%Paper: ewp-game/9506001
%From: Eric Rasmusen <erasmuse@rasmusen.bus.indiana.edu>
%Date: Wed, 14 Jun 95 13:37:26 -0500
%Date (revised): Wed, 14 Jun 95 13:57:27 -0500

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                     \begin{center}
            \begin{large}
    {\bf A Model of Negotiation, Not Bargaining }\\
             \end{large}
                     \vskip 15pt
                   May 9, 1995 \\
 

                    \bigskip
                    Eric Rasmusen \\
                    \vspace*{ 1in}
                    {\it Abstract} \\
\end{center}
  Bargaining models ask how a surplus is split between two parties in 
bilateral monopoly. Much of real-world negotiation involves 
complications to the original split which may or may not increase the 
welfare of both parties.  The parties must decide which complications 
to propose, how closely to  examine the other side's proposals, and 
when to accept them.   This type of negotiation raises welfare, 
rather than reducing it.  This paper models negotiation as a 
two-period  auditing game, and find a variety of plausible 
equilibria, some of which can be pareto-ranked.  Expectations are 
highly important, and precommitment can increase welfare 
substantially. 


 

          \vskip .3in
\begin{small}
          \noindent 

\hspace*{20pt}	  	  Indiana University
School of Business, Rm. 456, 

  10th Street  and Fee Lane,
  Bloomington, Indiana, 47405-1701.
  Office: (812) 855-9219.   Fax: 812-855-3354. Internet: 
Erasmuse@indiana.edu.\\ 

   %    Draft: 1.1 (Draft 1.1, xxx ).
 

\vskip 1in

  I would like to thank  Peter Cramton, Wolfgang Pesendorfer, J. Mark 
Ramseyer, and participants in seminars at  Indiana University, Kyushu 
University, and the 1994 Stanford meeting of the  American Law and 
Economics Association for helpful comments.  Not  all of those I 
thank     have  seen this draft, and they bear no responsibility for 
errors. 


            \end{small}

 \newpage

%------------------------------------------------------- \bigskip
\noindent
\begin{center}
{   1.  INTRODUCTION} 

 \end{center}

  Ordinary people often have  difficulty appreciating the idea of 
gains from trade in a  competitive market.  If one person sells and 
the other person buys, they think,  it must be that one of them  has 
gained a victory over the other; every transaction has a winner and a 
loser.  One of the central ideas in economics  is that this is  not 
true,   and in competitive markets, or even monopolized markets, one 
cannot really speak of a winner and a loser. Both      sides benefit. 


When it comes to bilateral monopoly or bargaining, however, the 
emphasis is different, even for professional economists. The 
standard paradigm is the splitting of a pie, whether the model uses 
cooperative game theory, as in Nash  (1950);   noncooperative game 
theory with complete information, as in Rubinstein (1982); or 
noncooperative game theory with incomplete information, as in the 
vast literature surved in Kennan \& Wilson (1993),  of which  Cramton 
(1984)  is a typical example. 

  Both bargainers will gain from the split, because if they fail to 
agree, the entire pie is lost, and in   this sense   there are gains 
from trade. Each person's bargaining effort, however,  is  devoted to 
trying to increase his own share   at the expense of the other, and 
in this  other sense   there is   a winner and loser. 

To the extent that agreement is not immediately reached,  such 
pie-splitting   is socially wasteful.     Bargaining is 
redistributive, not productive; a process of  rent-seeking,   not 
value creation.  The lesson for policy seems to be that institutions 
should be designed to eliminate bargaining. In this view,  salesmen, 
brokers, and   lawyers   may be useful to     initiate 
transactions,  but  time spent  on negotiations is time  wasted. 


    One contribution of the incomplete information models of 
bargaining is to show that bargaining is not quite so useless as 
this.  Without bargaining, it may be unclear whether the  pie 
actually exists: if the seller values an object more than the buyer, 
then the trade should not take place, and the bargaining process may 
be useful to reveal this. Even if it is common knowledge that the 
buyer values the object more than the seller, however,  costly 
bargaining will still take place and agreement will sometimes fail to 
be reached.\footnote{See Kennan \& Wilson, p. 67 for a numerical 
example of costly bargaining when both parties know there exist gains 
from trade.}


         Much dealmaking, however, does not fit into even  the 
extended paradigm of splitting a pie of uncertain size, because the 
process of coming to an agreement is   not   so much  about setting a 
price as about setting the terms of the  agreement.   This is 
especially true about deal-making's most time-consuming aspects, 
which are not  repetitions of   ``I want a higher price''    and ``I 
want a lower price,'' but  complicated suggestions and amendments, 
preceded and succeeded by careful investigation of  their 
implications.    I will call this process {\it negotiation}, in 
contrast to   {\it bargaining}.



 

  Example abound.   If I wish to build a house,   and I   negotiate 
with the contractor, we do not just talk about the price; and   our 
talk about the details of the contract is not  just redistributive. 
We  could eliminate a huge amount of negotiation if the government 
required that   the  house  be of a  single  standard design, but 
that would not be efficient.  I have  my  particular preferences 
about  the windows, woodwork, floor type,   color,  and time of 
completion, and  the  contractor has his own individual costs for 
each feature.   Much of my concern will not be about whether I can 
extract a good price at the expense of the contractor, but whether I 
am agreeing  to buy   features  of the house that  I really want. If 
I ask the contractor  to paint the house purple, and I find later I 
do not like a purple house, it is no consolation that  he  made no 
profit off the paint. 


 Labor contracting is dominated by negotiation. 

It may be that a  union and  an employer have agreed upon a wage, but 
that does not end the collective bargaining.  The employer may also, 
for example,  offer   an extension to the  health insurance benefits, 
in exchange for a wage concession. Possibly, the benefit to the union 
is greater than the lost wages, in which case the change   would 
benefit both sides. Or, maybe the  workers could do better by not 
accepting the new insurance. The union negotiator's uncertainty is 
not over the minimum offer that the company will accept, but over 
whether the health insurance  helps both sides. 


 Mergers and acquisitions are notoriously complicated deals. Suppose 
that company A  is selling off a division to  company B. They have 
agreed on a price, which  both of them would be willing to accept, 
but now company A asks that a clause be added to the deal under which 
it would   buy  back a certain amount of the output of the division 
each year at a specified price. The  clause might benefit company B, 
or   might hurt it.   Again, the uncertainty is not over the minimum 
that company A would accept, but over whether the contract benefits 
both sides or not. 

It may be quite clear that the clause benefits company A by exactly 
five   million 

dollars, but it may be unclear what the cost of the clause is to 
company B. 


 On a less grand level, suppose that Mr.  Smith is selling  a load of 
lumber to Mr. Jones. The deal has been made, but then Smith says, 
``Throw in an extra \$50.00 and I'll  deliver the lumber to your 
house.  It's no big deal for me, and you'll save a lot of effort.'' 
Jones's uncertainty is not over the cost of delivery to Smith, which 
is of no interest to him. Rather, it is over the cost to himself, 
which might be either the cost of his own effort or the cost of 
hiring a deliveryman.This  cost  might be \$80.00, in which case 
Smith is right that both parties would benefit from the proposal, or 
it might be \$30.00, in which case Smith gains but Jones loses, and, 
furthermore,   the proposal is inefficient. 


