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\begin{LARGE}

\begin{center}

 {\bf The Old, 3rd edition. Section 12.5

 \end{center}

 \end{LARGE}



 

 



\noindent

  October 29, 2003. Eric Rasmusen,

Erasmuse@indiana.edu.   Http:

//mypage.iu.edu/$\sim$erasmuse.   



I've improved the model of this section, making it yield more results with less complexity. But here is the old model in case anybody wants to use it. 



 \bigskip

\noindent

 {\bf 12.5 Incomplete Information}





\noindent

   Instant agreement has characterized even the multiperiod games of

complete information discussed so far.  Under incomplete information,

knowledge can change over the course of the game and bargaining can

last more than one period in equilibrium, a result that might be

called inefficient but is certainly realistic. Models with complete

information have difficulty explaining such things as strikes or

wars, but if over time an uninformed player can learn the type of the

informed player by observing what offers are made or rejected, such

unfortunate outcomes can arise.  The literature on bargaining under

incomplete information is vast. For this section, I have

chosen to use a model based on the first part of Fudenberg \& Tirole

(1983), but it is only a particular example of how one could

construct such a model, and not a good indicator of what results are

to be expected from bargaining.







 \begin{center}

{\bf  Bargaining with Incomplete Information }

 \end{center}

 {\bf Players}\\

  A seller, and a buyer called Buyer$_{100}$ or Buyer$_{150}$ depending

on

his type.





\noindent

 {\bf The Order of Play }\\

 0 Nature picks the buyer's type, his valuation of the object being

sold, which is $b = 100$ with probability $\gamma$ and $b = 150$ with

probability $(1-\gamma$).  \\

 1 The seller offers price $p_1$.\\

 2 The buyer accepts or rejects $p_1$ (acceptance ends the game).\\

 3 The seller offers a second price $p_2$.\\

 4 The buyer accepts or rejects $p_2$.



\noindent

 {\bf Payoffs}\\

 $\pi_{seller} $

\begin{tabular}{ll}

  = & $\left\{

\begin{tabular}{ll}

$ p_1$  & if $p_1$ is accepted\\

$\delta p_2 $ & if  $p_2$ is accepted\\

0 & if no offer is accepted\\

\end{tabular}

\right.$

\end{tabular}



\noindent

 $\pi_{buyer} $

\begin{tabular}{ll}

  = & $\left\{

\begin{tabular}{ll}

$ b - p_1$  & if $p_1$ is accepted\\

$ \delta (b - p_2) $  & if  $p_2$ is accepted\\

 0 & if no offer is accepted\\

\end{tabular}

\right.$

\end{tabular}\\





 The order of play is shown on the next page. Let us work back from the

end of it. If $p_2$ is accepted, the buyer's payoff is $ \delta (b -

p_2) $,

rather than $ \delta b - p_2 $, because the present value of cash

paid in the second period is less than that of cash paid in the first

period.  Consumption in the second period provides less pleasure, but

payment provides less pain.  Let us set $\delta = 0.9$ for the

numerical computations.



 As described, the game tells of an encounter between one buyer

and one seller. Another interpretation is that the game is between a

continuum of buyers and one seller, in which case the analysis works

out the same way, but the verbal translation is different. As noted

in Section 3.2, models with a continuum of buyers are sometimes

easier to understand, because the two types of buyers can be

interpeted as different fractions of the population, and mixed

strategies can be interpreted as fractions of the total population

using different pure strategies.



 If this were a game of complete information, then in equilibrium

the seller, who has the last-mover advantage, would choose $p_1 =

100$ or $p_1 = 150$ and the buyer would accept. If the buyer did not

accept, the seller would offer the same price again, and the buyer

would accept that second offer.  With incomplete information,

however, the probability that the buyer is a Buyer$_{100}$ determines

whether the equilibrium is pooling at a low price, or separating at a

high price with some offers rejected.



