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Default rules and equilibrium selection of contract terms
ELSEVIER
Default Rules and Equilibrium Selection of Contract Terms M O R T E N HVIID
Department of Economics University of Warwick Coventry, U.K. E-mail: [email protected]
T h e p a p e r analyses the effect o f default rules on the selection o f c o n t r a c t terms given asymmetric information. It is shown that even when costs o f c o n t r a c t i n g a r o u n d the default are zero, the f o r m o f the default rule may m a t t e r because it may serve as a m e a n s for selection between m u l t i p l e equilibria. T h e result is illustrated in a simple signalling game.
I. Introduction T h e r e has b e e n a r e c e n t surge o f interest in how law makers a n d the courts should c o m p l e t e i n c o m p l e t e contracts. A n u m b e r o f r e c e n t p a p e r s 1 have analyzed the effect o f default rules 2 on equilibrium terms o f contracts when the c o n t r a c t i n g p a r d e s have private information. ~ T h e m a i n p o i n t o f these p a p e r s is that default rules can be used as instruments for forcing the contracting parties to reveal private i n f o r m a t i o n by asking for contract terms that differ from the defaults. T h e aim o f this p a p e r is to focus directly on the effects o f the default rule on which equilibrium contract terms are selected and, in particular, to suggest that default rules can also serve as a m e a n s o f avoiding ineffi-
An earlier version of this p a p e r entitled "Completion of contracts by the legal system: examples where default rules matter" was presented at the Arne Ryde Symposium o n The Economic Analysis of Law, University of Lund, Sweden, August 19-21, 1993, as well as at seminars at Sussex a n d W a r ~ c k . I would like to thank the participants for constructive comments. In particular, I would like to thank Marcel Canoy, N o r m a n Ireland, Imelda Maher, Scott Masten, Joe McCahery, Catherine Waddams, George yon Wangenheim, Mike Waterson, a n d the anonymous referees for many helpful comments a n d discussions. The usual disclaimer applies. 1See Ayres a n d Gertner (1989,1992), Bebchuk a n d Shavell (1991), Johnston (1990), Perloff (1981) a n d Schwartz (1990). ~A default is best seen as either what the courts will decide or are expected to decide o n a particular issue if nothing is specified about this in the contract. SThis line of research contrasts with the traditional view on the use of default rules to fill gaps in contracts, in which the role of the default rules was to provide the terms that a hypothetical complete contract would have contained. See Cooter a n d Ulen (1988, p. 233).
International Review of Law a n d Economics 16:233-245, 1996 © 1996 by Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010
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Equilibrium selection of contract terms
cient information revelation. In the latter case the contract is left incomplete deliberately. 4 An important point about default rules is that contrary to immutable rules, they can be overruled by a provision put into the contract by the contracting parties. Because a default rule is only effective if nothing else is specified in the contract, a particular default rule may prompt the contracting parties to negotiate away from the default rather than letting it stand. In the presence of asymmetric information, the suggestion of an alternative to the default rule may reveal part or all of the privately held information. As an example, consider the compensation you get if a letter posted in a postbox is not delivered. In the U.K. the default rule is zero compensation. To get any form of compensation, you need at least proof of posting. To get a high level of damages, you have to self-select by opting for recorded delivery. Your action reveals the importance you attach to delivery. Thus, a suitably designed default rule may lead to information revelation by giving the contracting parties an incentive to replace the default. If that is the case, the contract (and hence the trade) will be based on better information. In the example above, the Royal Mail is able to choose more appropriate safeguards for delivery. This often is desirable in terms of efficiency but will at the same time typically have an effect on the distribution of the surplus created by the transaction. This view of default rules as information-forcing devices has been at the center of the recent papers on default rules. 5 Most of these analyze the effects of the default rule on the incentives for information transmission between asymmetrically informed agents when the uninformed makes a one-off offer of contract terms. By selecting sufficiently unpalatable default rules the contracting parties will choose alternative contract terms, which implements a separating equilibrium 6 in what is a screening model. 7 Only Johnston (1990) has briefly considered the alternative, a signaling model where the informed makes the first offer, s In contrast, this paper focuses directly on the role of the default rule for equilibrium selection in situations where asymmetric information is present. 9 Games with asymmetric information typically have multiple sequential equilibria, some separating, some pooling, 1° and some hybrids of the first two. To select an outcome among these equilibria, recourse is usually made to one of the equilibrium refinements, n which typically manages to remove all but one separating equilibrium and all pooling equilibria. Alterna'~rhe point that contracting parties may consciously choose to leave the contract incomplete is made in Hermalin a n d Katz (1993). SSee references in note 1 above. The arguments are summarized in Baird et al. (1994) a n d Trebilcock (1993). 6Games of asymmetric information are usually modeled by assigning a type to each possible realization of the information. As an example, consider the case where the private information is about whether the buyer has a high or low valuation of the service in question. We then talk about the high valuation a n d low valuation type of buyer. A separating equilibrium is then defined as an equilibrium in which each type of the informed agent takes a different action, thereby revealing its true type to the uninformed agent before the latter takes an action. 7Screening a n d signaling games are analyzed in, a m o n g others, Baird et al. (1994). Fudenberg a n d Tirole (1991), Gibbons (1992), a n d Rasmusen (1994). 8Johnston (1990) is mainly c o n c e r n e d with screening models a n d sketches a signaling model only on pp. 634-636 a n d in appendix C. 9,4.similar discussion for immutable rules can be f o u n d in Aghion a n d Hermalin (1990). They focus on cases where the separating equilibrium fails to exist a n d show how an immutable rule can increase welfare (or lower transactions costs) by selecting a particular pooling equilibrium. A similar a r g u m e n t could be made for default rules. l°A pooling equilibrium is an equilibrium where the informed takes the same action regardless of type. Thus, no information is revealed by the equilibrium actions. HA good discussion of equilibrium refinements is Cho a n d Sobel (1990).
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tively, the default rule might serve as a means of selecting a particular outcome. Clearly, if the aim is to ensure information revelation and hence a separating equilibrium, the default simply has to be unpalatable to one or both parties, implying negotiation away from the default. But, there are cases where a particular pooling equilibrium Pareto dominates the signaling equilibria. 1~ By mimicking its terms the default rule can serve as a means of implementing this preferred pooling equilibrium. This implies an opposite role for the default rule to that identified so far in the literature, as a means of hiding rather than revealing information. Furthermore, even with zero costs o f contracting away from the default, the level of transaction costs 13 incurred differs between the different equilibria. This implies that the choice of default rule matters because it can help the selection of the equilibrium contract with the lowest transaction costs. Section II sets up a simple formal signaling model to illustrate the role o f the default rule in equilibrium selection, and the implied levels of transactions costs are identified. Section III contains conclusions.
II. The Importance of Default Rules in Signaling Games The possibility of signaling occurs in models where a player with private information chooses an observable act before another player to whom the information is relevant. The points of the paper are illustrated in a simple signaling game, where the reliability of the seller is not known to the buyer. To allow comparison rather than opting for greater realism, the model has been kept as close as possible to other papers in the literature, notably Ayres and Gertner (1989).14 The assumption that the seller makes the take-it-or-leave-it offer is maintained. The probability o f a breach of the contract depends on the level o f care, k, which the seller expends. There are two possible types of sellers, a reliable (type H) and an unreliable (type L), such that, for a given level of care the unreliable type has a higher probability of breach. Assume that the probability of breach, bi(k ) i= L, H, is given by b,(k) = 1 - 0i. ~ / k ,
i=L,H
k
(1)
where 0i measures the effect of care in reducing the probability of breach. For a given level o f care, k, the reliable type, 14, reduces this probability by more than the unreliable type, i.e., 0~/> 0L. Assume that the level of care is verifiable, in which case the level of care can be used as a signal of reliability. Finally, assume that there is some default level of care, k d, which the parties can contract away from, but in doing so, they may possibly reveal their true type. T h e buyer has a known valuation, V,, of a completed contract. It is important not to have full insurance, otherwise, the buyer would not care about the reliability of the seller. For simplicity, assume that the compensation in case of breach is fixed by the law at the level D (
12In the sense that both reliable a n d unreliable sellers prefer the pooling equilibrium. lSFor a definition of transaction costs, see Milgrom a n d Roberts (1992, pp. 28-30). 14It is worth pointing out that neither their model, nor the model presented in the appendix of this p a p e r should be taken as a realistic representation of an actual contractual situations. Both models are examples chosen for their ability to make a simple point in a simple way. tSD < Vcould have been obtained endogenously by assuming a risk-averse seller, but this would complicate the analysis without adding anything substantial.
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Equilibrium selection o f contract terms
The following order of events is assumed: 1.
2.
