Bargaining with One-Sided Asymmetric Information and Nonstationary Behavioral Types

Transcription

1 Bargaining with One-Sided Asymmetric Information and Nonstationary Behavioral Types Dilip Abreu 1 David Pearce 2 Ennio Stacchetti 2 1 Princeton 2 NYU October 2015

2 Setting - Two Player Asymmetric Information Bargaining - The Seller: Known Reservation Value (normalized to 0) - The Buyer: Unknown Reservation Value - Investigate using reputational tools 1

3 Overview and Motivation - Rubinstein (1982): -horizon non-cooperative bargaining model with unique SPE. But... no delays. - Perhaps uncertainty causes delay? Players with private information may use delay to signal patience, low valuations... - Rubinstein (1985): model where the seller is unsure about the buyer s time preference (axiomatic), r1 B > r 2 B > 0 are the possible buyer s discount rates. Multiplicity of SE. However refinements imply virtually no delay. 2

4 - Uncertainty about valuations, investigated partly in the bargaining literature itself, and partly in the durable-goods monopoly setup [the Coase conjecture]. - Coase (1971), Stokey (1981), Bulow (1982), Fudenberg, Levine & Tirole (1985) - Gul, Sonnenschein and Wilson (1986): in the gap case, all SE are Coasean, almost all types trade near time 0 at price near the lowest buyer s valuation. - Gul and Sonnenschein (1988) in an alternating offers bargaining model with one-sided asymmetric information get results similar to Gul, Sonnenschein and Wilson (1986) 3

5 - The presumption in a 1-sided asymmetric information model is that the outcome is Coasean (when private information is about valuations or discount rates): uninformed player does virtually as poorly as if she faced for sure the strongest opponent (lowest valuation, most patient) in the distribution. - We investigate if reputation perturbations allow us to escape the Coasean result and lead to outcomes that are more sensitive to the primitives of the model. - In a natural static model this is not the case. - But in models with more sophisticated dynamic behavioral types this is the case. - In APS (2015) we perturb a model where uncertainty is about discount rates with a natural class of dynamic behavioral types. We now identify a class of canonical types for a model with uncertainty about valuations. 4

6 Abreu, Pearce and Stacchetti (TE (2015)) - APS (2015) perturb a one-sided asymmetric information bargaining model by adding behavioral types for both players. - Player A of known discount rate bargains with player B of unknown discount rate (as in Rubinstein (1985)). - APS (2015) considers two types of perturbations: one with (Myerson/Abreu-Gul) persistent types and the other allowing B, the informed player, to delay making his counterdemand. 5

7 - Model with only persistent types produces equilibrium with Coasean outcome. - But, model with (modestly) dynamic behavioral types produces a much more intuitive equilibrium where outcome is sensitive to parameters of the problem in a continuous way. - While APS (2014) demonstrate that it is important to consider a rich set of perturbations, they restrict attention to a specific set of perturbations. What happens without these restrictions? - Here we investigate the canonical types for a one-sided asymmetric information model where the buyer s valuation is private information. - Canonical means equilibrium outcome (essentially) robust to introduction of additional types. 6

8 Model 1: Stationary Behavioral Types S = rational seller (S) with reservation value 0. B k = rational buyer (B) with reservation value v k : v 1 > v 2 > 0. Payoffs for outcome (p, t): pe rt for S, and (v k p)e rt for B k. - Seller offers a price p S 0. - After observing price p S, player B can accept it or make a counter-offer p B [0, p S ). - Players can change their offers only at times {1, 2, 3,...}, but can accept an outstanding offer at any time t [0, ). - Hybrid model of time introduced in Abreu and Pearce (2007) 7

9 - Static Behavioral types: S, B (0, v 1 ) finite. Type p S S (similarly, type p B B) always offers the same price p S and accepts a price p B if and only if p B p S (p B p S ). Assume that min S > min B. - Player i {S, B} is behavioral with probability z i and rational with probability 1 z i, β k = P[v k B is rational] k = 1, 2 π S (p S ) = P[p S S is behavioral] p S S π B (p B ) = P[p B B is behavioral] p B B. 8

