A note on efficient signaling of bargaining power

99920009

Note

A note on e½cient signaling of bargaining power

Mehmet Bac*

Bilkent University, Department of Economics, Bilkent, Ankara, 06533 Turkey (e-mail: bac@bilkent.edu.tr)

Received: July 1996/Final version: August 1999

Abstract. Strategic delay and restricted o¨ers are two modes of signaling bar-gaining power in alternating o¨ers barbar-gaining games. This paper shows that when both modes are available, the best signaling strategy of the ``strong'' type of the informed player consists of a pure strategic delay followed by an o¨er on the whole pie. There is no signaling motivation for issue-by-issue bargaining when the issues are perfectly substitutable.

Key words: Alternating o¨ers bargaining, incomplete information, sequential equilibrium, delay

1. Introduction

The informed player in a standard alternating o¨ers bargaining game has two instruments to signal his bargaining power. The ®rst and the most well-known is what Admati and Perry (1987) have termed strategic delay, a strategy for the strong type that prescribes remaining silent for a su½ciently long period of time and an o¨er that the weak (impatient or high-valuation) type cannot bene®cially imitate. The second signaling instrument is a restricted o¨er, i.e., an o¨er restricted to a portion X of the pie which, if accepted, leaves the por-tion 1 ÿ X to the next period when the uninformed player takes his turn to make an o¨er. If X is chosen low enough, the weak type of the informed player prefers making his complete information o¨er on the whole pie, hence separation occurs after the ®rst round of o¨ers. This result, shown in Bac and Ra¨ (1996), also produces some motivation for bargaining in sequence over

* Acknowledgement: I am grateful to an associate editor and especially a referee whose correc-tions, suggestions and comments greatly improved the paper. Remaining errors are mine.

several issues within a single negotiation because the corresponding bargain-ing outcome is completed in two steps: an agreement on X, followed by an agreement on 1 ÿ X the next period.

This paper extends the strategies in the standard incomplete information bargaining game to include both strategic delay and restricted o¨ers. Our purpose is to investigate the e½cient signaling strategy of the informed player, more precisely, to characterize the combination of strategic delay and restri-cted o¨ers in a separating bargaining equilibrium. The equilibrium concept is sequential equilibrium, re®ned with an intuitive condition on beliefs o¨ the equilibrium path.

We consider the ®rst-round separating equilibria of the bargaining game between a seller and an informed buyer, starting with the buyer's o¨er. In such equilibria the uninformed player learns the type of the informed player after observing the opening o¨er. Many strategies of the informed player can gen-erate this outcome, i.e., there are many combinations of strategic delay, prices and o¨er restrictions that the strong buyer type can adopt but the weak buyer type cannot. The analysis identi®es a tradeo¨ between these instruments: The strong buyer type can reduce strategic delay costs by further restricting his o¨er, but the ®rst mover advantage stipulates that the o¨er be as unrestricted as possible. We show that strategic delay always dominates in this tradeo¨: In the unique ®rst-round separating equilibrium, the strong buyer type delays his o¨er for a su½ciently long period of time, and then makes an o¨er on the whole pie. Interpreting the restricted o¨ers outcome as an issue-by-issue bargaining procedure, we conclude that when players are risk neutral and strategic delay is available, there is no signaling motivation for bargaining in sequence over perfectly substitutable issues within a single negotiation. 2. E½cient signaling in separating bargaining equilibria

A seller S owns a perfectly divisible pie. It is common knowledge that the seller's valuation of the pie is zero, while the buyer's valuation b is his private knowledge and can take two values, H (high) or L (low) with H > L > 0. The high- (low-) valuation buyer is denoted BH…BL†, and the probability that the

seller assigns in period t to b ˆ H is denoted pt.

We study an alternating o¨er bargaining game that begins in period zero, where it is the informed player B's turn to make an o¨er. A player whose turn is to make an o¨er when X of the pie remains chooses a delay D and an o¨er …X; P† where P is a price (a nonnegative number) and X UX is an amount of the pie. If the responder accepts this o¨er, the buyer obtains X and the seller obtains P. If X < 1 (the o¨er is ``restricted'') or the responder rejects the o¨er, a minimal delay occurs and then it is the responder's turn to make an o¨er. Rubinstein's (1985) bargaining game corresponds to X ˆ 1 and D ˆ 0, Admati and Perry's game corresponds to X ˆ 1 in all o¨ers with D V 0, whereas in the bargaining game studied by Bac and Ra¨, X A f0:5; 1g and D ˆ 0 are assumed.

