1 / 34

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.

kyros
Download Presentation

Sequential Bargaining (Rubinstein Bargaining Model)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related