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\begin{LARGE}
 \begin{center}
{\bf 9 Adverse Selection}

 \end{center}
   \end{LARGE}


This is designed for one 75-minute lecture using {\it Games and Information}. Probably I
have more material than I will end up covering.

 This is just for sections 9.1 and 9.6. 

 October 7, 2006



\newpage


 \begin{LARGE}

  
 
 

\begin{center}
 {\bf  Production Game VI: Adverse Selection }
 \end{center}
 {\bf Players}\\
The principal and the agent.

\noindent
 {\bf The Order of Play}
 \begin{enumerate}
\item[(0)]
Nature chooses the agent's ability $a$,  observed by the agent but not by the
principal, according to distribution $F(a)$.
 \item[(1)] The principal offers the agent one or more wage contracts $w_1(q),
w_2(q), \ldots$
\item[(2)] The agent accepts one contract or rejects them all.
 \item[(3)]
Nature   chooses a value for the state of the world, $\theta$, according to
distribution $G(\theta)$.  Output is then $q = q(a,\theta) $.
 \end{enumerate}

\noindent
 {\bf Payoffs}\\
If the agent rejects all contracts, then $\pi_{agent} = \overline{U}(a)$, which
might or might not vary with  his type, $a$;  and $\pi_{principal} = 0$.\\
Otherwise, $\pi_{agent} = U(w,a)$ and $\pi_{principal} = V(q - w)$.

 

    Under adverse selection, it is not the worker's effort, but his ability,
that is noncontractible. 

No uncertainty---- Either high or low output might be observed in equilibrium,
unlike under moral hazard.

Offering multiple contracts can be
an improvement over offering a single contract-- perhaps 
  a flat-wage contract for   low-ability agents and an incentive
contract for  high-ability agents.


\newpage

Production Game VIa puts specific functional forms into the game to illustrate
how to find an   equilibrium.

\begin{center}
 {\bf Production Game VIa: Adverse Selection  with Particular Parameters  }
  \end{center}
 {\bf Players}\\
 The principal and the agent.

\noindent
{\bf The Order of Play}
 \begin{enumerate}
 \item[(0)] Nature chooses the agent's ability $a$, unobserved by the principal,
according to   distribution $F(a)$, which puts probability $0.9$ on low ability,
$a=   0$, and  probability  0.1 on   high ability, $a=     10$.

\item[(1)]
The principal offers the agent one or more wage contracts \\$W_1= (w_1(q=0),
w_1(q=10)  ), W_2=(w_2(q=0), w_2(q=10)  )  \ldots$

\item[(2)]
 The agent accepts one contract or rejects them all.

\item[(3)]
Nature chooses a value for the state of the world, $\theta$, according to
distribution $G(\theta)$, which puts equal weight on 0  and 10.  Output is then
$q = Min(  a+\theta,10) $.
  \end{enumerate}

\noindent {\bf Payoffs}\\
If the agent rejects all contracts, then depending on his type his reservation
payoff is either  $\overline{U}_{Low} =3$ or $\overline{U}_{High}  =2$ and the
principal's payoff is  $\pi_{principal}= 0$. \\
Otherwise, $U_{agent} = w$ and $V_{principal} =  q - w $.

 
\newpage



\noindent
 A separating   equilibrium  is 


\begin{tabular}{ll}
  Principal:    Offer  &  $W_1=\{w_1(q=0)=3, w_1(q=10)=3  \}$,\\
 &  $  W_2= \{w_2(q= 0)=0, w_2(q=10)=3  \}$\\
 & \\
 Low  agent: &    Accept   $ W_1$   \\
& \\
High   agent:  & Accept    $W_2$   \\
 \end{tabular}

     As usual, this is a weak equilibrium. Both Low and High agents are
indifferent about whether they accept or reject their   contract.  The
equilibrium indifference of the agents  arises from the open-set problem; if the
principal were to specify a  wage  of 3.01 for $W_2$, for example,  the  high-
ability agent  would no longer be indifferent about accepting it instead of
$W_1$.


\newpage



    In hidden-action models, the principal tries to construct a contract which
will induce the agent to take the single appropriate action. In hidden-knowledge
models, the principal tries to make different actions attractive to different
types of agent, so the agent's choice depends on the hidden information.  

\noindent
 (1) {\bf Incentive compatibility} (the agent picks the desired contract and
actions).

\noindent
(2) {\bf Participation} (the agent prefers the contract to his reservation
utility).

In a model with hidden knowledge, the incentive compatibility constraint is
customarily  called the {\bf self-selection constraint}.

There can be one  IC constraint and one Part. constraint for each type of agent.


  

 Here, 
  what action  does  the principal desire from each type of agent?

 The
agents do not choose effort, but they do choose whether or not to work for the
principal, and which contract to accept.

