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 \noindent
  July 17, 1999
  

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 \noindent
 {\bf 9.5 Other Equilibrium Concepts: Wilson Equilibrium and Reactive
Equilibrium} 

 \noindent
  In Insurance Game III , any pooling contract is vulnerable to a
cream-skimming contract that draws away the $Safe$s, but this is a
little strange, because it seems that after that happens the
now unprofitable old pooling contract (which was soaking up the
$Unsafe$s) would be withdrawn. The game tree does not reflect this,
nor does the Nash equilibrium concept.

 One way to obtain a pure strategy equilibrium is to redefine the
equilibrium concept.  C. Wilson (1980) suggests that the pooling
equilibrium is legitimate because a principal (an uninformed player) who was 
thinking about introducing the new contract would realize that it
would be unprofitable once the old contract was withdrawn.

 \noindent
 {\it A {\bf Wilson equilibrium} is a set of contracts such that when
the agents (informed players) choose among them so as to maximize
profits,\\
 (1) All contracts make nonnegative profits; and\\
 (2) No new contract (or set of contracts) could be offered that
would make positive profits even after all contracts that would make
negative profits as a result of its entry were withdrawn.}

  The Wilson equilibrium is the same as the Nash separating
equilibrium if that exists, and otherwise it is the pooling contract
most preferred by the $Safe$s. In figure 9.6, the Wilson equilibrium
is the same as the Nash equilibrium, the separating pair ($C_3,
C_5$). In figure 9.7, where no Nash equilibrium exists, the Wilson
equilibrium is the zero-profit pooling contract, $F'$.  It is on the
line $\omega F'$, so  it satisfies part (a) of the definition. It
provides the fullest insurance of any zero-profit pooling contract,
so that no new pooling contract would be more attractive, and while some
new separating contract might be profitable if the $Unsafe$s stayed
with $F'$, any such contract would cause $F'$ to be withdrawn and
would be unprofitable thereafter. 

 The idea of Wilson equilibrium can also be incorporated into the
game by modifying the game tree instead of redefining the equilibrium
concept, as suggested by Fernandez \& Rasmusen (unpublished), to obtain the
Wilson outcome as the perfect equilibrium of the modified game. 

\noindent
 {\bf Wilson Equilibrium}\\
{\it  (1)Principals simultaneously offer contracts, called ``old
contracts.'' \\
 (2) Principals may simultaneously offer other contracts, called ``new
contracts.'' \\
 (3) Principals may simultaneously withdraw any old contracts.\\
  (4) Agents choose from among the remaining old and new contracts,
and trading occurs.\\}

 In the perfect Bayesian equilibrium of this game, the principals
offer the contracts that form a Wilson equilibrium in move (1).  The
approach of changing the game tree may seem more complicated than
changing the equilibrium concept, but that is because it clearly
delineates the somewhat vague intuition behind the equilibrium
concept. Making use of the Wilson concept is not just a technical
assumption: it is assuming that the market has a particular
structure.

\bigskip

  Riley (1979b) uses reasoning similar  to Wilson's to justify
his concept of ``reactive equilibrium.''  Under this concept, an
equilibrium is a set of contracts such that though some new contract
might be profitable, that new contract would itself become
unprofitable if a second new contract were introduced. More formally,
following Engers \& Fernandez (1987), 

 \noindent {\it A {\bf reactive equilibrium} is a set of contracts
$S$ yielding nonnegative profits such that for any nonempty set of
contracts $S'$ (the defection), where $S \cup S'$ is closed, there
exists a closed set of contracts $S'' $ (the reaction) such that:\\
  (1) $S'$ incurs losses when only these three sets are tendered,
and\\
  (2) $S'' $ does not incur losses when these three sets are
tendered, whether or not other contracts are also
offered.} 


   In both figure 9.6 and   9.7, the reactive equilibrium is the
separating pair ($C_3,C_5$). That pair yields zero profits, and while
in figure 9.7 there is a profitable deviation ($C_6$), that deviation
would become unprofitable if a cream-skimming contract were added as
a reaction. Moreover, condition (2) of the definition is met, because
if the reactive cream-skimming contract is chosen carefully, no
additional contracts can be added which make it unprofitable, given
that $C_6$ continues to be offered.

 A separating reactive equilibrium always exists because any pooling
contract disrupting it could be reacted against: reactive equilibrium
makes constructive use of the nonexistence of a pooling equilibrium.
The Wilson concept is based on withdrawing contracts in response to
deviation, whereas the reactive concept is based on adding them.  As
a result, when Nash equilibrium does not exist, the Wilson concept
favors a pooling equilibrium, while the reactive concept favors a
separating equilibrium.



\bigskip
 \noindent
  {\bf Wilson Equilibrium and Reactive Equilibrium} (from the Signalling Chapter 10)

\noindent
  As in Insurance Game III, it is possible to go beyond Nash
equilibrium to find an equilibrium of some other kind for Education
VII.  Is it reasonable to say that a pooling equilibrium could
always be broken by a contract which draws away the $High$'s?  After
the $High$s departed, the old pooling contract, which would still
soak up all the $Low$s, would be unprofitable.  In figure 10.2, if
$C_5$ is withdrawn after $C_6$ is offered, the $Low$s prefer $C_6$ to
the zero they obtain from unemployment, and $C_6$ becomes a pooling
contract.  This is irrelevant to the question of whether $C_5$ is a
Nash equilibrium, but it might lead one to doubt the wisdom of the
equilibrium concept.

  Under the concept of the {\bf Wilson equilibrium } from section
9.5, the pooling equilibrium is legitimate, because an employer
thinking about introducing the new equilibrium-breaking contract
would realize that the new contract would be unprofitable once the
old contract was withdrawn.  {\bf Reactive equilibrium} can also be
applied, and generates a separating equilibrium.  Under its
reasoning, the separating equilibrium cannot be broken by a pooling
contract, because the pooling contract would in turn be broken by a
second separating contract.  The Wilson $C_5$ and the reactive
$(C_3,C_4)$ are the two clear candidates for equilibrium in figure
10.2.  Alternately, we could restructure the model so that the worker
moves first--- the assumption in sections 10.1 and 10.2.  While
avoiding the existence problem, that introduces the need to think
about out-of-equilibrium beliefs, and it really models a different
situation, in which workers cannot change their education in response
to employers' contracts.


\bigskip
\noindent
 {\bf N9.5} 
   {\bf Other Equilibrium Concepts: Wilson and Reactive Equilibrium}
  \begin{itemize}
 \item
  Engers \& Fernandez (1987) show how to transform a simultaneous
move game into a sequential move game such that the reactive
equilibrium of the original game is one of the perfect Bayesian
equilibria of the transformed game.
 \item
 In adverse-selection games it often matters whether the informed
player or the uninformed player offers the contract.  Wilson and
reactive equilibrium are important when the uninformed player offers
the contract, since it is only he who runs the risk of receiving
something unexpected in the transaction and he might then wish to withdraw an
offer. The issues involved are the same as in the difference between
screening and signalling, which will be discussed at length in
chapter 10.
  \end{itemize}



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