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Sequential Bargaining (Rubinstein Bargaining Model)

Sequential Bargaining (Rubinstein Bargaining Model). Two players divide a cake S Each in his turn makes an offer, which the other accepts or rejects. The game ends when someone accepts The players alternate in making offers There is a discount rate of δ. Y. N. t = 2. 1.

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Sequential Bargaining (Rubinstein Bargaining Model)

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  1. Sequential Bargaining (Rubinstein Bargaining Model) • Two players divide a cake S • Each in his turn makes an offer, which the other accepts or rejects. • The game ends when someone accepts • The players alternate in making offers • There is a discount rate of δ

  2. Y N t = 2 1 (x,y) ε S Sequential Bargaining (Rubinstein Bargaining Model) 1 denote these stages by 1/2 t = 1 (x,y) ε S 2 Y i.e. 1 makes an offer, 2 accepts or rejects (x,y) N 2 (x,y) ε S 1 (δx, δy) etc.

  3. histories: 2/1 1/2 2/1 1/2 Sequential Bargaining (Rubinstein Bargaining Model) 1/2 Strategies δ δ2 δ3 δ4

  4. histories: 2/1 1/2 Sequential Bargaining (Rubinstein Bargaining Model) 1/2 Strategies t = 1 δ t = 2 δ2 t = 3

  5. 2/1 1/2 Sequential Bargaining (Rubinstein Bargaining Model) 1/2 payoffs t = 1 δ t = 2 δ2 t = 3

  6. 2/1 1/2 2/1 1/2 Sequential Bargaining (Rubinstein Bargaining Model) 1/2 Nash Equilibria δ δ2 δ3 δ4

  7. 2/1 1/2 2/1 1/2 Sequential Bargaining (Rubinstein Bargaining Model) 1/2 Subgame Perfect Equilibria δ δ2 δ3 δ4

  8. 2/1 1/2 2/1 1/2 Sequential Bargaining (Rubinstein Bargaining Model) Subgame Perfect Equilibria 1/2 δ δ2 δ3 δ4

  9. 2/1 1/2 2/1 1/2 Sequential Bargaining (Rubinstein Bargaining Model) Subgame Perfect Equilibria 1/2 δ δ2 δ3 δ4

  10. 2/1 1/2 2 can ensure this payoff by making this offer Sequential Bargaining (Rubinstein Bargaining Model) Subgame Perfect Equilibria 1/2 ? Can be supported as an equilibrium payoff Can be supported as an equilibrium payoff

  11. 2/1 1/2 2 will not agree to less 1 cannot take more Sequential Bargaining (Rubinstein Bargaining Model) Subgame Perfect Equilibria 1/2

  12. 2/1 1/2 Sequential Bargaining (Rubinstein Bargaining Model) Subgame Perfect Equilibria 1/2

  13. 2/1 1/2 Sequential Bargaining (Rubinstein Bargaining Model) Subgame Perfect Equilibria 1/2 using similar arguments

  14. 2/1 1/2 Similarly the only possible (SPE) payoff for 2 in 2/1 is Sequential Bargaining (Rubinstein Bargaining Model) Subgame Perfect Equilibria 1/2

  15. 2/1 1/2 Sequential Bargaining (Rubinstein Bargaining Model) Subgame Perfect Equilibria 1/2 Check that it is a SPE !!

  16. 2/1 1/2 1/2 Sequential Bargaining (Rubinstein Bargaining Model) Subgame Perfect Equilibria Graphically 1/2

  17. 2/1 1/2 Sequential Bargaining (Rubinstein Bargaining Model) Subgame Perfect Equilibria 1/2 Show that there is a unique SPE, and that it’s payoff is:

  18. 2/1 (a,b) (a,b) 1/2 2 2 2/1 Sequential Bargaining (Rubinstein Bargaining Model) 1/2 Bargaining with an Outside Option a+b < 1 δ δ2 δ3

  19. 2/1 (a,b) (a,b) 1/2 2 2 2/1 Sequential Bargaining (Rubinstein Bargaining Model) Bargaining with an Outside Option 1/2 δ δ2 δ3

