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\begin{large}
{\bf Forward Induction as Confusion over the Equilibrium
Being Played Out}\
\end{large}
26 January 1991/11 September 2007 \\
\bigskip
Eric Rasmusen\\
{\it Abstract}\\
\end{center}
The Nash equilibrium of a game depends on it being common knowledge
among
the players which particular Nash equilibrium is being played
out. This common knowledge arises as the result of some unspecified
background process. If there are multiple equilibria, it is important
that all players agree upon which one is being played out. This
paper models a situation where there is noise in the background
process, so that players sometimes are unknowingly at odds in their
opinions on which equilibrium is being played out. Incorporating
this possibility can reduce the number of equilibria in a way similar
but not identical to forward induction and the intuitive criterion.
\begin{small}
\hspace*{8pt}
Eric Rasmusen, Dan R. and Catherine M. Dalton Professor, Department of
Business
Economics and Public Policy, Kelley School of
Business, Indiana University. Visitor (07/08), Nuffield College, Oxford
University. Office: 011-44-1865 554-163 or (01865) 554-163. Nuffield
College,
Room C3, New Road, Oxford, England, OX1 1NF.
Erasmuse@indiana.edu. \url{http://www.rasmusen.org}.
Copies of this paper can be found at:
\url{http://www.rasmusen.org/papers/rasmusen-subgame.pdf}.
\hspace*{.2in} I would like to thank Emmanuel Petrakis and
participants in the University of Chicago Theory Workshop and the
University of Toronto for helpful comments.
\end{small}
%---------------------------------------------------------------
\newpage
\begin{center}
{\bf 1. Introduction}
\end{center}
The idea here will be: if players observe actions by player Smith that
are compatible with Nash equilibrium $E_1$ but not with Nash
equilibrium $E_2$, they should believe that Smith will continue to play
according to equilibrium $E_1$, even if they themselves were earlier
intending to play according to equilibrium $E_2$.
\noindent
{\bf Background: Equilibrium in the Expensive-Talk Game}
Let us use the
Expensive-Talk Game (also known as the Money-Burning Game) as an
example for discussing equilibrium concepts.
\bigskip
\noindent
{\bf Expensive-Talk Game I}\\
1. The Man chooses Talk and say ``The strategies chosen will be ({\it
Fight, Fight})'' at cost $c<2$, or choose Silence at cost 0, observed
by the Woman. \\
2. The Man and Woman simultaneously choose $Fight$ or $Ballet$, which
add the amounts in Table 1 to their payoffs. \\
\begin{center}
\begin{tabular}{lllcc}
& & &\multicolumn{2}{c}{\bf Woman}\\
& & & Fight &Ballet \\
& & & & \\
& &Fight& {\bf 3,1} & 0,0 \\
&{\bf Man}& & & \\
& & Ballet & 0,0 & {\bf 1,3} \\
& & & & \\
\multicolumn{5}{l}{\it Payoffs to: (Man, Woman).} \\
\multicolumn{5}{c}{(Nash equilibrium payoffs are in boldface.)} \\
\multicolumn{5}{c} {\bf Table 1: The Battle of the Sexes }\\
\end{tabular}
\end{center}
Move (2) is the game known as
the Battle of the Sexes, a coordination game in which the two
players wish to choose the same action but have different preferences
over which action to choose. The players are a man who wishes to go
to a prizefight and a woman who wishes to go to a ballet, both of
whom would also like to attend an event together. The game has
two pure-strategy equilibria, (F:{\it Fight, Fight}) and (B:{\it
Ballet},
{\it Ballet}), and a mixed-strategy equilibrium $M$,
in which each player picks his preferred action with probability .75.
Even in the Battle of the Sexes by itself, there are several
arguments that can narrow down the number of equilibria.
1. Exclude mixed-strategy equilibria, as being more complicated and
special. This excludes equilibrium $M$.
2. Exclude pareto-dominated equilibria, since the players would endeavor
to avoid them. This excludes equilibrium $M$, which has an expected
payoff of xxx.
3. Exclude asymmetric equilibria, in which one player has a higher
payoff than the other, since they would be harder to coordinate upon.
This excludes equilibria $F$ and $B$.
None of these principles are compelling, but for our
illustration we will adopt principle (1) and exclude mixed-strategy
equilibria, as is commonly done when pure-strategy equilibria exist.
\footnote{
If $c$ took a small value, M could be used as a
punishment to support peculiar Nash equilibria such as
TBB in the Expensive-Talk Game. The equilibrium with that outcome is
Man: TB, Woman:($B|T,M|S)$. The man is willing to bear the cost of
$Talk$ because if he deviated, he would punished with MM in the
subgame, which is even worse than BB for him if for $c<1.75$.}
The Expensive-Talk Game precedes the Battles of the Sexes with a move
in which the man can choose $Talk$ to try to convince the woman that he
will pick $Fight$ in the Battle of the Sexes subgame. This has the
following Nash equilibria (all of which are subgame perfect):
E1: (Man: S, B|S, B|T) (Woman: B|S, B|T). Outcome: SBB.
E2: (Man: S, F|S, F|T) (Woman: F|S, F|T). Outcome: SFF.
E3: (Man: S, F|S, B|T) (Woman: F|S, B|T). Outcome: SFF.
E4: (Man: T, B|S, F|T) (Woman: B|S, F|T). Outcome: TFF.
Seen as Bayesian games, though, in which players are required to behave
rationally in light of their prior beliefs as updated using Bayes's
Rule, the statements of the equilibria are incomplete. In addition to
the strategies listed, prior to the game starting the players
implicitly hold the belief that with probability 100\% the particular
equilibrium is the one the other player will play out. Thus, the
priors for E1 are:
Man: With probability 1, the Woman will respond to S by choosing B.
\\
Woman: With probability 1, the Man will choose S, after which he
will choose B.
But what is the woman to believe if she observes the man choosing Talk?
That is an impossible event, in light of her prior, and Bayes's Rule
provides no way to update a prior of 0 or 1. Thus, to fully specify the
equilibrium, the modeller must specify the woman's posterior after she
observes $Talk$. Here, an out-of-equilibrium belief that supports the
equilibrium is
Woman's out-of-equilibrium belief: If the man chooses S, then with
probability one he will choose B.
Not all beliefs support the equilibrium. An equally rational belief,
but one that does not support the equilibrium, is
Woman's out-of-equilibrium belief: If the man chooses S, then with
probability .5 he will choose B.
The concept of perfect bayesian equilibrium, in which players are
required to behave rationally in light of their prior beliefs and in
which out-of-equilibrium beliefs are specified by the modeller, is
usually applied to games of incomplete information, in which the priors
are about an initial move by Nature choosing the type of a player. It is
equally applicable to games of complete information. We implicitly
assume, however, that the out-of-equilibrium belief is whatever belief
will support the specified subgame-perfect Nash equilibrium.
In some games, however, including the Expensive-Talk Game, people
feel uncomfortable with the out-of-equilibrium beliefs.
\underline{FI equilibria.} Only SFF is an equilibrium. (1) SBB is
not an FI outcome, because it is supported only by the equilibrium
(SB,B), and in that equilibrium, TB is dominated. If we drop TB, then
B is no longer a perfect strategy for the woman: if the man chooses
TF, she must rationally respond with F, so SBB is not self-enforcing
when dominated strategies are dropped. (2) TFF is not an FI outcome
because SB is dominated in it. If we drop SB, however, then if the
man picks S, that indicates to the woman that the man has chosen SF,
so she will respond with F. Thus, TFF is not self-enforcing when
dominated strategies are dropped. (3) SFF is an FI outcome because if we
drop the man's TF, TB, or SB, or the woman's ($B|S,F|T)$ or B, that
does not stop SFF from being self enforcing.
The content of the announcement is unimportant, only its cost. Even
if the man spends 1.5 to say ``The strategies to be played out are
({\it Ballet, Ballet })'' the woman will believe that he means to
convey to her that the equilibrium is ({\it Fight, Fight}); It's not
what you say, it's how you say it. But as we have just seen, silence
can then convey the message as effectively as the announcement and
more cheaply.\footnote{The argument for the effectiveness of silence
in this kind of game can be found in Ben-Porath \& Dekel
(unpublished) and Van Damme (1989).} Given that the man can guarantee
a favorable outcome by {\it Talk}, he would refrain from talking only
if talking were unnecessary, which is true only if he thinks that the
woman will pick $Fight$ anyway. If he thinks that, he will choose
$Fight$ himself and the woman will also wish to choose $Fight$. So
the man's silence also communicates that he will choose $Fight$. The
key to the success of the ``strong, silent type'' is that he have the
the $option$ of sending a costly message; It's not what you say; it's
whether you can say it.