 All four of these examples involve   the conflict between  {\it 
creating value} and {\it claiming value} which is emphasized in the 
informal literature on bargaining (a survey of which can be found in 
Sebenius (1992)).    Bargaining theorists in economics     think of 
negotiation as   multi-attribute  bargaining:  the splitting of 
several pies simultaneously rather than the splitting of just one 
pie,  but essentially the same problem.    In   incomplete 
information models,  bargaining over several attributes is similar to 
bargaining over one attribute  because the strategic problem is the 
same:  When player $A$ proposes a contract modification to player 
$B$, how can $B$ determine the value of that modification to $A$? 
It may  indeed be that expectations and signalling make it quite 
significant that the surplus to be split is first separated into two 
pies instead of one, because,  as Bac \& Raff  (1994) show, a 
bargainer may convey information about himself by  which pie he 
concentrates his efforts upon. But the idea of creating value is 
different. 

 

  This difference has been recognized but  not   modelled. At the 
start of their survey of bargaining models,  John Kennan and Robert 
Wilson list   three costs of bargaining: 

\begin{quotation}
 \begin{small}
  ``Costly delays and failures to agree when gains from trade exist 
represent two kinds of inefficiencies; a third is that  an agreement 
is inefficient  if its terms fail to realize all the potential gains 
from trade, as in the case that a firm's contract with a union 
specifies inefficient work rules or numbers of 
workers.''\footnote{Kennan \& Wilson (1993, p. 46.}
 \end{small}
\end{quotation}
 Kennan and Wilson go on to survey the bargaining literature at 
length, but while costly delays and failures to agree receive ample 
attention in the next fifty-eight pages, failure to realize  all the 
potential gains from trade does not resurface.


 This neglect of negotiation can be explained, perhaps, because it 
requires a different style of model. 

    The approach I will use below is   not the standard bargaining 
approach of  cooperative game theory or of   incomplete information 
games. Instead, I   use  an auditing model,   in which information is 
complete  but  one player takes an action which the other player can 
either audit or let pass. The two players have completed their basic 
bargain, and are negotiating on possible extra clauses to add to the 
contract.   The extra clauses   might benefit both players, creating 
value, or   just  the proposer, claiming value under the pretence of 
creating value.  The other player must decide whether to trust the 
offer or  inspect it carefully.  It will be costly both to propose 
and to inspect clauses, so this will be a model of both  the {\it 
contract-writing costs} which are already   well-known,  and  of {\it 
contract-reading costs}, which are perhaps much larger. The strategic 
problem   is different now:   When player $A$ proposes a contract 
modification to player $B$, how can $B$ determine the value of that 
modification to $B$ himself? 


      The auditing literature is completely distinct from the 
bargaining literature.  One strand of the literature,  which began 
with Baiman \& Demski (1980) examine  the incentives for high effort 
by a worker whose income is observed and can be used as the basis for 
an audit investigation.   Another strand, which began with Townsend 
(1979),  investigates the mechanism design question of how a provider 
of capital can elicit truthful reports of the financial condition of 
the user of capital. Since the problem is one of mechanism design, 
the players can contract in advance as to  penalties for lying and 
bonuses for telling the truth.\footnote{Townsend (1979) does not 
consider the use of random auditing strategies, an important omission 
corrected by    Border  \& Sobel  (1987) and 

Mookherjee \& Png (1989).  }  Neither of these types of models is 
quite suitable for  modelling negotiation, since   there is no 
variable equivalent to income which could be used as the basis for an 
audit and there is usually  no advance contracting  about penalties 
for lying. 

 

         The closest     model to the one in this paper  is  Katz 
(1990). Katz  is not concerned with negotiation {\it per se}, but 
with the legal rules involving the fine print  in contracts.   The 
courts must decide how much of the fine print in contracts to 
enforce. If they enforce none of it, they must pecify how the 
contract binds the parties, since the  writing in the contract itself 
has been  abandoned. If they enforce all of it, the each    party 
tothe contract  must read its terms   carefully or abandon writing 
detailed contracts  in favor of   short but ambiguous ones which 
leave much to   legal default rules and the  courts. 
Contract-reading costs are important,  but  the legal rules should be 
designed to induce the parties to monitor what each inserts into the 
contract. 


   Section 2 of the  present paper will construct the model of 
negotiation without solving for the equilibrium.   Sections 3 and  4 
will find the equilibria in the simple  cases when precommitment is 
possible--- in Section 3, when  a player can precommit to  honesty; 
and in Section 4, when he can precommit to inspecting all clauses 
carefully. Section 5  contains the bulk of the analysis,  setting out 
the six possible equilibria and  showing why four of them are 
plausible. Section 6 summarizes the results and draws general 
lessons. 



 


 

\begin{center}
 {2. THE MODEL}
 \end{center}
 

Two parties, the Offerer and the Accepter,  are trying to agree  on 
the details of  a contract.  They have  already agreed to a basic 
contract, splitting a surplus 50-50.  This basic    contract will not 
affect the model, and, indeed, it  is included only to emphasize that 
some parts of the deal may not require negotiation.  Let us say that 
the  original   deal generates  a surplus of $2\overline{\pi}$, 
where $\overline{\pi}$ could equal zero, and let us assume that the 
surplus is split equally between the two players by the bargaining 
process.   We will denote the per-round  expected  payoffs of the 
subgame starting with the offer of a new clause  by $\pi_{1o}, 
\pi_{1a} ,  \pi_{2o}$, and $\pi_{2a}$ .

A sincere clause yields the Offerer  $x_s$ if accepted, and a 
misleading one yields him $x_m$. The  cost to him of making an offer 
is  $c_s>0$  or  $c_m>0$,  depending on the type of clause, where 
either of these  might be  the greater cost,   depending on whether 
it is more costly to discover and propose a mutually beneficial 
sincere clause or to disguise a one-sided clause so that it is 
misleading.  Assume that
 \begin{equation} \label{e00}
   x_m-c_m>x_s-c_s>0. 

 \end{equation}
 The first inequality in (\ref{e00}) says that
  the Offerer  would prefer   a misleading   clause to a sincere one, 
given the cost of offering them, if  both clauses had equal 
probabilities of acceptance. the second inequality says that either 
type of clause helps the Offerer, if accepted. 

 


The Accepter's benefit  is $y_s>0$ from a sincere clause  and 
$-y_m<0$  from a misleading one.    He cannot costlessly identify a 
clause that is offered. Instead, he   can accept  it outright, 
reject it, or inspect it at cost $c_i$ and discover  whether it is 
sincere or misleading. 



\pagebreak
 The order of play is thus
    \begin{enumerate}
\item[(0)]
 Offerer and Accepter split the surplus of $2\overline{\pi}$ equally.

 

\item[(1a)]  The first round of offers occurs. The Offerer offers a 
sincere  clause at cost $c_s $, or a misleading clause at cost 
$c_m$,  or ends the game  by making no offer at all. 


\item[(1b)]  The Accepter inspects the clause at cost $c_i$, 
discovering whether it is sincere or misleading, or does not inspect. 


\item[(1c)]  The Accepter accepts or rejects the clause. 


\item[(2a)]  The second  round of offers occurs.  The Offerer offers 
a    sincere  clause at cost $c_s $, or a misleading clause at cost 
$c_m$,  or ends the game by making no offer at all.

\item[(2b)]  The Accepter inspects the  second clause at cost $c_i$, 
discovering whether it is sincere or misleading, or does not inspect. 


\item[(2c)]  The Accepter accepts or rejects the  second clause. 




\item[(3)]  The contract is finalized, payoffs are received, and the 
effect of the clauses is discovered. 

   \end{enumerate}
 

The Offerer has the option of not offering any  clause at all, for a 
payoff of 0. 

 

Please note that this is a game of complete information. There do not 
exist different ``types'' of Offerers, and there is nothing for the 
Accepter  to learn about the Offerer during the game, although the 
Accepter can learn about the Offerer's choice of clause. Because of 
this , it would make very little difference if the two players 
switched roles in mid-game, so that one player was the Offerer in the 
first round and the other was the Offerer in the second round.   The 
properties of the game would be essentially unchanged; each of the 
possible  equilibrium outcomes described below would continue to 
exist. 


