Bargaining and Cooperation in Strategic Form Games

Transcription

1 Bargaining and Cooperation in Strategic Form Games Sergiu Hart July 2008 Revised: January 2009 SERGIU HART c 2007 p. 1

2 Bargaining and Cooperation in Strategic Form Games Sergiu Hart Center of Rationality, Dept. of Economics, Dept. of Mathematics The Hebrew University of Jerusalem SERGIU HART c 2007 p. 2

3 Joint work with Andreu Mas-Colell Universitat Pompeu Fabra, Barcelona SERGIU HART c 2007 p. 3

4 Joint work with Andreu Mas-Colell Universitat Pompeu Fabra, Barcelona Center for Rationality DP-484, May 2008 ("Cooperative Games in Strategic Form") Revised: January 2009 SERGIU HART c 2007 p. 3

5 Joint work with Andreu Mas-Colell Universitat Pompeu Fabra, Barcelona Center for Rationality DP-484, May 2008 ("Cooperative Games in Strategic Form") Revised: January SERGIU HART c 2007 p. 3

6 Introduction SERGIU HART c 2007 p. 4

7 Introduction PROGRAM : SERGIU HART c 2007 p. 4

8 Introduction PROGRAM : GIVEN: game in strategic form SERGIU HART c 2007 p. 4

9 Introduction PROGRAM : GIVEN: game in strategic form AIM: bargaining and cooperation SERGIU HART c 2007 p. 4

10 Introduction PROGRAM : GIVEN: game in strategic form AIM: bargaining and cooperation Nash (1953): 2-person ("variable-threat") SERGIU HART c 2007 p. 4

11 Introduction PROGRAM : GIVEN: game in strategic form AIM: bargaining and cooperation Nash (1953): 2-person ("variable-threat") Harsanyi (1959, 1963): N-person SERGIU HART c 2007 p. 4

12 Introduction PROGRAM : GIVEN: game in strategic form AIM: bargaining and cooperation Nash (1953): 2-person ("variable-threat") Harsanyi (1959, 1963): N-person But: the players are not the original players (each coalition has a "player") SERGIU HART c 2007 p. 4

13 Introduction SERGIU HART c 2007 p. 5

14 Introduction The standard approach: SERGIU HART c 2007 p. 5

15 Introduction The standard approach: (1) Derive a COALITIONAL GAME from the strategic form SERGIU HART c 2007 p. 5

16 Introduction The standard approach: (1) Derive a COALITIONAL GAME from the strategic form (2) Apply a COOPERATIVE SOLUTION to the COALITIONAL GAME SERGIU HART c 2007 p. 5

17 Introduction The standard approach: (1) Derive a COALITIONAL GAME from the strategic form (2) Apply a COOPERATIVE SOLUTION to the COALITIONAL GAME OR (2 ) Apply a NONCOOPERATIVE BARGAINING PROCEDURE to the COALITIONAL GAME SERGIU HART c 2007 p. 5

18 Bargaining and Cooperation SERGIU HART c 2007 p. 6

19 Bargaining and Cooperation STRATEGIC FORM SERGIU HART c 2007 p. 6

20 Bargaining and Cooperation STRATEGIC FORM COOPERATIVE OUTCOMES SERGIU HART c 2007 p. 6

21 Bargaining and Cooperation STRATEGIC FORM? COOPERATIVE OUTCOMES SERGIU HART c 2007 p. 6

22 Bargaining and Cooperation STRATEGIC FORM? COOPERATIVE OUTCOMES COALITIONAL FORM SERGIU HART c 2007 p. 6

23 Bargaining and Cooperation STRATEGIC FORM? COOPERATIVE OUTCOMES (1) COALITIONAL FORM SERGIU HART c 2007 p. 6

24 Bargaining and Cooperation STRATEGIC FORM? COOPERATIVE OUTCOMES (1) (2) (2 ) COALITIONAL FORM SERGIU HART c 2007 p. 6

25 Bargaining and Cooperation STRATEGIC FORM COOPERATIVE OUTCOMES (1) (2) (2 ) COALITIONAL FORM SERGIU HART c 2007 p. 6

26 Introduction The standard approach: (1) Derive a COALITIONAL GAME from the strategic form (2) Apply a COOPERATIVE SOLUTION to the COALITIONAL GAME OR (2 ) Apply a NONCOOPERATIVE BARGAINING PROCEDURE to the COALITIONAL GAME SERGIU HART c 2007 p. 7