 The incomplete information game is a screening model  like those we

saw in chapter 11 in the sense that the uninformed player moves first.

Screening models with a continuum of types do not have pooling

equilibria, but the assumption that there are only two types will

permit one in this model.  If the buyer made the offers, instead of

the seller, this would be a signalling game, in which

out-of-equilibrium beliefs would be more important. The game is not a

pure screening model, however,  because it lasts two periods

and the uninformed player does have a chance to make his second move

after the informed player has moved and possibly, thereby, has revealed

information. This gives it the character of mixed screening and

signalling.













\bigskip

\noindent

 {\bf A Case of Many Low-Valuation Buyers: $\gamma =

0.5$}



\noindent

    We will start by assuming that $\gamma = 0.5$, so the probability

is fairly high that the buyer has the valuation 100.



\noindent

{\bf   Equilibrium (Pooling) } \\

 In the first period,

 $p_1= 100$,

 Buyer$_{100}$ accepts  $p_1 \leq 100$, and

 Buyer$_{150}$ accepts  $p_1 \leq 105$.

  In the second period,

 $p_2 = 100$,

  Buyer$_{100}$ accepts $p_2 \leq 100$, and Buyer$_{150}$ accepts $p_2

\leq 150$. The seller's beliefs out of equilibrium are that if a buyer

rejects $p_1= 100$, he is Buyer$_{100}$ with probability $\gamma$

(passive conjectures). The outcome is that $p_1 =100$ and the buyer

accepts.





  Let us test that this is a perfect Bayesian equilibrium. As always,

 work back from the end. Both types of buyers have a dominant

strategy for the last move: accept any offer below $b$. Given the

parameters, the seller should not raise $p_2$ above 100, because with

probability $\gamma=0.5$ he would lose a profit of 100 and his

potential revenue is no greater than 150.



  Buyer$_{150}$, although looking ahead to $p_2=100$, is willing to pay

more than 100 in period one because of discounting. His payoff is the

same from accepting $p_1 = 105$ as from accepting $p_2 =100$, because

his

nominal surplus of 50 from accepting the lower price is discounted

to a utility value of 45. Buyer$_{100}$, however, is never willing

to pay more than 100, and discounting is irrelevant because he has no

surplus to look forward to anyway.



   The seller knows that even if $p_1=105$ he will still sell to

Buyer$_{150}$, but if he tries that and finds no buyer in the first

period, he has delayed receipt of his payment, which is discounted.

Since $100 > 97.5 (=[1- \gamma] \cdot [105] + \gamma \cdot \delta

\cdot [100]$), the seller prefers the safe present price of 100 to

the alternative of a gamble between a present 105 and a future 100.



  Out-of-equilibrium beliefs are specified for the equilibrium, but

they do not really matter.  Whatever inference the seller may draw if

the buyer refuses $p_1=100$, the inference never induces the buyer to

change his actions. It might be that the seller believes a refusal

indicates that the buyer's value to be 150, so $p_2=150$, but that

does not change the buyer's incentive to accept $p_1=100$.













\bigskip

\noindent

 {\bf A Case with Few Low-Valuation Buyers:

$\gamma=0.05$}



\noindent

 If the proportion of low-valuation buyers is as small as

$\gamma=0.05$, the equilibrium is separating and in mixed strategies.





\noindent

 {\bf   Equilibrium  (Separating, in mixed strategies) }  \\

In the first period,  $p_1= 150$,

 Buyer$_{100}$ accepts  $p_1 \leq 100$, and Buyer$_{150}$ accepts $p_1$

with probability $m(p_1)$, where



 $\left\{

 \begin{tabular}{lll}

 $m =$& 1 & if $p_1 \leq 105$.\\

  $m = $& $\alpha$  & if  $105 <p_1 \leq 150$ (where $0 \leq \alpha

\leq 0.89$)\\

  $m = $& 0 & if $p_1 > 150$.\\

\end{tabular}

\right.$





 In the second period,    $p_2 = 150$ if the seller believes

that he faces a Buyer$_{100}$ with probability less than

$\frac{1}{3}$, and otherwise $p_2= 100$.