This line of reasoning strikes some people as so unintuitive as to
condemn the whole idea of forward induction, but the strangeness of
the result may be due to its lack of robustness with respect to
asymmetric information. If with some probability the rules of the
game do not allow the man to make announcements, the woman can
interpret silence as indicating that the man {\it cannot} talk,
rather than that he feels no need to talk, in which case TFF remains
an equilibrium. In the extended game, if Nature sends the message to
both that the equilibrium is for the man to talk if he is able and to
be silent otherwise and for the both players to choose $Ballet$ if
the man is silent, then the man cannot induce the woman to choose
$Fight$ by staying silent. A similar argument for the equilibrium
status of TFF could be based on the woman not knowing the man's
preference exactly, assigning some probability between 0.5 and 1 that
the man prefers the prizefight. A TFF equilibrium would then exist in
which the man chooses $Talk$ if and only if he prefers the
prizefight. In the extended game, if the woman received the message
for that TFF equilibrium, she would interpret the man's silence to
mean that he preferred the prizefight; if he actually prefers the
ballet, he must talk. Thus, with even a little asymmetric
information it is hard to rule out TFF and forward induction only
rules out SBB.\footnote{This argument can be found in footnote X of
Van Damme (1989).}
2. Give my new definition.
1. Determine the pure-strategy Nash equilibria of the game. Here, their
outcomes are E1: SFF, E2: SBB, and E3: TFF. In full, these include
out-of-eq. beliefs too.
E1: (Man: S, F|S, F|T, OOB: T means both players will choose F) (
Woman: F|S, F|T, OOB: T means both players will choose F).
2. Nature begins the game by sending each player a message of which
equilibrium is to be played out: E10, E20, etc.
3. With probability (1-epsilon), all players hear
the same message. With probability epsilon, the messages differ. We
will not restrict the way they might differ, because we will look for
equilibria robust to all possibilities. For example, we could assume
that if E10 is the high-probability message, then
A. With probability epsilon, Players 1, 2, 3 hear E10 but Player 4 hears
E20, E30, or E40 with 1/3 probability each.
or
B. With probability epsilon, Players 1 hears E10, Players 2 and 3 hear
E20, and Player 4 hears E30.
4. We will only consider Nash equilibria of the metagame in which
players play out the equilibrium they hear. (This is not the same as
assuming that in the main game they play the equilibrium they hear:
doing so must be a Nash equilibrium of the metagame too.) Such
equilibria definitely exist, but other equilibria exist too, such as
``Play E40 regardless of the message you hear.''
5. If one of the set E10, E20, etc. now fails to be Nash equilibria in
the metagame for some possible epsilon-specification, drop it.
6. Iterate using the surviving set E11, E21, etc.
\underline{Confusion-proof Equilibria in ET I.} SFF is the only
equilibrium. (1)
Suppose SBB were a confusion-proofness equilibrium and the man deviated
from it by
picking T. The woman concludes that either she misheard
(and nature chose TFF), or the man misheard (and
nature chose SBB). In either case, the man will pick F, so the woman
does also. The man's deviation was profitable. (2) Having disposed of
SBB, suppose the equilibrium outcomes are TFF and SFF. If Nature
sends the message TFF to each player, but the man remains silent,
the woman deduces that he received SFF and she chooses F. The man
can safely choose F himself, and the deviation has been profitable to
the man, so TFF is broken as an equilibrium outcome.\footnote{I ruled
out mixed-strategy equilibria, but that is not a necessary part of the
concept. Suppose we allowed tehm. Then, there are five additional Nash
equilibria, (T, M|S, M|S, B|T, B|T), (T, M|S, M|S, F|T, F|T), (S, F|S,
F|S, M|T, M|T), (S, B|S, B|S, M|T, M|T), (S, M|S, M|S, M|T, M|T). CP
will knock out the Silence equilibria. CP doesn't help more, even with
iteration. FI probably does. }
\bigskip
\noindent
Expensive-Talk Game II: Incomplete Information\\
1. Nature chooses F or B for the man, unobsreved by woman.\\
2. The Man chooses Talk, at cost $c$, or Silence, at cost 0. \\
3. The Woman chooses $Fight$ or Ballet. \\
We clearly need PBE for this.
Consider variant 3, the transition between the two, if these seem
unreasonable. The order of moves is:
\bigskip
\noindent
Expensive-Talk Game III: Post-Start Asymmetric Information\\
1. Man chooses F or B, unobsreved by woman.\\
2. The Man chooses Talk, at cost $c$, or Silence, at cost 0. \\
3. The Woman chooses $Fight$ or $Ballet$. \\
In choosing a particular strategy combination to be a game's
equilibrium, three criteria are generally accepted. First, the
strategy combination ought to be Nash--- every player's strategy
should be a best response to the other players' strategies. Second,
it ought to be subgame perfect---for every subgame the relevant
portions of the strategy combination should be Nash. Third, it should
be a perfect Bayesian equilibrium---remaining Nash when the players
follow Bayes Rule and some set of out-of-equilibrium beliefs assigned
by the modeller.
The Nash property is the most fundamental of these three criteria,
and it is attractive because of its consistency---every player's
behavior and beliefs are consistent with every other player's. But
there can be multiple Nash equilibria, and with multiple equilibria
this consistency is a less compelling argument for Nash equilibrium.
In equilibrium X a player may not be able to profit by deviating from
action $a_x$, and in equilibrium Y he may not be able to profit by
deviating from $a_y$, but how is he to know that the other players
are playing X and not Y? Anyone who finds Nash equilibrium plausible
must believe in some unmodelled background process by which the
players come to know which particular equilibrium is being played out.
When there is just one Nash equilibrium, the background process can
be simply that players realize that only one combination of
strategies is consistent, but when there are many equilibria the
background process is more mysterious. Games with multiple
equilibria are common. Signalling games, in particular, are prone to
multiple equilibria, because the concept of perfect Bayesian
equilibrium allows many out-of-equilibrium beliefs, and a host of
equilibrium refinements have been developed to reduce the number of
equilibria (see Kohlberg [unpublished] or Van Damme [1987] for
references). But multiple equilibria are a problem even in games of
symmetric information, where the multiplicity does not arise from
ignorance of the players' ``types,'' but from ignorance of what
actions they have taken in the course of the game. The most common
examples are coordination games, in which the players wish to
coordinate with each other by choosing the same actions. It would be
desirable to find an equilibrium refinement that would have bite
whether the information is asymmetric or not, and the presence of the
problem in games of symmetric information suggests that it is not
just a problem of out-of-equilibrium beliefs about types.
The great contribution of Harsanyi (1967) was to suggest that
games of incomplete information, in which a player is not sure
whether he is playing in game X or game Y, could be remodelled as
games of complete information in which Nature moves first and selects
X or Y with known probabilities. Could the same be done for a game in
which the players are uncertain over whether they are playing out
equilibrium X or equilibrium Y? To do so would contradict a
fundamental assumption of Nash equilibrium, that the equilibrium
being played out is common knowledge and all the players' strategies
are consistent, and if the probabilities of both X and Y are
substantial this would not lead to sensible results. But we could
hope that an equilibrium would be robust to having a little bit
of uncertainty about whether it is X or Y that is being played,
especially since we are somewhat hazy on how it becomes common
knowledge which particular equilibrium is to be played out. This,
after all, is the hope behind the ``tremble'' justification for
perfect equilibrium, and it is easier to imagine players being
confused over the difficult problem of deciding which equilibrium is
being played than to imagine them blundering by choosing obviously
unprofitable actions. As with trembles in actions, uncertainty over
the equilibrium might reveal some Nash equilibria not to be robust,
thus providing a way to refine the equilibrium concept. If the
modeller believes that there is any uncertainty in the minds of
players as to which equilibrium is being played out, his preferred
equilibria should be robust to the presence of that uncertainty
Besides the Tremble and Asymmetric Information approaches the
refinements there is a third approach, the Equilibrium Forcing approach.
In it, a player chooses an action to indicate to the other players that
he intends to play a particular strategy, in an effort to force them to
change their expectations.
The present article takes the Asymmetric Information line of attack. To
represent the
background process of equilibrium selection, two initial moves by
Nature will be added to the start of whatever game is under
consideration: a first move in which an equilibrium X is selected and
a second in which each player is sent a message announcing the
equilibrium. With a small probability, the message is not X but some
other equilibrium. The players must therefore take into account the
possibility of confusion over which equilibrium is being played out.
