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\noindent
23 November  2005.   Eric Rasmusen, Erasmuse@indiana.edu.
Http://www.rasmusen.org.

\begin{LARGE}
\begin{center}
 {\bf 4 Dynamic Games with Symmetric Information}
  \end{center} \end{LARGE}

 \noindent
 { \bf 4.1 Subgame Perfectness}

 \noindent
 In this chapter we will make heavy use of the extensive form to study games
with   moves that occur in sequence.  We start in Section 4.1 with a refinement
of the Nash equilibrium concept called perfectness that incorporates sensible
implications of the order of moves. Perfectness is illustrated in  Section 4.2
with a game of entry deterrence. Section 4.3 expands on the idea of perfectness
using the example of nuisance suits,  meritless lawsuits   brought   in the
hopes of obtaining a settlement out of court. Nuisance suits show the importance
of a threat being made credible and  how sinking costs early  or  having
certain nonmonetary payoffs  can benefit a player. This example will also be
used to discuss the open-set problem of weak equilibria in games with continuous
strategy spaces, in which a player offering a contract chooses its terms to make
the other player  indifferent about accepting or rejecting. The last perfectness
topic will be renegotiation: the idea that when there are multiple perfect
equilibria,   the players  will coordinate on equilibria that are pareto optimal
in subgames but not in   the   game as a whole.

\bigskip \noindent
{\bf The Perfect  Equilibrium of      Follow the Leader I  }

\noindent

     Subgame perfectness is an equilibrium concept based on the ordering of
moves and the distinction between an equilibrium path and an equilibrium. The
{\bf equilibrium path} is the path through the game tree that is followed in
equilibrium, but the equilibrium itself is a strategy profile, which includes
the players' responses to other players' deviations from the equilibrium path.
These off-equilibrium responses are  crucial to decisions on the equilibrium
path. A threat, for example, is a promise to carry out a certain action if
another player deviates from his equilibrium actions, and it has an influence
even if it is never used.

  Perfectness is best introduced with an example.  In Section 2.1, a flaw of
Nash equilibrium was revealed in the game {  Follow the Leader I}, which has
three pure strategy Nash equilibria  of which only one is reasonable.  The
players are Smith and Jones, who choose disk sizes. Both their payoffs are
greater if they choose the same size  and greatest if they coordinate on   {\it
Large}.  Smith moves first, so his strategy set is \{{\it Small, Large}\}.
Jones' strategy is more complicated,   because it must specify an action for
each information set, and Jones's  information set depends on what Smith chose.
A typical element of Jones's  strategy set is $(Large, Small)$, which specifies
that he chooses $Large$ if Smith chose $Large$, and $Small$ if Smith chose
$Small.$ From the strategic form we found the following three Nash equilibria.

 $$\left[ \begin{tabular}{ccc}
 {\bf Equilibrium} &  {\bf  Strategies} & {\bf Outcome}\\ $E_1$ &\{{\it   Large,
(Large, Large)}\} & Both pick $Large$.  \\ $E_2$ &         \{{\it Large, (Large,
Small)}\} & Both pick $Large$.\\ $E_3$ &       \{{\it Small, (Small, Small)}\} &
Both pick $Small$. \end{tabular} \right]$$

 Only Equilibrium $E_2$ is reasonable, because the order of the moves should
matter to the decisions players make.  The problem with the strategic form, and
thus with simple Nash equilibrium, is that it ignores who moves first.  Smith
moves first, and it seems reasonable that Jones should be allowed--- in fact
should be required--- to rethink his strategy after Smith moves.

\includegraphics[width=150mm]{fig04-01.jpg}

\begin{center}
 {\bf Figure 1:    {\it   Follow the Leader I} } \end{center}

 Consider Jones's  strategy of $(Small, Small)$ in equilibrium $E_3$. If Smith
deviated from equilibrium by choosing {\it Large},  it would be unreasonable for
Jones to stick to the response {\it Small}. Instead, he should also choose {\it
Large}. But if Smith expected a response of $Large$,  he would have chosen  {\it
Large} in the first place, and $E_3$ would not be an equilibrium.  A similar
argument shows that it would be irrational for Jones to choose ({\it Large,
Large}),  and we are left with $E_2$ as the unique equilibrium.

We say that equilibria $E_1$ and $E_3$ are Nash equilibria but not ``perfect''
Nash equilibria. A strategy profile is a perfect equilibrium if it remains an
equilibrium on all possible paths, including not only the equilibrium path but
all the other paths, which branch off into different ``subgames.''

\noindent
 {\it A {\bf subgame} is a game consisting of a node which is a singleton in
every player's information partition, that node's successors, and the payoffs at
the associated end nodes.}\footnote{Technically, this is a {\it proper} subgame
because of the information qualifier, but no economist is so ill-bred as to use
any other kind of subgame.}

\noindent
{\it A strategy profile is a {\bf subgame perfect Nash equilibrium} if (a) it is
a Nash equilibrium for the entire game; and (b) its relevant action rules are a
Nash equilibrium for every subgame.}

   The extensive form of   {   Follow the Leader I}  in Figure 1 (a reprise of
Figure  1 from Chapter 2) has three subgames: (1) the entire game, (2) the
subgame starting at node $J_1$, and (3) the subgame starting at node $J_2$.
Strategy profile $E_1$ is not a subgame perfect equilibrium because it is only
Nash in subgames (1) and (3), not in subgame (2). Strategy profile $E_3$ is not
a subgame perfect equilibrium  because it is only Nash in subgames (1) and (2),
not in subgame (3). Strategy profile $E_2$ is perfect because it is Nash in all
three subgames.

