Procedures we can use to resolve disagreements

Transcription

1 February 15th, 2012

2 Motivation for project What mechanisms could two parties use to resolve disagreements? Default position d = (0, 0) if no agreement reached Offers can be made from a feasible set V Bargaining mechanism Φ maps (V, d) to (x.y ) V Famous theoretical solutions Nash Bargaing solution Kalai-Smorodinsky solution What simple mechanism could be used to reach good solution?

3 What makes a good bargaing solution? A1 - Pareto Optimal Otherwise both players could be better off For instance, just choose the default point A2 - Symmetric Otherwise can be unfair For instance, player A chooses his favourite point A3 - Invariant under linear transformations Otherwise depends on how we represent cardinal utility

4 Nash Bargaining solution

5 Kalai-Smorodinsky solution

6 Another property of a good bargaing solution Figure: Property 1: - Whole frontier relevant

7 Structure of Presentation SECTION 1 - Models in discrete time Offers alternate stochastically Links with the models of Rubinstein, Stahl and Sjostrom SECTION 2 - Model in continuous time Offers arrive according to a Poisson process Links with Rubinstein model SECTION 3 - Links with theory How do these models fit into theoretical work on bargaining solutions?

8 Discrete time bargaining models Rubinstein Model Offers alternate deterministically Players have disount factors β 1, β 2 Infinite number of rounds, linear utility set Stahl Model Offers alternate deterministically. Finite number of rounds Players have disount factors β 1, β 2 Sjostrom Model Offers alternate stochastically Finite number of rounds

9 Model of Sjostrom General Two players {A,B} with a bargaining problem (V n, d) Finite horizon of N rounds If round N is reached, then d is implemented Timing In round n, we toss a coin One of the players is selected (each with probability 1 2 ) Player selected makes an offer v V n If other player accepts the offer v is implemented If other player rejects, then we proceed to round (n+1)

10 Solution of Sjostrom model graphically Figure: Offers that players make

11 Extension of Sjostrom model Offers follow a symmetric Markov Process ɛ 1 ɛ (1) 1 ɛ ɛ Timing In round (n+1), we toss a weighted coin Player who proposed last time is selected to propose again with probability ɛ Other player is selected with probability (1 ɛ) If the offer is accepted, v is implemented If the offer is rejected, we proceed to round (n+2)

12 Solution of this model graphically

13 Link with Rubinstein Model Add β 1 and β 2 by shrinking V n each round Set ɛ = 1 N As N, we reach Rubinstein solution (x, y ) = ( 1 β 2 1 β 1 β 2, β 2 1 β 1 1 β 1 β 2 ) When Nlog(β 1 ) 0 and Nlog(β 2 ) 0 Set ɛ = 1 N As N, we reach fixed solution (x m, y m) Note β 1 β 2 We approach Rubinstein setting with no discount factor

14 SECTION 2: - Continuous Time model General Two players, finite time horizon [0,T] We consider V = {(v A, v B ) va 2 + v B = 1} If time T is reached, d = (0,0) is implemented Timing Offers arrive at a Poisson process of rate λ When an offer arrives, a player is chosen to make an offer Each player selected with probability 0.5 If offer accepted, game ends If offer rejected, game continues

15 Results graphically

16 Results numerically As λ T 0, initial offer converges to (x c, y c ) y c = 1 2 exp ( 2 tan 1 ( 5 7 ) 7 π 7 ) x c = 1 y c Link with discrete time model (xc, yc ) = (xm, ym) The continuous time model implements same solution as discrete time model But we do not need many rounds, or very rare trembles

17 General results Here obtained closed form solution in particular case Pareto frontier is strictly decreasing, (d A, d B ) = (0,0) Pareto frontier defined by ψ(v B ) = v A In this case ψ(v B ) = 1 vb 2 For general case, we must solve this system Solve differential equation to find b(a) δb δa = ψ (b)(b a) ψ(a) ψ(b) Find v where b(v ) = v This v is utility player B recieves Initial conditions: (a, b) = (0, 1)

18 Method to reach results Consider reservation utility g A (t) is amount B offers to A at time t f A (t) is amount A offers to himself at time t T [ ] g A (t) = e (t s) fa (s) + g A (s) ds 2 Differentiate with respect to t t g A (t) = T t [ ] e (t s) fa (s) + g A (s) ds e t t) g A(t) + f A (t) 2 2 g A (t) = g A(t) f A (t) 2

19 Intuition why same as discrete time

20 First step to find closed form Take two equations from above Divide these equations f A (t) = f A(t) g A (t) 2 ψ (g A (t))(g A (t)) = ψ(g A(t)) ψ(f A (t)) 2 b = g A (t) ; a = f A (t) δb δa = ψ (b)(a b) ψ(b) ψ(a)

21 Propose one bargaining solution Φ Arguably the extension of the Rubinstein solution No discount factor, many offers, almost deterministic Arises as solution of bargaining model Independently arises as solution of simple model Satisfies axioms and property Sensitve to changes in Pareto frontier What is it?

22 Φ is the continuous Raiffa soltuion Discrete Raiffa solution Introduced by Raiffa in the 1950s Implemented by tossing a fair coin (Sjostrom) Continuous Raiffa solution Axiomatized by Livne in the 1980s Intermediate solutions Proposed and axiomatized by Diskin, Koppel and Samet Games and Economic Behaviour (2011) Propose another implementation in discrete time

23 Comparison between K-S and continuous Raiffa

24 Another implementation (Diskin et al) General N rounds At round N default is implemented Timing in other rounds Toss a weighted coin With probability ɛ game ends, and default allocation implemented With probability 1 ɛ an offer is made Players are selected to propose offers with equal probability.

25 Further work 1. Incomplete information Use continuous time model Much easier than discrete time models Players uncertain whether there will be another offer 2. Experiment Test whether property proposed is true Are alternatives on extreme of bargaining set relevant?

26 Conclusion Discrete time implementation In the limit, reaches continuous Raiffa solution Shows link with Rubinstein model Continuous time implementation Same solution arises from very simple model Can be implemented quickly Closed form solutions Closed form is very messy, even in simple case Maybe why it is not popular for applications?