 The low-ability agent's expected output
is 0.5(0) + 0.5(10)= 5, compared to a reservation payoff of 3, so the principal
will want to hire him if  $Ew  \leq 5$. 

 The high-ability agent's expected output is 0.5(10) + 0.5(10)= 10,
compared to a reservation payoff of 2, so the principal will want to hire the
high-ability agent  if $Ew  \leq 5$. 


\newpage

    
The participation constraints are, if we let $\pi_i(W_j)$ denote the
expected payoff an agent of type $i$ gets from contract $j$,
 \begin{equation} \label{e9.02}
 \begin{array}{l} \pi_L(W_1)  \geq \overline{U}_{Low}; \;   \;   0.5  w_1(0)  +
0.5w_1(10) \geq 3  \\
  \\ \pi_H(W_2)  \geq \overline{U}_{High};  \;  \;   0.5 w_2(10)  + 0.5w_2(10)
\geq  2.     \\
 \end{array}
 \end{equation}

 Clearly  the contracts in our conjectured equilibrium, $W_1 = ( 3,3)$ and $W_2
= ( 0,3)$,  satisfy the participation constraints.  In the equilibrium, the low-
and the high-output wages both  matter to the low-ability agent,   but only the
high-output wage matters to the high-ability agent. Both agents, however, end up
earning  a wage of 3 in each state of the world, the only difference being that
contract $W_2$ would be a very risky contract for the low-ability agent despite
being riskless for the high-ability agent.     principal would like to  make
$W_1$   risk-free, with the same wage in each state of the world.


    In our separating equilibrium, the participation constraint is binding for
the ``bad'' type but not for the ``good'' type.

This is typical of adverse selection
models (if there are more than two types it is the participation constraint of
the worst type that is binding, and no other).


\newpage 

The participation constraint is binding for
the ``bad'' type but not for the ``good'' type.


  The principal makes the bad
type's contract unattractive for two reasons. First, if he pays less, he keeps
more. 

Second, when  the bad type's contract is  less attractive, the good type
can be more cheaply lured away to a different contract. The principal can never
extract all the gains from trade from the good type unless he gives up on making
either of his  contracts acceptable to the bad type.


 Another typical feature of this equilibrium is that the low-ability agent's
contract not only drives him down to his participation constraint, but is
riskless.


\newpage

 

\noindent
The    self-selection constraints  are
\begin{equation} \label{e9.01}
 \begin{array}{l}
 \pi_L(W_1)  \geq \pi_L(W_2); \;   \;    0.5  w_1(0)  + 0.5w_1(10)  \geq 0.5
w_2(0)  + 0.5w_2(10)    \\
\\
\pi_H(W_2)  \geq \pi_H(W_1); \;   \;   0.5 w_2(10)  + 0.5w_2(10)  \geq   0.5
w_1(10)  + 0.5w_1(10)    \\
 \end{array}
\end{equation}

The first inequality in (\ref{e9.01}) says that the    contract $W_2$ has to
have a low enough expected return for the low-ability agent to deter  him from
accepting it. The second inequality says that the   wage contract $W_1$  must
be less attractive than $W_2$ to the high-ability agent. The conjectured
equilibrium contracts $W_1 = ( 3,3)$ and $W_2 = (0, 3)$ do this, as can be seen
by substituting their values into the constraints:
\begin{equation} \label{e9.01a}
 \begin{array}{l} \pi_L(W_1)  \geq \pi_L(W_2); \;   \;    0.5  (3)  + 0.5(3)
\geq  0.5  (0) + 0.5(3)    \\ \\
  \pi_H(W_2)  \geq \pi_H(W_1); \;   \;   0.5 (3)  + 0.5 (3)  \geq   0.5 (3)  +
0.5(3)    \\
  \end{array}
   \end{equation}

\newpage

The self-selection constraint is binding for the good type but not for the bad
type.

 This, too, is typical of adverse selection models.

 The principal wants the
good type to reveal his type by choosing the appropriate  to the good type as
the bad type's contract.   It does not have to be {\it more} attractive though
(here notice the open-set problem), so the principal will minimize his salary
expenditures and  choose two contracts equally attractive to the good type. In
so doing, however, the principal will have chosen a contract for the good type
that is strictly worse for the bad type, who cannot achieve so high an output so
easily.

   All that
remains to check is whether the principal could increase his payoff.

 He cannot,
because he makes a profit from either contract,  and having driven  the  low-
ability agent  down to his   reservation payoff and the high-ability agent down
to the minimum payoff needed to achieve separation, he cannot further reduce
their pay.

\newpage

\noindent
 {\bf Competition and Pooling}

 

 Although it is true, however,   that the   participation constraints must be
satisfied for agents who   accept the contracts,   it is not always the case
that they accept different contracts in equilibrium.