This provides an opportunity for their beliefs about the equilibrium
of the original game to change and to be manipulated by deviations.
The new refinement's implications are close but not identical to
those of the existing idea of
``forward induction,'' an Equilibrium Forcing idea. Elon Kohlberg \&
Mertens (1986) introduced forward
induction as part of the refinement they call ``stability,'' and it
is used in the ``intuitive criterion'' which In-Koo Cho \& David Kreps
(1987)
apply to signalling games. Eric Van Damme (1989) and a variety of
unpublished studies---by Kohlberg, Okuno-Fujiwara \& Andrew
Postlewaite, Martin
Osborne, and Ben-Porath \& Dekel--- have shown that forward induction
has interesting implications even without the other criteria that
compose ``stability'' and that forward induction can reduce the
number of equilibria even in games of symmetric information.
Forward induction has been commonly defined in terms of iterated
deletion of dominated equilibrium strategies but commonly justified
in terms of logical deductions made by the players. The intuition is
that after player Smith takes an action that could benefit him if and
only if player Jones held belief Y, Jones, realizing this, will adopt
belief Y. Such an intuition violates the fundamental assumption of
Nash equilibrium---that the information structure and the identity of
the equilibrium to be played out are common knowledge among the
players. Defining forward induction in terms of iterated dominance
skirts around that issue; $FI'$ will meet it head on.
\bigskip
%---------------------------------------------------------------
\noindent
{\bf The Problem to be Addressed: The Expensive-Talk Game and the
Twice-Repeated Battle of the Sexes}
To show the problem, I will lay out two variants of the
Battle of the Sexes, a coordination game in which the two
players wish to choose the same action but have different preferences
over which action to choose. The players are a man who wishes to go
to a prizefight and a woman who wishes to go to a ballet, both of
whom would like to attend an event together. They make their choices
simultaneously, and the payoffs are shown in Table 2. The game has
two pure-strategy equilibria, ({\it Fight, Fight}) and ({\it Ballet},
{\it Ballet}).\footnote{There is also a mixed-strategy equilibrium,
in which each player picks his preferred action with probability .75.
If we denote the equilibrium mixed strategy for the Ballet-Fight
subgame as M, then if $c$ took a small value, M could be used as a
punishment to support peculiar perfect Bayesian equilibrium such as
TBB in the Expensive-Talk Game. The equilibrium with that outcome is
Man: TB, Woman:($B|T,M|S)$. The man is willing to bear the cost of
$Talk$ because if he deviated, he would punished with MM in the
subgame, which is even worse than BB for $c<1.75$.}
\begin{center}
\begin{tabular}{lllcc}
& & &\multicolumn{2}{c}{\bf Woman}\\
& & & Fight &Ballet \\
& & & & \\
& &Fight& {\bf 3,1} & 0,0 \\
&{\bf Man}& & & \\
& & Ballet & 0,0 & {\bf 1,3} \\
& & & & \\
\multicolumn{5}{l}{\it Payoffs to: (man, woman).} \\
\multicolumn{5}{c}{(Nash equilibrium payoffs are in boldface.)} \\
\multicolumn{5}{c} {\bf Table 2: The Battle of the Sexes.}\\
\end{tabular}
\end{center}
\begin{small}
\begin{center}
\begin{tabular}{lllcccc}
& & &\multicolumn{4}{c}{\bf Woman}\\
& & & F &B & F$|$T, B$|$S & B$|$T, F$|$S \\
& & & & & & \\
& &Silence, Fight (SF)& {\bf 3,1} & 0,0 & 0,0 & {\bf 3,1}
\\
& & & & & & \\
& & Silence, Ballet (SB) & 0,0 & {\bf 1,3} & 1,3 & 0,0 \\
&{\bf Man}& & & & & \\
& &Talk, Fight (TF)& 3-c,1 & -c,0 & {\bf 3-c,1} & -c,0
\\
& & & & & & \\
& & Talk, Ballet (TB) & -c,0 & 1-c,3 & -c,0 & 1-c,3 \\
& & & & & & \\
\multicolumn{6}{l}{(Pure-strategy perfect equilibria when $c=1.5$
are in boldface.)} & \\
& \\
\multicolumn{6}{c} {\bf Table 3: The Expensive-Talk Game} & \\
\end{tabular}
\end{center}
\end{small}
{\it Variant 1: The Expensive-Talk Game}. This variant precedes the
simultaneous-move Battle of the Sexes with a single costly
announcement by the man. If he chooses to talk, his announcement
costs him $c=1.5$ units of payoff (in contrast to the ``cheap-talk''
of Farrell [1987] in which $c=0$).
The perfect equilibrium pure-strategy outcomes
are SFF, SBB, and TFF.
The problem is with the SBB equilibrium. In it, the woman expects the
man to be silent, because it would be irrational for him to uselessly
incur a cost, given that his message would not persuade her that he
really mean to choose $Fight$. But what if he {\it does} choose {\it
Talk}? What should the woman think? Within the model this is an
impossible event. When something impossible within a person's frame of
reference occurs in the real world, ordinarily one does not simply blink
and go on as if no miracle had occurred. Rather, one rethinks one's
frame of reference.
One reaction would be for the woman to question whether she really heard
a message or was hallucinating. Another would be to question whether the
man was rational. A third would be to suppose that the man had chosen
$Talk$ inadvertently, by a ``tremble''. All three of these would lend
some support-- but not confidence--- to $Ballet$ as a best response
for the Woman.
A fourth reaction would be that the man was trying to indicate to her
that he was trying to shift her expectation for his second-period action
to $Fight$ and that he expected to succeed in that attempt, so he was
going to choose $Fight$ himself. In that case, $Fight$ is the woman's
best response.
A fifth reaction would be that the man mistakenly thought that TFF was
the equilibrium both players were supposed to be playing out. Thus, he
thought that if he chose $Silence$ then the woman would choose $Ballet$,
but if he chose $Talk$ she would choose $Fight$. If that is what he
believes, the woman had better choose $Fight$.
The fourth idea--- Equilibrium Forcing-- is the idea behind Forward
Induction as conventionally defined. The fifth idea-- Incomplete
Information about Equilibrium Expectations-- is what I want to pursue in
this paper.
\underline{Discussion}.
The content of the announcement is unimportant, only its cost. Even
if the man spends 1.5 to say ``The strategies to be played out are
({\it Ballet, Ballet })'' the woman will believe that he means to
convey to her that the equilibrium is ({\it Fight, Fight}); It's not
what you say, it's how you say it. But as we have just seen, silence
can then convey the message as effectively as the announcement and
more cheaply.\footnote{The argument for the effectiveness of silence
in this kind of game can be found in Ben-Porath \& Dekel
(unpublished) and Van Damme (1989).} Given that the man can guarantee
a favorable outcome by {\it Talk}, he would refrain from talking only
if talking were unnecessary, which is true only if he thinks that the
woman will pick $Fight$ anyway. If he thinks that, he will choose
$Fight$ himself and the woman will also wish to choose $Fight$. So
the man's silence also communicates that he will choose $Fight$. The
key to the success of the ``strong, silent type'' is that he have the
the $option$ of sending a costly message; It's not what you say; it's
whether you can say it.\footnote{ This line of reasoning strikes some
people as so unintuitive as to
condemn the whole idea of forward induction, but the strangeness of
the result may be due to its lack of robustness with respect to
asymmetric information. If with some probability the rules of the
game do not allow the man to make announcements, the woman can
interpret silence as indicating that the man {\it cannot} talk,
rather than that he feels no need to talk, in which case TFF remains
an equilibrium. In the extended game, if Nature sends the message to
both that the equilibrium is for the man to talk if he is able and to
be silent otherwise and for the both players to choose $Ballet$ if
the man is silent, then the man cannot induce the woman to choose
$Fight$ by staying silent. A similar argument for the equilibrium
status of TFF could be based on the woman not knowing the man's
preference exactly, assigning some probability between 0.5 and 1 that
the man prefers the prizefight. A TFF equilibrium would then exist in
which the man chooses $Talk$ if and only if he prefers the
prizefight. In the extended game, if the woman received the message
for that TFF equilibrium, she would interpret the man's silence to
mean that he preferred the prizefight; if he actually prefers the
ballet, he must talk. Thus, with even a little asymmetric
information it is hard to rule out TFF and forward induction only
rules out SBB. This argument can be found in footnote X of
Van Damme (1989).}
\bigskip
\noindent
{\it Variant 2: The Twice-Repeated Battle of the Sexes}. Let the
Battle
of the Sexes be repeated twice, with no possibility of announcements.