 The term {\bf sequential rationality} is often used to denote the idea that a
player should maximize his payoffs at each point in the game, re- optimizing his
decisions at each point and taking into account the fact that he will re-
optimize in the future.  This is a  blend of the economic ideas of ignoring sunk
costs and rational expectations. Sequential rationality is so standard a
criterion for  equilibrium  now that often  I will speak of ``equilibrium''
without the qualifier  when I wish  to refer to an equilibrium that satisfies
sequential rationality in the sense of being a ``subgame perfect equilibrium''
or, in a game of asymmetric information, a ``perfect Bayesian equilibrium.''

 One reason why perfectness (the word ``subgame'' is usually left off) is a good
equilibrium concept  is because it represents the idea of sequential
rationality.   A second reason is that a weak Nash equilibrium is not robust to
small changes in the game.  So long as he is certain that Smith will not choose
{\it Large}, Jones is indifferent between the never-to-be-used responses ({\it
Small } if $Large$) and ({\it Large} if $Large$).  Equilibria $E_1$, $E_2$, and
$E_3$ are all weak Nash equilibria because of this. But if there is even a small
probability that Smith will choose {\it Large}--- perhaps by mistake--- then
Jones would prefer the response ({\it Large} if {\it Large}), and equilibria
$E_1$ and $E_3$ are no longer valid. Perfectness is a way to eliminate some of
these less robust weak equilibria. The   small probability of a mistake is
called a {\bf tremble}, and  Section 6.1 returns to this  {\bf trembling hand}
approach   as one way to extend the notion of perfectness to games of asymmetric
information.

For the moment, however, the reader should note that the tremble approach is
distinct from   sequential rationality. Consider  Figure 2's   { Tremble Game}.
This game has three Nash equilibria, all weak: {\it (Out, Down)},  {\it (Out,
Up)},  and {\it (In, Up)}. Only  {\it (Out,  Up)} and {\it (In, Up)} are subgame
perfect, because  although $Down$ is weakly Jones's   best response to Smith's
$Out$, it is inferior if Smith chooses $In$. In  the subgame starting with
Jones's   move, the only subgame perfect equilibrium is for Jones to choose
$Up$.  The possibility of trembles, however, rules out {\it (In, Up)} as an
equilibrium.   If Jones has even an infinitesimal chance of trembling and
choosing $Down$,   Smith will choose $Out$ instead of $In$.  Also,  Jones will
choose $Up$, not $Down$, because if Smith trembles and chooses $In$, Jones
prefers $Up$ to $Down$.  This leaves only   {\it (Out, Up)} as an equilibrium,
despite the fact that it is weakly pareto dominated by {\it (In, Up)}.

\includegraphics[width=150mm]{fig04-02.jpg}

\begin{center} {\bf Figure 2:   The     Tremble Game:     Trembling Hand Versus
Subgame Perfectness  } \end{center}

  \bigskip \noindent
 {\bf  4.2 An Example of Perfectness: Entry Deterrence I}

\noindent We turn now to a game in which perfectness plays a role just as
important as in   {   Follow the Leader I}   but in which the players are in
conflict.  An old question in industrial organization is whether an incumbent
monopolist can maintain his position by threatening to wage a price war against
any new firm that enters the market. This idea was heavily attacked by Chicago
School economists such as McGee (1958) on the grounds that a price war would
hurt the incumbent more than collusion with the entrant.  Game theory can
present this reasoning very cleanly.  Let us consider a single episode of
possible entry and price warfare, which nobody expects to be repeated. We will
assume that even if the incumbent chooses to collude with the entrant,
maintaining a duopoly is difficult enough that   market revenue drops
considerably from the monopoly level.

\begin{center}
{\bf  Entry Deterrence I }\\
 \end{center}
 {\bf Players}\\ Two firms, the entrant and the incumbent.

 \noindent
 {\bf The Order of Play} \vspace{-18pt} \begin{enumerate} \item[1] The entrant
decides whether to {\it Enter} or {\it Stay Out.} \item[2] If the entrant
enters, the incumbent can $Collude$ with him, or $Fight$ by cutting the price
drastically. \end{enumerate}

\noindent {\bf Payoffs}\\ Market profits are 300 at the monopoly price and 0 at
the fighting price.  Entry costs are 10. Duopoly competition reduces market
revenue to 100, which is split evenly.

\begin{center} {\bf Table   1: Entry Deterrence I}

 \begin{tabular}{lllccc} &       &             &\multicolumn{3}{c}{\bf
Incumbent}\\ &       &             &  {\it   Collude}    &   &  {\it Fight}
\\ &   & {\it Enter}      &     {\bf 40,50} & $\leftarrow$  &  $-10,0$ \\ & {\bf
Entrant:} &&$\uparrow$& & $\downarrow$ \\ &  &    {\it Stay Out }     &
$0,300$  & $\leftrightarrow$  & {\bf 0,300} \\
 \end{tabular} \end{center}
\vspace{-24pt}
 {\it Payoffs  to:  (Entrant, Incumbent). Arrows show how a player can increase
his payoff.   }

\bigskip

  The strategy sets can be discovered from the order of play. They are \{{\it
Enter}, {\it Stay Out}\} for the entrant, and \{{\it Collude} if entry occurs,
{\it Fight} if entry occurs\} for the incumbent.  The game has the two Nash
equilibria indicated in boldface in Table   1, ({\it Enter}, {\it Collude}) and
({\it Stay Out}, {\it Fight }).  The equilibrium ({\it Stay Out}, {\it Fight})
is weak, because the incumbent would just as soon {\it Collude} given that the
entrant is staying out.


