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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 $105 150$.\\
\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}