 

 \newpage

  \begin{center}
  {   3.  \\
 THE VALUE OF A REPUTATION FOR HONESTY: \\
 WHAT  IF THE OFFERER CAN PRECOMMIT?    }
 \end{center}

 What happens if the Offerer can precommit to offer only sincere 
clauses? The Accepter will then not bother to inspect, and will 
accept both clauses,  so  the   payoffs   in each round are 

 \begin{equation} \label{e0a}
 \pi_o   =    x_s-c_s 

 \end{equation}
 and 

\begin{equation} \label{e0b}
 \pi_a  =     y_s, 

 \end{equation}
 for a total social surplus over the two rounds of 

\begin{equation} \label{e0c}
   \pi_{1o}  +\pi_{1a}  +\pi_{2o}  +   \pi_{2a}  =   2(x_s  + y_s 
-c_s )  . 

 \end{equation}
  The social surpluses in this and the  scenarios below  are 
tabulated later in Table 1 for comparison.   The social surplus in 
this scenario is the highest possible, because reading contracts 
carefully is unnecessary and the sincere clauses are always offered.

   It is hard to see how someone could literally commit to offering 
only sincere clauses, but in some cases  plausible ways to  reach the 
same outcome are available. If the Offerer repeatedly    negotiates 
over time, with one or different Accepters, he may wish to preserve a 
reputation for sincerity. If he ever offered a misleading contract, 
it would be accepted, but if he tried to enforce it, he could lose 
his reputation.   Or, it might be possible to include in the contract 
a provision that if the Accepter thinks a clause is misleading, a 
court or an  impartial arbitrator with a reputation for honesty  can 
void the contract.  In practice, of course, it may be difficult for 
an outsider to determine whether a clause is misleading. 


 

\begin{center}
   {   4. \\
 THE VALUE OF A REPUTATION FOR READING CONTRACTS CAREFULLY: \\
 WHAT   IF THE ACCEPTER CAN PRECOMMIT?  }
 \end{center}

 What happens if the Accepter can precommit to inspect? The Offerer 
is then willing to offer two sincere clauses, and the   payoffs in 
each round are 

  \begin{equation} \label{e0d}
 \pi_o   =    x_s-c_s 

 \end{equation}
 and 

\begin{equation} \label{e0e}
 \pi_a  =     y_s - c_i, 

 \end{equation}
 for a total social surplus  over the two rounds of 

\begin{equation} \label{e0f}
  \pi_{1o}  +\pi_{1a}  +\pi_{2o}  +   \pi_{2a}  =  2(x_s  + y_s -c_s 
-c_i) . 

 \end{equation}
   The Accepter  is willing to precommit to inspection without any 
sort of side payment if   $y_s - c_i \geq 0$. 

 

    The Accepter may be able to do even better, however.\footnote{ I 
thank Wolfgang Pesendorfer for pointing this out to me. }
   Suppose  he can precommit to inspect with  auditing probability 
$\alpha$; e.g., he  precommits to inspect with  a probability of 90 
percent. 

This is cheaper in expected value than the probability of 100 percent 
used above, and might still deter the Offerer from attempting to 
sneak by a misleading clause. It will deter him if  $\alpha$ is 
chosen so that
  \begin{equation} \label{e0g}
 \pi_o (Sincere)   =    x_s-c_s  \geq  \pi_o (Misleading)   = 
(1-\alpha)x_m + \alpha (0) -c_m, 

 \end{equation}
 which requires that
   \begin{equation} \label{e0h}
\alpha \geq    \frac{(x_m -  c_m)-(x_s -  c_s)}{x_m}. 

 \end{equation}
 Assumption (\ref{e00}) ensures that the right-hand side of ({e0h}) 
is between zero and one. 

 If $\alpha$ is set at  $\alpha^*$, the cheapest level which makes 
inequality (\ref{e0h}) true, then
 the   payoffs in each round are 

  \begin{equation} \label{e0i}
 \pi_o   =    x_s-c_s 

 \end{equation}
 and 

\begin{equation} \label{e0j}
 \pi_a  =     y_s -  \left( \frac{x_s -c_s + c_m}{x_m} \right)c_i, 

 \end{equation}
 for a total social surplus  over the two rounds of 

\begin{equation} \label{e0k}
  \pi_{1o}  +\pi_{1a}  +\pi_{2o}  +   \pi_{2a}  =  2(x_s  + y_s -c_s 
-    \left( \frac{x_s -c_s + c_m}{x_m} \right)c_i  ) . 

 \end{equation}
 

The use of  a probability $\alpha$   does not imply that the 
equilibrium is in  mixed strategies; something quite different is 
going on here.   Precommitment to an auditing probability is distinct 
from a mixed strategy, because the Accepter must inspect with 
positive probability   even though he knows that in equilibrium the 
Offerer  will never offer a misleading clause.\footnote{ For more on 
the distinction, see Rasmusen (1994), pp. 81-83. }  Without 
precommitment, if    the Accepter announced he was following the 
strategy just described, it would not be an equilibrium. If the 
Offerer believed the announcement and only offered sincere clauses, 
the Accepter would change his mind and reset the  inspection 
probability to zero when the time came to pay the inspection cost. 

 


 Precommitment might take the form of paying for contract inspection 
in advance of the negotiation, by hiring an in-house lawyer, for 
example, and being careful to not have other uses for his time. An 
interesting alternative that would reach the same result would be for 
the Offerer to pay for the lawyer, bundling together the offer of a 
new clause and the inspecting of the clause.  ``He who pays the piper 
calls the tune,'' however, and such an arrangement might not be 
trusted by the Accepter. 


The usual substitute for precommitment, reputation, might not work. 
The reputation here would be a reputation for reading contracts 
carefully. If the Accepter can show Offerers that he does read 
contracts carefully, he can maintain such a reputation, but that may 
not be possible. Since only sincere clauses are offered in 
equilibrium,   neither the diligent contract-reading Accepter nor the 
deviant non-reading  Accepter  would ever find a misleading contract. 
The problem of verifying that the Accepter is following his 
equilibrium behavior becomes especially acute when that behavior is 
to inspect with probability $\alpha$ less than one; in each 
negotiation round, the Accepter could fail to inspect, and claim that 
his failure was  a matter of chance. 


 If the Accepter can somehow commit to inspect every offer, however, 
the social surplus is almost as high as  when the Offerer can commit 
to offering sincere clauses. The only difference is the cost of 
inspection. 

 


 \begin{center}
 5. THE EQUILIBRIA WITHOUT PRECOMMITMENT 

     \end{center}

     Let us now assume that neither player can precommit to any 
action.  The Offerer has free choice of clauses: he  can offer two 
misleading clauses, two sincere  ones, or one sincere and one 
misleading. 



  Any equilibrium must have the following properties: 

 \begin{enumerate}
 \item[(a)]
  The Offerer does not offer a misleading clause with probability 
one.  If he did, then the Accepter would never inspect or accept, and 
so the Offerer would incur cost $c_m$ for no benefit. 

   \item[(b)]
 The Offerer does not offer a sincere clause with probability one. If 
he did, the Accepter would never inspect the clause, and so the 
Offerer would prefer to deviate to offering a misleading clause to 
obtain $x_m$ instead of $x_s$. 

  \item[(c)]
 The Accepter does not have probability one of  accepting without 
inspection.    If he did, the Offerer would only offer misleading 
clauses. 

 \item[(d)]
 The Accepter does not have probability one of inspecting.  If he 
did, the Offerer would never offer misleading clauses, making the 
inspection pointless. 