\includegraphics[width=150mm]{fig04-03.jpg}

\begin{center}
{\bf Figure 3:  Entry Deterrence I}\end{center}

    A piece of information has been lost by condensing from the extensive form,
Figure 3, to the strategic form, Table   1: the fact that the entrant gets to
move first.  Once he has chosen {\it Enter}, the incumbent's best response is
{\it Collude}.  The threat to fight is not credible and would be employed only
if the incumbent could bind himself to fight, in which case he never does fight,
because the entrant chooses to stay out.  The equilibrium ({\it Stay Out}, {\it
Fight})  is Nash but not subgame perfect, because if the game is started after
the entrant has already entered, the incumbent's best response is {\it Collude}.
This does not prove that collusion is inevitable in duopoly,    but it is the
equilibrium for {  Entry Deterrence I.}

     The trembling hand interpretation of perfect equilibrium can be used here.
So long as it is certain that the  entrant will not enter, the incumbent is
indifferent between {\it Fight} and {\it Collude}, but if there were even a
small probability of entry--- perhaps because of a lapse of good judgement by
the entrant--- the incumbent would prefer {\it Collude} and the Nash equilibrium
would be broken.

 Perfectness rules out threats that are not credible.  {  Entry Deterrence I }
is  a good example because if a communication move were added to the game tree,
the incumbent might tell the entrant that entry would be followed by fighting,
but the entrant would ignore this noncredible threat.  If, however, some means
existed by which the incumbent could precommit himself to fight entry, the
threat would become credible. The next section will look at one context,
nuisance lawsuits,  in which such precommitment might be possible

\bigskip

\noindent {\bf Should the Modeller Ever Use Nonperfect Equilibria?}

\noindent A game in which a player can commit himself to a strategy can be
modelled in two ways:

\noindent 1  As a game in which nonperfect equilibria are acceptable, or\\ 2  By
changing the game to replace the action {\it Do X} with {\it Commit to Do X} at
an earlier node.

 An example of (2) in   { Entry Deterrence I }  is to reformulate the game so
the incumbent moves first, deciding in advance whether or not to choose {\it
Fight} before the entrant  moves.  Approach (2) is better than (1) because if
the modeller wants to let players commit to some actions and not to others,  he
can do this by carefully specifying the order of play. Allowing equilibria to be
nonperfect forbids such discrimination and   multiplies the number of
equilibria.     Indeed, the problem with subgame perfectness is not that it is
too restrictive but that it still allows too many strategy profiles to be
equilibria in games of asymmetric information.  A subgame must start at a single
node and not cut across any player's information set, so often the only subgame
will be the whole game and  subgame perfectness does not restrict equilibrium at
all.   Section 6.1  discusses perfect Bayesian equilibrium and other ways to
extend the perfectness concept to games of asymmetric information.

 \bigskip \noindent {\bf   4.3 Credible Threats, Sunks Costs, and the Open-set
Problem in the Game of Nuisance Suits}

    Like the related concepts of sunks costs and rational expectations,
sequential rationality is a simple idea with tremendous power. This section will
show that power in another simple game, one which models nuisance suits. We have
already come across one application of game theory to law, in the Png (1983)
model of Section 2.5. In some ways, law is particularly well suited to analysis
by game theory because the legal process is so concerned with conflict and the
provision of definite rules to regulate that conflict. In what other field could
an article be titled:  ``An Economic Analysis of Rule 68,''  as Miller (1986)
does in his   discussion of  the  federal rule of procedure that  penalizes a
losing  litigant who   had  refused to accept a settlement offer.   The growth
in the area can be seen by comparing the overview in the Ayres's (1990) review
of the first edition of the present book with the entire book by Baird, Gertner
\& Picker (1994).     In law, even more clearly  than in business,    a major
objective is to avoid inefficient outcomes by  restructuring the rules, and
nuisance suits are one of the inefficiencies that a good policy maker hopes to
eliminate.

     Nuisance suits  are lawsuits  with  little chance of success, whose only
possible purpose seems to be the hope  of a   settlement out of court. In the
context of entry deterrence   people commonly think large size is an advantage
and    a large incumbent will threaten a small  entrant,   but   in  the context
of nuisance suits  people commonly think   large size is a disadvantage and a
wealthy corporation is vulnerable to extortionary litigation.   {  Nuisance
Suits I}  models the essentials of the situation:   bringing  suit  is costly
and  has little chance of success, but because defending the suit is also costly
the defendant might pay a generous amount to settle it out of court.  The model
is similar to the Png Settlement Game of Chapter 2 in many respects, but here
the model will be one of symmetric information and we will make explicit the
sequential rationality requirement that was implicit the discussion in Chapter
2.

 \begin{center} {\bf     Nuisance  Suits I: Simple Extortion }
 \end{center} {\bf Players}\\
 A  plaintiff and a defendant.