The players know the outcome of the first repetition when
choosing their moves for the second.
The perfect equilibrium pure-strategy outcomes are BB-BB, BB-FF, FF-
FF, and FF-BB.
Here, Equilibrium Forcing will eliminate BB-FF and FF-BB, while
Incomplete Information about Equilibrium Expectations will not eliminate
any of the equilibria.
\bigskip
{\it The Expensive-Talk Game}. The FI and $FI'$ outcomes are SFF.
This game
will illustrate two steps of iteration of the extended game, and the
idea that the absence of a message can be just as meaningful as a
message.
\underline{FI equilibria.} Only SFF is an equilibrium. (1) SBB is
not an FI outcome, because it is supported only by the equilibrium
(SB,B), and in that equilibrium, TB is dominated. If we drop TB, then
B is no longer a perfect strategy for the woman: if the man chooses
TF, she must rationally respond with F, so SBB is not self-enforcing
when dominated strategies are dropped. (2) TFF is not an FI outcome
because SB is dominated in it. If we drop SB, however, then if the
man picks S, that indicates to the woman that the man has chosen SF,
so she will respond with F. Thus, TFF is not self-enforcing when
dominated strategies are dropped. (3) SFF is an FI outcome because if we
drop the man's TF, TB, or SB, or the woman's ($B|S,F|T)$ or B, that
does not stop SFF from being self enforcing.
\underline{$FI'$ equilibria.} SFF is the only equilibrium. (1)
Suppose SBB were an $FI'$ equilibrium and the man deviated from it by
picking T. The woman concludes that with probability .5 she misheard
(and nature chose TFF), and with probability .5 the man misheard (and
nature chose SBB). In either case, the man will pick F, so the woman
does also. The man's deviation was profitable. (2) Having disposed of
SBB, suppose the equilibrium outcomes are TFF and SFF. If Nature
sends them message TFF to each player, but the man remains silent,
the woman deduces that he received SFF and she chooses F. The man
can safely choose F himself, and the deviation has been profitable to
the man, so TFF is broken as an equilibrium outcome.
\underline{FI equilibria.} Only BB-FF and FF-BB remain. Consider
BB-BB, which gives the man a payoff of 1+1. The strategy for the man
of playing Fight in the first round and Ballet in the second is
dominated in this equilibrium; it could not profit him even if the
woman also deviated in the second round. But when that strategy is
eliminated, then if the man plays Fight the woman can conclude that
he is playing the strategy of Fight in the first round and Fight in
the second round. She will respond by switching to Fight in the
second round, and the outcome will be FB-FF, which gives the man a
payoff of 0+3. The man's deviation is profitable, so BB-BB cannot be
an equilibrium outcome. Since the game is symmetric, FF-FF cannot be
an equilibrium outcome either.\footnote{I have taken this example
from Van Damme (1989).}
BB-FF, on the other hand, is an FI equilibrium. Deviation by the man
in the first round could never be profitable, because he obtains his
desired outcome in the second round even without deviation. Deviation
by the woman in the first round is unprofitable because she would
have to give up the BB payoff of that round.
\underline{$FI'$ equilibria.} BB-BB, BB-FF, FF-FF, and FF-BB are all
$FI'$ equilibria. Suppose that BB-BB were the equilibrium chosen by
Nature in the extended game, and that both players received the
message without garbling, but the man deviates to Fight in the first
round. The woman's equilibrium interpretation of this is that the man
heard either FF-FF or FF-BB from Nature, with equal probability.
The expected value of her responding with Fight is therefore 0.5(1)
+ 0.5(0), and the expected value of responding with Ballet is 0.5(0)
+ 0.5(3). She responds with Ballet, and the man's deviation from
BB-BB is unprofitable to him. Parallel reasoning shows that FF-FF is
an $FI'$ equilibrium.
\underline{Discussion.} This example distinguishes two intuitions
that are at work in forward induction. The $FI'$ intuition is that
players are uncertain over which equilibrium is being played out, and
a deviation shows lack of synchronization. The FI intuition is that
players might be trying to disrupt the normal play of the game. If
BB-BB is the agreed equilibrium, the man might nonetheless deviate
with Fight in the first round, an action which conveys the message,
``I know we were supposed to play BB-BB, but I prefer FF, and I think
I can make you choose Fight in the second round. To show my
conviction, I have played Fight in the first round, an action which
would be worse than useless if it did not convince you to switch to
Fight. Maybe I am wrong and you will choose Ballet anyway, but I
myself am choosing Fight again.''
It does not matter why the man has this belief that he can induce the
woman to switch to Fight; whatever his reasoning, if he himself is
convinced by it he will be choosing
Fight in the second round, and that makes Fight the best response for
the woman.
In $FI'$ the player is taking strategic advantage of
uncertainty in the background process that chooses the equilibrium to
be played out, whereas in FI the player is taking strategic advantage
of the background process by which players decide how to respond to
what are known to be deviations from rationality.
%---------------------------------------------------------------
\newpage
\bigskip
\begin{center}
{\bf 3. Uncertainty over the Equilibrium to be Played Out }
\end{center}
Let us first define a ''metagame''.
Metagame $(F, C)$ is a game which precedes the original game with a
move in which Nature uses a distribution function $F(C)$ to select one
of the $N \geq 1$ strategy combinations $C$ with positive
probability and send a message to each player describing that single
chosen strategy combination. With infinitesimal probability
$\varepsilon$, however, a player is sent a message chosen from one of
the other $(N-1)$ strategy combinations using distribution $G$. (we
could make $\varepsilon$, and $G$ different for each $F$ if we
wanted).
Nash assumption: An equilibrium is a strategy combination in some
metagame in which no player has incentive to deviate from the
strategies in his message, given that he expects the other players not
to deviate.
A metagame could have some non-Nash strategy S as the only one in C,
but S would not be an equilibrium, because some player would deviate
from it. The $G$ assumption plays no role in Nash equilibrium. (or does
it-- what about weak domination?)
FI assumption: If a strategy combination $S$ is an equilibrium, then for
any given metagame $(F', C')$, $S$
could be included in $C'$ and $F'$ adjusted so that no player has
incentive to deviate from the strategies in his message, given that he
expects the other players not to deviate.
If we start with Silence, Prizefight, Prizefight as the only
element of C, and add
Talk, Prizefight, Prizefight, then under any F, the man will want to
deviate to SPP. So TPP is not an equilibriumj.
If we start with C= (SPP, TPP) and add SBB, the man will want to
deviate to TPP. So SBB is not an equilibrium.
If we start with C = (SBB, TPP) and add TFF, TFF is still viable. This
is true for any metagame if we are allowed to choose F (or even not, in
this case), so TFF is an eq.
Use the Van Damme example, with just two players in an expensie talk BS
game, in this section.
Then, do the twic-repeated PD.
Global games should be discussed. It is a stronger concept, working even
in a static game. There, though, it works only becuase of a continuou
strategy space, I think.
$FI'$ distinguishes between out-of-equilibrium moves and
out-of-every-equilibrium moves, a distinction also made by the
concept of ``forward induction''. Kohlberg (1989) says that forward
induction requires players to make ``deductions based on the
opponents' rational behavior in the past'' (p. 5), and that it is a
special case of the principle that ``a self-enforcing norm must be
robust to the elimination of a strategy which is certain not to be
employed where that norm is established'' (p. 9). He discusses a
variety of criteria that could go into the idea of forward induction,
of which the most basic is:
\noindent
{THE FORWARD-INDUCTION REQUIREMENT (FI)}: {\it A self-enforcing
outcome must remain self-enforcing when a strategy is deleted which
is inferior (i.e., not a best reply) at every equilibrium with that
outcome.}
FI and $FI'$ both deal with what happens when an action is
observed which is inferior at the current equilibrium. But making
deductions and deleting inferior strategies are not necessarily the
same idea. In some games $FI$ and $FI'$ reach the same conclusions,
but in others they do not. In Joint Embezzlement they differ. FI
does not refine perfect Bayesian equilibrium at all there, while
$FI'$ eliminates equilibria $E_{2a}$ and $E_{2b}$, in which the
outcome is $Fire$. $E_2a$ and $E_{2b}$ are the only equilibria with
that outcome, and the boss's $Hire$ is not a best reply, but if
$Hire$ is deleted from the game, it makes no difference to the
equilibrium. FI therefore has no effect.