A belief system for S is a function that speci®es a probability to the event b ˆ H given any relevant history of the game. An outcome of the bargaining game involving N V 1 agreements is denoted ……X1; P1; t1†; . . . ; …XN; PN; tN††

where P_iˆ1N XiU 1, XiV 0 and ti is the date at which agreement on the

agree to trade the whole pie at some ®nite date and that perpetual disagree-ment is a potential outcome. We consider only pure strategies. The outcome ……X1; P1; t1†; . . . ; …XN; PN; tN†† yields the utility US ˆPiˆ1N dtiPi to the seller,

and UbˆPiˆ1N dti‰bXiÿ PiŠ to the type-b buyer, where d, with 0 < d < 1, is

the common discount factor.

Our equilibrium concept is sequential equilibrium (see Rubinstein (1985) for a formal de®nition in the bargaining context), to which we add two restri-ctions, also imposed by Admati and Perry. The ®rst is a tie-breaking assump-tion: if a player obtains the same payo¨ by making fewer o¨ers, then he makes fewer o¨ers. The second restriction concerns beliefs o¨ the equilibrium path:1 Fix an equilibrium path and consider a deviation by B after some bar-gaining history h. Call that deviation ``bad'' for type b0 _{if the highest payo¨}

b0 _{can so obtain (for any belief the seller might have after observing this}

deviation) is lower than his continuation equilibrium payo¨. The restriction requires that the seller's beliefs put zero probability on type b0_{if there is a type}

b for whom the deviation in question is not bad. We shall use the term equi-librium for a sequential equiequi-librium satisfying the above restrictions.

Under complete information, our bargaining game has a unique subgame perfect equilibrium (SPE) where the buyer makes the o¨er …1; bd=…1 ‡ d††. That is, the whole pie is shared immediately and the buyer's corresponding utility is b=…1 ‡ d†.2 Subsequently we assume that the seller's initial belief as-signs positive probability to each type of the buyer. In this case, the following lemma, which can be shown using the arguments in Grossman and Perry (1986), provides bounds to the equilibrium utilities of the players determined by the SPE of the complete information version.

Lemma 1. In an equilibrium, after a history in which the amount of the pie remaining is X,

(i) S never accepts an o¨er …X; P† with P < XdL=…1 ‡ d†;

(ii) B never accepts an o¨er …X; P† with P > XH=…1 ‡ d† and always accepts an o¨er …X; P† with P UXL=…1 ‡ d†.

The following lemma extends Lemma 2.2 in Admati and Perry; the proof follows their arguments.

Lemma 2. Consider a history h that ends in period t with an o¨er …X; P† by B. If ptˆ 0 (respectively, ptˆ 1), then in equilibrium S accepts …X; P† if

and only if acceptance yields S a utility of at least dXL=…1 ‡ d† (respectively, dXH=…1 ‡ d††, where X is the amount of the pie remaining when B makes his o¨er.

Proof. Consider the case ptˆ 0. By Lemma 1, in equilibrium S accepts

…X; P† only if she obtains at least the utility dXL=…1 ‡ d†. We need to show that S accepts …X; P† if doing so yields her a utility of at least dXL=…1 ‡ d†. Suppose S rejects such a pair …X; P†. Then, there is a continuation equil-ibrium in which her utility (by the tie-breaking rule) exceeds dXL=…1 ‡ d†.

1 See Assumption (A2) in Admati and Perry for a formal de®nition.

Denote by UM

S the supremum of S's equilibrium utilities in continuation

games following a history in which X of the pie remains and S believes with probability one that b ˆ L and she has just rejected an o¨er, discounted to the ®rst period in which an o¨er is subsequently accepted. By assumption, UM

S > XL=…1 ‡ d†. Choose a continuation equilibrium in which S obtains

the utility U@

s belonging to the set de®ned above. Denote the outcome of this

equilibrium by ……X@

1 ; P@1 ; t1†; . . . ; …Xz@; P@z ; tz††.

Suppose that S makes the ®rst accepted o¨er …X@

1 ; P@1 † at date t1V t ‡ 1

(the case where BL makes the ®rst accepted o¨er is similar). The

corre-sponding utilities of S and B, discounted to date t1, can be written as US@ ˆ

P_Z iˆ1dtiÿt1P@i and UB@ ˆ P_Z iˆ1dtiÿt1‰bXi@ÿ P@i Š ˆ P_Z iˆ1dtiÿt1bXi@ÿ US@.