\noindent {\bf The Order of Play} \vspace{-18pt} \begin{enumerate} \item[1 ] The
plaintiff decides whether to bring suit against the defendant  at cost $c$.
 \item[2] The plaintiff makes a  take-it-or-leave-it settlement offer of $s>0$.
\item[3] The defendant accepts or rejects the settlement offer. \item[4] If the
defendant rejects the offer, the plaintiff decides whether to give up or   go to
trial  at a cost $p$ to himself and $d$ to the defendant.
 \item[5] If the case goes to trial, the plaintiff wins amount $x$ with
probability $\gamma$ and otherwise wins nothing. \end{enumerate}

\noindent {\bf Payoffs}\\
 Figure 4 shows the payoffs.  Let $\gamma x < p$, so the plaintiff's expected
winnings    are less than his marginal cost of going to trial.

\includegraphics[width=150mm]{fig04-04.jpg}

\begin{center}
 {\bf Figure 4  The Extensive Form for   Nuisance Suits } \end{center}

\noindent
The perfect equilibrium is

\noindent
 Plaintiff: {\it Do nothing}, {\it Offer $s$}, {\it Give up}\\
 Defendant: {\it Reject}\\
 Outcome: The plaintiff does  not bring a suit.

The  equilibrium settlement offer $s$ can be any  positive  amount. Note that
the equilibrium specifies actions at all four nodes of the game, even though
only the first is reached in equilibrium.

 To  find a perfect equilibrium the modeller starts at the end of the game tree,
following the  advice of Dixit \& Nalebuff (1991, p. 34) to ``Look ahead and
reason back.''   At node $P_3$, the plaintiff will choose {\it Give up}, since
by assumption $\gamma x-c-p < -c$. This is because the  suit is brought only in
the hope of settlement, not in the hope of winning at trial. At node $D_1$, the
defendant, foreseeing that the plaintiff will give up, rejects any positive
settlement offer. This makes the plaintiff's offer at $P_2$ irrelevant, and,
looking ahead to a payoff of $-c$ from choosing $Sue$ at $P_1$, the plaintiff
chooses {\it Do nothing}.

 Thus, if nuisance suits are brought, it must be for some reason other than the
obvious one, the plaintiff's hope of extracting a settlement offer from a
defendant who wants to avoid   trial costs. This is fallacious because the
plaintiff himself bears trial costs and hence cannot credibly make the threat.
It is fallacious even if the defendant's legal costs would be much higher than
the plaintiff's ($d$ much bigger than $p$), because the relative size of the
costs does not enter into the argument.

     One might wonder how risk aversion affects this conclusion. Might not the
defendant settle because he is more risk averse than the plaintiff?  That is a
good question, but   {   Nuisance Suits I}  can be adapted to    risk-averse
players with very little change.   Risk would enter  at the trial stage, as a
final move by Nature to decide who wins. In   {  Nuisance Suits I},     $\gamma
x$  represented the expected value of the  award. If both the defendant and the
plaintiff are equally risk averse,  $\gamma x$ can still represent the expected
payoff from the award--- one simply interprets $x$ and 0  as the utility of the
cash award and the utility of an award of 0,  rather than  as the actual cash
amounts. If the players have different degrees of risk aversion,   the expected
loss to the defendant is not the same as the expected gain to the plaintiff, and
the payoffs must be adjusted. If the defendant is more risk averse, the payoffs
from {\it Go to trial} would change to $(- c-p+\gamma x, -\gamma x -y - d)$,
where $y$ represents the extra disutility of risk to the defendant. This,
however, makes no difference to the equilibrium. The  crux of the game is that
the plaintiff is unwilling to go to trial because of the cost to himself, and
the cost to the defendant, including the cost of bearing risk,  is irrelevant.

 If nuisance suits are brought,  it must therefore be for some more complicated
reason. Already, in Chapter 2, we looked at one reason for litigation to reach
trial  in the {\it Png Settlement Game:} incomplete information. That is
probably  the most important explanation  and it has been much studied, as can
be seen from the  surveys  by Cooter \& Rubinfeld (1989) and Kennan \& R. Wilson
(1993).  In   this section, though,    let us    confine ourselves to
explanations   where  the probability of the suit's success is common knowledge.
Even then, costly threats might be credible because of  sinking costs
strategically (Nuisance Suits II), or  because of the nonmonetary payoffs
resulting from going to trial (Nuisance Suits III) .

\bigskip

\noindent {\bf    Nuisance  Suits II
Using Sunk Costs Strategically}

\noindent
 Let us now modify the game, following the inspiration of Rosenberg \& Shavell
(1985),  so that the plaintiff can pay his lawyer the amount $p$ in advance,
with no refund if the case settles.   This inability to obtain a refund actually
helps the plaintiff, by changing the payoffs from the game  so   his payoff from
{\it Give up} is $-c-p$, compared to $-c-p+\gamma x$ from {\it Go to trial}.
Having sunk the legal costs, he   will  go to trial if $\gamma x  > 0$--- that
is, if he has any chance of success at all.\footnote{ Have a figure with the
previous NS I Extensive Form and the new one side by side, with the differences
highlighted. }

 This, in turn, means that the  plaintiff  would only prefer settlement to trial
if $s> \gamma x$. The defendant would prefer settlement to trial if $s < \gamma
x + d$, so there is a positive {\bf settlement range} of $[\gamma x, \gamma x
+d]$ within which both players are willing to settle.   The exact amount of the
settlement depends on the bargaining power of the parties, something to be
examined in chapter 11. Here, allowing the plaintiff to make a take- it-or-
leave-it offer means that $s= \gamma x +d $ in equilibrium, and if $\gamma x +d
> p + c$, the nuisance suit will be brought  even though $\gamma x  <  p + c$.
Thus, the plaintiff is bringing the suit only because  he can extort $d$, the
amount of the defendant's legal costs.