The seller offers the buyer a contract containing a price, P, and possibly a level of care, k. If the level of care is not specified in the contract, the default rule will fix it at k a. The buyer accepts or rejects the contract given updated beliefs about the reliability o f the seller based on the proposed contract.
The expected surplus o f the buyer accepting a contract is given by (1 - E[b(k) ]) " V + E[b(k) ] " D -
P
(2)
where E['] is the expectations operator. The m a x i m u m price the buyer is willing to accept given the expected probability of breach E[ b(k)] based on the level o f care either explicitly n a m e d in the contract or implied by the default rule is, from (2) P(k) = V- E[b(k)] • (V-
D)
(3)
The more reliable the seller, the more a buyer is willing to pay for a contract. 16 The profit of a seller of type i setting a price Pi and level of care ki is given by IIi (Pi, k~) = P~ - (1 - 0~. V ~ ) "
D - ki
(4)
From (4) is it clear that a contract would offer the m a x i m u m price given in (3). We can then use (3) in (4) to write profits as a function of k alone I I , ( k ) = V - E [ b ( k ) ] • ( V - D) - (1 - 0 i • ~ / ~ ) " D - k,
(5)
If there was no uncertainty about 9, the profit-maximizing level of care of a type i is
07. v
ki* = - - ' ~
, i = L,H
(6)
In the following, we will refer to this as the first-best level of care. Note that in the case of perfect information, type L will take less care and hence have a higher probability o f breach than type H. Type H is thus the reliable seller. With asymmetric information about the type o f seller, the unreliable type L has an incentive to act as a reliable type H, whereas the reliable type would prefer to be distinguishable from the unreliable type. These incentives give rise to the possibility o f two types of equilibria: separating equilibria and pooling equilibria. These are analyzed in tum. Initially only Perfect Bayesian Nash Equilibria, in which actions are optimal given beliefs, and beliefs are f o u n d using Bayes rule whenever possible, are considered. Separating equilibrium
In a separating equilibrium, the H type has to take an action to distinguish itself from the L type. As the choice of k is the only one that implies different costs to different types, a separating contract must include an explicit level o f k. Further, as the buyer can make the correct inference about the true type from the (k, P) combination offered in 16From (3) we can also see the importance of assuming V> D, since, if V= D, the price that the buyer will pay is i n d e p e n d e n t both of the level of care a n d the type of the seller.
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the contract, the L type chooses its full information level of care and price. From (3) and (6) these are:
v k~. -
4
(7)
P~L = D + - ~ • V. ( V -
D)
(8)
and the profits of type L are
0[
II s L = ~"
v~
(9)
W h e n type H c h o o s e s kH, the buyer correctly infers that this is a reliable seller and, from (3), is willing to pay the price
ph=D+ 0H'VTH" (V-D)
(10)
T h e level of k~ is yet to be determined. Recall that in equilibrium,/?H has to be such that s /~H) to ( k L, s ~ ) . Using (9) and (5) this incentive compattype L does not prefer (kn, ibility constraint can be written as (IC-L)
02" 4 V2 1> Or/" "V/-kH • (V - D) + 0L" V " k ~ " D - kH
(11)
To derive type/-/'s incentive compatibility constraint, assume that an H type is believed to be an L type. T h r o u g h E[b(k~)] in (3) this affects the maximal price the buyer is willing to accept. Choosing k to maximize profits given this belief, we find the optimal level of care taken, kH, as kH = (0/~ • V+ (OH - 0/) • D)/2. In this case profits are given by 1 IIH(/~H) = 5 "
(0L" V + (OH -
0 L ) . D) 2
(12)
To support the separating equilibrium, we must specify what the buyer will believe if she is offered a contract with a level of care different from one of the two proposed equilibrium levels. T h e Perfect Bayesian Nash Equilibrium concept places no restriction on this. We follow the literature by assuming that out-of-equilibrium beliefs are such that observing any other level of k than h?H implies that the seller is believed to be of type L. Given this, for incentive compatibility, type H must not prefer (kr/, WL) tO (k~, WH) (/C-H)
OH. V'-kH. V -
1 kH~>~ • (0 L • V+(O H - OL). 19)2
(13)
A separating Perfect Bayesian Nash Equilibrium consists of (7), (8), and (10) and a level of kH that satisfied both the incentive compatibility constraints (11) and (13). We get 17 PROPOSITION 1. There exists a continuum of separating equilibria. COROLLARY.I f D < (Or/+ OL) • V / (2" OH) , all separating equilibria will involve the reliable seller taking more care than under full information. tTAll proofs are collected in the appendix.
Equilibrium selection of contract terms
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Pooling equilibrium In a pooling equilibrium, the contract need not specify k explicitly if the level specified by the default rule is part of a pooling equilibrium. Assume that the probability that the true type is H is given by k. Then, the buyers expectation of the seller's reliability is = h" 0n + (1 - h) • 0L. To load the result toward the reliable type, I concentrate on showing that there exists a pooling equilibrium in which both types choose type H's preferred level of care k. The buyer's expectation of the probability of breach is b (k) -1 - 0" ~]k. Given this belief we can rewrite the profit of type H given in (5) as IIH(k) = V - (1 - 0 ~ / ~ ) ( V - D) - (1 - OHV~)D - -k
(5')
Maximizing (5') with respect to k, the pooling level of care is found to be 1 = 4 " ( ~" V+ (OH -- 0 ) " D)2
(14)
Hence, using (3) the price in the pooling equilibrium is 1
P=D+~.O.(O.
V+(OH--O).D).(V-
D)
The profits to the two types are given by 1
HH = ~ ( 0 " --
V+(0 H -
g).D) z
1
IIL=~'(g"
V+(OH-g).D).(O.V-
(0H+0-- 20L)'D)
(151
As above, we assume that the out-of-equilibrium beliefs are such that a deviation from the pooling equilibrium strategy implies that the seller is type L for sure. We get PROPOSITION2. A pooling equilibrium in which both types choose type H's preferred level of care
given in (14) exist. The existence of a continuum of separating equilibria as well as pooling equilibria has been established by adopting the out-of-equilibrium beliefs that, if the seller deviated from the proposed equilibrium, it must be type L. For some deviations, this is not a reasonable belief, because the L type has no incentives to choose this deviation. With reasonable beliefs, some of the equilibria in Propositions 1 and 2 are not good predictions of an outcome of the game. To refine the set of equilibria, we adopt the requirement that the equilibrium should satisfy the intuitive criterion of Cho and Kreps [see Gibbons (1992), p. 239]. This criterion requires that zero probability is attached to a type who could never gain by the deviation regardless of which type it is believed to be.
Refinement of equilibria Consider first the separating equilibria PROPOSmON 3. If beliefs are required to satisfy the intuitive criterion, there is a unique separating
equilibrium. Further,
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(/) if D t> (0n + 0L) • W(2 • 0/4) the e q u i l i b r i u m level o f kH is
k~/=~. : (ii) if D < (OH+ 0L) • V/(2 • OH), 1
k~/=~ [ ( 0 / 4 V - ( 0 / 4 - 0L)D+ " ~ / ( 0 / 4 - 0L)(V-- D)((OH+OL)V-- ( 0 / 4 - 0L)D))] 2 (16) Refinements o f out-of-equilibrium beliefs remove all b u t the e q u i l i b r i u m in which the reliable type incurs the least cost o f separation. T h e r e are two subcases. In case (z) the full i n f o r m a t i o n level o f care, k~/, o f the reliable type satisfies the incentive compatibility constraint o f the unreliable type, ensuring that the latter will n o t mimic the reliable type. Thus, in this case, asymmetric i n f o r m a t i o n causes n o distortion. It is so costly for the L type to mimic the action o f an H type that even if the H type ignores the signaling p r o b l e m a n d chooses its full i n f o r m a t i o n level o f care, the L type will p r e f e r n o t to mimic. In case (i0 where the two types are n o t too dissimilar, the equilibrium level o f care is the lowest level o f care, k/4 (>k~), satisfying the incentive compatibility constraints. In this case, the H type has to choose an upwards biased level o f care. This bias comes a b o u t because the seller c a n n o t credibly disclose its true type directly to the buyer. 18 T u r n i n g to the p o o l i n g equilibrium, we get a m o r e drastic result. PROPOSITION 4. The pooling equilibrium in proposition 2 fails the intuitive criterion. In general, for this simple class o f signaling games, the p r o p o s e d e q u i l i b r i u m refinem e n t will rule o u t all p o o l i n g equilibria. Based on the refinement, o n e c o u l d argue that the separating equilibrium is the best p r e d i c t i o n o f an o u t c o m e in this game.