Buyer$_{100}$ accepts $p_2 \leq 100$,  and

Buyer$_{150}$ accepts $p_2 \leq 150$.   The outcome is

that $p_1 =150$, which is sometimes accepted by

Buyer$_{150}$;  $p_2=150$, which   is accepted by

Buyer$_{150}$; and  Buyer$_{100}$ never accepts an

offer.



 The observed outcome is simple --- the price always stays

at 150, and some buyers accept in each period  while other

buyers never accept --- but the equilibrium strategies

are quite complicated. As we will see, the equilibrium  is

not even fully determined, because the mixing

probability $\alpha$ can take any of a continuum of

values.



 The strategies in the second-period game are simple enough. In the

second period, the buyer accepts if the price is less than his

valuation and the seller trades off a safe 100 against a gamble

between 0 and 150. He is indifferent between them if

 \begin{equation}  \label{e11.11}

  100 = 0 \cdot Prob({\rm Buyer}_{100}) + 150 \cdot [1-Prob({\rm

Buyer}_{100})],

 \end{equation}

  which yields a critical value of $Prob$(Buyer$_{100}$) = $\frac{1}{3}

$.  If neither type of buyer accepted first-period offers, the second

period belief would be $Prob$(Buyer$_{100}$) = $\gamma$, which we

assumed to be 0.05, so the second-period price would be 150.



  The first-period strategies are more complicated. The first period

strategy of Buyer$_{150}$ is not the pure strategy of accepting the

offer $p_1=150$, because if he always accepted in the first period the

seller would lower the price in the second period, knowing that a buyer

who rejected the first-period offer must be a Buyer$_{100}$.

Anticipating the price drop, Buyer$_{150}$ would refuse $p_1=150$, which

contradicts the reason for the drop.



  In equilibrium, it must be that after a refusal in the first period

the seller puts a high enough probability on Buyer$_{150}$ that he

decides to keep the price high in the second period.  For the seller to

want to keep $p_2= 150$, the probability that a buyer who refused the

first-period offer is a Buyer$_{150}$ must be at least $\frac{2}{3} $,

from equation (\ref{e11.11}).  If both $p_1$ and $p_2$ are equal to 150,

the buyer will be indifferent as to when he accepts, so he is willing to

follow a mixed strategy.  We can calculate the mixing probability

$m(150)$ by finding the value that makes the seller willing to keep the

price at 150 in the second period. The probability that a buyer is a

Buyer$_{150}$ is equal to $1-\gamma$ in the first period, but a

Buyer$_{150}$ only rejects the first-period offer with probability $1-

m(150)$.  Therefore, in the second period, using Bayes' Rule,

  \begin{equation}  \label{e11.12}

 Prob({\rm Buyer}_(100)) = \frac{\gamma }{ \gamma + [1-m(150)]

[1-\gamma]}.

  \end{equation}

  Plugging in $\gamma = 0.05$ and $Prob$(Buyer$_{100}$) = $\frac{1}{3}$,

equation (\ref{e11.12}) can be solved to yield $m(150) = 0.89$

(rounded). The calculation has ensured that if the Buyer$_{150}$ accepts

the first offer with probability 0.89, the probability that a second-

period buyer has valuation 100 is $\frac{1}{3}$.  The value $\alpha=

0.89$ is the maximum equilibrium probability that a Buyer$_{150}$

refuses to buy in the first period, but a smaller value for $\alpha$

would support an equilibrium {\it a fortiori} since the probability that

a refuser had valuation 150 would be even greater than $\frac{2}{3}$.



   There is a continuum of equilibria, which differ in their values for

$\alpha$.  Two values are focal points, 0 and 0.89. The value of 0 is a

pure strategy, and has the attraction of simplicity. The value of 0.89

is Pareto efficient, because it is the highest equilibrium probability

that the buyer accepts immediately, which avoids the lost utility from

delay.  The qualitative difference between the two equilibria is that in

the pure-strategy equilibrium no buyers accept the offer in the first

period.