27 Introduction SERGIU HART c 2007 p. 8

28 Introduction How to define the COALITIONAL GAME? SERGIU HART c 2007 p. 8

29 Introduction How to define the COALITIONAL GAME? TU (transferable utility) (von Neumann and Morgenstern) SERGIU HART c 2007 p. 8

30 Introduction How to define the COALITIONAL GAME? TU (transferable utility) (von Neumann and Morgenstern) NTU (non-transferable utility) (Aumann: α-, β-) SERGIU HART c 2007 p. 8

31 Introduction How to define the COALITIONAL GAME? TU (transferable utility) (von Neumann and Morgenstern) NTU (non-transferable utility) (Aumann: α-, β-) "C-GAMES" (Shapley and Shubik): the coalitional form is "self-evident" and "uncontroversial" example: market games SERGIU HART c 2007 p. 8

32 Introduction How to define the COALITIONAL GAME? TU (transferable utility) (von Neumann and Morgenstern) NTU (non-transferable utility) (Aumann: α-, β-) "C-GAMES" (Shapley and Shubik): the coalitional form is "self-evident" and "uncontroversial" example: market games Question: Is the "factorization" through the coalitional game appropriate? SERGIU HART c 2007 p. 8

33 This paper SERGIU HART c 2007 p. 9

34 This paper Propose a specific ("simple" and "natural") bargaining procedure: SERGIU HART c 2007 p. 9

35 This paper Propose a specific ("simple" and "natural") bargaining procedure: The PC PROCEDURE ("Proposer Commitment") SERGIU HART c 2007 p. 9

36 This paper Propose a specific ("simple" and "natural") bargaining procedure: The PC PROCEDURE ("Proposer Commitment") Study its outcomes and implications in various setups SERGIU HART c 2007 p. 9

37 This paper Propose a specific ("simple" and "natural") bargaining procedure: The PC PROCEDURE ("Proposer Commitment") Study its outcomes and implications in various setups Get general conclusions on the "program" SERGIU HART c 2007 p. 9

38 The Model N-person GAME IN STRATEGIC FORM G SERGIU HART c 2007 p. 10

39 The Model N-person GAME IN STRATEGIC FORM G N = (finite) set of players SERGIU HART c 2007 p. 10

40 The Model N-person GAME IN STRATEGIC FORM G N = (finite) set of players For each i N: A i = (finite) set of actions u i : A R = payoff function SERGIU HART c 2007 p. 10

41 The Model N-person GAME IN STRATEGIC FORM G N = (finite) set of players For each i N: A i = (finite) set of actions u i : A R = payoff function Notations: x i (A i ): mixed action of player i N z S (A S ): correlated action of coalition S N (where A S = i S Ai ) SERGIU HART c 2007 p. 10

42 The PC Procedure The PC PROCEDURE ("Proposer Commitment") SERGIU HART c 2007 p. 11

43 The PC Procedure The PC PROCEDURE ("Proposer Commitment") (cf. Hart and Mas-Colell, Econometrica 1996) SERGIU HART c 2007 p. 11

44 The PC Procedure The PC PROCEDURE ("Proposer Commitment") (cf. Hart and Mas-Colell, Econometrica 1996) Let 0 ρ < 1 be a fixed parameter (the probability of "REPEAT"). SERGIU HART c 2007 p. 11

45 The PC Procedure The PC PROCEDURE ("Proposer Commitment") (cf. Hart and Mas-Colell, Econometrica 1996) Let 0 ρ < 1 be a fixed parameter (the probability of "REPEAT"). In each round: The set of "ACTIVE" players is S N SERGIU HART c 2007 p. 11