10 War of Attrition Against Two Types: after S demands p S S and B demands p B B with p B < p S, and assuming updated beliefs are (ẑ S, ẑ B, ˆβ 1, ˆβ 2 ): In [0, T 1 ) between S and B 1, and in [T 1, T 2 ) between S and B 2 λ S k = r(v k p S ) k = 1, 2, λ B = rp B p S p B p S p B ẑ i (0) = ẑi 1 µ i, i = S, B, where µs µ B = 0 ẑ S (0)e λs 1 T 1+λ S 2 (T 2 T 1 ) = 1, [ẑ B (0) + ˆα 2 ]e λb T 1 = 1, ẑ B (0)e λb T 2 = 1 NOTE: as ẑ B 0, T 1 remains roughly constant, but T 2 ẑ S 1 µ S e λs 2 T 2 1 = ẑ B e λ BT 2 1 µ S ẑs ẑ B 1 e (λ B λ S )T 2 0 as T 2 if λ B > λ S 2. 9

11 For small (ẑ S, ẑ B ), the outcome of the WOA is almost completely determined by v 2. B is strong if λ S 2 < λb. When B is strong, S concedes to p B immediately with probability µ S close to 1, so payoffs are close to (p B, v 1 p B, v 2 p B ). What determines a players s strength? High rate of concession λ i and (high initial reputation z i ). λ B is large when p S is large and/or when p B is small. Note that λ B > λ S k p B < v k p S. 10

12 Balanced Counter Offer for B k : p k(p S ) = v k p S p k(p S ) = min {p S, min {p B B p B > p k(p S )}} As in Abreu and Gul (2000), when the z s are small, for each p S S, B 1 and B 2 will choose to mimic the balanced counter demand p2 (p S). Hence, S will mimic with probability close to 1 the type ps = argmax ps S p2(p S ) v 2 2. The outcome is Coasean: it is as if S is dealing with B 2 only. p B v k p k(p S ) v k /2 v k p S 11

13 Model 2: Temporal Types In asymmetric information bargaining, often informed players delay making offers in order to signal strength. Cramton RESTUD paper opens with this wonderful quote: Panmunjon, Korea-(UPI)- The American general and the North Korean general glared at each other across the table and the only sound was the wind howling across the barren hills outside their hut. Maj. Gen. James Knapp, negotiator for the United Nations Command, was waiting for Maj. Gen. Ri Choonsun of the Democratic People s Republic of North Korea to propose a recess. They sat there, arms folded, for hours. Not a word. Finally, Gen. Ri got up, walked out and drove away. -Evening Bulletin, Philadelphia (11 April 1969) 12

14 Model 2: Temporal Types APS (2015) enlarge the set of behavioral types in the simplest way to allow for the possibility that B makes his initial counter demand with delay. Behavioral Types: S for player S as before, and [0, T ] B for player B: a type (t, p B ) makes his first counter demand p B at time t and accepts immediately any demand p S such that p S p B. Behavioral types are opportunistic: if S changes her initial offer at time s, before B has made his (first) counter demand, the behavioral type (t, p B ) becomes type (s, p B ), where p B = min B. π B (p B, t) = probability density of type (p B, t) conditional on B being behavioral. Equilibrium outcome is sensitive to the parameters of the problem in a continuous way. In particular, for a range of parameters, the outcome is non-coasean. 13

15 Behavioral Types: Non-Stationary Types Could S do better with access to a richer set of types? S of known valuation 0. B with probability β k has valuation v k, k = 1,..., K. - finite S {p S : [0, ) [ [0, T ] B ] [0, v 1 ]} for player S. - [0, T ] B with finite B (0, v 1 ) for player B. Let p B = min B. - type p S makes an initial offer p S (t) for each time t. After B makes a counteroffer p B > p S at some time t, type p S modifies his future offers to ps f (s) for s > t. - type (t, p B ) makes first counteroffer p B at time t and accepts immediately any price p S such that p S p B. - (t, p B ) is opportunistic: if S reveals rationality at s < t, then (t, p B ) becomes (s, p B ). 14