   \end{enumerate}

 These properties still leave a multiplicity of equilibria, depending 
on the expectations of the two players and the parameters of the 
model. 

 

 

 \begin{quotation}
 {\it EQUILIBRIUM 1: Mixing    Each Round.} 

 \begin{enumerate}
 \item[ \underline{Offerer:} \hspace*{9pt}] In each round,  offer 
the sincere clause  with probability  $p_s$ and the  misleading 
clause with probability $1-p_s$. 

\item[ \underline{ Accepter:} ]  In each round, accept   without 
inspection  with probability $p_a$ and inspect  with probability 
$1-p_a$. 

 \end{enumerate}
  \end{quotation}



  Let the probability with which the Offerer offers the sincere 
clause be $p_s$ and the probability with which the Accepter accepts 
without inspection be $p_a$.  The one-round payoffs to the Offerer 
are
 \begin{equation} \label{e1}
 \pi_o(sincere) = -c_s + p_a x_s + (1-p_a)x_s
 \end{equation}
 and 

\begin{equation} \label{e2}
  \pi_o(misleading) = -c_m +p_a x_m + (1-p_a )\cdot 0,
 \end{equation}
    since the misleading clause will be rejected if the Accepter 
chooses to inspect.  If there is a mixed strategy equilibrium, the 
two pure-strategy payoffs must be equated,  so
\begin{equation} \label{e2a}
   -c_s + p_a x_s + (1-p_a)x_s
   = -c_m +p_a x_m, 

 \end{equation}
 and 

 \begin{equation} \label{e3}
 p_a^* =\frac{x_s - c_s + c_m}{x_m}.
 \end{equation}
 


 

The one-round payoffs of the Accepter are
 \begin{equation} \label{e5}
 \pi_a(accept) =  p_s y_s - (1-p_s )y_m
 \end{equation}
 and 

\begin{equation} \label{e6}
  \pi_a(inspect) = -c_i + p_s y_s - (1-p_s )\cdot 0
  \end{equation}
 since the misleading clause will be rejected if the Accepter 
inspects it. 

 If there is a mixed strategy equilibrium, the two pure-strategy 
payoffs must be equated, so 

 \begin{equation} \label{e5a}
    p_s y_s - (1-p_s )y_m
  = -c_i + p_s y_s, 

  \end{equation}
 and 

  \begin{equation} \label{e7}
 p_s^* =1-\frac{c_i}{y_m}.
 \end{equation}
   For the probabilities in  (\ref{e3}) and (\ref{e7}) to remain 
between zero and one requires that
 \begin{equation} \label{e9d}
    x_s   -c_s + c_m  \geq 0,
 \end{equation}
  which is guaranteed by assumption  (\ref{e00}), and 

  \begin{equation} \label{e9e}
   c_i \leq y_m. 

 \end{equation}
 If   assumption (\ref{e9e}) is false, Equilibrium 1 does not exist. 
If   $c_i > y_m $, the Accepter is not willing to inspect in moves 
(1b) and (2b), even though  such inspection is needed for the Offerer 
to be willing to offer a sincere clause. 


 In addition, the Accepter has the option of rejecting without 
inspection, for a payoff of 0. Comparing the payoff of 0 with the 
payoff in (\ref{e6}) from the pure strategy of inspecting ($-c_i + 
p_s y_s$), it is apparent that for  him  to  refrain from outright 
rejection requires that 

  \begin{equation} \label{e8}
   p_s y_s \geq c_i, 

 \end{equation}
 or, substituting for the equilibrium level of $p_s$ from (\ref{e7}), 

 \begin{equation} \label{e9}
     \left(  1- \frac{c_i}{y_m}   \right) y_s \geq c_i
 \end{equation}
 If condition (\ref{e9}) is false, then equilibrium 1 does not exist. 
Note that condition (\ref{e9}) requires that $c_i \leq y_s$. 
Inspection must be  cheap enough relative to the value of a sincere 
clause  that the Acceptor is willing to undertake the amount of 
inspection needed to give the Offeror incentive to sometimes offer 
the sincere clause. 


 The equilibrium payoffs for the entire subgame are twice the 
per-round payoffs using the equilibrium mixing probabilities, so they 
equal, from (\ref{e1}), 

 \begin{equation} \label{e9a}
 \pi_{1 o}^* + \pi_{2 o}^*    = 2(   x_s -c_s) 

 \end{equation}
 and, from (\ref{e6}) and (\ref{e7}), 

 \begin{equation} \label{e9b}
 \pi_{1 a}^* + \pi_{2 a}^*  =   2(y_s - c_i  \left(1 + \frac{ 
y_s}{y_m} \right)  ), 

 \end{equation}
 for a total social surplus of 

\begin{equation} \label{e9c}
  \pi_{1 o}^* + \pi_{2 o}^*+ \pi_{1 a}^* + \pi_{2 a}^* = 2(  x_s  + 
y_s -c_s -c_i - \frac{c_iy_s}{y_m}). 

 \end{equation}

   The surplus in (\ref{e9c})  is less than the surplus when the 
Accepter can precommit to inspecting every offer. 


 

 

 \begin{quotation}
{\it EQUILIBRIUM  2: No Offers.} 

  \begin{enumerate}
 \item[ \underline{Offerer:} \hspace*{9pt}]    Do not offer either 
clause.
  \item[ \underline{ Accepter:} ] Reject any clause that is offered.
 \item[ \underline{   Out-of-equilibrium belief:}] If the Offerer 
deviates and offers a clause,  the Accepter believes it   is 
sincere    with  a probability  $\beta$ of  no more than  $ Max 
\{\frac{y_s}{y_s + y_m}, \frac{c_i}{y_s} \} $. 

  \end{enumerate}
 \end{quotation}

 Equilibrium 2 is an equilibrium because the Offerer has no incentive 
to offer clauses if the Accepter always rejects, and the Accepter has 
no incentive to inspect or accept given his beliefs. This dilemma is 
similar to the situation in some signalling and coordination games, 
but here unlike in those games (e.g., Cho \& Kreps (1987)  Van Damme 
(1989)), the intuitive criterion and forward induction have no 
bite.\footnote{One change to the model which could make a big 
difference is incomplete information. If even a small number of 
honest Offerers will never make misleading offers (i.e., $c_m=\infty$ 
for them), then Equilibrium 2  breaks down, because an offer observed 
out-of-equilibrium would have to be a sincere clause offered by one 
of these honest Offerers. Incomplete information, however, brings in 
other complications; if we also added a number of dishonest Offerers 
for whom $c_m=0$, then the out-of-equilibrium offer might come from 
one of them, and Equilibrium 2 is revived.} 




 The out-of-equilibrium beliefs needed to sustain Equilibrium 2 are 
obtained as follows. 

 If $\beta$ is the  Accepter's subjective probability that the 
Offerer's out-of-equilibrium offer is of a sincere clause, then the 
Accepter's single-round subgame payoff is 

   \begin{equation} \label{e10}
    \pi_a (accept) = \beta  y_s  - (1-\beta) y_m, 

 \end{equation}
 which  yields us $\beta = \frac{y_s}{y_s + y_m}$ for the value of 
$\beta$ which makes the Accepter prefer to reject and receive the 
payoff of 0. 


It must also be true that the Accepter does not think it worthwhile 
to inspect. The payoff from inspecting, given the belief $\beta$, is 

 \begin{equation} \label{e11}
    \pi_a (inspect) = \beta  y_s  - (1-\beta)(0) - c_i, 

 \end{equation}
 which gives us $\beta =\frac{c_i}{y_s} $ for the value of $\beta$ 
which makes the  Accepter prefer to reject and receive the payoff of 
0. 


