Dynamic Games, II: Bargaining

Transcription

1 Dynamic Games, II: Bargaining Econ 400 University of Notre Dame Econ 400 (ND) Dynamic Games, II: Bargaining 1 / 16

2 Bargaining In bargaining games, we are generally trying to model agents haggling back-and-forth over how to split the value or surplus from trade. They typically have the following ingredients: There s a pie /value/surplus of fixed value. We usually normalize it to 1, so the players are trying to get the largest share they can. The players can choose any split, (s,1 s). There s an extensive form that describes who makes offers and counter-offers when, and under what circumstances bargaining breaks down and the players walk away from the trading opportunity. If trade breaks down, the players get their outside option payoffs, r i, which represent what the player can get on his own. We usually assume that r 1 +r 2 < 1 in two-players bargaining games, so that trade would be efficient if it can be arranged. Econ 400 (ND) Dynamic Games, II: Bargaining 2 / 16

3 The Dictator Game This is a very stark example of a bargaining game: Player one proposes a split of the pie, (t,1 t), with t going to player 1. Player two accepts or rejects. If player 2 accepts, they split the pie. Otherwise, player 2 gets r 2 and player 1 gets r 1, where r 1 +r 2 < 1. Econ 400 (ND) Dynamic Games, II: Bargaining 3 / 16

4 The Dictator Game This is a very stark example of a bargaining game: Player one proposes a split of the pie, (t,1 t), with t going to player 1. Player two accepts or rejects. If player 2 accepts, they split the pie. Otherwise, player 2 gets r 2 and player 1 gets r 1, where r 1 +r 2 < 1. What s the subgame perfect Nash equilibrium, especially when r 1 = r 2 = 0? Is this a good prediction? Econ 400 (ND) Dynamic Games, II: Bargaining 3 / 16

5 Discounting Suppose an agent gets a payoff u for each date t = 0,1,2,3,...,T. Then the discounted payoff when an agent has discount factor δ is given by the sum S = u +δu +δ 2 u +...+δ T u where δ < 1. Econ 400 (ND) Dynamic Games, II: Bargaining 4 / 16

6 Alternating Offer Bargaining But suppose that rather than accept t = r 2, the second player makes a counteroffer. To make it interesting, we imagine that there are costs to delay, so that the players payoffs after a counteroffer are discounted by δ, giving us an extensive form: In particular, if r 1 = r 2 = 0, what is the SPNE? Econ 400 (ND) Dynamic Games, II: Bargaining 5 / 16

7 Alternating Offer Bargaining Or... Econ 400 (ND) Dynamic Games, II: Bargaining 6 / 16

8 Alternating Offer Bargaining If we set r 1 = r 2 = 0, a pattern begins to emerge: The game ends in the first round, with the first player offering t 3 1 t 1 1 = 1 t 2 1 = 1 δ = 1 δ(1 δ) = 1 δ +δ2 t 4 1 = 1 δ(1 δ(1 δ)) = 1 δ +δ 2 δ 3... t T 1 = 1 δ +δ 2 δ 3 +δ 4...±δ T 1 = Σ T t=0( δ) t Econ 400 (ND) Dynamic Games, II: Bargaining 7 / 16

9 Alternating Offer Bargaining If we set r 1 = r 2 = 0, a pattern begins to emerge: The game ends in the first round, with the first player offering t 3 1 t 1 1 = 1 t 2 1 = 1 δ = 1 δ(1 δ) = 1 δ +δ2 t 4 1 = 1 δ(1 δ(1 δ)) = 1 δ +δ 2 δ 3... t T 1 = 1 δ +δ 2 δ 3 +δ 4...±δ T 1 = Σ T t=0( δ) t But these are somewhat confusing because of the... terms. Econ 400 (ND) Dynamic Games, II: Bargaining 7 / 16

10 Geometric Sums But how do we compute a sum like S = y +xy +x 2 y +...+x T y? First multiple S by x: Then xs = xy +x 2 y +x 3 y +...+x T+1 y S = y +xy +x 2 y +x 3 y +...+x T y xs = xy x 2 y x 3 y... x T y - x T+1 y S(1 x) = y - x T+1 y Or S(T) = 1 xt+1 y 1 x Econ 400 (ND) Dynamic Games, II: Bargaining 8 / 16