  20. 2/1 (a,b) (a,b) 1/2 2 2 2/1 Sequential Bargaining (Rubinstein Bargaining Model) Bargaining with an Outside Option 1/2 δ δ2 δ3

  21. 2/1 (a,b) (a,b) 1/2 2 2 2/1 Sequential Bargaining (Rubinstein Bargaining Model) Bargaining with an Outside Option 1/2 δ δ2 δ3

  22. 2/1 (a,b) (a,b) 1/2 2 2 2/1 Sequential Bargaining (Rubinstein Bargaining Model) Bargaining with an Outside Option 1/2 δ δ2 δ3

  23. 2/1 (a,b) (a,b) 1/2 2 2 2/1 Sequential Bargaining (Rubinstein Bargaining Model) Bargaining with an Outside Option 1/2 δ δ2 δ3

  24. 2/1 (a,b) (a,b) 1/2 2 2 2/1 Sequential Bargaining (Rubinstein Bargaining Model) Bargaining with an Outside Option 1/2 δ δ2 δ3

  25. 2/1 (a,b) (a,b) 1/2 2 2 2/1 1/2 b Sequential Bargaining (Rubinstein Bargaining Model) Bargaining with an Outside Option 1/2 δ 1 δ2 δ3 1 1/2

  26. Compare this with the Nash Bargaining Solution of 2/1 disagreement pt. (a,b) (a,b) 1/2 2 2 2/1 Sequential Bargaining (Rubinstein Bargaining Model) Bargaining with an Outside Option 1/2 δ δ2 (1+b)/2 δ3 b (1-b)/2

  27. 2/1 (a,b) (a,b) 1/2 2 2 Outside Option 1 2/1 1/2 b 1 1/2 Sequential Bargaining (Rubinstein Bargaining Model) Bargaining with an Outside Option 1/2 δ δ2 Nash Bargaining Solution δ3

  28. 2/1 (a,b) (a,b) 1/2 2 2 Outside Option 1 2/1 1/2 b 1 1/2 Sequential Bargaining (Rubinstein Bargaining Model) Bargaining with an Outside Option 1/2 So where is the disagreement point ?? δ Nash Bargaining Solution δ2 • The Nash Bargaining solution • increases with b • The Outside Option equilibrium • remains constant for small b δ3

  29. p p p p 1-p 2/1 1-p (a,b) (a,b) (a,b) (a,b) 1/2 0 0 0 0 2/1 1-p 1-p Sequential Bargaining (Rubinstein Bargaining Model) Bargaining with random breakdown of negotiations 1/2 after an offer is rejected, Nature breaks down the negotiations with probability p negotiations continue with probability 1-p No need to have a discount rate !!

  30. p p p p 1-p 2/1 1-p (a,b) (a,b) (a,b) (a,b) 1/2 0 0 0 0 2/1 1-p 1-p Sequential Bargaining (Rubinstein Bargaining Model) Bargaining with random breakdown of negotiations 1/2

  31. p p p p 1-p 2/1 1-p (a,b) (a,b) (a,b) (a,b) 1/2 0 0 0 0 2/1 1-p 1-p Sequential Bargaining (Rubinstein Bargaining Model) Bargaining with random breakdown of negotiations 1/2

  32. p p p p 1-p 2/1 1-p (a,b) (a,b) (a,b) (a,b) 1/2 0 0 0 0 2/1 1-p 1-p Sequential Bargaining (Rubinstein Bargaining Model) Bargaining with random breakdown of negotiations 1/2 The payoff of player 2 :

  33. p p p p 1-p 2/1 This coincides with the Nash Bargaining Solution of 1-p (a,b) (a,b) (a,b) (a,b) 1/2 0 0 0 0 2/1 1-p b 1-p a Sequential Bargaining (Rubinstein Bargaining Model) Bargaining with random breakdown of negotiations 1/2

  34. p p p p 1-p 2/1 This coincides with the Nash Bargaining Solution of 1-p (a,b) (a,b) (a,b) (a,b) 1/2 0 0 0 0 2/1 1-p b 1-p a Sequential Bargaining (Rubinstein Bargaining Model) Bargaining with random breakdown of negotiations 1/2 END

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