  Even though the plaintiff can now extort a settlement, he does it at some cost
to himself, so an equilibrium with nuisance suits will require that
 \begin{equation}   \label{e4.1}
 -c -p  + \gamma x + d  \geq 0
 \end{equation}
 If inequality (\ref{e4.1}) is false, then, even if the plaintiff could extract
the maximum possible settlement of  $s=\gamma x +d $,  he would not do so,
because he would then have to pay $c+p$ before reaching the settlement stage.
This implies that a  totally meritless suit (with $\gamma =0$), would not be
brought unless the defendant had higher legal costs than the plaintiff ($d >p$).
If inequality (\ref{e4.1}) is satisfied, however, the following strategy profile
is a perfect  equilibrium:

\noindent
 {\bf Plaintiff:} {\it Sue}, {\it Offer $s=\gamma x +d$}, {\it Go to trial}\\
 {\bf Defendant:} {\it Accept   $s \leq \gamma x +d$}\\
 {\bf  Outcome:} Plaintiff  sues and offers  to settle, to which the defendant
agrees.

 An obvious counter to the plaintiff's ploy  would be for the defendant to also
sink his costs, by  paying $d$ before the settlement negotiations, or even
before the plaintiff decides to file suit. Perhaps this is one reason why
large corporations use in house counsel,  who are paid a salary regardless of
how many hours they work, as well as outside counsel, hired by the hour. If  so,
nuisance suits   cause a social loss---the wasted time of the lawyers, $d$---
even if nuisance suits are never brought, just as aggressor nations cause social
loss in the form of world military expenditure even if they never start a
war.\footnote{Nonrefundable lawyer's fees, paid in advance, have traditionally
been acceptable, but a New York court recently ruled they were unethical. The
court thought that such fees unfairly restricted the client's ability to fire
his lawyer, an example of how ignorance of game theory can lead to confused
rule-making. See ``Nonrefundable Lawyers' Fees, Paid in Advance, are Unethical,
Court Rules,'' {\it Wall Street Journal}, January 29, 1993, p. B3, citing {\it
In the matter of Edward M. Cooperman, Appellate Division of the Supreme Court,
Second Judicial Department, Brooklyn, 90-00429.}}

Two problems, however, face the defendant   who tries to sink the cost $d$.
First, although it   saves  him $\gamma x$ if it deters the plaintiff from
filing suit, it  also means  the defendant must pay the full amount $d$. This
is worthwhile  if the plaintiff has all the bargaining power, as in   {
Nuisance  Suits II}, but it  might not be if $s$ lay in the middle of the
settlement range because the plaintiff was not able to make a take-it-or-leave-
it offer. If   settlement negotiations resulted in  $s$  lying     exactly in
the middle of the settlement range, so $s = \gamma x + \frac{d}{2}$, then it
might not be worthwhile for the defendant to sink $d$ to deter nuisance suits
that would settle for $\gamma x + \frac{d}{2}$.

 Second, there is an asymmetry in  litigation: the plaintiff has the choice of
whether to bring suit or not. Since it is the plaintiff who has the initiative,
he can sink $p$  and make the settlement offer before the defendant has the
chance to sink $d$. The only way for the defendant to avoid this is to pay $d$
well in advance, in which case the expenditure is wasted if no possible suits
arise. What the defendant would like best would be to buy legal insurance which,
for a small premium, would pay all defense costs in future suits that might
occur. As we will see in Chapters 7 and 9, however, insurance of any kind faces
problems arising from asymmetric information. In this  context,   there is the
``moral hazard'' problem, in that once the defendant is insured he has less
incentive to avoid causing harm to the plaintiff and provoking a lawsuit.

\bigskip \noindent
{\bf The Open-Set Problem in   {  Nuisance  Suits II} }

  {  Nuisance  Suits II}  illustrates  a technical point that arises in a great
many games with continuous strategy spaces and causes  great distress to novices
in game theory.       The equilibrium in   {  Nuisance  Suits II}  is  only  a
weak Nash equilibrium. The plaintiff proposes    $s =\gamma x + d$, and the
defendant  has the same payoff from accepting or rejecting, but in equilibrium
the defendant accepts the offer  with probability one, despite his indifference.
This seems   arbitrary, or even silly. Should not  the plaintiff  propose a
slightly  lower settlement to give the defendant a strong incentive to accept it
and avoid the risk of having to go to trial?  If the parameters are such that $s
=\gamma x + d= 60$, for example, why does the plaintiff risk holding out for 60
when he might be rejected and  most likely receive  0 at trial, when he could
offer 59 and give the  defendant a strong incentive to accept?