 

 The equilibrium payoffs are zero in each round in Equilibrium 2. 
Existence of Equilibrium 2, unlike existence of Equilibrium 1, 
requires  no special assumptions    beyond those made in setting up 
the model in Section 2. 



 


 \begin{quotation}
{\it EQUILIBRIUM  3: Just One Offer.}
 \begin{enumerate}
 \item[ \underline{Offerer:} \hspace*{9pt}] Offer a sincere clause in 
the first round with probability  $p_s^*= 1-\frac{c_i}{y_m}.$ Do not 
offer a second clause. 

    \item[ \underline{ Accepter:} ]  Accept the  first clause with 
probability $p_a^*=\frac{x_s -c_s + c_m}{x_m}.$ Reject the second 
clause if it is offered.
       \item[ \underline{   Out-of-equilibrium belief:}] If the 
Offerer offers a second clause, the Accepter believes it    is 
sincere  with  a probability  $\beta$ of  no more than  $ Max 
\{\frac{y_s}{y_s + y_m}, \frac{c_i}{y_s} \} $. 

   \end{enumerate}
  \end{quotation}

 Equilibrium 3 is a combination of Equilibria 1 and 2, with different 
behavior in each round. Equilibrium payoffs are half those in 
Equilibrium  1.  Assumptions (\ref{e9}) and (\ref{e9e}) must be true 
for Equilibrium 3 to exist, for the same reasons as in Equilibrium 1. 

 


 


 \begin{quotation}
{\it EQUILIBRIUM  4:  Customary Delay.}
  \begin{enumerate}
 \item[ \underline{Offerer:} \hspace*{9pt}]  Offer a sincere  clause 
in the first round with probability of no more than  $Max 
\{\frac{y_s}{y_s + y_m}, \frac{c_i}{y_s} \} $, and otherwise offer a 
misleading clause.  Offer a sincere clause in the second round with 
probability $p_s^*= 1-\frac{c_i}{y_m},$  and otherwise offer a 
misleading clause. 

  \item[ \underline{ Accepter:} ]  Reject the first clause offered. 
Accept the second clause with probability $p_a^*=\frac{x_s-c_s + 
c_m}{x_m}.$
  \end{enumerate}
\end{quotation}

 Equilibrium 3 is also a combination of Equilibria 1 and 2, with 
different behavior in each round, but here the equilibrium payoffs 
are  less than half those in Equilibrium  1, because a wasteful offer 
must be made in Round 1. 

 The Offerer is willing to incur the cost of offering a  first-round 
clause which is sure to be rejected because  by doing so he prolongs 
the negotiations until the second round, when his offer may be 
accepted. 

 


   Equilibrium  subgame payoffs are 

\begin{equation} \label{e12}
   \pi_{1 o}^* + \pi_{2 o}^*   =  -c_m +      (x_s -c_s) 

 \end{equation}
and,  taking half of Equilibrium 1's payoff,  (\ref{e9b}), 

  \begin{equation} \label{e13}
   \pi_{1 a}^* + \pi_{2 a}^*   =     y_s - c_i -  \frac{c_i 
y_s}{y_m}, 

 \end{equation}
 for  a total of 

\begin{equation} \label{e14}
   \pi_{1 o}^* + \pi_{2 o}^* +   \pi_{1 a}^* + \pi_{2 a}^*    = 
x_s + y_s     -c_s    -c_m  -   c_i -  \frac{c_i y_s}{y_m}. 

 \end{equation}

Assumptions (\ref{e9}) and (\ref{e9e}) must be true for  Equilibrium 
4  to exist,  for the same reasons as in Equilibrium 1. Since    the 
Offerer must incur the contract-writing cost   twice, while only one 
clause is possibly accepted, for  Equilibrium 4  to exist requires 
that   expression (\ref{e12}) be positive, a stronger condition than 
assumed so far. 


Existence of  Equilibrium 4 requires one other assumption: that 

  $c_s  \geq c_m$. If this is false, then  the  Offerer would deviate 
to offer the cheaper sincere clause in Round 1 with probability 1, 
which condition (b) showed is impossible in any equilibrium. 


 




 


 \begin{quotation}
 {\it EQUILIBRIUM  5: History Dependence.} 

 \begin{enumerate}
  \item[ \underline{Offerer:} \hspace*{9pt}]   Offer a sincere clause 
in the first round with probability  $p_{1s}^*= \frac{y_m - 
\left[y_s -c_i -\frac{c_i y_s}{y_m}\right]}{y_m -  \left[y_s -c_i 
-\frac{c_i y_s}{y_m}\right] - c_i}$.   If  the first clause is 
accepted, offer a sincere clause in the second round with probability 
$p_{2s}^*= 1-\frac{c_i}{y_m}.$ If the   first clause is rejected,  do 
not offer a clause in the second round. 

  \item[ \underline{ Accepter:} ]  Accept the  first clause with 
probability     $   p_{1a}^* = \frac{ (x_s -c_s  + c_m)  + (x_s -c_s) 
}{  (x_s -c_s  + x_m)}$.   If the first clause was inspected and 
found to be misleading,   reject   any  second-round offer.    If the 
first clause was not inspected  or not found  to be misleading , 
accept the  second clause with probability     $   p_{2a}^* = 
\frac{x_s-c_s + c_m}{x_m}.$ 

  \item[ \underline{   Out-of-equilibrium belief:}] If the Offerer 
offers a second clause after the first was rejected,  the Accepter 
believes  the second clause   is sincere  with  a probability 
$\beta$ of  no more than  $ Max \{\frac{y_s}{y_s + y_m}, 
\frac{c_i}{y_s} \} $. 

  \end{enumerate}
  \end{quotation}


  Here, the two rounds are connected in a more complicated way.  If 
the game reaches the second round without it being revealed that the 
first clause was misleading, the Offerer's second-round expected 
payoff will be  $x_s -c_s$, as in Equilibrium 1.   If the game 
reaches the second round after  it is  revealed that the first clause 
was misleading, the Offerer's second-round expected payoff will be 0, 
because the Accepter will stop listening to him.     Hence, the 
overall subgame payoffs  from  the Offerer's  two   first-round 
actions  are: 

       \begin{equation} \label{e15}
  ( \pi_{1 o}^* + \pi_{2 o}^* )|(Sincere \; in \; the\; first \; 
round)  =    -c_s  +  x_s  +   (x_s -c_s) 

 \end{equation}
 and 

 \begin{equation} \label{e16}
  ( \pi_{1 o}^* + \pi_{2 o}^* )|(Misleading \; in \; the\; first \; 
round)  =    -c_m   + p_a [x_m+(x_s -c_s)]  +   [1-p_a][0]. 

 \end{equation}
 Equating (\ref{e15}) and (\ref{e16})  to solve for the 
mixed-strategy equilibrium yields
  \begin{equation} \label{e17}
       -c_s  +  x_s  +   (x_s -c_s) 

   =    -c_m   + p_a [x_m+(x_s -c_s)] , 

 \end{equation}
 so 

\begin{equation} \label{e18}
    p_{1a}^* =1- \frac{  (x_s -c_s) }{  (x_s -c_s  + x_m)}. 

  \end{equation}
 This is greater than the $p_a^*$ found in Equilibrium 1--- there is 
a greater likelihood that the  Accepter will accept the first clause. 
Because the Acceptor  would break  off negotiations  before  the 
second round after discovering that the first offer was misleading, 
it is now   less tempting for the    Offerer  to offer a misleading 
clause. 