Pareto ranking of equilibria F r o m (16) n o t e that in the separating equilibrium with a distorted level o f care, the cost o f signaling i m p l i e d by the distortion is i n d e p e n d e n t o f the probability distribution over types given by h. Further, this cost can be increased by lowering D. O n the o t h e r h a n d , the cost o f p o o l i n g does d e p e n d on the probability distribution a n d this cost disappears in the limit where all the probability mass is p l a c e d o n the " g o o d " type, i.e., when X a p p r o a c h e s one. T h e r e f o r e , an interesting case arises when the seller is very likely to be reliable a n d when signaling implies a distorted level o f care. In this case, the p o o l i n g e q u i l i b r i u m level o f care is close to the full information equilibrium level. O n the o t h e r hand, d u e to the distortion in level o f care, the separating equilibrium level o f care for the reliable seller is potentially m u c h h i g h e r than in the full i n f o r m a t i o n case. This implies that for some p a r a m e t e r values, the payoff to botl t ~ e s o f sellers is h i g h e r in the p o o l i n g equilibrium than in the separating equilibrium . ' ~ XaThus, the signalingproblem would disappear if, for instance, there existed a government agency that could validate claimsby the H type about its reliability. lt~"he unreliable type can never be worse off in a pooling equilibriumsince it is on average thought to be more reliablethan in the separatingequilibrium.If the poolinglevelof care is too high, the unreliabletypecan alwaysdeviate to its separatinglevelof care and get the associatedpayoffs.
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Equilibrium selection of contract terms
PROPOSITION5. For sufficiently similar types and )~ close enough to one, the pooling equilibrium
in proposition 2 dominates the separating equilibrium of proposition 3. Even so, the pooling equilibrium is ruled out by the equilibrium refinement, a result that is quite general. Typically, sufficiently strong equilibrium refinements will rule out pooling equilibria. 2° This leaves a role for the default rule, because although a particular separating equilibrium could not be implemented by a default rule, a pooling equilibrium could. If the default level of care coincided with the level o f care in a Pareto-dominating pooling equilibrium, neither type of sellers has an incentive to offer a contract that includes the level of care. Further, it is unlikely that any deviation from this should imply out-of-equilibrium beliefs that the seller is reliable. This implies that when there exist Pareto-dominating pooling equilibria, one of these could be implemented by a suitable choice o f default rule, i.e., one that specified the pooling equilibrium level of care. If, on the other hand, a default rule was chosen from which at least one o f the types would wish to depart, separation would occur. In this case the buyer will correctly infer that a contract without level of care k must have been offered by the " b a d " type and act accordingly, upsetting the pooling equilibrium. An important point illustrated by proposition 5 is that the default rule matters for the equilibrium contract once the contracting game has the form o f a signaling game. To see this, consider the particular transaction costs associated with "using" the private information. In signaling games, the informed is unable to communicate its information directly. Any public statement about what its information is will not be believed as it is simply cheap talk. 21 Still, the full information case where such disclosure is possible offers the best benchmark. Consider the pooling and partial-pooling equilibria in a signaling game. These equilibria involve, at most, partial use of the private information. This implies that the strategy choice in these equilibria is suboptimal relative to the case of perfect information and, hence, the payoff is lower for at least some types. This is a transactions cost arising from not being able to communicate one's private information directly and will be referred to a pooling cost. A separating equilibrium typically involves the types who are most anxious to reveal who they are, taking a distorted action that other types cannot profitably mimic. This implies lower profit for these types relative to the full information cost. Again, there is a cost associated with not being able to communicate the private information directly to the uninformed. This is also a transaction cost and will be referred to as a separation cost. W h e n a pooling equilibrium dominates a separating equilibrium, it is because the pooling costs are lower than the separation costs. Hence, the pooling equilibrium entails the lower transactions costs. As we saw, this equilibrium will not emerge unless the default rule offers the same level of care as that implied by the pooling equilibrium. From this follows that even when the costs of contracting are zero, the default rule matters, as different defaults can lead to different
2°Although generally used, such refinements are not universally accepted, as they rely on using local arguments about how deviations are interpreted. See Mailath et al. (1993). alThat is, any statement that is not a legally binding promise. Both a reliable and an unreliable seller have an incentive to claim to be reliable if such a claim cannot be enforced. This contrasts with the model by Bebchuk and Shavell (1991), in which the buyers can make a statement about their actual type, which need not be correct, but which can be entered into the contract. Furthermore, the courts can ex post determine the true type of buyer. These assumptions about the possibility of information transfer enable the authors to avoid any problems with screening or signaling and the costs associated with these activities. The assumption that a third party can verify the statements is, at least in some cases, heroic.
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levels o f transactions costs. This suggests an active role for the choice o f default rule to minimize the transaction costs associated with the use of private information. A related point is that a marginal revision of a default rule may have drastic effects. If the current default can implement a pooling equilibrium, but the revised default c a n n o t because at least one type will want to contract away from the revised default, then the possibility m i n o r revision can lead to a change from a pooling equilibrium to the separating equilibrium and, hence, in our example, a drasdc change in the level of care. This is an illustration of why care must be exercised when selecting a default rule in cases where the contracting parties have private information. HI. Conclusion
T h e paper has argued that the effects of default rules on equilibrium selection should not be ignored. T h e paper broadens the focus of Ayres and Gertner (1989, 1992) from considering information revealing separating equilibria to include nonrevealing pooling equilibria and argues that there are cases where the latter equilibrium leads to lower transactions costs. Furthermore, the implementation of such an equilibrium was shown to d e p e n d crucially on the default rule. This is in contrast to Ayres and Gertner (1992) who presented a case where, if it is costless to contract a r o u n d the default, the default rule does not add any distortions to the contracting equilibrium over and above what is caused by asymmetric information. In their model, a monopolist seller makes a take-it-or-leave-it offer to a buyer. The latter is assumed to have private information about her valuation of the contract. Since revealing a high valuation leads to both a higher level o f compensation in case of breach (and lower probability o f breach) and a higher price for the contract, the incentives to reveal information are not clear. Because of the asymmetry in information, the first-best o u t c o m e cannot be implemented. In the best possible separating equilibrium, the high-valuation type, who is harmed the most by a breach o f the contract, will be offered its first-best contract at a high price; whereas, the low-damages type will be offered a distorted contract. Consider adding a default rule as an alternative to the separating equilibrium. The seller has two reasons for ignoring the default and implementing a separating equilibrium via a m e n u of contract terms. To maximize revenue it needs to identify the high-valuation type. To minimize costs it needs to identify the type who suffers the greater actual damage, as this type will d e m a n d a bigger compensation. These two types are one and the same. Hence, independently of how m u c h weight a default rule places on actual damages, the seller always wants to identify the same type by contracting away from the default, and if there are no costs of doing so, the default rule does not matter. T h e Ayres and Gertner (1992) result is model specific and arises because they consider a screening game. In many interesting cases, the u n i n f o r m e d will make sorae decision after an observable action by an informed player, adding a signaling aspect to the model. In this class o f games the default rule may matter even when direct costs of contracting are zero, because it may implement or destroy a pooling equilibrium when such an equilibrium has the lowest level of transactions costs. Apart from the particular example analyzed in Section II, there are other cases in which a particular default rule may force separation even if pooling is preferred. An example is the case where any signal from the seller to the buyer also signals to other agents in the market, such as regulators or actual/potential competitors. If the seller prefers to send opposite signals to the two audiences, the conflicting incentives may lead
Equilibrium selection of contract terms
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to a p r e f e r e n c e for a p o o l i n g equilibrium in which n o t h i n g is revealed to e i t h e r audience. T h e r e are two ways in which the result o f the p a p e r can be read. It may be seen as providing an a r g u m e n t for a wide a n d clever use o f default rules to select equilibria. O n the o t h e r h a n d , sufficient i n f o r m a t i o n is n o t likely to be available in most cases, a n d default rules would have to be so special that the users would n e e d precise i n f o r m a t i o n to accurately p r e d i c t what the default rule looks like in any given case. A m o r e reasonable conclusion t h e n is that a l t h o u g h the m a i n usefulness o f the default rules is saving on transaction costs, if asymmetric i n f o r m a t i o n is e x p e c t e d to be i m p o r t a n t , the default rule may have u n e x p e c t e d side-effects by influencing which equilibrium is selected.
Appendix: proofs PROOF OF PROPOSITION 1. Define the following two functions:
f(k) = k - (0/4" V - (0/4 - 0L)" D ) . ~ / k + z 0~. v
g(k) = k -
OH • V. "V/'k+ ~ . (0 L • V+ ( 0 n -
~
0t)" D) 2
For a separating equilibrium to exist we n e e d s i m u l t a n e o u s l y f ( k ) >t 0 a n d I> g(k). Both
j(k) and g(k) are convex functions o f k, g(0) > f(0) > 0, a n d b o t h f a n d g go to infinity for k going to infinity. Construct the difference between the functions
1 g(k) - f(k) = ~ . (OH - 0 / ) . ( 2 . 0 L. V+ ( 0 . - 0L). D ) . D - (0 H - 0L)" D . ~ ' k T h e two functions only cross once, at/~ = (2 • 0L • V+ (OHm 0L) " D ) z / 1 6 > 0, a n d ~(k) must cross f(k) from above. T h e m i n i m u m o f f ( k ) is at k ~ = (0H V - ( O n - OL)D)~/4, a n d the m i n i m u m of g(k) is at/~Hin= 0 ~ . Vz/4 > kr~r L in , the full f o r m a t i o n choice o f an H-type.