One  answer  is that no other   equilibrium exists besides   $s=60$. Offering 59
cannot be part of an equilibrium  because it is dominated by offering 59.9;
offering 59.9 is dominated by offering 59.99, and so forth. This is known as the
{\bf open-set problem}, because the set of offers that the defendant strongly
wishes to accept is open and   has no maximum--- it is bounded at 60, but a set
must be bounded {\it and closed} to guarantee that a maximum exists.

 A second answer is that under the assumptions of rationality and Nash
equilibrium the  objection's premise is false because  the plaintiff bears no
risk whatsoever in offering $s=60$. It is fundamental to Nash equilibrium that
each player  believe that the others will follow equilibrium behavior. Thus, if
the equilibrium  strategy profile says that the defendant will accept $s \leq
60$, the plaintiff can offer 60 and believe it will be accepted. This is really
just to say that a weak Nash equilibrium is still a Nash equilibrium, a point
emphasized in chapter 3 in connection with mixed strategies.

A third answer is that the problem is an artifact of using a model with a
continuous strategy space, and it disappears if the strategy space is made
discrete.  Assume that $s$ can only take values in multiples of 0.01, so it
could be 59.0, 59.01, 59.02, and so forth, but not 59.001 or 59.002.    The
settlement part of the game will now have two perfect equilibria. In the strong
equilibrium E1, $s=59.99$ and the defendant accepts any offer $s <60$. In the
weak equilibrium E2, $s=60$ and the defendant accepts any offer $s \leq 60$. The
difference is trivial, so the   discrete strategy space  has made the model more
complicated without any extra insight.\footnote{A good example  of the ideas of
discrete money values and sequential rationality is in Robert Louis Stevenson's
story, ``The Bottle Imp''  (Stevenson [1987]).  The   imp grants the wishes of
the bottle's owner  but will seize his soul if he dies in possession of  it.
Although the  bottle cannot be  given away, it can be sold, but only at a price
less than that for which it was purchased.}

 One can also specify a more complicated bargaining game to  avoid the issue of
how exactly the settlement is determined. Here one could say that the settlement
is not proposed by the plaintiff, but simply emerges with a value  halfway
through the settlement range, so $s= \gamma  x + \frac{d}{2}$. This seems
reasonable enough, and it adds a little extra realism to the model at the cost
of a little extra complexity.   It avoids the open-set problem, but only by
avoiding being clear  about how $s$ is determined. I call this kind of modelling
{\bf blackboxing}, because it is as if at  some point in the game,    variables
with certain values go into a black box and  come out the other side with values
determined by an exogenous process.   Blackboxing is perfectly acceptable as
long as it neither  drives nor obscures the point the model is making.   {
Nuisance  Suits III}  will illustrate this method.

  Fundamentally, however, the point to keep in mind is  that games are models,
not reality. They are meant to clear away the unimportant details of a real
situation and  simplify  it down to the essentials. Since a model is trying to
answer a question, it should focus on what answers that question. Here, the
question is why nuisance suits might be brought, so it is proper to exclude
details of the bargaining if they are irrelevant to the answer.   Whether a
plaintiff offers 59.99 or 60, and whether a rational person accepts an offer
with probability 0.99 or 1.00, is part of the unimportant detail, and whatever
approach is simplest should be used. If the modeller really thinks that these
are important matters,   they can indeed be modelled, but they are not important
in this context.

 One source of concern over the open-set problem, I think, is that perhaps that
the payoffs are not quite realistic, because    the players should derive
utility from hurting ``unfair'' players.   If  the plaintiff makes a settlement
offer of 60, keeping the entire savings from avoiding the trial for himself,
everyday experience tells us that the defendant  will indignantly refuse the
offer. Guth, Schmittberger \&   Schwarze (1982) have found in experiments that
people turn down bargaining offers they perceive as unfair, as one might expect.
If   indignation is truly  important, it can be explicitly incorporated into the
payoffs, and if that is done, the  open-set problem returns. Indignation is not
boundless, whatever people may say.  Suppose that accepting a settlement offer
that benefits the plaintiff more than the defendant gives a disutility of $x$ to
the defendant because of his indignation at his unjust treatment. The plaintiff
will then offer to settle for exactly $60-x$, so the equilibrium is still weak
and the defendant is still indifferent  between accepting  and rejecting the
offer. The open-set problem persists, even after realistic emotions are added to
the model.

I have spent so much time on the open-set problem  not because it is important
but because it arises so often and is a sticking point for people unfamiliar
with modelling. It is not a problem that disturbs experienced modellers, unlike
other basic issues we have already encountered---for example, the issue of how a
Nash equilibrium comes to be common knowledge among the players--- but it is
important to understand why it is not important.

\bigskip \noindent
 {\bf   {  Nuisance  Suits III}: Malice   }

    One of the most common  misconceptions about game theory, as about economics
in general, is   that it ignores non-rational and non-monetary motivations.
Game theory does   take the basic motivations of the players  to be  exogenous
to the model, but those motivations are crucial to the outcome and they often
are not  monetary, although payoffs are always given numerical values. Game
theory does not call somebody irrational who prefers leisure to money or who is
motivated by the desire to be world dictator.  It does require the  players'
emotions to be carefully gauged to determine exactly how the actions and
outcomes affect the players'  utility.