46 The PC Procedure The PC PROCEDURE ("Proposer Commitment") (cf. Hart and Mas-Colell, Econometrica 1996) Let 0 ρ < 1 be a fixed parameter (the probability of "REPEAT"). In each round: The set of "ACTIVE" players is S N The action of each "INACTIVE" player j / S is fixed at some b j A j. SERGIU HART c 2007 p. 11

47 The PC Procedure The PC PROCEDURE ("Proposer Commitment") (cf. Hart and Mas-Colell, Econometrica 1996) Let 0 ρ < 1 be a fixed parameter (the probability of "REPEAT"). In each round: The set of "ACTIVE" players is S N The action of each "INACTIVE" player j / S is fixed at some b j A j. Let ω = (S, b N\S ) (the "STATE"). Start in state (N, ) (everyone is "active"). SERGIU HART c 2007 p. 11

48 The PC Procedure SERGIU HART c 2007 p. 12

49 The PC Procedure In each state ω = (S, b N\S ) SERGIU HART c 2007 p. 12

50 The PC Procedure In each state ω = (S, b N\S ) 1. A "PROPOSER" k in S is selected at random SERGIU HART c 2007 p. 12

51 The PC Procedure In each state ω = (S, b N\S ) 1. A "PROPOSER" k in S is selected at random 2. The proposer k chooses z S (A S ) (a "PROPOSED AGREEMENT") SERGIU HART c 2007 p. 12

52 The PC Procedure In each state ω = (S, b N\S ) 1. A "PROPOSER" k in S is selected at random 2. The proposer k chooses z S (A S ) (a "PROPOSED AGREEMENT") x k (A k ) (a "THREAT") SERGIU HART c 2007 p. 12

53 The PC Procedure In each state ω = (S, b N\S ) 1. A "PROPOSER" k in S is selected at random 2. The proposer k chooses z S (A S ) (a "PROPOSED AGREEMENT") x k (A k ) (a "THREAT") 3. If all players in S AGREE to z S then a S A S is selected according to the distribution z S, and the procedure ENDS: the N-tuple of actions (a S, b N\S ) A is played in G SERGIU HART c 2007 p. 12

54 The PC Procedure In each state ω = (S, b N\S ) SERGIU HART c 2007 p. 13

55 The PC Procedure In each state ω = (S, b N\S ) 4. If at least one player in S REJECTS z S then SERGIU HART c 2007 p. 13

56 The PC Procedure In each state ω = (S, b N\S ) 4. If at least one player in S REJECTS z S then with probability ρ, REPEAT (the state remains ω = (S, b N\S )); with probability 1 ρ, the proposer k becomes INACTIVE: SERGIU HART c 2007 p. 13

57 The PC Procedure In each state ω = (S, b N\S ) 4. If at least one player in S REJECTS z S then with probability ρ, REPEAT (the state remains ω = (S, b N\S )); with probability 1 ρ, the proposer k becomes INACTIVE: b k A k is selected according to the distribution x k (the threat is realized) the new state is ω = (S\{k}, (b N\S, b k )) SERGIU HART c 2007 p. 13

58 The PC Procedure In each state ω = (S, b N\S ) 4. If at least one player in S REJECTS z S then with probability ρ, REPEAT (the state remains ω = (S, b N\S )); with probability 1 ρ, the proposer k becomes INACTIVE: b k A k is selected according to the distribution x k (the threat is realized) the new state is ω = (S\{k}, (b N\S, b k )) 5. Start a new round (i.e., go back to step 1). SERGIU HART c 2007 p. 13

59 Outcomes and Equilibria SERGIU HART c 2007 p. 14

60 Outcomes and Equilibria The procedure ends with probability 1 (since ρ < 1) SERGIU HART c 2007 p. 14

61 Outcomes and Equilibria The procedure ends with probability 1 (since ρ < 1) The procedure ends with an N-tuple of actions a A being played in the original strategic game G SERGIU HART c 2007 p. 14