16 Indifference Curves for B k p p 1 (t) = v 1 w 1 e rt Indifference Curve for B 1 at utility level w 1 p 2 (t) = v 2 w 2 e rt t 15

17 Background Lemmas Let K (ε, M) = {z R 2 ++ z i ε, i {S, B} and z S /M z B Mz S }. Lemma 1 Suppose S offers some p S S. For any M > 1 and δ > 0 there exists ε > 0 such that if B counteroffers v 2 /2 (forever) at some t, and (z S (t), z B (t)) K (ε, M) and B is known to be behavioral or type B 2, then in the subsequent WOA, µ S > 1 δ. Lemma 2 Similar conclusion follows if B makes the stationary counteroffer v 1 /2 even if B might be of type B 1. Intuition clear but takes some work to establish when p S may be highly non-stationary. 16

18 Upper Bound for S s payoff Any offer by S leads to some equilibrium IC s for B 1 (& B 2 ): v 1 w 1 p 1( ) v 2/2 (upper) bound for p 2( ) t 12 v 1 w 1 v 1 Now e rt 12 (v 1 v ) = w 1 Also v 1 w 1 v 2 2 v 1 v 2 2 w 1 t 12 0 Upper bound for seller (given basic IC s) max β 1 (v 1 w 1 ) + (1 β 1 )e rt v 12 2 w 1 2 S.T. v 1 v 2 2 w 1 v 1 2 Simplifies to: β 1 v 1 + γw 1 where γ = [ β 1 + (1 β 1 ) Hence, γ < 0 β 1 > v 2 2v 1. v 2 /2 v 1 v 2 /2 ]. 17

19 Optimal Solution p 1 v 1 /2, p 2 = v 2 /2, t 1 = 0 & t 2 solves (v 1 p 1 ) = e rt 2(v 1 p 2 ) Possible that p 1 = p 2 & t 2 = 0 If & only if β 1 < v 2 /(2v 1 ). Otherwise p 1 = v 1 /2. The program focuses on a subset of the constraints imposed in equilibrium. Our main result: For any solution to the program, there is a strategy S can announce which (if available to imitate) will yield her an expected payoff essentially the same as the optimized value of the program. Henceforth, assume β 1 > v 2 2v 1 p 1 = v 1 /2, p 2 = v 2 /2 and t 2 > t 1 = 0. (When β 1 v 2 /2v 1 it is optimal not to separate and S can trivially attain the optimum). 18

20 Seller s Dynamic Posture p v 1/2 p S (t) v 2/2 v 2/2 t 12 p S (t) = v k wk Seρt t [t k 1,k, t k,k+1 ] k = 1,..., K wk S w k (but smaller.) ρ r ɛ Red Dots One time benefit available to B if B does not counteroffer. t 19

21 p p S ( ) v 2/2 v 2/2 f freezing parameter t f B p S ( ) stops declining at t f B if there has been a counter offer p B( ) earlier and p B ( ) p S (t f B ) f. One time benefit small but large relative to ɛ and f. t 20

22 Equilibrium in subgame after S offers p S p Almost entire mass of B1 at ps(t1) ps v2/2 v2/2 Almost entire mass of B2 at ps(t2) p1 p2 t1 t12 t2 Overview: We will show that there is a unique equilibrium in this subgame. In this equilibrium, B k accepts the seller s offer p k (t k ) at time t k with probability approaching 1 as z B, z S 0. Thus, in this subgame the seller obtains close to the maximal payoff attained in the ideal program. t 21

23 p Almost entire mass of B1 at ps(t1) ps v2/2 v2/2 Almost entire mass of B2 at ps(t2) p1 p2 t1 We first present this equilibrium, turning later to the issue of uniqueness. t12 t2 According to Lemma X, in any equilibrium S will adhere to p S until B makes a counteroffer. Hence B k can attain at least the utility level represented by the indifference curve p k. t 22

