1. Bargaining Theory MIT 14.126 Game Theory

2. Bargaining Theory Cooperative (Axiomatic) –Edgeworth –Nash Bargaining –Variations of Nash –Shapley Value Non-cooperative –Rubinstein-Stahl (*) (complete info) –Asymmetric info Rubinstein, Admati-Perry, Crampton, Gul, Sonenchein, Wilson; Abreu and Gul –Non-common priors Posner, Bazerman, Yildiz 2 (*)

3. Rubinstein-Stahl Model N = {1,2} X = feasible expected-utility pairs (x,y ∈ X ) U i (x,t) = δ i t x i D = (0,0) ∈ X disagreement payoffs g is concave, continuous, and strictly decreasing

4. Timeline T = {0,1,…, t,…} At each t, if t is even, –Player 1 offers some x –Player 2 Accepts or Rejects the offer –If the offer is Accepted, the game ends yielding x –Otherwise, we proceed to t + 1 if t is odd, –Player 2 offers some y –Player 2 Accepts or Rejects the offer –If the offer is Accepted, the game ends yielding y –Otherwise, we proceed to t + 1

6. Theorem [OR 122.1] The following is the unique subgame-perfect equilibrium: –player 1 always offers x*; –player 2 accepts an offer x iff x 2 ≥ x* 2 ; –player 2 always offers y*; –player 1 accepts an offer y iff y 1 ≥ y* 1 ;

7. Proof (it is a SPE) Use single deviation principle: 1.If player 2 rejects an offer x at t, she will get y 2 * at t+1. Hence, Accept iff x 2 ≥δ 2 y* 2 = x* 2 is optimal at t. 2.At t, it is optimal for 1 to offer x* = argmax {x 1 |x 2 ≥ x* 2 }.

8. “Extensive-form rationalizability” [FT 4.6] Definition: In a multistage game with observable actions, action a i t is conditionally dominated at stage t given history ht iff, in the subgame starting at h t, every strategy for player i that assigns positive probability to a i t is strictly dominated. Theorem: In any perfect-information game, every subgame-perfect equilibrium survives iterated elimination of conditionally dominated strategies.

10. A usual generalization of Rubinstein’s Model At any t, 1.A player i is recognized with probability p t i ; 2.Player i offers some x; 3.The other player 1.either Accepts, bargaining ends with payoffs δ t x, or 2.Rejects the offer, when we proceed to t+1. [There may also be a deadline t, when the game automatically ends, yielding (0,0).]

11. SPE Take X = {(x 1,x 2 )|x 1 +x 2 ≤1}. Iterated elimination of conditionally dominated strategies yields a unique vector V t of continuation values at each t where Any SPE is payoff equivalent to the following SPE. At any t, the recognized player i gives δV j t+1 to the other player j and keeps 1−δV j t+1 for himself, and the offer is barely accepted.

12. Proof Write S* for the strategies that survive IECDS. Write and for the max and min payoff for i at t over S*, and Then,

13. Proof, continued If there is deadline at t, then Δ t = 0, and thus Δ t = 0 for all t. If infinite horizon, then Δ t ≤δn for all n, hence Δ t = 0. Write S t = V t 1 + V t 2.

14. A usual generalization of Rubinstein’s Model At any t, 1.A player i is recognized with probability p t i ; 2.Player i offers some x; 3.The other player 1.either Accepts, bargaining ends with payoffs δ t x, or 2.Rejects the offer, when we proceed to t+1. [There may also be a deadline t, when the game automatically ends, yielding (0,0).]

15. Components of complete information Rubinstein, Stahl Rubinstein 85, Admati-Perry, Gul- Sonnenchein-Wilson and many others CPANo CPA CK No CK

16. Without a common prior At any t, 1.A player i is recognized by Nature; 2.Player i offers some x; 3.The other player 1.either Accepts, bargaining ends with payoffs δ t x, or 2.Rejects the offer, when we proceed to t+1. [There may also be a deadline t, when the game automatically ends, yielding (0,0).] Each i believes that he will be recognized at t with probability p t i.

17. Game Tree In 1’s mind: In 2’s mind:

18. Game Tree

19. Example t = 2. p 1 1 = p 1 2 = 1 δ>1/2.

20. Example, continued V 1 = (1,1) S 1 = 2.

21. Optimism, Agreement-Disagreement The level of optimism for t: y t = p t 1 + p t 2 –1. There is a disagreement regime at t iff δS t+1 > 1. –No agreement; V t = δV t+1 ; S t = δS t+1 There is an agreement regime at t iff δS t+1 ≤ 1. –They agree; –Recognized i gets 1-δV j t+1, the other j gets δV j t+1 ; –V t i = p t i (1-δV j t+1 ) + (1-p t i )δV i t+1 = p t i (1-δS t+1 ) + δV i t+1 –S t = (1+y t ) (1-δS t+1 ) + δS t+1 = 1 + y t (1-δS t+1 ).

22. Lemma If S t+1 ∈ [1,1/δ], then S t ∈ [0,2-δ] ⊂ [1,1/δ]. Proof: 1-δS t+1 ∈ [0,1-δ] and y t ∈ [0,1]. S t = 1 +y t (1-δS t+1 ) ∈ [0,1+ 1-δ] = [0,2-δ].

23. Immediate agreement Definition: 1 < 2 L(δ) ≤ 1/δ. Theorem: Given any t*, assume that y t ≥ 0 for each t ≤ t*. Then, there is an agreement regime at each t < t* – L(δ) – 1.