62 Outcomes and Equilibria The procedure ends with probability 1 (since ρ < 1) The procedure ends with an N-tuple of actions a A being played in the original strategic game G SP EQUILIBRIUM: a (subgame-)perfect equilibrium in Stationary strategies SERGIU HART c 2007 p. 14

63 Outcomes and Equilibria Given an SP EQUILIBRIUM σ = (σ i ) i N and a state ω = (S, b N\S ): SERGIU HART c 2007 p. 15

64 Outcomes and Equilibria Given an SP EQUILIBRIUM σ = (σ i ) i N and a state ω = (S, b N\S ): ζ S ω (AS ) is the expected outcome starting at ω SERGIU HART c 2007 p. 15

65 Outcomes and Equilibria Given an SP EQUILIBRIUM σ = (σ i ) i N and a state ω = (S, b N\S ): ζ S ω (AS ) is the expected outcome starting at ω ζ S ω,k (AS ) is the expected outcome starting at ω with proposer k SERGIU HART c 2007 p. 15

66 Outcomes and Equilibria Given an SP EQUILIBRIUM σ = (σ i ) i N and a state ω = (S, b N\S ): ζ S ω (AS ) is the expected outcome starting at ω ζ S ω,k (AS ) is the expected outcome starting at ω with proposer k ζ S ω = 1 S k S ζs ω,k SERGIU HART c 2007 p. 15

67 Outcomes and Equilibria Given an SP EQUILIBRIUM σ = (σ i ) i N and a state ω = (S, b N\S ): ζ S ω (AS ) is the expected outcome starting at ω ζ S ω,k (AS ) is the expected outcome starting at ω with proposer k ζ S ω = 1 S... k S ζs ω,k SERGIU HART c 2007 p. 15

68 Outcomes and Equilibria Given an SP EQUILIBRIUM σ = (σ i ) i N and a state ω = (S, b N\S ):... SERGIU HART c 2007 p. 16

69 Outcomes and Equilibria Given an SP EQUILIBRIUM σ = (σ i ) i N and a state ω = (S, b N\S ):... Zω,k (ζ) is the set of "acceptable" (given ζ) proposals of k that maximize k s payoff SERGIU HART c 2007 p. 16

70 Outcomes and Equilibria Given an SP EQUILIBRIUM σ = (σ i ) i N and a state ω = (S, b N\S ):... Zω,k (ζ) is the set of "acceptable" (given ζ) proposals of k that maximize k s payoff Proposition. The SP EQUILIBRIA are characterized by: SERGIU HART c 2007 p. 16

71 Outcomes and Equilibria Given an SP EQUILIBRIUM σ = (σ i ) i N and a state ω = (S, b N\S ):... Zω,k (ζ) is the set of "acceptable" (given ζ) proposals of k that maximize k s payoff Proposition. The SP EQUILIBRIA are characterized by: ζ S ω,k Z ω,k (ζ) for all ω and k. SERGIU HART c 2007 p. 16

72 Basic Results SERGIU HART c 2007 p. 17

73 Basic Results Proposition. An SP EQUILIBRIUM always exists. SERGIU HART c 2007 p. 17

74 Basic Results Proposition. An SP EQUILIBRIUM always exists. Proposition. The SP EQUILIBRIUM outcomes become Pareto efficient as ρ 1. SERGIU HART c 2007 p. 17

75 Two-Person Games SERGIU HART c 2007 p. 18

76 Two-Person Games N = {1, 2} D := co {(u 1 (a), u 2 (a)) : a A} q i := min a j A j max a i A i u i (a i, a j ) SERGIU HART c 2007 p. 18

77 Two-Person Games N = {1, 2} D := co {(u 1 (a), u 2 (a)) : a A} q i := min a j A j max a i A i u i (a i, a j ) Proposition. Let G be a two-person strategic game such that (D, q) is a pure bargaining problem. As ρ 1 the SP equilibrium outcomes converge to the Nash bargaining solution of (D, q). SERGIU HART c 2007 p. 18