\noindent
{\it 3.1. Three Players in the Game: Joint
Embezzlement}
A boss must decide whether to $Hire$ or $Fire$ two workers, Smith
and Jones. Smith and Jones then simultaneously choose whether to
$Work$ or $Steal$. Figure 1 shows the payoffs for the entire game in
extensive form, and Table 1 shows the payoffs in the Smith-Jones
subgame. If the boss chooses $Fire$, the payoffs are (0,0,0), and
the strategies of Smith and Jones are irrelevant. If the boss chooses
$Hire$, then whether Smith and Jones work or steal does matter. If
both work, then output is high and the workers receive wages, for
payoffs of (3, 2, 2). If both steal, their theft is successful, for
payoffs of $(-2,4,4)$. If Smith works and Jones steals, then output
is moderate, Smith gets his wage plus a small bonus from turning in
Jones, and Jones goes to jail, for payoffs of $(-1, 3, -6)$.
\begin{center}
{\bf Figure 1: Joint Embezzlement.}
\end{center}
\begin{center}
\begin{tabular}{llrcc}
& & &\multicolumn{2}{c}{\bf Jones}\\
& & & Work & Steal \\
& & & & \\
& & Work& {\bf 2,2} & $3,-6$ \\
&{\bf Smith}& & & \\
& & Steal & $-6,3$ & {\bf 4,4} \\
& & & & \\
\multicolumn{4}{l}{\it Payoffs to: (Smith, Jones).} &\\
\end{tabular}
{\bf Table 1: The Coordination Subgame from Joint Embezzlement.}
\end{center}
An ``equilibrium'' is a strategy combination: one strategy for each
player, chosen according to some rule favored by the modeller. The
usual rule is that the strategy combination be a ``perfect Bayesian
equilibrium'': the strategies are best responses to each other, the
players follows Bayes's rule when possible, with beliefs specified by
the modeller where Bayes' rule does not apply, and players'
strategies must remain best responses regardless of the past history
of the game. An ``equilibrium outcome'' is a path through the game
tree generated by an equilibrium. There may be multiple equilibria,
but only one can be played out in a given realization of the game;
let us denote the equilibrium being played out as the ``realized
equilibrium''.
The perfect Nash equilibria for Joint Embezzlement
are:\footnote{There are also two non-perfect Nash equilibria:\\
\hspace*{18pt}$E_3:(Fire,Work,Steal)$ \\
\hspace*{18pt}$E_4:(Fire,Steal,Work)$ \\
}
$E_1: (Hire,Work,Work)$ with payoffs (3,2,2) \\
$E_{2a}: (Fire,Steal,Steal)$ with payoffs (0,0,0)\\
$E_{2b}$:{\it (Fire, Work with probability 1/9, Work
with probability 1/9})\\
To reduce the number of equilibria further one needs to go beyond
the generally accepted equilibrium concepts. The idea that will be
modelled below is that $E_1$ should be the only equilibrium, because
if the workers think that the realized equilibrium is $E_{2a}$ or
$E_{2b}$ (which have the same outcome), but then observe the boss
choosing $Hire$, each worker should become worried that the other
worker is going to play according to what the boss seems to think is
the realized equilibrium, $E_1$.\footnote{ This is a different
problem from that of a boss with employees who collude to avoid a
prisoner's dilemma subgame (e.g., Tirole, 1986). In Joint
Embezzlement, the workers' problem is to coordinate on a Nash
equilibrium, which is self-enforcing. Non-binding contracts between
Smith and Jones might be important if there were no communication
move, but if there is, then making the contract binding has no
marginal effect. When Smith and Jones are in a tournament against
each other, on the other hand, and they wish to collude,
communication makes no difference. A non-binding contract would be
useless, but making the contract binding would have a big marginal
effect, since then both workers could trust each other to exert low
effort.}
Suppose it is common knowledge that $E_2$ is the realized
equilibrium of Joint Embezzlement that is being played out. If the
boss makes the off-equilibrium move of $Hire$, what is Smith to
think? The boss choosing $Hire$ is a zero-probability event. One
interpretation Smith might make is that the boss made a random
mistake, in which case Smith's beliefs about the remainder of the
game are unchanged. This is the response known as ``passive
conjectures'' in signalling games. Alternatively, Smith might make a
different interpretation: that he himself is confused and the
equilibrium being played out by the other players is $E_1$, not
$E_2$. If it is impossible for the boss to move accidentally, this is
plausible, because the boss would be strictly worse off choosing
$Hire$ if the realized equilibrium were $E_2$. The boss's move is a
credible indicator of his confidence in $E_1$. If Smith believes
this, he believes that Jones will pick $Work$, so $Work$ is Smith's
own best response. Jones can either use the same reasoning or know
that Smith is following it; either way, Jones will pick $Work$ too.
If Smith and Jones behave according to this logic, the boss will
certainly pick $Hire$ and $E_1$ is the only equilibrium that will
ever be observed.
It seems here that out-of-equilibrium behavior changes a player's
belief about which equilibrium is realized, which is possible only if
he does not assign a prior of 100 percent to a single strategy
combination being the realized equilibrium. A fundamental assumption
behind Nash equilibrium is that the structure of the game and the
identity of the realized equilibrium are common knowledge, an
assumption which is violated if we specify that players put positive
probability on more than one equilibrium being realized. Harsanyi
(1967) showed that this assumption need not prevent the modeller from
introducing asymmetric information about the structure of the game:
the modeller simply adds a move at the start of the game in which
Nature randomly chooses the game's structure, observed by some but
not all players, using probabilities that are themselves common
knowledge. Something similar can be done here to allow opinions to
differ on which equilibrium is realized.
Let us follow Harsyani's approach of reformulating a conventionally
unanalyzable game into one that can be analyzed using standard
methods. First, let us construct an ``extended game.'' The modeller
begins by deciding which strategy combinations he considers to be
candidates for equilibrium, using whatever criteria he finds
plausible. Usually he will choose the set of perfect Bayesian
equilibria, but other criteria might also be available; e.g., mixed
strategies might be ruled out as implausible. Denote the outcomes of
the strategies that survive this process as the set of {\it plausible
equilibrium outcomes,} $E_i, i = 1,\ldots m$.
The {\it extended game} is the original game preceded by two moves
by Nature. In the first move, Nature picks equilibrium outcome $E_i$
with probability $f_i$, where $\sum_i f_i = 1$, unobserved by the
players. In the second move, Nature sends a separate private message
to each player $j$. With probability $(1-\sum_k \epsilon_{ijk})$ the
message is $E_i$, and with probability $\epsilon_{ijk}$ it is $E_k$,
where $k \neq j$ and $\sum_k \epsilon_{ijk}$ is an arbitrarily small
probability.\footnote{The natural extension to games with continua of
equilibrium outcomes is to make $f_i$ into a density $f(i)$ and
$\epsilon_{ijk}$ into a function $\epsilon_j(ik)$ that integrates
over $i$ and $k$ to an arbitrarily small probability.}
ASSUMPTION 1: Player $i$ believes that player $j$ will follow the
equilibrium revealed to $j$ by Nature, X, until $j$ discovers that
some other player has deviated from X. This belief is common
knowledge.
If any of the plausible equilibria fail to be subsets of perfect
Bayesian equilibrium outcomes in the extended game, drop them and
begin again with the smaller revised set of plausible equilibria.
Iterate this process until all $m$ equilibrium outcomes are subsets
of perfect Bayesian equilibrium outcomes of the extended game.
{\it An $FI'$ equilibrium is a strategy combination for the original
game that generates an outcome which is a subset of a perfect
Bayesian equilibrium outcome of the extended game once iteration has
proceeded as far as possible.}
The values of $m$, $f_i$ and $\epsilon_{ijk}$ are chosen by the
modeller. They represent the modeller's prior knowledge about the
probability of different equilibrium outcomes. The particular values
chosen affect the set of $FI'$-equilibria in some but not all games,
as will be seen later. In the absence of special information, the
natural values are the uniform ones: $f_i = 1/m$ and $\epsilon_{ijk}
= \epsilon/(m-1)$ (so that $\sum_k \epsilon_{ijk}= \epsilon$).
Unless otherwise specified, these uniform values will be used in the
examples.
Since $\epsilon_{ijk}$ is arbitrarily small, it leaves behavior
unaffected except for the potential to change players' beliefs about
which equilibrium is realized. Its only
effect is to allow Bayes' rule to interpret actions that would
otherwise be out-of-equilibrium. Unlike the case where players makes
mistakes because of ``trembling hands,'' $FI'$-equilibrium does not
allow Bayes' rule to interpret every possible action, only actions
that are equilibrium actions for {\it some} plausible equilibrium.
