rational) and  x i = 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

15. Networks and Coalitions MasterAE2 - M2 14 Shapley Value

16. Networks and Coalitions MasterAE2 - M2 15 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)?

17. Networks and Coalitions MasterAE2 - M2 16 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.

18. Networks and Coalitions MasterAE2 - M2 17 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.

19. Networks and Coalitions MasterAE2 - M2 18 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) More Definitions

20. Networks and Coalitions MasterAE2 - M2 19 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.

21. Networks and Coalitions MasterAE2 - M2 20 Theorem A unique value function exists which satisfies Axioms 1-3, for games with finite carriers; this is the Shapley Value.

22. Networks and Coalitions MasterAE2 - M2 21 The Shapley value i: a partner N: the set of partners V: the characteristic function n: the size of N S: the size of coalition S

23. Networks and Coalitions MasterAE2 - M2 22 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: V S (T) = 1 if S C T V S (T) = 0 otherwise For each unanimity game V S, there is a unique value which satisfies the three axioms:  i (V S ) = 0 if i N\S  i (V S ) = 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

24. Networks and Coalitions MasterAE2 - M2 23 Applications

25. Networks and Coalitions MasterAE2 - M2 24 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?

26. Networks and Coalitions MasterAE2 - M2 25 Airport Cost Game How to share the cost of constructing the runway among the three planes?

27. Networks and Coalitions MasterAE2 - M2 26 Airport Cost Game How to share the cost of constructing the runway among the three planes?

28. Networks and Coalitions MasterAE2 - M2 27 Airport Cost Game How to share the cost of constructing the runway among the three planes?

29. Networks and Coalitions MasterAE2 - M2 28 Airport Cost Game How to share the cost of constructing the runway among the three planes?

30. Networks and Coalitions MasterAE2 - M2 29 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 = N 1 U N 2 U … U N m C i construction necessary for type i: C 1 <…<C m

31. Networks and Coalitions MasterAE2 - M2 30 For a coalition SCN, the runway necessary has to be long enough for the largest plane of the coalition. ==> cost sharing game: C(S) = C j (S) With j(S) = Max {j | S N j = O} Application of the Shapley value ==> allocation of costs:  j = C 1 /n + (C 2 -C 1 )/(n-n 1 ) +…+ (C j -C j-1 )/(n-n j-1 ) Application to the Birmingham Airport (investment and policy pricing in 1968-1969) ==> The real fees appear to be similar to the Shapley value.

32. Networks and Coalitions MasterAE2 - M2 31 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

33. Networks and Coalitions MasterAE2 - M2 32 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 V K (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 w i the weight of voter i, i.e. the number of votes

34. Networks and Coalitions MasterAE2 - M2 33 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.

35. Networks and Coalitions MasterAE2 - M2 34 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!

36. Networks and Coalitions MasterAE2 - M2 35 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.

37. Networks and Coalitions MasterAE2 - M2 36 Section 2 Shapley Value with Coalition Structure

38. Networks and Coalitions MasterAE2 - M2 37 Coalition Structure Value Players belong to coalitions in a structure:  = (B 1,…, B m ) 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).

39. Networks and Coalitions MasterAE2 - M2 38 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?

40. Networks and Coalitions MasterAE2 - M2 39 Axiom 1: Relative efficiency  (v,  )(B k ) = v(B k ) Axiom 2: Symmetry (anonimity) Axiom 3: Additivity  i  (v + w,  )(N) =  i  (v,  ) +  i  (w,  ) Axiom 4: Null-player condition  i  (v,  ) = 0 if i is a dummy

41. Networks and Coalitions MasterAE2 - M2 40 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 B k and all i B k by:  i  (v,  ) =  i  (v|B k ) The restriction of the value is the value of the restriction of the game. Intuitive when there are no externalities.

42. Networks and Coalitions MasterAE2 - M2 41 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).

43. Networks and Coalitions MasterAE2 - M2 42 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:

44. Networks and Coalitions MasterAE2 - M2 43 Axiomatic Approach Axiom 1: Efficiency (N is a carrier)  (v,  )(N) =   i  (v,  ) = v(N) Axiom 2: Anonymity (symmetry) Axiom 3: Additivity  i  (v + w,  ) =  i  (v,  ) +  i  (w,  ) Axiom 4: Inessential Game For each coalition of the structure  (v,  )(B k ) = v(B k )

45. Networks and Coalitions MasterAE2 - M2 44 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 P i denotes the random set of predecessors of i.

46. Networks and Coalitions MasterAE2 - M2 45 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.

47. Networks and Coalitions MasterAE2 - M2 46 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.

48. Networks and Coalitions MasterAE2 - M2 47 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.

49. Networks and Coalitions MasterAE2 - M2 48 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. S it = a + b 1 Pit + b 2 A it + b 3 y it + u it 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.

50. Networks and Coalitions MasterAE2 - M2 49 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: France- Germany statistically. –They grouped together the countries which are more likely to vote together.

51. Networks and Coalitions MasterAE2 - M2 50 Second Week: Strategic Approach Normal Form Games

52. Networks and Coalitions MasterAE2 - M2 51 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.

53. Networks and Coalitions MasterAE2 - M2 52 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)   

54. Networks and Coalitions MasterAE2 - M2 53 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 (  ’,  )

55. Networks and Coalitions MasterAE2 - M2 54 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.

56. Networks and Coalitions MasterAE2 - M2 55 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

57. Networks and Coalitions MasterAE2 - M2 56 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 member Each cartel member maximizes the sum of its partners’ payoffs.   F (k) utility of each non-member Each frange member maximizes its own payoff.

58. Networks and Coalitions MasterAE2 - M2 57 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 stability  C (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

59. Networks and Coalitions MasterAE2 - M2 58 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.

60. Networks and Coalitions MasterAE2 - M2 59 Application 1: Stable cartel in a Cournot oligopoly

61. Networks and Coalitions MasterAE2 - M2 60 Two properties of positive externalities: (P1). (P2) and

62. Networks and Coalitions MasterAE2 - M2 61

63. Networks and Coalitions MasterAE2 - M2 62 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 x i units of public good for a cost in private goods c(x) = cx2, c > 0 Amount of public good: X =  i = 1,…,n x i Benefit from consuming the public good: g(X) Player i’s net payoff is:  i(x 1,…,x n ) = g(X) – c(x i )

64. Networks and Coalitions MasterAE2 - M2 63 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

65. Networks and Coalitions MasterAE2 - M2 64 Application 2: IEA (International Environmental Agreement)

66. Networks and Coalitions MasterAE2 - M2 65 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:

67. Networks and Coalitions MasterAE2 - M2 66 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 = (a 1,…,a l ) with l > n. The new rule  : On each « location », an alliance is formed by the inhabitants.

68. Networks and Coalitions MasterAE2 - M2 67 Application 1 Yi and Shin IJIO (2000): “Endogenous Formation of Research Coalition with Spillovers »

69. Networks and Coalitions MasterAE2 - M2 68 Exclusive Membership

70. Networks and Coalitions MasterAE2 - M2 69 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.

71. Networks and Coalitions MasterAE2 - M2 70  rule: a coalition is formed by all the players who have announced the same wish, whether or not this wish can be realized.

72. Networks and Coalitions MasterAE2 - M2 71  rule: a coalition is formed if and only if all the members have announced the same coalition.

73. Networks and Coalitions MasterAE2 - M2 72 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

74. Networks and Coalitions MasterAE2 - M2 73 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.

75. Networks and Coalitions MasterAE2 - M2 74 Third Week Strategic Approach Extensive Form Games