78 Transferable Utility SERGIU HART c 2007 p. 19

79 Transferable Utility Definition. G is a STRATEGIC TU GAME if for every state (S, b N\S ) there exists a real number v(s, b N\S ) such that any c = u S (z S, b N\S ) (for some z S (A S )) that is Pareto efficient for S and individually rational satisfies SERGIU HART c 2007 p. 19

80 Transferable Utility Definition. G is a STRATEGIC TU GAME if for every state (S, b N\S ) there exists a real number v(s, b N\S ) such that any c = u S (z S, b N\S ) (for some z S (A S )) that is Pareto efficient for S and individually rational satisfies c i = v(s, b N\S ) i S SERGIU HART c 2007 p. 19

81 Transferable Utility Definition. G is a STRATEGIC TU GAME if for every state (S, b N\S ) there exists a real number v(s, b N\S ) such that any c = u S (z S, b N\S ) (for some z S (A S )) that is Pareto efficient for S and individually rational satisfies c i = v(s, b N\S ) i S Proposition. Let G be a strategic TU game. Then the SP equilibrium outcomes are given by an "expected marginal contribution in a random order" formula... [read the paper] SERGIU HART c 2007 p. 19

82 Equilibria with Fixed Threats SERGIU HART c 2007 p. 20

83 Equilibria with Fixed Threats Definition. An SP equilibrium σ has FIXED THREATS (f k ) k N A if, for each player k, the threat of k (whenever k is the proposer) is f k in all states (S, f N\S ) (i.e., along the "backward induction equilibrium path"). SERGIU HART c 2007 p. 20

84 Equilibria with Fixed Threats Definition. An SP equilibrium σ has FIXED THREATS (f k ) k N A if, for each player k, the threat of k (whenever k is the proposer) is f k in all states (S, f N\S ) (i.e., along the "backward induction equilibrium path"). In this case, define an NTU coalitional game (N, V G,σ ) by V G,σ (S) = {c R S : c u S (z S, f N\S ) for some z S (A S )} for every coalition S N. SERGIU HART c 2007 p. 20

85 Equilibria with Fixed Threats Proposition. Let (N, V G,σ ) be derived from the strategic game G Let σ be a fixed-threat equilibrium Suppose that (N, V G,σ ) is a TU game SERGIU HART c 2007 p. 21

86 Equilibria with Fixed Threats Proposition. Let (N, V G,σ ) be derived from the strategic game G Let σ be a fixed-threat equilibrium Suppose that (N, V G,σ ) is a TU game Then the payoffs induced by σ equal the SHAPLEY VALUES of (N, V G,σ ) and its subgames SERGIU HART c 2007 p. 21

87 Equilibria with Fixed Threats Proposition. Let (N, V G,σ ) be derived from the strategic game G Let σ be a fixed-threat equilibrium Suppose that (N, V G,σ ) is a TU game Then the payoffs induced by σ equal the SHAPLEY VALUES of (N, V G,σ ) and its subgames NTU: MASCHLER OWEN VALUES SERGIU HART c 2007 p. 21

88 Fixed Threats SERGIU HART c 2007 p. 22

89 Fixed Threats When can one get the fixed-threat property? SERGIU HART c 2007 p. 22

90 Games with Damaging Actions When can one get the fixed-threat property? Definition. d k A k is a DAMAGING ACTION of player k if u i (d k, a N\k ) u i (a) for every a A and every player i k. SERGIU HART c 2007 p. 22

91 Games with Damaging Actions When can one get the fixed-threat property? Definition. d k A k is a DAMAGING ACTION of player k if u i (d k, a N\k ) u i (a) for every a A and every player i k. Definition. A strategic game G is a D-GAME if every player k N has a damaging action. SERGIU HART c 2007 p. 22

92 Games with Damaging Actions SERGIU HART c 2007 p. 23

93 Games with Damaging Actions Proposition. Let G be a strategic TU game which is a d-game. SERGIU HART c 2007 p. 23

94 Games with Damaging Actions Proposition. Let G be a strategic TU game which is a d-game. Then there exists a fixed-threat SP equilibrium of the PC procedure where each player k uses a damaging d k action as threat. SERGIU HART c 2007 p. 23