If they do not, they do
not need to satisfy self-selection constraints.

  \noindent
  {\it If all types of agents  choose  the same strategy in all states, the
equilibrium is {\bf pooling}.  Otherwise, it is {\bf separating}.}

 The distinction between pooling and separating is different from the
distinction between equilibrium concepts. 

   A model might have multiple Nash
equilibria, some pooling and some separating.

 Moreover, a single equilibrium---
even a pooling one--- can include several contracts, but if it is pooling the
agent always uses the same strategy, regardless of type. 

If the agent's
equilibrium strategy is mixed, the equilibrium is pooling if the agent always
picks the same mixed strategy, even though the messages and efforts would differ
across realizations of the game.


\newpage

 

    The possibility of a pooling equilibrium reveals one more step we need to
take to establish that the proposed separating equilibrium in Production Game
VIa is really an equilibrium:

 {\bf Would the principal do better by offering a
pooling contract instead, or a separating contract  under which one type of
agent does not participate? }

 All of my derivation above was to show that the
agents would not deviate from the proposed equilibrium, but it might still be
that the principal would deviate.

\newpage

 First, would the principal prefer pooling?  

Then all that is necessary is that
the contract as cheaply as possible induce both types of agent to participate.

Here, that would require that we make the contract barely acceptable to the type
with the lowest ability and highest reservation payoff, the low-ability agent.


The contract $(3,3)$ offered by itself would do that, but it would not increase
profits over $W_1$ and $W_2$ in our equilibirum above. 

Either pooling or
separating would yield profits of $0.9 (0.5 (0-3) + 0.5 (10-3))  + 0.1 (0.5
(10-3) + 0.5 (10-3))  = 2.5.$

\newpage

 Second, would the principal prefer a separating contract that ``gave up'' on
one type of agent? 

The principal would not want to drive away the high-ability
agent, of course,  though he could do so by offering a high wage for $q=0$ and a
low wage for $q=10$, because the high-ability agent has both greater output and
a lower reservation payoff (if we had  $\overline{U}_{High}  =11$ then the
outcome would be different). 

   But if the principal did not have to offer a
contract that gave the low-ability agent his reservation payoff of 3, he could
be more stingy towards the high-ability agent. 

If there were no low-ability
agent, the principal would offer a contract such as $(0,2)$ to the high-ability
agent, driving him down to his reservation payoff and increasing the profits
from hiring him. 

Here, however, there are not enough high-ability agents for
that to be a good strategy for the principal.

 His payoff would decline to
 $0.9 (0)  + 0.1 (0.5 (10-2) + 0.5 (10-2))  = 0.8, $ a big decline from 2.5.

 If
99\% of the agents were high-ability, instead of 10\%, things would have turned
out differently.
 

 \newpage

 


\begin{center}
 {\bf  Production Game VII: Adverse Selection and Moral Hazard COMBINED }
 \end{center}
  {\bf Players}\\
The principal and the agent.

\noindent
 {\bf The Order of Play}
 \begin{enumerate}
\item[(0)]
Nature chooses the state of the world $s$,  observed by the agent but not by the
principal, according to distribution $F(s)$, where the state $s$ is Good with
probability 0.5 and Bad with probability 0.5.
 \item[(1)] The principal offers the agent a wage contract $w(q)$.
\item[(2)] The agent accepts or rejects the contract.
\item[(3)] The agent chooses   effort level $e$.
 \item[(4)]
   Output is   $q = q(e,s) $. where $ q(e, good) =3e$ and  $q(e,bad) =e$.
 \end{enumerate}

\noindent
 {\bf Payoffs}\\
If the agent rejects all contracts, then $\pi_{agent} = \overline{U}=0$ and
$\pi_{principal} = 0$.\\
Otherwise, $\pi_{agent} = U(e,w,s)= w-e^2$ and $\pi_{principal} = V(q - w)= q-
w$.


Thus, there is no uncertainty,  both principal and agent are risk neutral in
money, and effort is increasingly costly.

\newpage


 In this model, the first-best effort depends on the state of the world. The two
social surplus maximization problems are
\begin{equation} \label{e50}
  \stackrel{Maximize}{e_g}\;\;\; 3e_g - e_g^2,
 \end{equation}
  which is solved by the optimal effort $e_g=1.5$ (and $q_g =4.5$) in the good
state, and
\begin{equation} \label{e51}
  \stackrel{Maximize}{e_g} \;\;\; e_b - e_b^2,
 \end{equation}
which is solved by the optimal effort $e_b=0.5$  (and $q_b =0.5$)in the  bad
state.

\newpage


 The problem is that the principal does not know what level of  effort and
output are appropriate.

 He does not want to require high output in both states,
because if he does, he will have to pay too high a salary to the agent to
compensate for the difficulty of attaining that output in the bad state.