 If the game reaches the second round without it being revealed that 
the first clause was misleading, the Accepter's second-round expected 
payoff will be  $ y_s -c_i -\frac{c_i y_s}{y_m}$, as in Equilibrium 
1.    The overall subgame payoffs   from   the Accepter's  two 
first-round actions  are:
     \begin{equation} \label{e19}
  ( \pi_{1 a}^* + \pi_{2 a}^* )|(Accept \; in \; the\; first \; 
round)  =      p_s   y_s    - (1-p_s) y_m  +  \left[y_s -c_i 
-\frac{c_i y_s}{y_m}\right]
 \end{equation}
 and 

 \begin{equation} \label{e20}
  ( \pi_{1  a}^* + \pi_{2 a}^* )|(Inspect \; in \; the\; first \; 
round)  = -c_i +   p_s  \left(y_s +  \left[y_s -c_i -\frac{c_i 
y_s}{y_m}\right]\right)  - (1-p_s)(0 + 0)
  \end{equation}
 Equating (\ref{e19}) and (\ref{e20})  to solve for the 
mixed-strategy equilibrium yields
  \begin{equation} \label{e21}
            p_s   y_s    - (1-p_s) y_m  +  \left[y_s -c_i -\frac{c_i 
y_s}{y_m}\right] = -c_i +   p_s  \left(y_s +  \left[y_s -c_i 
-\frac{c_i y_s}{y_m}\right]\right) 

 \end{equation}
so 

 \begin{equation} \label{e22}
   p_{1s}^* = 1- \frac{    c_i}   {y_m -  \left[y_s -c_i -\frac{c_i 
y_s}{y_m}\right]  }
    \end{equation}
 This is smaller than the value of $p_s^*$ in Equilibrium 1, which 
was $1 - \frac{c_i}{y_m}$; in Equilibrium 5 the sincere clause is 
offered less often in the first round.   The reason is that the 
Accepter would have to break off negotiations after discovering a 
misleading clause, so inspection is not so tempting an alternative to 
the Accepter as in Equilibrium 1. 

 

  The payoffs in Equilibrium 5 will be 

\begin{equation} \label{e22a}
   \pi_{1 o}^* + \pi_{2 o}^*   =  2    (x_s -c_s) 

 \end{equation}
and 

  \begin{equation} \label{e22b}
   \pi_{1 a}^* + \pi_{2 a}^*   =      (p_{1s}^*y_s - c_i) +  p_{1s}^* 
\left(  y_s -c_i -   \frac{c_i y_s}{y_m} \right).
  \end{equation}
 This payoff is lower than the payoff in Equilibrium 1 because (a) 
$p_{1s}^*$ is lower in Equilibrium 5, so the first-round payoff of 
$p_sy_s -c_i$ is lower, and (b) the second-round payoff is multiplied 
by $p_s <1$ in Equilibrium 5. 

 The total is
\begin{equation} \label{e22c}
   \pi_{1 o}^* + \pi_{2 o}^* +   \pi_{1 a}^* + \pi_{2 a}^*    =   2 
(x_s -c_s)  + (p_{1s}^* y_s - c_i) +  p_{1s}^* \left(  y_s -c_i - 
\frac{c_i y_s}{y_m} \right). 

\end{equation}
    Since the payoff to the Offeror is the same as in Equilibrium 1, 
and the payoff to the Accepter is lower, Equilibrium 5 has a lower 
social surplus. 


\newpage
 \begin{quotation}
{\it EQUILIBRIUM 6: Peculiar History Dependence.} 

  \begin{enumerate}
     \item[ \underline{Offerer:} \hspace*{9pt}] Offer a sincere 
clause in the first round with probability  $p_{1s}^*=   \frac{(y_m - 
\left[y_s -c_i -\frac{c_i y_s}{y_m}\right] ) -  \left[y_s -c_i 
-\frac{c_i y_s}{y_m}\right]    - c_i}{   y_m -  \left[y_s -c_i 
-\frac{c_i y_s}{y_m}\right]}
$.   If  the first clause is accepted, offer a sincere clause in the 
second round with probability   $p_{2s}^*= 1-\frac{c_i}{y_m}.$  If 
the   first clause is  inspected and found to be sincere,  do not 
offer a clause in the second round. 

   \item[ \underline{ Accepter:} ]   Accept the  first clause with 
probability     $    p_{1a}^* = \frac{c_m}{ x_m -x_s + c_s}   . $. 
If the first clause was inspected and found to be sincere,   reject 
any  second-round offer.    If the first clause was not inspected  or 
not found  to be sincere,  accept the  second clause with probability 
$   p_{2a}^* = \frac{x_s-c_s + c_m}{x_m}.$ 

   \item[ \underline{   Out-of-equilibrium belief:}] If the Offerer 
offers a second clause after the first was found to be sincere,  the 
Accepter believes   the second clause   is sincere  with  a 
probability  $\beta$ of  no more than  $ Max \{\frac{y_s}{y_s + y_m}, 
\frac{c_i}{y_s} \} $. 

   \end{enumerate}
  \end{quotation}

 The analysis is parallel to that of Equilibrium 5. 

  If the game reaches the second round without   the first clause 
having been revealed to be sincere, the Offerer's second-round 
expected payoff will be  $x_s -c_s$, as in Equilibrium 1.   If the 
game reaches the second round after    the first clause has been 
revealed to be  sincere, the Offerer's second-round expected payoff 
will be 0, because the Accepter will stop listening to him. 
Hence, the overall subgame payoffs  from  the Offerer's  two 
first-round actions  are: 

       \begin{equation} \label{e24}
  ( \pi_{1 o}^* + \pi_{2 o}^* )|(Sincere \; in \; the\; first \; 
round)  =    -c_s  +  x_s  +   p_a(x_s -c_s)  + (1-p_a)  (0) 

 \end{equation}
 and 

 \begin{equation} \label{e25}
  ( \pi_{1 o}^* + \pi_{2 o}^* )|(Misleading \; in \; the\; first \; 
round)  =    -c_m   +  p_a [x_m+(x_s -c_s)]  +   [1-p_a][ 0 +  (x_s 
-c_s)   ]. 

 \end{equation}
 Equating (\ref{e24}) and (\ref{e25})  to solve for the 
mixed-strategy equilibrium yields
 \begin{equation} \label{e27}
    p_{1a}^* = \frac{c_m}{ x_m -x_s + c_s}   . 

  \end{equation}
 


 If the game reaches the second round without   the first clause 
having been revealed to be sincere, the Accepter's second-round 
expected payoff will be  $ y_s -c_i -\frac{c_i y_s}{y_m}$, as in 
Equilibrium 1.    The overall subgame payoffs   from   the Accepter's 
two   first-round actions  are:
     \begin{equation} \label{e28}
  ( \pi_{1 a}^* + \pi_{2 a}^* )|(Accept \; in \; the\; first \; 
round)  =      p_s   y_s    - (1-p_s) y_m  +  \left[y_s -c_i 
-\frac{c_i y_s}{y_m}\right]
 \end{equation}
 and 

 \begin{equation} \label{e29}
  ( \pi_{1  a}^* + \pi_{2 a}^* )|(Inspect \; in \; the\; first \; 
round)  = -c_i +   p_s   (y_s +  0 ) - (1-p_s)(0 +  \left[y_s -c_i 
-\frac{c_i y_s}{y_m}\right] ).
  \end{equation}
 Equating (\ref{e28}) and (\ref{e29})  to solve for the 
mixed-strategy equilibrium yields
  \begin{equation} \label{e31} 

   p_{1s}^* = 1-  \frac{  \left[y_s -c_i -\frac{c_i y_s}{y_m}\right] 
+ c_i}{   y_m -  \left[y_s -c_i -\frac{c_i y_s}{y_m}\right]}
     \end{equation}
    This is an even lower probability of a sincere offer than in 
Equilibrium 5.
   The payoffs in  Equilibrium 6 can be easily computed, but since 
they are lengthy expressions of no great interest they will not be 
presented here. 