95 Games with Damaging Actions Proposition. Let G be a strategic TU game which is a d-game. Then there exists a fixed-threat SP equilibrium of the PC procedure where each player k uses a damaging d k action as threat. What about NTU? SERGIU HART c 2007 p. 23

96 c-games SERGIU HART c 2007 p. 24

97 c-games A "C-GAME" is a game where the coalitional function is "self-evident" and "uncontroversial" (Shapley and Shubik) SERGIU HART c 2007 p. 24

98 c-games and Market Games A "C-GAME" is a game where the coalitional function is "self-evident" and "uncontroversial" (Shapley and Shubik) For example: economic models of exchange without externalities (MARKET GAMES) SERGIU HART c 2007 p. 24

99 c-games and Market Games A "C-GAME" is a game where the coalitional function is "self-evident" and "uncontroversial" (Shapley and Shubik) For example: economic models of exchange without externalities (MARKET GAMES) Is a market game a D-GAME? SERGIU HART c 2007 p. 24

100 c-games and Market Games A "C-GAME" is a game where the coalitional function is "self-evident" and "uncontroversial" (Shapley and Shubik) For example: economic models of exchange without externalities (MARKET GAMES) Is a market game a D-GAME? Yes: "Keep your endowment" is a damaging action SERGIU HART c 2007 p. 24

101 c-games and Market Games A "C-GAME" is a game where the coalitional function is "self-evident" and "uncontroversial" (Shapley and Shubik) For example: economic models of exchange without externalities (MARKET GAMES) Is a market game a D-GAME? Yes: "Keep your endowment" is a damaging action Is this an optimal threat in equilibrium? SERGIU HART c 2007 p. 24

102 c-games and Market Games A "C-GAME" is a game where the coalitional function is "self-evident" and "uncontroversial" (Shapley and Shubik) For example: economic models of exchange without externalities (MARKET GAMES) Is a market game a D-GAME? Yes: "Keep your endowment" is a damaging action Is this an optimal threat in equilibrium? In the TU case: YES (previous Proposition) SERGIU HART c 2007 p. 24

103 c-games and Market Games A "C-GAME" is a game where the coalitional function is "self-evident" and "uncontroversial" (Shapley and Shubik) For example: economic models of exchange without externalities (MARKET GAMES) Is a market game a D-GAME? Yes: "Keep your endowment" is a damaging action Is this an optimal threat in equilibrium? In the TU case: YES (previous Proposition) In the NTU case: NOT NECESSARILY! SERGIU HART c 2007 p. 24

104 Market Games Are Not c-games SERGIU HART c 2007 p. 25

105 Market Games Are Not c-games Example. SERGIU HART c 2007 p. 25

106 Market Games Are Not c-games Example. A pure exchange economy (market) SERGIU HART c 2007 p. 25

107 Market Games Are Not c-games Example. A pure exchange economy (market) 4 commodities: b, c, f, g 3 traders: 1, 2, 3 SERGIU HART c 2007 p. 25

108 Market Games Are Not c-games Example. A pure exchange economy (market) 4 commodities: b, c, f, g 3 traders: 1, 2, 3 Initial endowments: e 1 = (0, 0, 1, 1) e 2 = (0, 1, 0, 0) e 3 = (1, 0, 0, 0) SERGIU HART c 2007 p. 25

109 Market Games Are Not c-games SERGIU HART c 2007 p. 26

110 Market Games Are Not c-games Utility functions: SERGIU HART c 2007 p. 26

111 Market Games Are Not c-games Utility functions: u 1 (b, c, f, g) = b u 2 (b, c, f, g) = b + c 1 u 3 (b, c, f, g) = 1 2 c + { } 1 + max b +b =b 2 min{b, f} + min{b, g} b,b 0 SERGIU HART c 2007 p. 26