1
f(k~ ~") = - ~ . ( 0 , , -
00 . (V-
1 g(k~in) = - 5 " (oH - 0L)" ( V -
D) . ( 0 ~ . ( V -
D ) + OL . ( V + D)) < 0
D)" (0 H" (V+ D) + 0 L" ( V -
D)) < 0
Finally, g(k~Hin) < f(k~Lin), implying that g a n d f c r o s s to the left o f ~ i . . This guarantees existence o f a separating equilibrium. Start at k~sin. H e r e g(/~Hin) < 0. Iff(k~z in) > 0 we have f o u n d an equilibrium. If not, since b o t h functions are increasing a n d f(k) > g(k) for k/>/~Hin we can increase k until we get to a p o i n t where f(k) > 0 > g(k). To completely characterize the set o f equilibria, note that t h e r e are two subcases, f f f = g ~> 0, we get the case illustrated in Figure 1. H e r e t h e r e will be two separate intervals for which f(k) >~ 0 and g(k) ~< 0. All these levels o f k can be s u p p o r t e d as a Perfect Bayesian Equilibrium by o u r out-of-equilibrium beliefs. F o r f = g < 0 we get the case illustrated in Figure 2. H e r e t h e r e is only o n e interval o f separating equilibria.
HVIID
243
f(k),g(k) f(k)
g(k)
I / hbria~//
"Qj
V
k
FIG. 1. The case where f(k) and g(k) intersect for f = g > 0. PROOF OF COROLLARY.Iff(k~/) < 0, the first-best of the H-type is outside and to the left of the interval of separating levels of k and hence not a possible equilibrium. Using (6) in (13) 1
f(/~n) = ~ • ( 0 H - Or)" (20/_/. D -
(0H+ 0 ~ " V). V<0,
from which we get D t> - - . ( 0 H0r) + V (A1) 2 • OH When D is close enough to V it is too costly for the L-type to mimic the H-type. PaooF OF PROPOSrrloN 2. Comparing (15) and (12) it is clear that type H has no incentive to deviate from the pooling equilibrium. We then only have to check that type L does not want to deviate. T h e incentive compatibility constraint is now 1
(/C-L)
1
4" (g" V+ (01-1 - 0)" D). (0. V - (0u + 0 -- 20r). D) 1> 4 " 0~" Vz
Rearrange to get (~2 _ 0 ~ ) . V z + 0 . 0 L • D . V -
( ( 0 u -- 0)" (01-/+ 0 -- 20/_)). D 2 >i 0
(A2)
Clearly the higher D is, the harder is it to satisfy condition (A2). By looking at the case where D --~ V, we can obtain the following sufficient condition for (A2) to hold: ( 0 H - 0/_)X~ + 0LX - ( 0 u -
0D/> 0
Define )~* as the value of k E [0,1] for which (A3) hold with equality. ~*_
- 0 L + "~V/40~ - 80/_10L+ 50~ 2 ( 0 / 4 - 0/4)
(A3)
244
Equilibrium selection of contract terms f(k),g(k)
Y
k
FIG. 2. The case where f(k) and g(k) intersect for f = g < 0. Hence, a sufficient condition for (A2) to hold is that )~ ~ [•*, 1]. Note that as 0L goes to zero, )~* goes to one and as 0 L goes to OH, )~* goes to zero. Also, )~* is decreasing in 01.. Hence, the closer the two types are, the more likely the pooling equilibrium is to dominate the separating equilibrium. PROOF OF PROPOSITION3. For the first case, the first-best o f the H-type is in the set of separating equilibria. Any deviation to this level of care must have been carried out by type H, since by (11), the L-type has no incentive to mimic regardless of the inference. As the first-best is the preferred level of care, this is the only separating equilibrium that survives the application o f the intuitive criteiron. For the second case, type H c a n n o t get its first-best. It will then want to get the one that is closest to it but still in the interval of separating equilibria where (11) hold. Intuitively, it is the level of care closest to the first-best level o f k for an H-type, which an L-type would not want to copy. As/~L'n < k~Hin, the equilibrium value o f kH is given by the larger o f the two solutions to J(k) = 0. PROOF OF PROPOSITION4. To satisfy the intuitive criterion, there must not be a k such that if the buyer, after observing k, believes that the true type is H a n d , given the belief, only type H prefers k to the pooling equilibrium. Thus it must not be the case that OH • ~v/-k • V -
1_ k I> ~ ( 0 . V + (0 M - 0 ) . D) 2
(A4)
1
4 " (~" V+ (0/_/- 0). D). (0. V+ (0 + 0L - 0/4)" D) />
0/4. X / h . ( v - D) +
X/k" D - k
(AS)
For k = 0~ • V/4, (A4) clearly holds. Further, i f k is close e n o u g h to 1, (AS) will also hold, breaking the equilibrium.
HVlID
245
PROOF OF PROPOSITION 5. Fix V a n d D. For sufficiently similar types, the separating e q u i l i b r i u m will involve a level of care greater t h a n the first-best and, hence, a profit lower than the first-best. Further, from (A1), the closer is 0L to On, the greater is the bias. Contrast that with the p o o l i n g e q u i l i b r i u m profits given in (15). For h a p p r o a c h i n g 1, this approaches the first-best profits of the H-type. Hence, for h close, to 1, profits for the H-type are higher in the pooling equilibrium. T h e L-type is clearly better off in the p o o l i n g e q u i l i b r i u m by construction.
References AGHION,Ph. ANDHERMALIN,B. (1990). Legal restrictions on private contracts can enhance efficiency. Journal of Law, Economics and Organization. 6:381--409. AYRES,I., ANDGERTNER,R. (1989). Filling gaps in incomplete contracts: An economic theory of default rules. Yale Law Journal. 99:87-130. AYe.Es,I., ANDGERTNER,R. (1992). Strategic contractual choice and the optimal choice of legal rules. Yale Law Journal. 101:729-773. BAIRD,D.G., GERTNER,R.H. ANDPICKER,R.C. (1994). Game Theory and theLaw, Harvard University Press, Cambridge, Massachusetts. BEBCHUK,L.A., ANDSHAVELL,S. (1991). Information and the scope of liabilityfor breach of contract: The rule of Hadley v. Baxendale. The Journal of Law, Economics and Organization. 7:284-312. CHO,I.-K.,ANDSOBEL,J. (1990). Strategic stability and uniqueness in signalling games.Journal ofEconomic Theory. 50:381-413. COOTER,R., ANDULEN,T. (1988). Law and Economics, Harper Colins Publishers, U.S.A. FUDENBERG,D., ANDTmOLE,J. (1991). Game Theory,MIT Press, Cambridge, Massachusetts. GIBBONS,R. (1992). A Primer in Game Theory. Harvester/Wheatsheaf, Henffordshire, England. HERMAtaN,B.E., ANDKATZ,M.L. (1993).Judicial modification of contracts between sophisticated parties: A more complete view of incomplete contracts and their breach. Journal of Law, Economics, and Organization. 9:230-255. JOHNSTON,J.J. (1990). Strategic bargaining and the economic theory of contract default rules. YaleLaw Journal. 100:615-664. MAILATH,G.J., OKUNO-FUJIWARA,M. ANDPOSTLEWMTE,A. (1993). Belief-based refinements in signaling games. Journal of Economic Theory. 60:241-276. MILGROM, P., AND ROBERTS,J.(1992). Economics, Organization and Management, Prentice-Hall International, London, England. PERLOFF,J.M. (1981). Breach of contract and the foreseeability doctrine of Hadley v. Baxendale.Journal of Legal Studies. 10:39--63. RASMUSEN,E. (1994). Gamesand Information, 2d edition, Blackwell, Oxford, England. SCHWARTZ,A. (1990). The myth that promises prefer supracompensatory remedies: An analysis of contracting for damage measures. Yale Law Journal. 100:369-407. SHAPIRO,C. (1989). Theories of oligopoly behavior. In R. SCHMALENSEEand R.D. WILUG(Eds.), Handbook of Industrial Organization 1:329-414. TREmLCOCK,M.J. (1993). The Limits of Freedom of Contracts. Harvard University Press, Cambridge, Massachusetts.