Assumption 1 says that every player believes that the other players
are deciding what is the realized equilibrium based on Nature's
message. If a player receives message $E_i$, his belief is that
$E_i$ is going to be played by the other players unless they have
received different messages. Since the probability they have
received different messages is arbitrarily low, he will play $X$ at
least until he sees a deviation by another player. Thus, the
behavior rules of the players under Assumption 1 are consistent with
each other. Assumption 1 may seem novel, but it is in fact a weakened
version of a standard but implicit assumption that Nash equilibrium
requires for Nash behavior to be utility-maximizing:
ASSUMPTION $1'$: Player $i$ believes that player $j$ will follow the
equilibrium, X, revealed to $j$ by Nature. This belief is common
knowledge.
The initial moves by Nature together with Assumption 1 represent
the background process by which the players arrive at common
knowledge of the realized equilibrium. The process could be evolutive
or eductive: it might be some kind of pre-game communication, or
history, or psychological drives towards focal points, and it
determines the probabilities $f_i$ with which Nature chooses each
equilibrium. Game theory has not yet determined this process, but
Nash equilibrium implicitly assumes that it exists, and existence is
all that we require. What is important is that somehow the process
selects one strategy combination to be the realized equilibrium, and
that the process contains a little noise. The ordinary concept of
Nash equilibrium effectively relies on Assumption 1 or 1$'$ and on
Nature choosing $\epsilon=0$, but this structure is concealed by
using using a reduced form: the modeller just picks a strategy
combination and asks whether any player would deviate unilaterally,
without inquiring as to why that strategy combination was chosen. The
approach I suggest brings this structure into the open, and what is
new is the addition of a little noise.
This is not to be confused with the introduction of ``cheap talk''
into a game. The purpose of introducing Nature's moves is not to see
what would happen if players could try to change the course of the
game by costless communication. Nature's moves are not literal
moves, but a heuristic to represent the background process by which
the players choose which equilibrium they are going to play out.
Allowing the players to ignore Nature's move, treating it as if it
were ``cheap talk'' by a genuine player, would defeat the purpose of
putting Nature's move into the game, since we would then require
either (a) a new representation for the background process or (b) a
return to the old assumption that the equilibrium being played out is
common knowledge. It would be like saying that the players in a
Bayesian game ought to be allowed to ignore the modeller when he
tells them what priors they ought to hold. The expanded game is not
so much adding assumptions to the game as replacing the assumption
that the equilibrium being played out is common knowledge with the
assumption that the background process represented by Nature's moves
is common knowledge. What is truly new is not Assumption 1 and
Nature's moves {\it per se}, but the possibility that Nature makes
different announcements to different players.
$FI'$-equilibrium is thus based on the idea that if each player
believes that every other player follows the rule of playing out the
current equilibrium ordained by the background process, he too should
be willing to follow the rule. It rules out equilibria in which a
player would profit by unilateral deviation. This would not reduce
the number of equilibria in a game if the background process were
perfectly coordinated, but if it is common knowledge that
occasionally players come to different beliefs about which
equilibrium is realized, then this has the potential to reduce the
number of equilibria, as will be seen in the case of the game Joint
Embezzlement.
%---------------------------------------------------------------
To apply $FI'$ to Joint Embezzlement, start with the two perfect
equilibria $E_1$ and $E_2$.\footnote{Strictly speaking, $E_2$ is one
equilibrium outcome, which results from the two equilibria $E_{2a}$
and $E_{2b}$.} Do they form a set of $FI'$ equilibria? The first
iteration of the extended game is:
(1) Nature chooses $E_1$ with probability 0.5 and $E_2$ with
probability 0.5.\\
(2) Nature sends messages to the boss, Smith, and Jones. In each
case the message is the equilibrium selected in move (1) with
probability $(1-\epsilon)$ and the unselected equilibrium with
probability $\epsilon$.\\
(3) The boss chooses $Hire$ or $Fire.$\\
(4) Smith and Jones simultaneously choose $Work$ or $Steal$.
$E_1$ is an $FI'$ equilibrium because the boss has no incentive to
deviate by choosing $Fire$ regardless of the effect of that deviation
on beliefs. If he chooses $Fire$, that might induce the workers to
choose ({\it Steal, Steal}) under the belief that the boss received
the message ``$E_2$'' from Nature, but that deduction gives the boss
no incentive to choose $Fire$.
$E_2$ is not an $FI'$ equilibrium. Suppose that it were, that
Nature had chosen $E_2$ to be the equilibrium, and that Nature sent
the message $E_2$ to all three players without garbling. If the boss
deviated by choosing $Hire$, how would Smith respond? Interpreting
$Hire$ as equilibrium behavior, Smith would know that one of two
things happened: Nature chose $E_1$ to be the equilibrium and sent
the true message to the boss (and most likely to Jones) but sent
Smith the garbled message $E_2$; or Nature chose $E_2$ to be the
equilibrium but sent the boss the garbled message $E_1$. Either the
boss or Smith has been sent a garbled message, with equal
probability. If the boss received a garbled message and $E_2$ is the
equilibrium, then Smith should choose $Steal$, because it is almost
certain that Jones will choose $Steal$, in which case $Steal$ yields
a payoff of 4 and $Work$ yields only 3. If the boss received a
correct message and $E_1$ is the equilibrium, then Smith should
choose $Work$, because it is almost certain that Jones will choose
$Work$, in which case $Work$ yields a payoff of 2 and $Steal$ yields
$-6$. Given the asymmetric losses and the equal probabilities of
mistakes by Smith and the boss, Smith should choose $Work$. The
boss, foreseeing this, would choose $Hire$, so equilibrium $E_2$ is
broken.
It was assumed above that the probabilities with which Nature
selects each equilibrium are equal, and that the probabilities of
garbled messages are the same for each player and for each message.
If Nature selects $E_1$ with probability .1
and $E_2$ with probability .9, then $E_2$ {\it is} an equilibrium.
In the extended game, if Smith observes $Hire$, he believes that
there is a 90 percent probability that the Boss mistakenly received
the message $E_1$, and only a 10 percent probability that he himself
received a garbled message. In other games, Nature's probabilities
do not matter.\footnote{ An example is the extended version of the
PhD Game in Section 4: the professor would accept the applying
student regardless of the probabilities of $E_1$ and $E_2$ and of
garbling. Those probabilities are irrelevant because it is not
Nature's choice of equilibrium that matters to the professor, but
Nature's message to the student. Even if the professor knows that it
is the student, and not himself, who received the garbled message,
the professor will go along with the equilibrium suggested by the
student's behavior.}
Whether this dependence on detail is an advantage or a disadvantage
of $FI'$ is open for debate. It explains why some examples are more
compelling than others, an advantage, but it makes the idea
context-dependent, so that it requires more thought to apply it. As
an example in which $FI'$ is particularly compelling and Nature's
probabilities matter less, consider the following modification of
Joint Embezzlement. Instead of one boss there are one hundred
bosses, all partners in the business, and if even a single boss
chooses $Fire$ instead of $Hire$, the payoffs are 0 for everyone. If
the equilibrium is supposed to be $E_2$, but one hundred bosses in a
row do the unthinkable and choose $Hire$, what is Smith to believe?
An equilibrium like $E_2$, in which Smith responds by choosing
$Steal$, seems unreasonable. Under $FI'$, Smith would believe that it
is he who received the garbled message, rather than every one of the
hundred bosses, and this would be true even if $E_1$ has a very low
probability of being selected by Nature. Thus, $FI'$ can
differentiate between the game with one boss and the game with one
hundred.\footnote{FI, discussed below, cannot make this distinction.}
%---------------------------------------------------------------
\newpage
\begin{center}
{\bf 4. Incomplete Information and Beliefs about Types }
\end{center}
Until recently, equilibrium refinements were discussed almost
solely in the context of games of incomplete information, in which
Nature makes an initial move and chooses the ``types'' of the
players. Each player has a prior belief on the types of the other
players that he updates into a posterior belief as events occur,
using Bayes' rule and the out-of-beliefs specified by the particular
perfect Bayesian equilibrium. Many refinements place restrictions on
the out-of-equilibrium beliefs. This makes it seem as if the
refinement is essentially about updating beliefs on types, but it is
not possible to restrict beliefs on types without restricting beliefs
on which equilibria are being played out. The examples earlier in
this paper show that the issue of beliefs about which equilibrium is
being played out arise whether information is asymmetric or not, and
suggest that the emphasis on updating beliefs about types clouds the
basic issue.