 \bigskip





\noindent

{\bf Out-of-equilibrium Behavior}



\noindent

The description just given describes only part of the

separating equilibrium. A full description would

specify each player's actions at each node of the game

tree given the past history of the game, including

out-of-equilibrium paths that start with deviations

such as $p_1=140$.    As we will see in describing the path

starting with the seller deviation $p_1=140$, there is

an overwhelming range of possible out-of-equilibrium

behavior.



    Consider what happens if  the seller offers a price of

140 in the first period. For the same reasons as we described for

$p_1=150$, the equilibrium cannot be in pure strategies.  The

equilibrium strategies in the out-of-equilibrium subgame are for the

Buyer$_{150}$  to mix between accepting and rejecting, and for the

seller

to mix between $p_2 = 100$ and $p_2 = 150$. Here, unlike on the

equilibrium path, the seller must also mix, because otherwise the

buyer would strongly prefer to accept 140 rather than wait for 150.

The seller is willing to mix only if he believes that there is

exactly a one-third probability that the buyer is a Buyer$_{100}$, so

the

buyer's strategy is $m(150) = 0.89$, as before.  Denote the seller's

mixing probability of $p_2=100$ by $\mu$.  It must take a value that

makes the buyer indifferent between accepting and rejecting, so

 \begin{equation}  \label{e11.13}

150 - p_1 = 0.9\mu (150 -100) + (1-\mu) \cdot 0,

\end{equation}

  which solves to $\mu =3\frac{1}{3} - p_1/45$, or $\mu =0.22$ for $p_1

= 140$.





 The consequences of this kind of deviation are

unimportant on the equilibrium path.

Out-of-equilibrium behavior is important when player

Smith deviates, because it might induce player Jones to change his

behavior in a way that makes Smith's deviation

profitable. This was the case in the entry deterrence

games of chapter 4 that were used to illustrate

perfectness:  if the entrant deviated from a strategy

combination in which he was supposed to stay out, the

incumbent would change his fighting behavior to

collusion, which would make the entrant's entry

profitable. Here, however, the seller deviating by

charging a price of $p_1=140$ does not call a bluff in any

way;  it is simply a blunder.   The complications in the

equilibrium description do not arise from the buyer's

immediate response, but in the seller's  second-period

response to his own deviation.   Descriptions of

equilibria  are customarily incomplete because they do

not specify a full strategy for each player, including

his way of responding to his own deviations from

equilibrium, but the off-equilibrium paths that are not

described  have no importance to the analysis.





\bigskip



 These calculations have been intricate and hard, though they provide

examples of the kind of care that must be taken with

out-of-equilibrium behavior. The most important lesson of this

model is that bargaining can lead to inefficiency. In the separating

equilibrium, some of the Buyer$_{150}$s delay their transactions until

the second period, which is inefficient since the payoffs are

discounted. Moreover, the Buyer$_{100}$s never buy at all, and the

potential gains from trade are lost.



 A more technical conclusion is that the price the buyer pays depends

heavily on the seller's equilibrium beliefs. If the seller thinks that

the buyer has a high valuation with probability 0.5, the price is 100,

but if he thinks the probability is 0.05, the price rises to 150. This

implies that a buyer is unfortunate if he is part of a group which is

believed to have high valuations more often; even if his own valuation

is low,  what we might call his bargaining power is low when he is part

of a high-valuing group.  Ayres (1991) found  that when  he hired

testers  to pose as   customers at car dealerships, their success

depended on their race and gender even though they were given

identical predetermined bargaining strategies to follow.  Since the

testers did as badly even when faced with salesmen of their own race and

gender, it seems likely that they were hurt by being members of groups

that usually can be induced to pay higher prices.



\end{document}