Default Rules and Equilibrium Selection of Contract Terms M O R T E N HVIID
Department of Economics University of Warwick Coventry, U.K. E-mail: [email protected]
T h e p a p e r analyses the effect o f default rules on the selection o f c o n t r a c t terms given asymmetric information. It is shown that even when costs o f c o n t r a c t i n g a r o u n d the default are zero, the f o r m o f the default rule may m a t t e r because it may serve as a m e a n s for selection between m u l t i p l e equilibria. T h e result is illustrated in a simple signalling game.
I. Introduction T h e r e has b e e n a r e c e n t surge o f interest in how law makers a n d the courts should c o m p l e t e i n c o m p l e t e contracts. A n u m b e r o f r e c e n t p a p e r s 1 have analyzed the effect o f default rules 2 on equilibrium terms o f contracts when the c o n t r a c t i n g p a r d e s have private information. ~ T h e m a i n p o i n t o f these p a p e r s is that default rules can be used as instruments for forcing the contracting parties to reveal private i n f o r m a t i o n by asking for contract terms that differ from the defaults. T h e aim o f this p a p e r is to focus directly on the effects o f the default rule on which equilibrium contract terms are selected and, in particular, to suggest that default rules can also serve as a m e a n s o f avoiding ineffi-
An earlier version of this p a p e r entitled "Completion of contracts by the legal system: examples where default rules matter" was presented at the Arne Ryde Symposium o n The Economic Analysis of Law, University of Lund, Sweden, August 19-21, 1993, as well as at seminars at Sussex a n d W a r ~ c k . I would like to thank the participants for constructive comments. In particular, I would like to thank Marcel Canoy, N o r m a n Ireland, Imelda Maher, Scott Masten, Joe McCahery, Catherine Waddams, George yon Wangenheim, Mike Waterson, a n d the anonymous referees for many helpful comments a n d discussions. The usual disclaimer applies. 1See Ayres a n d Gertner (1989,1992), Bebchuk a n d Shavell (1991), Johnston (1990), Perloff (1981) a n d Schwartz (1990). ~A default is best seen as either what the courts will decide or are expected to decide o n a particular issue if nothing is specified about this in the contract. SThis line of research contrasts with the traditional view on the use of default rules to fill gaps in contracts, in which the role of the default rules was to provide the terms that a hypothetical complete contract would have contained. See Cooter a n d Ulen (1988, p. 233).
International Review of Law a n d Economics 16:233-245, 1996 © 1996 by Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010
0144-8188/96/$15.00 SSDI 0144-8188(95)00023-2
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Equilibrium selection of contract terms
cient information revelation. In the latter case the contract is left incomplete deliberately. 4 An important point about default rules is that contrary to immutable rules, they can be overruled by a provision put into the contract by the contracting parties. Because a default rule is only effective if nothing else is specified in the contract, a particular default rule may prompt the contracting parties to negotiate away from the default rather than letting it stand. In the presence of asymmetric information, the suggestion of an alternative to the default rule may reveal part or all of the privately held information. As an example, consider the compensation you get if a letter posted in a postbox is not delivered. In the U.K. the default rule is zero compensation. To get any form of compensation, you need at least proof of posting. To get a high level of damages, you have to self-select by opting for recorded delivery. Your action reveals the importance you attach to delivery. Thus, a suitably designed default rule may lead to information revelation by giving the contracting parties an incentive to replace the default. If that is the case, the contract (and hence the trade) will be based on better information. In the example above, the Royal Mail is able to choose more appropriate safeguards for delivery. This often is desirable in terms of efficiency but will at the same time typically have an effect on the distribution of the surplus created by the transaction. This view of default rules as information-forcing devices has been at the center of the recent papers on default rules. 5 Most of these analyze the effects of the default rule on the incentives for information transmission between asymmetrically informed agents when the uninformed makes a one-off offer of contract terms. By selecting sufficiently unpalatable default rules the contracting parties will choose alternative contract terms, which implements a separating equilibrium 6 in what is a screening model. 7 Only Johnston (1990) has briefly considered the alternative, a signaling model where the informed makes the first offer, s In contrast, this paper focuses directly on the role of the default rule for equilibrium selection in situations where asymmetric information is present. 9 Games with asymmetric information typically have multiple sequential equilibria, some separating, some pooling, 1° and some hybrids of the first two. To select an outcome among these equilibria, recourse is usually made to one of the equilibrium refinements, n which typically manages to remove all but one separating equilibrium and all pooling equilibria. Alterna'~rhe point that contracting parties may consciously choose to leave the contract incomplete is made in Hermalin a n d Katz (1993). SSee references in note 1 above. The arguments are summarized in Baird et al. (1994) a n d Trebilcock (1993). 6Games of asymmetric information are usually modeled by assigning a type to each possible realization of the information. As an example, consider the case where the private information is about whether the buyer has a high or low valuation of the service in question. We then talk about the high valuation a n d low valuation type of buyer. A separating equilibrium is then defined as an equilibrium in which each type of the informed agent takes a different action, thereby revealing its true type to the uninformed agent before the latter takes an action. 7Screening a n d signaling games are analyzed in, a m o n g others, Baird et al. (1994). Fudenberg a n d Tirole (1991), Gibbons (1992), a n d Rasmusen (1994). 8Johnston (1990) is mainly c o n c e r n e d with screening models a n d sketches a signaling model only on pp. 634-636 a n d in appendix C. 9,4.similar discussion for immutable rules can be f o u n d in Aghion a n d Hermalin (1990). They focus on cases where the separating equilibrium fails to exist a n d show how an immutable rule can increase welfare (or lower transactions costs) by selecting a particular pooling equilibrium. A similar a r g u m e n t could be made for default rules. l°A pooling equilibrium is an equilibrium where the informed takes the same action regardless of type. Thus, no information is revealed by the equilibrium actions. HA good discussion of equilibrium refinements is Cho a n d Sobel (1990).
HVIID
235
tively, the default rule might serve as a means of selecting a particular outcome. Clearly, if the aim is to ensure information revelation and hence a separating equilibrium, the default simply has to be unpalatable to one or both parties, implying negotiation away from the default. But, there are cases where a particular pooling equilibrium Pareto dominates the signaling equilibria. 1~ By mimicking its terms the default rule can serve as a means of implementing this preferred pooling equilibrium. This implies an opposite role for the default rule to that identified so far in the literature, as a means of hiding rather than revealing information. Furthermore, even with zero costs o f contracting away from the default, the level of transaction costs 13 incurred differs between the different equilibria. This implies that the choice of default rule matters because it can help the selection of the equilibrium contract with the lowest transaction costs. Section II sets up a simple formal signaling model to illustrate the role o f the default rule in equilibrium selection, and the implied levels of transactions costs are identified. Section III contains conclusions.
II. The Importance of Default Rules in Signaling Games The possibility of signaling occurs in models where a player with private information chooses an observable act before another player to whom the information is relevant. The points of the paper are illustrated in a simple signaling game, where the reliability of the seller is not known to the buyer. To allow comparison rather than opting for greater realism, the model has been kept as close as possible to other papers in the literature, notably Ayres and Gertner (1989).14 The assumption that the seller makes the take-it-or-leave-it offer is maintained. The probability o f a breach of the contract depends on the level o f care, k, which the seller expends. There are two possible types of sellers, a reliable (type H) and an unreliable (type L), such that, for a given level of care the unreliable type has a higher probability of breach. Assume that the probability of breach, bi(k ) i= L, H, is given by b,(k) = 1 - 0i. ~ / k ,
i=L,H
k
(1)
where 0i measures the effect of care in reducing the probability of breach. For a given level o f care, k, the reliable type, 14, reduces this probability by more than the unreliable type, i.e., 0~/> 0L. Assume that the level of care is verifiable, in which case the level of care can be used as a signal of reliability. Finally, assume that there is some default level of care, k d, which the parties can contract away from, but in doing so, they may possibly reveal their true type. T h e buyer has a known valuation, V,, of a completed contract. It is important not to have full insurance, otherwise, the buyer would not care about the reliability of the seller. For simplicity, assume that the compensation in case of breach is fixed by the law at the level D (
12In the sense that both reliable a n d unreliable sellers prefer the pooling equilibrium. lSFor a definition of transaction costs, see Milgrom a n d Roberts (1992, pp. 28-30). 14It is worth pointing out that neither their model, nor the model presented in the appendix of this p a p e r should be taken as a realistic representation of an actual contractual situations. Both models are examples chosen for their ability to make a simple point in a simple way. tSD < Vcould have been obtained endogenously by assuming a risk-averse seller, but this would complicate the analysis without adding anything substantial.
236
Equilibrium selection o f contract terms
The following order of events is assumed: 1.
2.