 Emotions are often important to lawsuits, and law professors tell their
students that when  the cases they study seem  to involve  disputes too trivial
to be worth taking to court, they can guess that the real motivations are
emotional. Emotions could enter in a variety of distinct ways.  The plaintiff
might simply like going to trial, which can be expressed as a value of $p<0$.
This would be true of many criminal cases,  because prosecutors like news
coverage and want   credit with the public for prosecuting certain kinds of
crime.  The    Rodney King  trials of 1992 and 1993 were of this variety;
regardless of the merits of the cases against the policemen who  beat Rodney
King, the prosecutors  wanted to go to trial to satisfy  the  public outrage,
and when the state prosecutors failed in the first trial, the  federal
government was happy to accept the cost of bringing suit in the second   trial.
A different motivation is that the  plaintiff might derive utility from the fact
of winning the case quite separately from the  monetary award, because  he wants
a public statement that he  is in the right. This is a   motivation in bringing
libel suits, or for a criminal defendant who wants to  clear his good name.

  A different emotional  motivation for going to trial is the desire to inflict
losses on the defendant, a  motivation we will  call ``malice,'' although it
might as inaccurately be called ``righteous anger.''    In this case, $d$ enters
as a positive argument in the plaintiff's utility function. We will construct a
model of this kind, called   {  Nuisance Suits III}, and    assume that $\gamma=
0.1$, $c=3$, $p=  14$, $d=50$, and $x=100$, and  that  the plaintiff receives
additional utility of  0.1 times the defendant's disutility. Let us also  adopt
the blackboxing technique discussed earlier and assume that the settlement $s$
is in the middle of the settlement range. The payoffs  conditional on suit being
brought are
\begin{equation} \label{e4.0} \pi_{plaintiff}(Defendant \; accepts) =    s - c +
0.1 s = 1.1s -3 \end{equation} and \begin{equation} \label{e4.0a} \begin{array}
{ll} \pi_{plaintiff}(Go\;to \;trial)  & = \gamma x - c -p + 0.1 (d + \gamma x)\\
& \\ &  = 10 -  3 - 14  + 6 = -1.\\ \end{array} \end{equation}
 Now, working back from the end in accordance with sequential rationality, note
that since the plaintiff's payoff from {\it Give Up} is $-3$, he will go to
trial if the defendant rejects the settlement offer.    The overall payoff from
bringing a suit that eventually goes  to trial is still $- 1$, which is worse
than the payoff of 0 from not bringing suit in the first place, but if $s$ is
high enough, the payoff from bringing suit and settling is higher still. If $s$
is greater than 1.82 ($=\frac{-1+3}{1.1}$, rounded), the plaintiff   prefers
settlement to trial, and if $s$ is greater than about  2.73 ($=\frac{0+ 3 }{1.1}
$, rounded), he prefers settlement to not bringing the suit at all.

In determining the settlement range, the relevant payoff is the expected
incremental payoff since   the suit was brought. The plaintiff will settle for
any $s  \geq  1.82 $, and the defendant will settle for any $s \leq \gamma x +
d=60$, as before. The settlement range is  $[1.82, 60]$, and   $s= 30.91$.  The
settlement offer is no longer the maximizing choice of a player, and hence is
moved to the outcome in the equilibrium description below.

\noindent
 {\bf  Plaintiff:} {\it Sue},   {\it Go to Trial} \\
 {\bf  Defendant:} {\it  Accept any $s \leq 60$}\\
 {\bf Outcome:} The plaintiff sues and offers $s=30.91$, and the defendant
accepts the settlement.

Perfectness is important here because the defendant would like to threaten never
to settle and be believed. The   plaintiff  would not bring suit given his
expected   payoff of $-1$ from bringing  a suit that goes to trial, so a
believable threat would be effective.  But such a threat is not believable. Once
the plaintiff does bring suit, the only Nash equilibrium in the remaining
subgame is for the defendant to accept his settlement offer. This is
interesting  because the plaintiff, despite his willingness to go to trial, ends
up settling out of court. When information is symmetric, as it is here, there is
a tendency for equilibria to be efficient. Although the plaintiff wants to hurt
the defendant, he also  wants to  keep his expenses low.   Thus, he  is willing
to hurt the defendant less  if it enables him to save on his own legal costs.

  \bigskip

 One final point before leaving these models is that much of the value of
modelling comes simply from setting up the rules of the game, which helps to
show what is important in a situation. One  problem that arises in setting up a
model of nuisance suits is deciding what a ``nuisance suit'' really is. In  the
game of Nuisance Suits, it  has been defined as a suit whose expected damages do
not repay the plaintiff's costs of going to trial. But having to formulate a
definition brings to mind another     problem that might be called the problem
of nuisance suits: that the plaintiff  brings suits he knows will not win unless
the court makes a mistake. Since the court might make a mistake with very high
probability, the games above  would not be appropriate models--- $\gamma$  would
be high, and the problem is not that the plaintiff's expected gain  from trial
is low, but that it is high.  This, too, is an important problem,  but  having
to construct a model shows that it is different.

  \bigskip \noindent
  {\bf  4.4 Recoordination to Pareto-Dominant Equilibria in Subgames: Pareto
Perfection }

   One simple  refinement of equilibrium that was mentioned in chapter 1 is to
rule out  any strategy profiles that are pareto dominated by Nash equilibria.
Thus, in the game of  {  Ranked Coordination}, the inferior Nash equilibrium
would be ruled out as an acceptable equilibrium. The idea behind this is that in
some unmodelled way the players discuss their situation and coordinate to avoid
the bad equilibria. Since   only Nash equilibria are discussed, the players'
agreements  are self-enforcing and this is a  more  limited suggestion than the
approach in cooperative game theory according to which the players   make
binding agreements.