11 Geometric Sums But how do we compute a sum like S = y +xy +x 2 y +...+x T y? First multiple S by x: Then xs = xy +x 2 y +x 3 y +...+x T+1 y S = y +xy +x 2 y +x 3 y +...+x T y xs = xy x 2 y x 3 y... x T y - x T+1 y S(1 x) = y - x T+1 y Or Note that S(T) = 1 xt+1 y 1 x S( ) = 1 1 x y Econ 400 (ND) Dynamic Games, II: Bargaining 8 / 16

12 Alternating Offer Bargaining Then using the geometric summation formula on t T 1 = 1 δ +δ2 δ 3 +δ 4...±δ T 1 = Σ T t=0 ( δ)t yields t T 1 = ΣT 1 t=0 ( δ)t = 1 ( δ)t 1+δ Econ 400 (ND) Dynamic Games, II: Bargaining 9 / 16

13 Alternating Offer Bargaining Then using the geometric summation formula on yields t T 1 = 1 δ +δ2 δ 3 +δ 4...±δ T 1 = Σ T t=0 ( δ)t t T 1 = ΣT 1 t=0 ( δ)t = 1 ( δ)t 1+δ If we take the limit as we allow the bargaining to go on indefinitely, T, and t 1 = 1 1+δ and the players payoffs are 1 1+δ, δ 1+δ Econ 400 (ND) Dynamic Games, II: Bargaining 9 / 16

14 Alternating Offer Bargaining So as δ 1, we get so the players split the value equally. 1 lim δ 1 t = lim δ 1 1+δ = 1 2 Econ 400 (ND) Dynamic Games, II: Bargaining 10 / 16

15 Alternating Offer Bargaining So as δ 1, we get 1 lim δ 1 t = lim δ 1 1+δ = 1 2 so the players split the value equally. Likewise, if δ = e R, so that interest compounds continuously at rate R over a short period of time of length, 1 lim 0 t = lim 0 1+e R = 1 2 Econ 400 (ND) Dynamic Games, II: Bargaining 10 / 16

16 Alternating Offer Bargaining So as δ 1, we get 1 lim δ 1 t = lim δ 1 1+δ = 1 2 so the players split the value equally. Likewise, if δ = e R, so that interest compounds continuously at rate R over a short period of time of length, 1 lim 0 t = lim 0 1+e R = 1 2 So if the players become very patient, or the time between offers goes to zero, the players split the pie equally: Bargaining power here comes from agenda control and the ability to waste resources credibly. If these are removed by allowing many offers and counter-offers, or the time between offers goes to zero, then the players split the pie equally in the SPNE. Econ 400 (ND) Dynamic Games, II: Bargaining 10 / 16

17 Alternating Offer Bargaining What s missing from this model of bargaining that is probably important in real applications? Econ 400 (ND) Dynamic Games, II: Bargaining 11 / 16

18 The Hold-Up Problem In many situations, opportunities to bargain and the inability to commit lead to inefficient outcomes, even when property rights are clearly defined. Econ 400 (ND) Dynamic Games, II: Bargaining 12 / 16

19 The Hold-Up Problem In many situations, opportunities to bargain and the inability to commit lead to inefficient outcomes, even when property rights are clearly defined. The student decides on an amount of education to get, e 0, which costs c 2 e2 The firm makes the student a wage offer, w 0 The student can accept the job, receiving a payoff of w c 2 e2, or reject, and open his own business that makes profits be. If the student accepts the job, the firm makes profits ae, where a > b, and otherwise the firm makes nothing. Econ 400 (ND) Dynamic Games, II: Bargaining 12 / 16

20 The Hold-Up Problem In many situations, opportunities to bargain and the inability to commit lead to inefficient outcomes, even when property rights are clearly defined. The student decides on an amount of education to get, e 0, which costs c 2 e2 The firm makes the student a wage offer, w 0 The student can accept the job, receiving a payoff of w c 2 e2, or reject, and open his own business that makes profits be. If the student accepts the job, the firm makes profits ae, where a > b, and otherwise the firm makes nothing. Sketch an extensive form for this game and solve for the subgame perfect Nash equilibrium. What is the efficient level of education? Is the student over- or under-educated in the SPNE relative to the efficient outcome? If the firm could tie its hands and commit to making a wage offer w = a (a/c), would the problem be resolved? Econ 400 (ND) Dynamic Games, II: Bargaining 12 / 16

21 Partnership in Firms Instead of paying a wage w, imagine that the firm offers a share 0 < s < 1 of the profits to the student contingent on getting a degree, like becoming a partner in a law firm or a medical practice. How high does the share have to be to get the student to take the job, and what level of education does he choose? Econ 400 (ND) Dynamic Games, II: Bargaining 13 / 16