We claim that in period t1 there is a deviation by BL in which instead

of accepting S's o¨er …X@

1 ; P@1 † he rejects it, waits until time tj and makes

a countero¨er …X;US† for which

US> dUSM; L XZ iˆ1 dtiÿt1_X@ i ! ÿ U@ SU dtjÿt1…LX ÿ US†; H XZ iˆ1 dtiÿt1_X@ i ! ÿ U@ SV dtjÿt1…HX ÿ US†:

The ®rst inequality implies that S should accept BL's deviant o¨er

…X;US†, the second inequality states that BLbene®ts from the deviation if S

accepts the deviant o¨er, the third inequality states that BL's deviation is

``bad'' for BH, and together they imply ptj ˆ 0, by the re®nement on beliefs.

The third inequality above can be written as dtjÿt1U S‡ H XZ iˆ1 dtiÿt1X@ i ÿ dtjÿt1X " # V U@ S: …1†

A necessary condition for (1) to hold isP_iˆ1Z dtiÿt1_X@

i > dtjÿt1X, which is

satis®ed for large values of tj. Since US< HX, the LHS of (1) is increasing in

tj and achieves a limiting value of H‰Piˆ1Z dtiÿt1Xi@Š. On the other hand, we

have H‰P_iˆ1Z dtiÿt1_X@

i Š ÿ US@V dHX=…1 ‡ d†, for otherwise BH would

bene-®cially deviate. Therefore there exist U@

S A …XL=…1 ‡ d†; USM†, US such that

US> dUSM and a value of tjsuch that (1) holds. The second inequality, which

states that BL bene®ts from the deviation, can be written as

U@ SV dtjÿt1US‡ L XZ iˆ1 dtiÿt1_X@ i ÿ dtjÿt1X " # : …2†

Note that the RHS of (2) is increasing in tj, and therefore the relevant

point to evaluate (2) is the value of tj satisfying (1) with equality when US@ ˆ

UM

evaluation yields the condition 1 ÿ L=H V dtjÿt1‡1_{1 ÿ L=H†, which is}

sat-is®ed because d < 1. Thus, all three conditions can simultaneously be satsat-is®ed. This contradicts the de®nition of UM

S .

The proof for the case pt ˆ 1 uses similar arguments but does not rely on

the re®nement for beliefs o¨ the equilibrium path. Q.E.D.

Lemma 2 has the following important implication: If B makes his opening o¨er …X; P† at date T when pT ˆ 0 (respectively, pT ˆ 1), in equilibrium S

accepts the o¨er if and only if acceptance yields her at least the discounted utility dL=…1 ‡ d† (respectively, dH=…1 ‡ d†) as viewed from date T.3

In what follows, we concentrate on the set of ®rst-round separating equi-libria (FRSE) and study the strong buyer's equilibrium signaling strategy. De®nition: A ®rst-round separating equilibrium is an equilibrium such that after the buyer's opening o¨er the seller either believes that b ˆ L with probability one or believes that b ˆ H with probability one.

We can now state and prove our main result.

Proposition: The game has a unique FRSE. In this equilibrium, BL's opening

o¨er is …1; dL=…1 ‡ d††, which he makes after a delay, and BH's opening o¨er is

…1; dH=…1 ‡ d††, which he makes immediately. S accepts both o¨ers.

Proof. Let ``o¨er …X; P† at date TH'' be the ®rst-round action of BH in a

FRSE. Thus, when S receives the o¨er …X; P†, she concludes pTH ˆ 1. By

Lemma 2, in any continuation equilibrium after such a history S can guar-antee herself the discounted utility dH=…1 ‡ d†. Therefore, after the history ``…X; P† is o¨ered at date TH'', if pTH ˆ 1, the maximum discounted utility

that BH can obtain in any continuation equilibrium is H=…1 ‡ d†. Clearly, in

equilibrium BH must choose THˆ 0 and o¨er …1; dH=…1 ‡ d†† without delay,

which is an o¨er that S always accepts. This is the unique opening o¨er of BH

in any FRSE.4

We claim that in any FRSE BLmakes an accepted opening o¨er. Suppose,

to show a contradiction, that there is a FRSE s in which BLmakes a rejected

opening o¨er at some date TL, and pTLˆ 0. Let there be N V 1 agreements

…Xi; Pi†iˆ1N dated T1; T2; . . . ; TN where T1> TLalong the path of s if B ˆ BL.