Rather, he must  solve the following problem:
 \begin{equation} \label{e52}
    \stackrel{Maximize}{q_g, q_b,w_g, w_b} \;\;\;\; [ 0.5 (q_g - w_g) + 0.5 (q_b-w_b)],
 \end{equation}
 where the agent has a choice between two forcing contracts,  $(q_g, w_g)$ and
$(q_b, w_b)$,  and the contracts must induce  participation and self selection.


\newpage

 The self-selection constraints are   based on efforts of $e=q/3$ for the good
state and $e=q$ for the bad state.


 In the good state, the agent must choose the good-state contract,so
 \begin{equation} \label{e55}
{\displaystyle  \pi_{agent} (q_g, w_g|good) =  w_g -     \left( \frac{q_g}{3}
\right)^2  \geq  \pi_{agent} (q_b, w_b|good) = w_b -      \left( \frac{q_b}{3}
\right)^2  }
 \end{equation}
  and    in the bad state  he must choose the bad-state contract,so
 \begin{equation} \label{e56}
\pi_{agent} (q_b, w_b|bad) =    w_b -    q_b^2  \geq \pi_{agent} (q_g, w_g|bad)
=    w_g -    q_g^2.
 \end{equation}

 The participation constraints    are
\begin{equation} \label{e53}
{\displaystyle  \pi_{agent} (q_g, w_g|good) =  w_g -      \left( \frac{q_g}{3}
\right)^2  \geq 0}
 \end{equation}
and
\begin{equation} \label{e54}
\pi_{agent} (q_b, w_b|bad) =    w_b -      q_b^2  \geq 0.
 \end{equation}

\newpage


The bad state's participation constraint will be binding, since  in the bad
state the agent will not be tempted by the  good-state contract's higher output
and wage. Thus, we can conclude from constraint (\ref{e54}) that
\begin{equation} \label{e57}
    w_b =    q_b^2.
 \end{equation}

 The good state's participation constraint will not be binding, since there the
agent will be left with an informational rent-- the principal must leave the
agent some surplus to induce him to reveal the good state.

\newpage

 The good state's self-selection constraint will be binding, since in the good
state the agent {\it will } be tempted to take the easier contract appropriate
for the bad state. Thus, we can conclude from constraint (\ref{e55}) that
\begin{equation} \label{e58}
\begin{array}{ll}
 w_g  &=  {\displaystyle   \left( \frac{q_g}{3}\right)^2   + w_b -     \left(
\frac{q_b}{3}   \right)^2 } \\
 & \\
 &{\displaystyle =    \left( \frac{q_g}{3}\right)^2   +   q_b^2 -     \left(
\frac{q_b}{3}   \right)^2,}   \\
\end{array}
\end{equation}
where the second step substitutes for $w_b$ from equation (\ref{e57}). The bad
state's self-selection constraint will not be binding, since the agent would
then not be tempted to produce a large amount for a large wage.

\newpage

Now let's return to the principal's maximization problem. Having found
expressions for $w_b$ and $w_g$ we can rewrite (\ref{e52}) as
 \begin{equation} \label{e59}
 {\displaystyle  \stackrel{Maximize}{q_g, q_b} \;\;\;\; [ 0.5 \left(q_g -   \left( \frac{q_g}
{3} \right)^2    - q_b^2   +  \left( \frac{q_b}{3}   \right)^2 \right) + 0.5 (q_b  -
q_b^2)] }
 \end{equation}
 with no constraints.  The first-order conditions are
 \begin{equation} \label{e60}
    {\displaystyle   0.5 \left(1 -   \frac{2q_g}{9} \right)    =0, }
 \end{equation}
 so $q_g =4.5$, and
  \begin{equation} \label{e61}
{\displaystyle   0.5 \left(-2   q_b  +     \frac{2q_b}{9}   \right) + 0.5 (1  - 2q_b )  =0,
}
 \end{equation}
so $q_b   \approx .26$. We can then find the wages that satisfy the constraints,
which are $w_g \approx 2.32$ and $w_b \approx 0.07$.

\newpage

 Thus, in the second-best world of information asymmetry, the effort in the good
state remains the first-best effort, but second-best effort in the bad state is
lower than   first-best. This results from the principal's need to keep the bad-
state contract from being too attractive in the good state. Bad-state output
and compensation must be suppressed.  Good-state output, on the other hand,
should be left at the first-best level, since the agent will not be tempted by
that contract in the bad state.

Also, observe that in the good state the agent earns an informational rent. As
explained earlier, this is because the good-state agent could  always earn a
positive payoff by pretending the state was bad and taking that contract, so any
contract that separates out the good-state agent (while leaving some contract
acceptable to the bad-state agent) must also have a positive payoff.


 




\end{LARGE}

\end{document}