 


\begin{center}
   6.  INTERPRETATION
     \end{center}


    This simple model of negotiation generates  a surprising number 
of plausible outcomes  supported by simple and reasonable beliefs 
which are listed in Table 1. 



     In Equilibrium 1, each player is uncertain as to what will 
happen, and each takes chances. The Offerer takes a chance that the 
Accepter may inspect, and the Accepter takes  a chance that the 
Offerer might have offered a misleading clause. 

  In Equilibrium 2, the players are   pessimistic. The Accepter does 
not trust the Offerer, and so the Offerer does not attempt to offer 
extra clauses. 

  In Equilibrium 3, it is customary for  the Offerer to  offer one 
good clause, perhaps, but certainly not two. 

 In Equilibrium 4, it is customary for  the Offerer to  start with a 
misleading offer,  before getting serious with his second  offer. The 
first offer would appear ritualistic, since everybody knows it will 
be refused, but expectations make it a necessary part of the process. 


  All four equilibria seem realistic in different contexts, and, as 

 Table 1 shows,  they can, together with the precommitment 
equilibria,  be pareto-ranked.  The best outcome is when the Offerer 
can be trusted to always offer only sincere clauses,  and the next 
best is when the Accepter can be trusted to always inspect the 
clauses offered to him. These are placed above the line in Table 1 
because they are really separate games from the equilibria of Section 
6, in which precommitment is impossible. Of the four plausible 
equilibria in Section 6, the best involved mixed strategies in each 
round but no grudges following the discovery of attempted deception. 
Next best is  when one offer is made, in the first round, which is 
better than when one possibly sincere offer is made, but only in the 
second round. Worst of all is when  an atmosphere of mistrust 
prevents the Accepter from bothering to even inspect any clauses that 
might be offered to him, and so none are. 


 

 \begin{center}
{ TABLE 1:     THE SOCIAL SURPLUS  } 

 

 \begin{tabular}{ll}
\hline
\hline
 EQUILIBRIUM  & TOTAL SURPLUS \\
  Honest Offerer & $2(x_s  + y_s -c_s )$  \\
 Careful Accepter & $2(x_s  + y_s -c_s - \alpha^*c_i)$   \\
 \hline
 1:  Mixing Each Round& $2(  x_s  + y_s -c_s -c_i - 
\frac{c_iy_s}{y_m})$ \\
 2. No Offers & 0 \\
 3. Just One Offer &  $x_s  + y_s -c_s -c_i - \frac{c_iy_s}{y_m}$ \\
 4. Customary Delay &$    x_s + y_s     -c_s    -   c_i -  \frac{c_i 
y_s}{y_m} -c_m$  \\
 \hline
\end{tabular}
\end{center}

 Equilibria 5 and 6  are excluded from Table 1 because  the  beliefs 
that support them are  rather  baroque,  like sunspot equilibria  but 
triggered by a public event involving the players' actions. 
Equilibrium 5 is plausible on its face, and seems to represent the 
situation where  the Accepter becomes offended on discovering a 
misleading clause and refuses to negotiate further. That  intuition, 
however, should really be the result of incomplete information, 
because it relies on the Accepter learning something about the 
Offerer's type, which does not happen in this model.  The 
arbitrariness of Equilibrium 5 is shown by the existence of its twin, 
Equilibrium 6. In Equilibrium 6, it is as if the Accepter becomes 
offended on discovering that the clause is sincere, and refuses to 
negotiate further. This is absurd, but no more arbitrary than the 
belief in Equilibrium 5. 


 

     From the analysis, a number of lessons can be drawn. 


{\it 1. Contract-reading costs matter as much as contract-writing 
costs. }    Although the  contract-writing costs $c_s$ and $c_m$ have 
an influence in this model,  it would not be fundamentally changed if 
they were set to zero.  As Table 1 shows, these costs are simply 
subtracted from the social surplus like any simple transactions cost. 
The contract-reading cost $c_i$ is much more important. In Table 1, 
it  has an indirect effect, via the subtraction of $\frac{c_i 
y_s}{y_m}$, as well as a direct effect. This effect is still 
continuous in $c_i$, so if contract-reading costs are small, it might 
seem that their effect on welfare is also low, but $c_i$ has a second 
impact:  it permits a variety of equilibria to exist. The 
contract-reading cost, in combination with unfortunate  expectations, 
can lead to Equilibria 2,3, or 4, in which  the benefit from at least 
one round of offers is lost. Because of this, the ultimate effect of 
a contract-reading cost of $c_i$ can be much greater than $c_i$; in 
the extreme case of Equilibrium 2, a small contract-reading cost can 
destroy the entire gains from trade. 


  The existence of contract-reading costs is, moreover, both 
realistic and hard to eliminate institutionally. 

   It is relatively easy to write  fifty new pages  for  a contract 
to provide for  extra contingencies, but it is quite difficult for 
the reader to be sure what those fifty pages contain. Standard, 
``boilerplate'' contracts are a good solution to the problem of 
contract-writing costs, but not to the problem of contract-reading 
costs. Boilerplate is easy to propose, but the accepting side of the 
contract still finds it difficult to know what the boilerplate 
contains. 


 

{\it 2.  Legal default rules, or even mandatory rules, can   overcome 
contract-reading costs. }  Although boilerplate contracts are no 
solution, because  unless they can somehow be guaranteed to be pure 
boilerplate with no additions, court or government-determined default 
rules can be a solution.  If such rules exist, the two parties can 
refrain from writing anything, leaving the binding clause to be the 
default clause determined by an unbiased third party. The default 
rule is  important  for much more than just saving the costs $c_i$ 
and $c_s$; even if the contract-reading  and contract-writing costs 
are small, the  default rule is important.\footnote{This result has 
previously been found in    models of strategic contracting, but 
under incomplete information--- see Ayres \& Gertner (1989, 1992) and 
Johnston (1990). } 


 One can go further and use this model to argue for mandatory 
contract rules, which override  whatever may be written in the 
contract. If it is practical for a court to determine that a clause 
is misleading, the best solution is for courts to refuse to enforce 
such clauses.  Modern courts try to do this, refusing to enforce 
suspicious fine printm, and that is the subject of the Katz (1990) 
article discussed in the Introduction. 


The disadvantage of  legal default and mandatory rules is, of course, 
that they reduce  the flexibility of the contract.  If different 
``sincere clauses'' are appropriate for  different contracts,  it is 
difficult for courts to choose which clause should be used. 


 


 


 {\it 3.  A reputation for honesty in negotiations is a valuable 
asset.}    The contracting parties are best off if the Offerer is 
inescapably honest. His honesty does not eliminate the 
contract-writing cost, but it does eliminate the need to inspect the 
contract and allows efficient clauses to be added. Any player who has 
established a reputation for honesty will  in this context, as in 
others, be an attractive business partner and will be offered  more 
attractive contracts. 


{\it 4. Negotiation increases social welfare, even if  it  is 
costly.}   Even if the parties cannot trust each other, negotiation 
still increases welfare. The parties have the option to refuse to 
negotiate, and if they do so, it is  in the hopes of creating 
surplus.  Lengthy dealmaking sessions are not necessarily 
inefficient: to the extent  that they add mutually beneficial details 
to the deal, they are efficient. 


{\it 5.   A mistrustful attitude in negotiations can be 
self-enforcing.}    If a model has multiple equilibria, that means 
that expectations are important to the outcome. If the Accepter 
expects the Offerer to offer only misleading clauses, he will not 
bother to inspect any clauses that are offered, and so the Offerer 
will offer none.   This is the worst case from the point of view of 
both parties, and changing the expectations --- however that might be 
done--- is  an important prerequisite to negotiations. 