But it is desirable to have an equilibrium refinement that can help
determine the equilibrium in games of both complete and incomplete
information. This section turns to three games of incomplete
information to show the effect of $FI'$. The first game, ``The PhD
Game,'' has both separating and pooling perfect bayesian equilibria.
$FI'$ eliminates the pooling equilibrium in much the same way as the
intuitive criterion does. The second game, ``The Beer-Quiche Game,''
has two pooling equilibria, one preferred by each type of player. In
this game, the intuitive criterion eliminates one equilibrium, but
$FI'$ does not. The third game, a signalling game with a continuum of
actions, has a continuum of separating and a continuum of pooling
equilibria. $FI'$ can reduce these to a single equilibrium.
Consider first ``the PhD Game'' from Rasmusen
(1989), in which a student who is either smart or stupid must decide
whether to apply to graduate school and a professor must decide
whether to admit students who apply.
(0) Nature chooses the type of the student to be either Smart
(probability 0.1) or Stupid (probability 0.9), unobserved by the
professor.\\
(1) The student decides to {\it Apply} to graduate school, at some
cost, or {\it Not Apply}.\\ \nopagebreak
\nopagebreak (2) If the student chose {\it Apply}, the professor
decides whether to {\it Accept} or {\it Reject} him.
\begin{center}
\begin{tabular}{llrcc}
& & &\multicolumn{2}{c}{\bf
Professor}\\
& & & Accept & Reject \\
& & & & \\
& & Smart& 10, 10 & $-1,0$ \\
&{\bf Applying Student}& & &
\\
& & Stupid & $-10,-10$ & {-1,0} \\
& & & & \\
\multicolumn{4}{l}{\it Payoffs to: (Student, Professor).} &\\
\multicolumn{4}{l}{\it If the Student chooses {\it Not Apply}, payoffs
are (0,0).} &\\
\multicolumn{5}{c}{\bf Table 4: The PhD Game}
\end{tabular}
\end{center}
The smart student would like to be accepted, and the professor would
like to accept him. For the stupid student it is a strictly dominant
strategy not to apply; he actually lowers his payoff by successfully
pretending to be smart and being admitted.
The game has two perfect Bayesian equilibrium outcomes. In the
separating equilibrium, $E_1$, the smart student chooses {\it Apply},
the stupid student chooses {\it Not Apply}, and the professor accepts
anyone who applies. In the pooling equilibrium, $E_2$, neither type
of student applies, and the professor would reject anyone who did
choose {\it Apply}, under any out-of-equilibrium belief that
specifies that the probability that a mistaken application is by a
stupid student is over 0.5.
$E_2$ is perverse, but its status as an equilibrium under Nash logic
is impeccable: no player has any incentive to deviate, and the result
is not due to a very contrived choice of out-of-equilibrium beliefs.
Passive conjectures, for example, would lead the professor to put a
probability of 0.9 that a student who deviates and applies is stupid.
$E_2$ is not, however, an $FI'$-equilibrium, because in the extended
game a student who chooses $Apply$ might be doing so under the belief
that the realized equilibrium is $E_1$. The first iteration of the
extended game, which treats both $E_1$ and $E_2$ as plausible, is:
(A1) Nature chooses $E_1$ with probability 0.5 and $E_2$ with
probability 0.5.\\
(A2) Nature sends messages to the student and the professor. In each
case the message is the equilibrium selected in move (A1) with
probability $(1-\epsilon)$ and the unselected equilibrium with
probability $\epsilon$.\\
(0) Nature chooses the type of the student to be either Smart
(probability 0.1) or Stupid (probability 0.9), unobserved by the
professor.\\
(1) The student decides to {\it Apply} to graduate school, at some
cost,
or {\it Not Apply}.\\
(2) If the student chose {\it Apply}, the professor decides whether
to {\it Accept} or {\it Reject} him.
If the student chooses {\it Apply}, the professor deduces that the
student heard $E_1$ from Nature and is acting accordingly---in which
case the move of {\it Apply} is a sure sign that the student is
smart. Note too that that this result is independent of the
probabilities used in moves (A1) and (A2). $E_2$ is not an
$FI'$-equilibrium, because in any specification of the extended game
that includes both $E_1$ and $E_2$, a student who chooses $Apply$ might
be doing so under the belief that the realized equilibrium is $E_1$.
This seems very different from the usual approach to refining
equilibrium, which is based on restricting the out-of-equilibrium
beliefs that might be held. Possibly the best-known refinement is the
``intuitive criterion'' of Cho \& Kreps (1987), which is defined for
games in which the first mover is trying to communicate private
information about his type, $t'$. The intuitive criterion says that
if he takes an out-of-equilibrium action $m'$ which could not
possibly be profitable were he of type $t$, the other players should
respond by assigning zero probability to type $t$. In the words of
Cho and Kreps (1987, p. 181), if the first player sends the
following message he should be believed:
\protect\samepage{
\begin{quotation}
``I am sending the message $m'$, which ought to convince you that I
know $t'$. For I would never wish to send $m'$ if I know $t$, while
if I know $t'$, and if sending this message so convinces you, then,
as you can see, it is in my interest to send it.''
\end{quotation}
}%end of sampeage
It might seem that this is quite different from $FI'$---that the
Cho-Kreps player is trying to communicate private information,
whereas under $FI'$ the first player is trying to change later
players' beliefs as to which equilibrium is realized. But the
difference is superficial. The Cho-Kreps player is not just trying
to communicate his private information: when he sends a message
impossible in a particular equilibrium, he is trying to convince the
other players to change their beliefs {\it about which equilibrium is
realized} in order to use their new belief about the equilibrium to
convey a new belief about his type. If he fails to convince other
players about the equilibrium, he will not change their beliefs about
his type.
In the PhD Game, the intuitive criterion rules out $E_2$ by
drastically restricting beliefs. It notes that the stupid student
would have no incentive to {\it Apply} even if this would change the
professor's out-of-equilibrium belief to one that would induce him to
{\it Accept}, and concludes that the professor's out-of-equilibrium
belief should be that an applying student is smart, which eliminates
$E_2$ as an equilibrium.\footnote{FI also rules out $E_2$, on the
more formal ground that if the equilibrium-dominated strategy of
({\it Apply} if stupid) is dropped, then a deviation to ({\it Apply}
if smart) becomes profitable.}
The intuitive criterion and $FI'$ reach the same result in this
example, but to accept the intuitive criterion one must believe that
some beliefs are self-evidently reasonable or that
equilibrium-dominated strategies are axiomatically disqualified.
Under $FI'$, on the other hand, the out-of-equilibrium belief
is the inevitable consequence of a less-than-perfect process for
deciding which equilibrium is being played out.
%---------------------------------------------------------------
\bigskip
A second example, the Beer-Quiche Game of Cho \& Kreps (1986), will
show that in some games of incomplete information, $FI'$ cannot
reduce the number of equilibria at all, even if one of the equilibria
is based on out-of-equilibrium beliefs that seem very unreasonable.
In this game, Player I might be either weak or strong. Player II
wishes to fight a duel only if Player I is weak, which has a
probability of .1. Player II also observes what Player I has for
breakfast, and he knows that weak players prefer quiche for breakast,
while strong players prefer beer. Player I wishes above all to avoid
a duel, regardless of his type. The payoffs are as shown in Figure
4.
\begin{center}
{\bf Figure 2: The Beer-Quiche Game.} . (see Kreps article fgiure
2.13).
\end{center}
This game has two perfect Bayesian equilibrium outcomes, both of
which are pooling. In $E_1$, Player I has beer for breakfast
regardless of type, and Player II chooses not to duel. This is
supported by the out-of-equilibrium beliefs that a quiche-eating
Player I is weak with probability over 0.5, in which case Player II
would choose to duel on observing quiche. In $E_2$, Player I has
quiche for breakfast regardless of type, and Player II chooses not to
duel. This is supported by the out-of-equilibrium beliefs that a
quiche-eating Player I is weak with probability over 0.5, in which
case Player II would choose to duel on observing quiche.
The intuitive criterion rules out $E_2$, on the grounds that if
Player I deviates and says ``I am having beer for breakfast, which
ought to convince you that I am strong. For I would never wish to
have beer for breakfast if I were weak, while if I am strong, and if
sending this message so convinces you, then, as you can see, is in my
interest to have beer for breakfast.''
$FI'$ cannot rule out $E_2$. If the plausible equilibrium outcomes
are $E_1$ and $E_2$, then in the extended game Nature chooses both of
these with positive probabilities whose exact values do not matter.