Denote by UR S…s† ˆ

P_N

iˆ1dTiÿT1Pi the corresponding utility of S, discounted

to date TL when she rejects BL's opening o¨er. Since pTLˆ 0, by lemma

2 UR

S…s† V dL=…1 ‡ d†. Since s is a FRSE, BH prefers not to imitate BL's

equilibrium strategy: H

1 ‡ dV dT1HA ÿ dTLUSR…s†

3 Or, if pTˆ 0 (respectively, pTˆ 1) after the buyer's opening o¨er at date T; S rejects the

opening o¨er if and only if acceptance yields her less than dL=…1 ‡ d† (respectively, dH=…1 ‡ d†) as viewed from date T. This means that at date T, after B's opening o¨er, in equilibrium S can guarantee herself the discounted utility dL=…1 ‡ d† if pTˆ 0 (respectively, dH=…1 ‡ d† if pTˆ 1).

4 Because of discounting, if B ˆ BHit is impossible to construct a FRSE involving two or more

agreements such that BH obtains H=…1 ‡ d† while S obtains at least dH=…1 ‡ d† as viewed from

where A ˆP_iˆ1N dTiÿT1_X

i. Consider the following deviation of BL at date t ˆ

0: Wait until date TD_{and make the opening o¨er …1; P† such that}

H 1 ‡ dV dT1HA ÿ dTLUSR…s† ˆ sT D …H ÿ P†; …3† dT1_{LA ÿ d}TL_UR S…s† U dT D …L ÿ P†; …4†

and P V dL=…1 ‡ d†. To see that such a pair …TD_{; P† exists, solve for d}TD

from (3) and substitute in (4). Then, (4) can be written as dTL_UR

S…s† V dT1AP, or

P U USR…s†

dT1ÿTL_A: …5†

Since UR

S…s† V dL=…1 ‡ d†, T1> TL and A U 1, there exists P V dL=…1 ‡ d†

such that (5) holds. Observing the deviant opening o¨er …1; P† at date TD_{, the}

seller must believe pTD ˆ 0 and, by lemma 2, accept …1; P†. This contradicts

our assumption that s (according to which BLmakes a rejected opening o¨er)

is a FRSE.

We now claim that in any FRSE BL's accepted opening o¨er involves

X ˆ 1. Suppose, on the contrary, that there is a FRSE s which generates the outcome ……X1; P1; T1†; . . . ; …XN; PN; TN†† if B ˆ BL, where X1< 1 and

P_N

iˆ1XiU 1. Let Ub…s† be B's utility discounted to date t ˆ 0, and let

UF S…s† ˆ

P_N

iˆ1dTtÿT1Pidenote S's continuation equilibrium utility according

to s, discounted to date T1 (®rst agreement date). We can write Ub…s† ˆ

dT1_{‰bA ÿ U}F

S…s†Š where A ˆ

P_N

iˆ1dTiÿT1Xi. To be a FRSE, s must satisfy the

following (no imitation) conditions:

UH…s† 1_{1 ‡ d}H V dT1‰HA ÿ USF…s†Š; …6†

UL…s† 1 dT1‰LA ÿ USF…s†Š > L ÿ_{1 ‡ d}dH ; …7†

and UF

S…s† V dL=…1 ‡ d†. The ®rst condition states that BH has no incentive

to imitate sL; the second is the corresponding condition for BL. In

equilib-rium, the delay T1must satisfy (6) with equality, which can be solved to yield

dT1_ˆ H

…1 ‡ d†…HA ÿ UF

S…s††: …8†

Now de®ne T…A; UF

S…s†† as the delay T1that solves (8) as a function of A and

the ``price'' UF

S…s† V dL=…1 ‡ d†. Using (8) in (7), we obtain a reduced form

expression for UL…s†: UL…s† ˆ H…LA ÿ U F S…s†† …1 ‡ d†…HA ÿ UF S…s††:

This expression is increasing in A because UL…s† < L=…1 ‡ d† (which is

implied by the fact that UF

deviate to the opening o¨er …1; UF

S…s†† at date T…1; USF…s††. This deviation is

``bad'' for BH, increases BL's discounted utility above UL…s†, and S accepts

the o¨er …1; UF

S…s†† because pTL ˆ 0 and USF…s† V dL=…1 ‡ d†.5 Therefore, in

any FRSE, X1ˆ A ˆ 1: BLmakes his opening o¨er on the whole pie.