The seller offers the buyer a contract containing a price, P, and possibly a level of care, k. If the level of care is not specified in the contract, the default rule will fix it at k a. The buyer accepts or rejects the contract given updated beliefs about the reliability o f the seller based on the proposed contract.
The expected surplus o f the buyer accepting a contract is given by (1 - E[b(k) ]) " V + E[b(k) ] " D -
P
(2)
where E['] is the expectations operator. The m a x i m u m price the buyer is willing to accept given the expected probability of breach E[ b(k)] based on the level o f care either explicitly n a m e d in the contract or implied by the default rule is, from (2) P(k) = V- E[b(k)] • (V-
D)
(3)
The more reliable the seller, the more a buyer is willing to pay for a contract. 16 The profit of a seller of type i setting a price Pi and level of care ki is given by IIi (Pi, k~) = P~ - (1 - 0~. V ~ ) "
D - ki
(4)
From (4) is it clear that a contract would offer the m a x i m u m price given in (3). We can then use (3) in (4) to write profits as a function of k alone I I , ( k ) = V - E [ b ( k ) ] • ( V - D) - (1 - 0 i • ~ / ~ ) " D - k,
(5)
If there was no uncertainty about 9, the profit-maximizing level of care of a type i is
07. v
ki* = - - ' ~
, i = L,H
(6)
In the following, we will refer to this as the first-best level of care. Note that in the case of perfect information, type L will take less care and hence have a higher probability o f breach than type H. Type H is thus the reliable seller. With asymmetric information about the type o f seller, the unreliable type L has an incentive to act as a reliable type H, whereas the reliable type would prefer to be distinguishable from the unreliable type. These incentives give rise to the possibility o f two types of equilibria: separating equilibria and pooling equilibria. These are analyzed in tum. Initially only Perfect Bayesian Nash Equilibria, in which actions are optimal given beliefs, and beliefs are f o u n d using Bayes rule whenever possible, are considered. Separating equilibrium
In a separating equilibrium, the H type has to take an action to distinguish itself from the L type. As the choice of k is the only one that implies different costs to different types, a separating contract must include an explicit level o f k. Further, as the buyer can make the correct inference about the true type from the (k, P) combination offered in 16From (3) we can also see the importance of assuming V> D, since, if V= D, the price that the buyer will pay is i n d e p e n d e n t both of the level of care a n d the type of the seller.
HVIID
237
the contract, the L type chooses its full information level of care and price. From (3) and (6) these are:
v k~. -
4
(7)
P~L = D + - ~ • V. ( V -
D)
(8)
and the profits of type L are
0[
II s L = ~"
v~
(9)
W h e n type H c h o o s e s kH, the buyer correctly infers that this is a reliable seller and, from (3), is willing to pay the price
ph=D+ 0H'VTH" (V-D)
(10)
T h e level of k~ is yet to be determined. Recall that in equilibrium,/?H has to be such that s /~H) to ( k L, s ~ ) . Using (9) and (5) this incentive compattype L does not prefer (kn, ibility constraint can be written as (IC-L)
02" 4 V2 1> Or/" "V/-kH • (V - D) + 0L" V " k ~ " D - kH
(11)
To derive type/-/'s incentive compatibility constraint, assume that an H type is believed to be an L type. T h r o u g h E[b(k~)] in (3) this affects the maximal price the buyer is willing to accept. Choosing k to maximize profits given this belief, we find the optimal level of care taken, kH, as kH = (0/~ • V+ (OH - 0/) • D)/2. In this case profits are given by 1 IIH(/~H) = 5 "
(0L" V + (OH -
0 L ) . D) 2
(12)
To support the separating equilibrium, we must specify what the buyer will believe if she is offered a contract with a level of care different from one of the two proposed equilibrium levels. T h e Perfect Bayesian Nash Equilibrium concept places no restriction on this. We follow the literature by assuming that out-of-equilibrium beliefs are such that observing any other level of k than h?H implies that the seller is believed to be of type L. Given this, for incentive compatibility, type H must not prefer (kr/, WL) tO (k~, WH) (/C-H)
OH. V'-kH. V -
1 kH~>~ • (0 L • V+(O H - OL). 19)2
(13)
A separating Perfect Bayesian Nash Equilibrium consists of (7), (8), and (10) and a level of kH that satisfied both the incentive compatibility constraints (11) and (13). We get 17 PROPOSITION 1. There exists a continuum of separating equilibria. COROLLARY.I f D < (Or/+ OL) • V / (2" OH) , all separating equilibria will involve the reliable seller taking more care than under full information. tTAll proofs are collected in the appendix.
Equilibrium selection of contract terms
238
Pooling equilibrium In a pooling equilibrium, the contract need not specify k explicitly if the level specified by the default rule is part of a pooling equilibrium. Assume that the probability that the true type is H is given by k. Then, the buyers expectation of the seller's reliability is = h" 0n + (1 - h) • 0L. To load the result toward the reliable type, I concentrate on showing that there exists a pooling equilibrium in which both types choose type H's preferred level of care k. The buyer's expectation of the probability of breach is b (k) -1 - 0" ~]k. Given this belief we can rewrite the profit of type H given in (5) as IIH(k) = V - (1 - 0 ~ / ~ ) ( V - D) - (1 - OHV~)D - -k
(5')
Maximizing (5') with respect to k, the pooling level of care is found to be 1 = 4 " ( ~" V+ (OH -- 0 ) " D)2
(14)
Hence, using (3) the price in the pooling equilibrium is 1
P=D+~.O.(O.
V+(OH--O).D).(V-
D)
The profits to the two types are given by 1
HH = ~ ( 0 " --
V+(0 H -
g).D) z
1
IIL=~'(g"
V+(OH-g).D).(O.V-
(0H+0-- 20L)'D)
(151
As above, we assume that the out-of-equilibrium beliefs are such that a deviation from the pooling equilibrium strategy implies that the seller is type L for sure. We get PROPOSITION2. A pooling equilibrium in which both types choose type H's preferred level of care
given in (14) exist. The existence of a continuum of separating equilibria as well as pooling equilibria has been established by adopting the out-of-equilibrium beliefs that, if the seller deviated from the proposed equilibrium, it must be type L. For some deviations, this is not a reasonable belief, because the L type has no incentives to choose this deviation. With reasonable beliefs, some of the equilibria in Propositions 1 and 2 are not good predictions of an outcome of the game. To refine the set of equilibria, we adopt the requirement that the equilibrium should satisfy the intuitive criterion of Cho and Kreps [see Gibbons (1992), p. 239]. This criterion requires that zero probability is attached to a type who could never gain by the deviation regardless of which type it is believed to be.
Refinement of equilibria Consider first the separating equilibria PROPOSmON 3. If beliefs are required to satisfy the intuitive criterion, there is a unique separating
equilibrium. Further,
HWID
239
(/) if D t> (0n + 0L) • W(2 • 0/4) the e q u i l i b r i u m level o f kH is
k~/=~. : (ii) if D < (OH+ 0L) • V/(2 • OH), 1
k~/=~ [ ( 0 / 4 V - ( 0 / 4 - 0L)D+ " ~ / ( 0 / 4 - 0L)(V-- D)((OH+OL)V-- ( 0 / 4 - 0L)D))] 2 (16) Refinements o f out-of-equilibrium beliefs remove all b u t the e q u i l i b r i u m in which the reliable type incurs the least cost o f separation. T h e r e are two subcases. In case (z) the full i n f o r m a t i o n level o f care, k~/, o f the reliable type satisfies the incentive compatibility constraint o f the unreliable type, ensuring that the latter will n o t mimic the reliable type. Thus, in this case, asymmetric i n f o r m a t i o n causes n o distortion. It is so costly for the L type to mimic the action o f an H type that even if the H type ignores the signaling p r o b l e m a n d chooses its full i n f o r m a t i o n level o f care, the L type will p r e f e r n o t to mimic. In case (i0 where the two types are n o t too dissimilar, the equilibrium level o f care is the lowest level o f care, k/4 (>k~), satisfying the incentive compatibility constraints. In this case, the H type has to choose an upwards biased level o f care. This bias comes a b o u t because the seller c a n n o t credibly disclose its true type directly to the buyer. 18 T u r n i n g to the p o o l i n g equilibrium, we get a m o r e drastic result. PROPOSITION 4. The pooling equilibrium in proposition 2 fails the intuitive criterion. In general, for this simple class o f signaling games, the p r o p o s e d e q u i l i b r i u m refinem e n t will rule o u t all p o o l i n g equilibria. Based on the refinement, o n e c o u l d argue that the separating equilibrium is the best p r e d i c t i o n o f an o u t c o m e in this game.