Suppose that Players I and II have both received message $E_1$ and
Player I is weak, so that $E_1$ tells him to choose beer. If Player I
deviates and chooses quiche, then Player II's conclusion will be that
Player I thinks the equilibrium is $E_2$, and that Player I is strong
with probability 0.9. Player II therefore will not duel. This would
make deviation profitable and rule out $E_1$, leaving $E_2$ as the
equilibrium. Suppose, on the other hand, Players I and II have both
received message $E_2 $ and Player I is strong, so that $E_2$ tells
him to choose beer. If Player I deviates and chooses beer, then
Player II's conclusion will be that Player I thinks the equilibrium
is $E_1$, and that Player I is strong with probability 0.9. Player II
therefore will not duel. This would make deviation profitable and
rule out $E_2$, leaving $E_1$ as the equilibrium. Thus, by choosing
the order of iteration, either $E_1$ or $E_2$ can be made to survive
as $FI'$ equilibria.
There is an approach similar in spirit to $FI'$ which can rule out
$E_2$.
Suppose that we construct an extended game in which Nature chooses
not between two realized equilibria, but between two games. In Game
X, Player II observes what Player I has for breakfast; in Game Y,
Player II does not. Nature tells each player which game is being
played, but with some small probability the game is X and Nature tells
Player I that the game is Y. In this case, Player I will have beer or
quiche for breakfast, depending on which breakfast he prefers.
Knowing that this is a possibility, Player II would interpret
out-of-equilibrium quiche as indicating a weak Player I and
out-of-equilibrium beer as indicating a strong Player I. This in
turn both supports the out-of-equilibrium beliefs necessary for $E_1$
and rules out the beliefs necessary for $E_2$.
%---------------------------------------------------------------
\bigskip
A third example will be presented to illustrate the application of
$FI'$ to a game of incomplete information with a
continuum of perfect Bayesian equilibria, to show how these can be
reduced to a single $FI'$ equilibrium. Let there be two types of
workers, of ability $a=1$ and $a= 2$, where 0.8 are of type 1 and 0.1
are of type 2. Workers of type $i$ acquire $s_i$ years of education
at cost $s_i/a$, where $s_i$ is a continuous non-negative variable.
Employers will pay a worker according to their estimate of his
ability, but they observe only education.
This game has a continuum of separating perfect Bayesian equilibrium
outcomes, in which $s_1^*=0$ and $s_2^* \in [1,2]$, and $w(s=0)=1$
and $w(s=s_2^*)=2$. An out-of-equilibrium belief that supports such
an outcome is that any worker who acquires an education level not
specified by the particular equilibrium is of type 1.\footnote{ There
is a continuum of equilibria that support each outcome, differing in
the out-of-equilibrium beliefs; e.g., if $s_2^*=1.5$ replace
Prob(type 1$|s_2 = 0.3) = 1$ with Prob(type 1$|s_2 = 0.3) = .99$.}
The type-1 workers will wish not to acquire education because the
benefit is a salary increase of 2-1, but the cost is at least 1/1.
The type-2 workers will wish to acquire education because the benefit
is 2-1 and the cost is no more than 2/2.
This game also has a continuum of pooling perfect Bayesian equilibrium
outcomes, in which $s_1=s_2 \in [0,0.2]$ and $w(s=s^*)=1.2$. An
out-of-equilibrium belief that supports such an outcome is that any
worker who acquires an education level not specified by the
particular equilibrium is of type 1 (again, there is a continuum of
equilibria supporting each equilibrium outcome) Any deviating worker
would deviate to $s=0$. The type-1 workers will acquire education
because the benefit is a salary increase of 1.2-1, and the cost is no
more than .2/1. The benefit is the same for type-2 workers, and the
cost is even less.\footnote{xxx There are probbaly semi-pooling
euqilibria too; I haven't thoght about it.}
The initial extended game is:
(1) Nature picks an equilibrium outcome. With probability 1/6 it is a
pooling equilibrium and $s_2^*$ takes values in $[0,0.2]$ with
uniform density. With probability 5/6 it is a separating equilibrium
and $s_2^*$ takes values in $[1,2]$ with uniform density.\\
(2) Nature announces an equilibrium outcome to each player. With
probability $(1-\epsilon)$ it is the outcome chosen in (1). With
probability $\epsilon$ it is not, and the announcement is drawn
randomly using the same distributions as in (1).\footnote{Note that
the probability of drawing the same outcome twice from a continuous
distribution is zero.}\\
(3) Nature chooses the worker to have ability $a=1$ with probability
0.8 and $a=2$ with probability 0.2.\\
(3) The worker chooses education $s$.\\
(4) Two employers compete for the services of the worker by
simultaneously offering him wages $w (e_1)$ and $w(e_2)$.
Suppose first that the message from Nature is the most attractive
pooling equilibrium, with $s^*=0$, but the worker picks $s=1$. The
employer will conclude that the worker is type 2 and thinks the
realized equilibrium is separating. The worker will be paid 2
instead of 1.2, and he will incur a cost of 1/2, so if he is indeed
type-2 he will benefit from this. Type-2 workers will deviate from
even the most attractive pooling equilibrium, ruling out that class
of equilibria. But we can also rule out every separating equilibrium
but $s_2^*=1$, because the employer will interpret a deviation to
$s_2=1$ as the playing out of an $s_2^*=1$ equilibrium, which leaves
the type-2 worker with the same wage but a lower signalling cost.
%---------------------------------------------------------------
\newpage
\begin{center}
{\bf 5. Concluding Comments}
\end{center}
My intent in this article has not been to persuade the reader to
adopt a new axiom that he must apply to the equilibrium of every
game. Rather, I hope to help him recognize the implications of
something he very likely accepts already: Nash equilibrium. If one
accepts Nash equilibrium when there are multiple equilibria, then one
accepts the existence of a process by which the players are apprised
of the equilibrium to be played out. If one furthermore accepts that
the process is imperfect, then one has accepted $FI'$ or something
very like it.
The simplest way to describe $FI'$ is as a requirement that players
in a game interpret actions as equilibrium actions whenever possible,
even if they must change their beliefs about what equilibrium is
being played out. Most refinements of equilibrium suggested in recent
years are based on what out-of-equilibrium beliefs seem intuitively
reasonable. Often the idea of weak dominance is invoked--- that a
strategy should work well even if the world is not exactly as the
player thinks it is. $FI'$ tries to provide a more fundamental
grounding for our intuition. It traces what seems intuitively
reasonable to the idea that when there are multiple equilibria the
identity of the equilibrium being played is only almost common
knowledge and the players are aware that their beliefs might be
inconsistent with those of other players. $FI'$ reduces the number of
equilibria in a way very similar to forward induction. This suggests
that the idea behind $FI'$ is also a justification for forward
induction, and, since the two concepts do not always lead to exactly
the same predicted equilibria, it suggests an improvement upon
forward induction as well.
The reader of a theory article sometimes turns the last page with
the feeling that the conclusions are all very well, but no
application for them ever existed or ever will exist. I have shown
that $FI'$ and FI differ in a certain kind of three-player dynamic
game. Are there any useful games of this kind? The situation does
not seem unusual: a dominant player sets the agenda and a number of
other players wish to pick the same response from a number of
possibilities. The specific example which inspired this article is
the model of exclusionary practices in Rasmusen, Ramseyer, and Wiley
(1990). In that model, a firm may require its customers to sign
exclusive-dealing contracts. If most customers sign, other firms drop
out of the market, so the individual is no worse off than if he had
not signed. Thus, there is a coordination game among customers, who
would jointly prefer not to sign the contract, but who will sign in
exchange for a small payment from the firm if each believes that the
others will sign. The game has two perfect equilibria, one in which
the contract is offered and signed, and one in which it is not
offered. The reasoning of $FI'$ says that if the firm, at some cost,
offers the contracts, each customer must consider whether the firm
has better knowledge of the equilibrium than he does. In this
example, the customers, with less at stake, would reasonably have a
greater chance than the firm of being confused over the equilibrium
being played out. In that case, the no-exclusion equilibrium
disappears, which suggests a reason for firms to use and anti-trust
authorities to worry about exclusion contracts. Thus, the choice of
equilibrium may matter to profitability and policy.
%---------------------------------------------------------------
\newpage
\noindent
{\bf References.}
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Potential for Self Sacrifice'' working paper, Dept of Economics, U. of
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{\it Journal of Economic Theory}, 48: 476-496.
%---------------------------------------------------------------
\end{raggedright}
\end{document}