Negotiation: Strategic Approach

Transcription

1 Negotiation: Strategic pproach (September 3, 007) How to divide a pie / find a compromise among several possible allocations? Wage negotiations Price negotiation between a seller and a buyer Bargaining Situation: / (i) Individuals are able to make mutually beneficial agreements (ii) There is a conflict of interest over the set of possible agreements (iii) Every agent can individually reject any proposal /

2 Before Nash (950, 953), the only solution proposed by economic theory is that the agreement should be: individually rational (i.e., better than full disagreement) Pareto optimal (i.e., no other agreement is strictly better for all agents) Nash suggests two kinds of solutions: ➊ The axiomatic approach: what properties should the solution satisfy? 3/ ➋ The strategic (non-cooperative) approach: what is the equilibrium outcome of a specific and explicit bargaining situation? Here, strategic approach ➋: The bargaining problem is represented as an extensive form game (alternating offers, perfect information) Explicit bargaining rules Two players bargain to share an homogeneous pie (surplus), whose size is normalized to n offer is a pair (x, x ) Set of all possible agreements (Pareto optimal offers): X = {(x, x ) + : x + x = } 4/ Examples: Sharing one euro: x i = amount of money for player i Price negotiation: x = price paid by the buyer (player ) to the seller (player ) Wage negotiation: x = profit of the firm (player ) Preferences: Player i prefers x = (x, x ) X to y = (y, y ) X iff x i > y i

3 Point of Departure: Ultimatum Game (continuous) First period: player offers x = (x, x ) X Second period: player ccepts () or ejects () the offer. If he rejects they both get 0 Extensive form: 5/ x (x, x ) (0, 0) Is every agreement a Nash equilibrium outcome? Unique SPNE: player proposes (,0) and player accepts every offer But the ultimatum game is usually not appropriate because player has all the bargaining power ssume that player can make a counter offer, that player should accept or reject 6/

4 7/ (x, x ) (y, y ) x y (0, 0) Now, it is player who has all the bargaining power Backward induction solution y = (0, ) and in the second period x = (0,), or x (0, ) and in the first period at every SPNE player obtains all the pie More generally, whatever the length of the game, the player who makes the last offer obtains all the pie But time is valuable, delay in bargaining is costly... Discount factor δ i (0, ) for player i 8/ (x, x ) x y (δ y, δ y ) (0, 0)

5 (x, x ) x y 9/ Backward induction: (δ y, δ y ) (0, 0) Subgame after player s rejection: unique SPNE: player proposes (0, ) and player accepts every offer payoff (0, δ ) Subgame after player s proposal: player accepts x δ and rejects x < δ player proposes (x, x ) = ( δ, δ ) in the first period Finite Horizon Bargaining x, x δ x, δ x j δ T x T, δt x T x x i x T 0,0 0/ i (j) = player if T is odd (even) i (j) = player if T is even (odd) Backward induction Check that if T = 3 then x = ( δ ( δ ), δ ( δ )) Check that if T = 4 then x = ( δ ( δ ( δ )), δ ( δ ( δ ))) Problem: the solution depends significantly on the exact deadline

6 Infinite Horizon Bargaining x, x δ x, δ x j δ t x t, δt x t x x i x t / i (j) = player if t is odd (even) i (j) = player if t is even (odd) emarks. Every subgame starting with player s offer is equivalent to the entire game Unique asymmetry in the game tree: player is the first to make an offer It is common knowledge that players only care about the final agreement x and the period at which this agreement is reached (very strong assumption) / The structure of the game is repeated, but it is not a repeated game ( end of the repetition )

7 Pure strategy of player : Sequence σ = (σ t ) t=, where σ t : X t X if t is odd σ t : X t {, } if t is even Pure strategy of player : Sequence τ = (τ t ) t=, where 3/ τ t : X t X if t is even τ t : X t {, } if t is odd Stationary strategies: do not depend on the period and on past offers Player : 4/ Player : σ t (x t ) = x σ t (x t ) = τ t (x t ) = y τ t (x t ) = if t is odd if x t x if t is even if x t < x if t is even if x t y if t is odd if x t < y ccepted offers at the SPNE: t, δ < y = x and x = y

8 Player in (odd) period t given those strategies: δ t x, δt x δ t y, δt y x y 5/ Equilibrium δ t x = δt y, i.e., x = δ y Symmetric reasoning for player y = δ x Hence ( x δ =, δ ) ( δ ) δ δ δ δ ( ) y δ ( δ ) δ =, δ δ δ δ Find a Nash equilibrium (specify the complete strategies, the outcome and the payoffs) that is not Pareto optimal. Explain why this Nash equilibrium is not a subgame perfect Nash equilibrium Proposition. (ubinstein, 98) The preceding stationary strategy profile, i.e., Player always offers x and accepts an offer x iff x y Player always offers y and accepts an offer x iff x x 6/ where ( x δ =, δ ) ( δ ) δ δ δ δ ( ) y δ ( δ ) δ =, δ δ δ δ is the unique subgame perfect Nash equilibrium of the alternating offer bargaining game with perfect information

9 Equilibrium Properties. Efficiency in the sense of Pareto (no delay) Patience of player i increases (δ i ) player i s share increases First-mover advantage: if δ = δ the first player gets as δ +δ >, but +δ 7/ emarks. If proposals are simultaneous in each period then every Pareto optimal share is a SPNE outcome If only one player is able to make offers then, at a SPNE, he obtains all the pie in the first period isk of Breakdown fter every rejection, negotiations terminate with probability α (0,) Even if players are very patient (assume δ = δ = ) there is a pressure to agree rapidly Payoffs when negotiations terminate: (b, b ) +, with b + b < 8/ x, x x, x x 3, x3 x N α α x N α α x 3 (b, b ) (b, b )

10 s in the basic model the unique SPNE is a stationary strategy profile Player always proposes x and accepts a proposal x iff x y Player always proposes y and accepts a proposal x iff x x Player at some period given this strategy: y, y x, x 9/ y N α α x (b, b ) Equilibrium y = α b + ( α) x Symmetric reasoning for player x = α b + ( α)y Hence ( x b + ( α) b = α ( ( α)( y b ) + b = α, ( α)( b ) ) + b α ), b + ( α) b α llocation when the probability of breakdown α 0: 0/ ( x b + b b, b + b ) b Each player gets his payoff in the event of breakdown (b i ) and we split equally the excess of the pie ( b b )

11 eferences Nash, J. F. (950): Equilibrium Points in n-person Games, Proc. Nat. cad. Sci. U.S.., 36, (953): Two Person Cooperative Games, Econometrica,, ubinstein,. (98): Perfect Equilibrium in a Bargaining Model, Econometrica, 50, /