Pareto ranking of equilibria F r o m (16) n o t e that in the separating equilibrium with a distorted level o f care, the cost o f signaling i m p l i e d by the distortion is i n d e p e n d e n t o f the probability distribution over types given by h. Further, this cost can be increased by lowering D. O n the o t h e r h a n d , the cost o f p o o l i n g does d e p e n d on the probability distribution a n d this cost disappears in the limit where all the probability mass is p l a c e d o n the " g o o d " type, i.e., when X a p p r o a c h e s one. T h e r e f o r e , an interesting case arises when the seller is very likely to be reliable a n d when signaling implies a distorted level o f care. In this case, the p o o l i n g e q u i l i b r i u m level o f care is close to the full information equilibrium level. O n the o t h e r hand, d u e to the distortion in level o f care, the separating equilibrium level o f care for the reliable seller is potentially m u c h h i g h e r than in the full i n f o r m a t i o n case. This implies that for some p a r a m e t e r values, the payoff to botl t ~ e s o f sellers is h i g h e r in the p o o l i n g equilibrium than in the separating equilibrium . ' ~ XaThus, the signalingproblem would disappear if, for instance, there existed a government agency that could validate claimsby the H type about its reliability. lt~"he unreliable type can never be worse off in a pooling equilibriumsince it is on average thought to be more reliablethan in the separatingequilibrium.If the poolinglevelof care is too high, the unreliabletypecan alwaysdeviate to its separatinglevelof care and get the associatedpayoffs.
240
Equilibrium selection of contract terms
PROPOSITION5. For sufficiently similar types and )~ close enough to one, the pooling equilibrium
in proposition 2 dominates the separating equilibrium of proposition 3. Even so, the pooling equilibrium is ruled out by the equilibrium refinement, a result that is quite general. Typically, sufficiently strong equilibrium refinements will rule out pooling equilibria. 2° This leaves a role for the default rule, because although a particular separating equilibrium could not be implemented by a default rule, a pooling equilibrium could. If the default level of care coincided with the level o f care in a Pareto-dominating pooling equilibrium, neither type of sellers has an incentive to offer a contract that includes the level of care. Further, it is unlikely that any deviation from this should imply out-of-equilibrium beliefs that the seller is reliable. This implies that when there exist Pareto-dominating pooling equilibria, one of these could be implemented by a suitable choice o f default rule, i.e., one that specified the pooling equilibrium level of care. If, on the other hand, a default rule was chosen from which at least one o f the types would wish to depart, separation would occur. In this case the buyer will correctly infer that a contract without level of care k must have been offered by the " b a d " type and act accordingly, upsetting the pooling equilibrium. An important point illustrated by proposition 5 is that the default rule matters for the equilibrium contract once the contracting game has the form o f a signaling game. To see this, consider the particular transaction costs associated with "using" the private information. In signaling games, the informed is unable to communicate its information directly. Any public statement about what its information is will not be believed as it is simply cheap talk. 21 Still, the full information case where such disclosure is possible offers the best benchmark. Consider the pooling and partial-pooling equilibria in a signaling game. These equilibria involve, at most, partial use of the private information. This implies that the strategy choice in these equilibria is suboptimal relative to the case of perfect information and, hence, the payoff is lower for at least some types. This is a transactions cost arising from not being able to communicate one's private information directly and will be referred to a pooling cost. A separating equilibrium typically involves the types who are most anxious to reveal who they are, taking a distorted action that other types cannot profitably mimic. This implies lower profit for these types relative to the full information cost. Again, there is a cost associated with not being able to communicate the private information directly to the uninformed. This is also a transaction cost and will be referred to as a separation cost. W h e n a pooling equilibrium dominates a separating equilibrium, it is because the pooling costs are lower than the separation costs. Hence, the pooling equilibrium entails the lower transactions costs. As we saw, this equilibrium will not emerge unless the default rule offers the same level of care as that implied by the pooling equilibrium. From this follows that even when the costs of contracting are zero, the default rule matters, as different defaults can lead to different
2°Although generally used, such refinements are not universally accepted, as they rely on using local arguments about how deviations are interpreted. See Mailath et al. (1993). alThat is, any statement that is not a legally binding promise. Both a reliable and an unreliable seller have an incentive to claim to be reliable if such a claim cannot be enforced. This contrasts with the model by Bebchuk and Shavell (1991), in which the buyers can make a statement about their actual type, which need not be correct, but which can be entered into the contract. Furthermore, the courts can ex post determine the true type of buyer. These assumptions about the possibility of information transfer enable the authors to avoid any problems with screening or signaling and the costs associated with these activities. The assumption that a third party can verify the statements is, at least in some cases, heroic.
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levels o f transactions costs. This suggests an active role for the choice o f default rule to minimize the transaction costs associated with the use of private information. A related point is that a marginal revision of a default rule may have drastic effects. If the current default can implement a pooling equilibrium, but the revised default c a n n o t because at least one type will want to contract away from the revised default, then the possibility m i n o r revision can lead to a change from a pooling equilibrium to the separating equilibrium and, hence, in our example, a drasdc change in the level of care. This is an illustration of why care must be exercised when selecting a default rule in cases where the contracting parties have private information. HI. Conclusion
T h e paper has argued that the effects of default rules on equilibrium selection should not be ignored. T h e paper broadens the focus of Ayres and Gertner (1989, 1992) from considering information revealing separating equilibria to include nonrevealing pooling equilibria and argues that there are cases where the latter equilibrium leads to lower transactions costs. Furthermore, the implementation of such an equilibrium was shown to d e p e n d crucially on the default rule. This is in contrast to Ayres and Gertner (1992) who presented a case where, if it is costless to contract a r o u n d the default, the default rule does not add any distortions to the contracting equilibrium over and above what is caused by asymmetric information. In their model, a monopolist seller makes a take-it-or-leave-it offer to a buyer. The latter is assumed to have private information about her valuation of the contract. Since revealing a high valuation leads to both a higher level o f compensation in case of breach (and lower probability o f breach) and a higher price for the contract, the incentives to reveal information are not clear. Because of the asymmetry in information, the first-best o u t c o m e cannot be implemented. In the best possible separating equilibrium, the high-valuation type, who is harmed the most by a breach o f the contract, will be offered its first-best contract at a high price; whereas, the low-damages type will be offered a distorted contract. Consider adding a default rule as an alternative to the separating equilibrium. The seller has two reasons for ignoring the default and implementing a separating equilibrium via a m e n u of contract terms. To maximize revenue it needs to identify the high-valuation type. To minimize costs it needs to identify the type who suffers the greater actual damage, as this type will d e m a n d a bigger compensation. These two types are one and the same. Hence, independently of how m u c h weight a default rule places on actual damages, the seller always wants to identify the same type by contracting away from the default, and if there are no costs of doing so, the default rule does not matter. T h e Ayres and Gertner (1992) result is model specific and arises because they consider a screening game. In many interesting cases, the u n i n f o r m e d will make sorae decision after an observable action by an informed player, adding a signaling aspect to the model. In this class o f games the default rule may matter even when direct costs of contracting are zero, because it may implement or destroy a pooling equilibrium when such an equilibrium has the lowest level of transactions costs. Apart from the particular example analyzed in Section II, there are other cases in which a particular default rule may force separation even if pooling is preferred. An example is the case where any signal from the seller to the buyer also signals to other agents in the market, such as regulators or actual/potential competitors. If the seller prefers to send opposite signals to the two audiences, the conflicting incentives may lead
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to a p r e f e r e n c e for a p o o l i n g equilibrium in which n o t h i n g is revealed to e i t h e r audience. T h e r e are two ways in which the result o f the p a p e r can be read. It may be seen as providing an a r g u m e n t for a wide a n d clever use o f default rules to select equilibria. O n the o t h e r h a n d , sufficient i n f o r m a t i o n is n o t likely to be available in most cases, a n d default rules would have to be so special that the users would n e e d precise i n f o r m a t i o n to accurately p r e d i c t what the default rule looks like in any given case. A m o r e reasonable conclusion t h e n is that a l t h o u g h the m a i n usefulness o f the default rules is saving on transaction costs, if asymmetric i n f o r m a t i o n is e x p e c t e d to be i m p o r t a n t , the default rule may have u n e x p e c t e d side-effects by influencing which equilibrium is selected.