24 p ps v2/2 v2/2 pb( ) f Very Long Phase } of WOA t t12 Consider p B ( ) and suppose it induces p S ( ) to freeze at a price v 2 /2. Recall from the basic analysis of reputational wars of attrition that if the density with which this counteroffer is made is bounded above 0 as z B, z S 0, then B is weak and must concede to the seller with strictly positive probability (indeed approaching 1) at the start of the reputational WOA. B k s equilibrium utility would then be (v k p S (t))e rt < (v k p k (t))e rt, a contradiction. Since such counteroffers are aggressive for both B 1 and B 2 they must be made with low densities going to zero as z S 0 and z B 0. t f B t 23

25 p ps v2/2 v2/2 f pb( ) t t f B t t12 Now consider p B ( ) that induces a freezing time tb f such that p S (tb f ) > v 2/2. In the equilibrium we display, counteroffers that induce tb f t are made, if at all, only by B 1. But such a counteroffer is aggressive for B 1. Thus, all such counteroffers are made with low or zero density (for small z s). Hence all counteroffers are made with zero density or density going to zero (as z S, z B 0). Hence in such an equilibrium, B k accepts the sellers offer p k (t k ) at t k with probability close to 1 as z S 0 and z B 0. It follows that in equilibrium, in this subgame, the seller obtains close to the maximal 24 payoff attained in the ideal program. t

26 Why does B 2 not make counteroffers for which t f B t? p a a c p S p c 1 b p c 2 v 2/2 v 2/2 d v 2/2 p B ( ) t t f B t t 12 Consider dynamic counteroffer p B ( ) such that p S ( ) freezes at t f B t. Then p 1 (t) p I (p B ( )) (1 µ) [ p S (t) p c 2(t) ] < (1 µ) [ p c 1(t) p c 2(t) ] < ac < cd. Thus B 2 does not mimic postures p B ( ) such that t f B t. They yield too low a payoff. t 25

27 p Uniqueness p S p 1 v 2 /2 p B v/2 + f f p B ( ) p 2 p 1 p 2 IC for S p 1 t t 12 t p1, p 2 that intersect ABOVE v 2/2 yield contradiction. B 2 has separating dynamic counteroffers for which p S (tb f ) = v f > v 2 2 = p B (tb f ). Profitable deviation for B 2. 26

28 p p S p 2 p 1 v 2/2 p B a p 2 p B ( ) p 1 IC for S t t 12 Suppose p1 & p 2 intersect so close to IC for S such that B 2 does not have a separating counteroffer. Then it must be that p1 & p 2 intersect below v 2 2. B 1 & B 2 must pool with probability close to 1 on dynamic counteroffers that start below a < v 2 2 and for which tb f t. But then S s payoff is < v 2 2! Contradiction. t 27

29 Lemma Consider p +, Ĝ + and p, Ĝ and h > l such that p + (s) h p (s) l s (v p + (s))e rs dĝ + = (v p (s))e rs dĝ. s Then d + e rs dĝ + > e rs dĝ d and p + p + (s)e rs dĝ + > p (s)e rs dĝ p. 28

30 p S p t t Corollary Suppose p1 ( ) and p 2 ( ) intersect at ( p, t) and consider the counteroffer p B ( ) at t < t such that p B (s) > p s t. Then p I (p B ( )) > p 2 (t). 29

31 The corollary follows directly from writing IC s in d e rs dĝ and p p(s)e rs dĝ space: w = vd p p v 1 w 1 v 2 w 2 1 d w 2 w 1 30

32 Final Remarks Reputational Perturbations seem like a very promising tool in the context of bargaining. Note TWO sided reputation formation and NON-asymptotic patience. This paper: Simplest perturbations lead to classic Coasean results consistent with the earlier literature. Allowing for non-stationary types yields very different conclusions Result not simply support dependent (as in Coase) but depends on the relative weights of different types. Seller achieves theoretical upper bound. Identifies canonical type. Technical Contribution: Exploration of models with more textured types. Quite a bit more complicated but finally tractable. Future work: Two sided. Big problem that was effectively abandoned in the 80 s. 31