Since X ˆ 1 in any FRSE, the analysis of the delay-price combination that BL should use in his FRSE opening o¨er is given by the case X ˆ 1 studied

in Admati and Perry. Lemma 3.1 in their paper shows that when X ˆ 1, BL's

equilibrium o¨er is …1; dL=…1 ‡ d††, after delay D_{ˆ T…1; dL=…1 ‡ d††, given}

by (8) for A ˆ 1 and UF

S…s† ˆ dL=…1 ‡ d†.

Let Pbˆ db=…1 ‡ d†. The unique FRSE is given as follows.

BH's strategy: When S retains X U 1, o¨er …X;XPH† with no delay;

respond to an o¨er …X; P† as follows: if X ˆ X and dP UXPH, accept;

if 0 < X < X and P ‡ d…X ÿ X†PHUXPH=d, accept and o¨er

…X ÿ X; …X ÿ X†PH† the next round without delay;

otherwise reject and o¨er the next round …X;XPH† without delay.

BL's strategy: When S retains X U 1, o¨er …X;XPL† after the delay

maxf0; T…X;XPL†g. Respond to an o¨er …X; P† as follows:

if X ˆ X and dP UXPL, accept;

if 0 < X < X and LX ÿ P ‡ dmaxf0; T…XÿX; …XÿX†PL†g_{‰…X ÿ X†P}

LŠ V

dmaxf0; T…X; XPL†g_XP

L, accept and o¨er …X ÿ X; …X ÿ X†PL† the next round

after the delay maxf0; T…X ÿ X; …X ÿ X†PL†g;

otherwise reject, and o¨er …X;XPL† the next round after the delay

maxf0; T…X;XPL†g.

S's strategy: Given the pie stock X U 1 and an o¨er …X; P† where X UX made after T units of delay,

(i) if p ˆ 1, accept the o¨er and countero¨er …X ÿ X; …X ÿ X†PH=d† if

P ‡ …X ÿ X†PHVXPH; otherwise reject the o¨er and countero¨er

…X;XPH=d† with no delay;

(ii) if p ˆ 0, accept the o¨er and countero¨er …X ÿ X; …X ÿ X†PL=d† if

P ‡ …X ÿ X†PLVXPL; otherwise reject the o¨er and countero¨er

…X;XPL=d† with no delay.

S's beliefs: Given the pie stock X U 1, if B makes an o¨er …X1; P1† after

delay T1such that (6) and (7) hold forPiXiUX andPidTiÿT1PiV PL, then

p ˆ 0. Otherwise p ˆ 1. 3. Conclusions

To recapitulate, in the alternating o¨ers, incomplete information bargaining game with linear buyer preferences ubˆ bX over portions X of the pie, the

only ®rst-round separating equilibrium outcome is …1; dH=…1 ‡ d†; 0† if b ˆ H and …1; dL=…1 ‡ d†; T…1; dL=…1 ‡ d††† if b ˆ L. That is, signaling bargaining power through strategic delay is a more e¨ective strategy than restricted

5 The tradeo¨ between strategic delay and o¨er restriction, which is mentioned in the introduc-tion, is apparent in (8) de®ning T…A; UF

S…s††. By choosing X1ˆ 1, hence A ˆ 1, BLcan reduce

o¨ers. While a restricted o¨er economizes on delay and the e¨ective price that the strong buyer pays for the whole pie, his relatively low delay cost and the cost of losing the ®rst-mover advantage on the remaining 1 ÿ X portion of the pie dominate the tradeo¨.

An important extension is to drop the assumption of constant marginal utility of the pie, which can be interpreted as allowing buyer types to have heterogeneous preferences over di¨erent portions of the pie. Since u…x ‡ y† < u…x† ‡ u…y† for any strictly concave utility function, introducing a decreasing marginal utility for the strong type of the informed player will strengthen the case for restricted o¨ers. We have investigated this question in Bac (2000) and shown that the answer depends on the nature of the ``pie'': If the pie is a consumption good, under certain conditions the strong buyer's FRSE strategy involves a restricted o¨er; if the pie is a durable good, then this strategy is unlikely to involve a restricted o¨er. Another interesting formulation of the bargaining problem would be to consider two pies (or ``issues'') on which the players may have sub-additive or super-additive preferences. Under complete information, Fershtman (1990) has shown that such di¨ering preferences over individual pies play a role in determining negotiation agendas. It would be interesting to see how multiple signaling modes interact under di¨ering pref-erence structures.

References

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