Appendix: proofs PROOF OF PROPOSITION 1. Define the following two functions:
f(k) = k - (0/4" V - (0/4 - 0L)" D ) . ~ / k + z 0~. v
g(k) = k -
OH • V. "V/'k+ ~ . (0 L • V+ ( 0 n -
~
0t)" D) 2
For a separating equilibrium to exist we n e e d s i m u l t a n e o u s l y f ( k ) >t 0 a n d I> g(k). Both
j(k) and g(k) are convex functions o f k, g(0) > f(0) > 0, a n d b o t h f a n d g go to infinity for k going to infinity. Construct the difference between the functions
1 g(k) - f(k) = ~ . (OH - 0 / ) . ( 2 . 0 L. V+ ( 0 . - 0L). D ) . D - (0 H - 0L)" D . ~ ' k T h e two functions only cross once, at/~ = (2 • 0L • V+ (OHm 0L) " D ) z / 1 6 > 0, a n d ~(k) must cross f(k) from above. T h e m i n i m u m o f f ( k ) is at k ~ = (0H V - ( O n - OL)D)~/4, a n d the m i n i m u m of g(k) is at/~Hin= 0 ~ . Vz/4 > kr~r L in , the full f o r m a t i o n choice o f an H-type.
1
f(k~ ~") = - ~ . ( 0 , , -
00 . (V-
1 g(k~in) = - 5 " (oH - 0L)" ( V -
D) . ( 0 ~ . ( V -
D ) + OL . ( V + D)) < 0
D)" (0 H" (V+ D) + 0 L" ( V -
D)) < 0
Finally, g(k~Hin) < f(k~Lin), implying that g a n d f c r o s s to the left o f ~ i . . This guarantees existence o f a separating equilibrium. Start at k~sin. H e r e g(/~Hin) < 0. Iff(k~z in) > 0 we have f o u n d an equilibrium. If not, since b o t h functions are increasing a n d f(k) > g(k) for k/>/~Hin we can increase k until we get to a p o i n t where f(k) > 0 > g(k). To completely characterize the set o f equilibria, note that t h e r e are two subcases, f f f = g ~> 0, we get the case illustrated in Figure 1. H e r e t h e r e will be two separate intervals for which f(k) >~ 0 and g(k) ~< 0. All these levels o f k can be s u p p o r t e d as a Perfect Bayesian Equilibrium by o u r out-of-equilibrium beliefs. F o r f = g < 0 we get the case illustrated in Figure 2. H e r e t h e r e is only o n e interval o f separating equilibria.
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f(k),g(k) f(k)
g(k)
I / hbria~//
"Qj
V
k
FIG. 1. The case where f(k) and g(k) intersect for f = g > 0. PROOF OF COROLLARY.Iff(k~/) < 0, the first-best of the H-type is outside and to the left of the interval of separating levels of k and hence not a possible equilibrium. Using (6) in (13) 1
f(/~n) = ~ • ( 0 H - Or)" (20/_/. D -
(0H+ 0 ~ " V). V<0,
from which we get D t> - - . ( 0 H0r) + V (A1) 2 • OH When D is close enough to V it is too costly for the L-type to mimic the H-type. PaooF OF PROPOSrrloN 2. Comparing (15) and (12) it is clear that type H has no incentive to deviate from the pooling equilibrium. We then only have to check that type L does not want to deviate. T h e incentive compatibility constraint is now 1
(/C-L)
1
4" (g" V+ (01-1 - 0)" D). (0. V - (0u + 0 -- 20r). D) 1> 4 " 0~" Vz
Rearrange to get (~2 _ 0 ~ ) . V z + 0 . 0 L • D . V -
( ( 0 u -- 0)" (01-/+ 0 -- 20/_)). D 2 >i 0
(A2)
Clearly the higher D is, the harder is it to satisfy condition (A2). By looking at the case where D --~ V, we can obtain the following sufficient condition for (A2) to hold: ( 0 H - 0/_)X~ + 0LX - ( 0 u -
0D/> 0
Define )~* as the value of k E [0,1] for which (A3) hold with equality. ~*_
- 0 L + "~V/40~ - 80/_10L+ 50~ 2 ( 0 / 4 - 0/4)
(A3)
244
Equilibrium selection of contract terms f(k),g(k)
Y
k
FIG. 2. The case where f(k) and g(k) intersect for f = g < 0. Hence, a sufficient condition for (A2) to hold is that )~ ~ [•*, 1]. Note that as 0L goes to zero, )~* goes to one and as 0 L goes to OH, )~* goes to zero. Also, )~* is decreasing in 01.. Hence, the closer the two types are, the more likely the pooling equilibrium is to dominate the separating equilibrium. PROOF OF PROPOSITION3. For the first case, the first-best o f the H-type is in the set of separating equilibria. Any deviation to this level of care must have been carried out by type H, since by (11), the L-type has no incentive to mimic regardless of the inference. As the first-best is the preferred level of care, this is the only separating equilibrium that survives the application o f the intuitive criteiron. For the second case, type H c a n n o t get its first-best. It will then want to get the one that is closest to it but still in the interval of separating equilibria where (11) hold. Intuitively, it is the level of care closest to the first-best level o f k for an H-type, which an L-type would not want to copy. As/~L'n < k~Hin, the equilibrium value o f kH is given by the larger o f the two solutions to J(k) = 0. PROOF OF PROPOSITION4. To satisfy the intuitive criterion, there must not be a k such that if the buyer, after observing k, believes that the true type is H a n d , given the belief, only type H prefers k to the pooling equilibrium. Thus it must not be the case that OH • ~v/-k • V -
1_ k I> ~ ( 0 . V + (0 M - 0 ) . D) 2
(A4)
1
4 " (~" V+ (0/_/- 0). D). (0. V+ (0 + 0L - 0/4)" D) />
0/4. X / h . ( v - D) +
X/k" D - k
(AS)
For k = 0~ • V/4, (A4) clearly holds. Further, i f k is close e n o u g h to 1, (AS) will also hold, breaking the equilibrium.
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PROOF OF PROPOSITION 5. Fix V a n d D. For sufficiently similar types, the separating e q u i l i b r i u m will involve a level of care greater t h a n the first-best and, hence, a profit lower than the first-best. Further, from (A1), the closer is 0L to On, the greater is the bias. Contrast that with the p o o l i n g e q u i l i b r i u m profits given in (15). For h a p p r o a c h i n g 1, this approaches the first-best profits of the H-type. Hence, for h close, to 1, profits for the H-type are higher in the pooling equilibrium. T h e L-type is clearly better off in the p o o l i n g e q u i l i b r i u m by construction.
References AGHION,Ph. ANDHERMALIN,B. (1990). Legal restrictions on private contracts can enhance efficiency. Journal of Law, Economics and Organization. 6:381--409. AYRES,I., ANDGERTNER,R. (1989). Filling gaps in incomplete contracts: An economic theory of default rules. Yale Law Journal. 99:87-130. AYe.Es,I., ANDGERTNER,R. (1992). Strategic contractual choice and the optimal choice of legal rules. Yale Law Journal. 101:729-773. BAIRD,D.G., GERTNER,R.H. ANDPICKER,R.C. (1994). Game Theory and theLaw, Harvard University Press, Cambridge, Massachusetts. BEBCHUK,L.A., ANDSHAVELL,S. (1991). Information and the scope of liabilityfor breach of contract: The rule of Hadley v. Baxendale. The Journal of Law, Economics and Organization. 7:284-312. CHO,I.-K.,ANDSOBEL,J. (1990). Strategic stability and uniqueness in signalling games.Journal ofEconomic Theory. 50:381-413. COOTER,R., ANDULEN,T. (1988). Law and Economics, Harper Colins Publishers, U.S.A. FUDENBERG,D., ANDTmOLE,J. (1991). Game Theory,MIT Press, Cambridge, Massachusetts. GIBBONS,R. (1992). A Primer in Game Theory. Harvester/Wheatsheaf, Henffordshire, England. HERMAtaN,B.E., ANDKATZ,M.L. (1993).Judicial modification of contracts between sophisticated parties: A more complete view of incomplete contracts and their breach. Journal of Law, Economics, and Organization. 9:230-255. JOHNSTON,J.J. (1990). Strategic bargaining and the economic theory of contract default rules. YaleLaw Journal. 100:615-664. MAILATH,G.J., OKUNO-FUJIWARA,M. ANDPOSTLEWMTE,A. (1993). Belief-based refinements in signaling games. Journal of Economic Theory. 60:241-276. MILGROM, P., AND ROBERTS,J.(1992). Economics, Organization and Management, Prentice-Hall International, London, England. PERLOFF,J.M. (1981). Breach of contract and the foreseeability doctrine of Hadley v. Baxendale.Journal of Legal Studies. 10:39--63. RASMUSEN,E. (1994). Gamesand Information, 2d edition, Blackwell, Oxford, England. SCHWARTZ,A. (1990). The myth that promises prefer supracompensatory remedies: An analysis of contracting for damage measures. Yale Law Journal. 100:369-407. SHAPIRO,C. (1989). Theories of oligopoly behavior. In R. SCHMALENSEEand R.D. WILUG(Eds.), Handbook of Industrial Organization 1:329-414. TREmLCOCK,M.J. (1993). The Limits of Freedom of Contracts. Harvard University Press, Cambridge, Massachusetts.