112 Market Games Are Not c-games SERGIU HART c 2007 p. 27

113 Market Games Are Not c-games The strategic game G (Scarf 1971): SERGIU HART c 2007 p. 27

114 Market Games Are Not c-games The strategic game G (Scarf 1971): Each player i distributes his endowment e i among the 3 players: SERGIU HART c 2007 p. 27

115 Market Games Are Not c-games The strategic game G (Scarf 1971): Each player i distributes his endowment e i among the 3 players: d i,j R 4 + is the bundle that i transfers to j e i = 3 j=1 di,j ( 3 ) j s final payoff is u j i=1 di,j SERGIU HART c 2007 p. 27

116 Market Games Are Not "c-games" SERGIU HART c 2007 p. 28

117 Market Games Are Not "c-games" Proposition. In every SP equilibrium: SERGIU HART c 2007 p. 28

118 Market Games Are Not "c-games" Proposition. In every SP equilibrium: The threat of player 1 in coalition {1, 2, 3} is to transfer 1 unit of good f to player 3 The threat of player 1 in coalition {1, 3} is to keep his endowment SERGIU HART c 2007 p. 28

119 Market Games Are Not "c-games" Proposition. In every SP equilibrium: The threat of player 1 in coalition {1, 2, 3} is to transfer 1 unit of good f to player 3 The threat of player 1 in coalition {1, 3} is to keep his endowment The fixed-threat property does not hold SERGIU HART c 2007 p. 28

120 Market Games Are Not "c-games" Proposition. In every SP equilibrium: The threat of player 1 in coalition {1, 2, 3} is to transfer 1 unit of good f to player 3 The threat of player 1 in coalition {1, 3} is to keep his endowment The fixed-threat property does not hold The coalitional form is not well-defined SERGIU HART c 2007 p. 28

121 Conclusions SERGIU HART c 2007 p. 29

122 Conclusions Market games are not really "c-games": SERGIU HART c 2007 p. 29

123 Conclusions Market games are not really "c-games": Defining the coalitional function as what a coalition can do with the total endowment of its members may not be adequate SERGIU HART c 2007 p. 29

124 Conclusions Market games are not really "c-games": Defining the coalitional function as what a coalition can do with the total endowment of its members may not be adequate The problem arises only in the NTU case and not in the TU case SERGIU HART c 2007 p. 29

125 Conclusions SERGIU HART c 2007 p. 30

126 Conclusions In general strategic games: SERGIU HART c 2007 p. 30

127 Conclusions In general strategic games: The way to obtain the coalitional form from the strategic form depends on the bargaining procedure ("the institutional setup") that is used SERGIU HART c 2007 p. 30

128 Conclusions In general strategic games: The way to obtain the coalitional form from the strategic form depends on the bargaining procedure ("the institutional setup") that is used There is no universal way to define a coalitional function from the strategic form SERGIU HART c 2007 p. 30

129 Conclusions In general strategic games: The way to obtain the coalitional form from the strategic form depends on the bargaining procedure ("the institutional setup") that is used There is no universal way to define a coalitional function from the strategic form Going from the strategic form to cooperative outcomes should be done directly, not via the coalitional form SERGIU HART c 2007 p. 30

130 Conclusions In general strategic games: The way to obtain the coalitional form from the strategic form depends on the bargaining procedure ("the institutional setup") that is used There is no universal way to define a coalitional function from the strategic form Going from the strategic form to cooperative outcomes should be done directly, not via the coalitional form in the general NTU case SERGIU HART c 2007 p. 30

131 Bargaining and Cooperation SERGIU HART c 2007 p. 31

132 Bargaining and Cooperation STRATEGIC FORM COOPERATIVE OUTCOMES COALITIONAL FORM SERGIU HART c 2007 p. 31

133 Bargaining and Cooperation STRATEGIC FORM COOPERATIVE OUTCOMES COALITIONAL FORM SERGIU HART c 2007 p. 31

134 C-Game SERGIU HART c 2007 p. 32

135 C-Game SERGIU HART c 2007 p. 32