{\it 6.   Frivolous offers can be part of rational, if inefficient, 
negotiations.}  If  it is expected that the first clause offered will 
be misleading, that too can be self-enforcing, if not quite so 
harmful.  The first clause offered will be dismissed without serious 
consideration, but it is still rational to offer it, because 
otherwise the negotiations cannot proceed to the second, serious 
round. 

 

{\it 7.   It is better to deal with someone who is on guard against 
you:  good fences make good neighbors.     }    Even if the Offerer 
cannot be trusted to be honest  in the absence of external 
influences, if the Accepter can commit to always inspecting the 
offers, welfare is almost as high. Somewhat counterintuitively, the 
Offerer actually prefers to deal with an Accepter who always inspects 
the clauses.  The Offerer need not worry about unfortunate 
expectations which would cause the Accepter to dismiss his offers 
without serious consideration. 

 


 {\it 8.  Corporate lawyers are worth their salary even if they never 
discover flaws in contracts.}  An immediate implication of lesson (6) 
is that  inspection is valuable even if it never reveals anything 
untoward.  The corporate counsel's veto of a contract term is like 
the atomic bomb, most useful when not used.   The purpose of the 
legal staff  is to deter the other side from trying to be sly or 
dishonest, and if the staff's lawyers are well enough respected, they 
will never   discover any dishonesty. 


   The model may also have something to say about two other subjects: 
contractual default rules and product quality. 


   The issue of contractual default rules arises when a contract 
fails to specify certain  details of the agreement or when it is 
claimed that the part of the contract which does specify them is 
invalid for one reason or another. 

 The 1990 article of A. Katz mentioned   in the introduction 
examined the question of when  a contractual party has  duty to read 
the fine print provided by the other party and came to no definite 
conclusion except that circumstances exist when  such a duty should 
not exist or when mandatory terms should replace the contractual 
terms.  In another information-based model,  Spier (1992) shows that 
contractual terms can be used by  one of the parties to signal his 
hidden characteristics.  Inefficient equilibria are commonly found in 
signalling games,   and such remains the case when the signal is  a 
contractual term or its absence. If, for example, some companies are 
of a type that will carry out an agreement and some are not, a 
company of the desirable type might choose not to offer a clause 
specifying damages  for breach,  even though such a clause might 
otherwise be desirable. Or, it might be that the necessary signal is 
to offer to strike such a clause from a standard-form contract, even 
though it is  an efficient clause. 

Thus, it might be efficient for a default rule to specify damages for 
breach, or even to have a mandatory rule that the parties cannot 
waive. 

 

 The present model of negotiation also has multiple equilibria, which 
can be pareto-ranked. In the  worst equilibrium (equilibrium 2), no 
offers are made. It thus becomes quite important what the default 
rules are. They will be preferred precisely because they are default 
rules, rather than rules tailored by the offerer to the particular 
situation. 

The negotiation model is not a signalling model. There is only one 
type of offerer, and the accepter's inferior information is only 
about the clause offered, not about the offerer.   This means that 
unlike in the models of Katz or Spier, no argument can be made for 
mandatory contract rules, only for default rules.  At the same time, 
since  the  combination of contract-reading costs and pessimistic 
expectations can create high transactions costs, the default rules 
can be very important.\footnote{For a  somewhat contrary view, see A. 
Schwartz  (1993), the  main point of which is that contracts cover so 
many different situations that no single  set of sensible  default 
rules can be found, and that a better approach is for defaults to 
established in specialized areas of the law, e.g. computer law or 
securities law.} 

 

       Product quality is an area of economic analysis to which the 
negotiation model could also be applied. In the classic paper of 

 Klein  \&   Leffler (1981), the basic problem is how a seller can, 
in the absence of warranties, persuade the buyer that the product is 
of high quality. 

Their solution   is for the seller to charge a premium over cost, 
which creates a stream of profits over time that the seller would be 
unwilling to jeopardize by deviating to low quality for one period of 
extraordinary profit.   The negotiation model could be adapted to a 
different set of circumstances, where  repeat purchases are not made 
but the buyer can inspect the product at    low cost.  The model 
suggests that expectations will determine the outcome, and that the 
equilibrium may be in mixed strategies or involve some rejection of 
inferior products before good products are produced. Product quality, 
however, is perhaps not so serious a problem as contract terms, 
because if the quality were the only problem, warranties could be 
provided. If it is the warranty itself, or some other contractual 
term, whose quality is in question because of what the fine print may 
say, the warranty solution  fails. Thus, the model's main 
contribution to the product quality literature may be to suggest why 
warranties will not work. 

 

 


      The present  model  is only a first step in the analysis of 
negotiation, and it  is simple enough that it can be extended in a 
number of directions. I will comment briefly on these extensions and 
speculate as to what they might show. 


 A first extension is  to  more than two rounds of offers. 

   It should be clear that the  two-round model can easily be 
extended to many rounds, and that the same sorts of equilibria would 
continue to exist in the $N$-round game.  The same mixing 
probabilities could be used in each round (equilibrium 1), there 
might be no offers made (equilibrium 2), only a limited numbers of 
offers might be made (equilibrium 3), no offers might be accepted 
until late in the game (equilibrium 4), or strange patterns of offers 
depending on history might comprise an equilibrium (equilibria 5 and 
6). The results seem robust in this direction,  and this extension is 
unlikely to yield anything  by itself, though  it might yield more 
when combined  with   incomplete information. 


 A second  extension is  to repetition of the game.  Two repetitions 
of a two-round game are different from one repetition of a 
100-round game because even with the two repetitions, the accepter 
discovers the nature  of the clauses costlessly after the first 
repetition, whereas in the 100-round game this is not discovered 
until it is too late  to take action.\footnote{Note, however, that if 
a clause is rejected, its nature is never discovered by the accepter. 
Thus, repetition is no solution to the pessimistic expectations that 
underlie the No Offers equilibrium 2. } Repetition introduces the 
possibility of reputation, as in the Klein \& Leffler (1981)  model 
mentioned above, and may reach the same results as the commitment 
equilibria.\footnote{A caveat: the fine print in contracts often 
concerns rare or end-game events--- liability for toxic waste spills, 
or who pays for arbitration expenses.  These, in fact, are the kinds 
of clauses for which inspection costs would be highest, since the 
rarity  of their application would make them less familiar to the 
accepter.   In such cases, the nature of the clause may ordinarily 
not be discovered  even after the deal is closed, or may be 
discovered only when the relationship is terminated, so repetition 
may not be useful.}

 A third extension is to incomplete information.  Suppose the 

offerer and the accepter do not know each other's costs of  offering 
each type of clause,  or of reading clauses.  If, for example, a 
certain proportion of  upright offerers have infinite moral costs of 
offering   misleading clauses,    then  learning  may  occur  during 
the negotiation process, since the discovery of a misleading clause 
in the first round would reveal to the accepter that he was not 
dealing with one of the upright offerers. Or, the model might specify 
situations where the  set of available clauses is restricted. In the 
present model, the offerer is free to offer either kind of clause in 
either round, but one can also imagine that he might, for example, 
have  only one welfare-improving clause available, so the    issue 
becomes  whether he offers it in the first, the second, or neither 
round.  Incomplete information or  restricted availability of 
welfare-improving clauses   is unlikely to  invalidate the lessons of 
this paper, but they may add new effects or equilibrium refinements 
that  could not arise  in the complete information game  that was 
analyzed here. 



\newpage

\bigskip
\begin{center}
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