 The  coordination idea can be taken further in various ways. One is to think
about coalitions of players coordinating on favorable equilibria, so that two
players might coordinate on an equilibrium even if a third player dislikes it.
Bernheim, Peleg, \& Whinston (1987) and Bernheim \& Whinston (1987) define   a
Nash strategy profile  as a {\bf coalition-proof Nash equilibrium}   if no
coalition of players could form a self-enforcing agreement to deviate from it.
They take the idea further by    subordinating it to the idea of sequential
rationality.  The natural way to do this is to require that no coalition  would
deviate in future subgames,  a notion  called by various names, including {\bf
renegotiation proofness}, {\bf recoordination} (e.g., Laffont \& Tirole [1993],
p. 460),  and  {\bf pareto perfection} (e.g., Fudenberg \& Tirole (1991a), p.
175).   The idea has been used extensively in the analysis of infinitely
repeated games, which are particularly subject to the problem of multiple
equilibria; Abreu, Pearce \& Stachetti (1986) is an example of this literature.
Whichever name is used,    the idea is distinct from the renegotiation problem
in the principal-agent models to be studied   in Chapter 8, which involves the
rewriting of  earlier binding contracts  to make new binding contracts.

  The best way to demonstrate the idea of pareto perfection is by an
illustration, the      {  Pareto Perfection Puzzle }, whose extensive form   is
shown in Figure 5.  In this game Smith chooses $In$ or {\it Outside Option 1},
which yields payoffs of 10 to  each player. Jones then chooses {\it Outside
Option 2}, which yields 20 to each player, or initiates either  a
coordination game or a prisoner's dilemma. Rather than draw the full subgames in
extensive form, Figure 5 inserts the payoff matrix for the subgames.

\includegraphics[width=150mm]{fig04-05.jpg}

\begin{center} {\bf Figure 5: The  Pareto Perfection   Puzzle} \end{center}

The {  Pareto Perfection   Puzzle}  illustrates the complicated interplay
between perfectness and pareto dominance.  The pareto-dominant strategy profile
is {\it (In,   Prisoner's Dilemma$|$In, any  actions   in the coordination
subgame,  the actions yielding (50,50) in the    {  Prisoner's Dilemma}  subgame
)}.  Nobody expects this strategy profile to be an equilibrium, since it is
neither perfect nor Nash.   Perfectness tells us that if the   {\it  Prisoner's
Dilemma}  subgame is reached, the payoffs will be (0,0),   and if the
coordination subgame is reached  they  will be  either (1,1) or (2,30).  In
light of this, the  perfect equilibria of the   {  Pareto Perfection Puzzle }
are:

E1:{ ($In$, outside option 2$| In$, the actions yielding (1,1) in the
coordination subgame,  the actions yielding (0,0) in the    { Prisoner's
Dilemma}  subgame)}. The  payoffs are (20,20).

E2: {  (outside option 1, coordination game$|   In$,  the actions yielding
(2,30) in the coordination subgame,  the actions yielding (0,0) in the    {
Prisoner's Dilemma}  subgame)}. The  payoffs are (10,10).

 If one applies    pareto dominance without  perfection, E1 will be the
equilibrium, since both players prefer it. If the players can recoordinate at
any point and change their expectations, however,   then if   play of the game
reaches the coordination subgame, the players will recoordinate on the actions
yielding (2,30). pareto perfection thus knocks out  E1 as an equilibrium. Not
only does it rule out   the pareto-dominant strategy profile that yields (50,50)
as an equilibrium,   it also  rules out the pareto-dominant perfect strategy
profile that yields (20,20) as    an equilibrium. Rather, the payoff is (10,10).
Thus, pareto perfection is   not the same thing as simply picking the pareto-
dominant perfect strategy profile.

    It is difficult to say which equilibrium is best here, since this is an
abstract game and we cannot call upon details from the real world to refine the
model. The approach of applying an equilibrium refinement is not as likely to
yield results as using the intuition behind the refinement. The intuition here
is that the players will somehow coordinate on pareto-dominant equilibria,
perhaps  finding open discussion helpful. If we ran an experiment on student
players using the {  Pareto Perfection Puzzle }, I would expect to reach
different equilibria depending on what communication is allowed.  If the players
are allowed to talk only before the game starts, it seems more likely that E1
would be the equilibrium, since players could agree to play it and would have no
chance to explicitly recoordinate later. If the players could talk at any time
as the game proceeded, E2 becomes more plausible. Real-world situations arise
with many different communications technologies, so there is no one right
answer.

\newpage

\begin{small}

\noindent {\bf  Notes}

\noindent {\bf N4.1} {\bf Subgame perfectness} 
\begin{itemize}
  \item The terms ``perfectness'' and     ``perfection''  are   used
synonymously. Selten (1965) proposed the equilibrium concept  in  an article
written in German. ``Perfectness'' is used in Selten (1975) and conveys an
impression of completeness more appropriate to the concept than the goodness
implied by ``perfection.''  ``Perfection,'' however, is more common.

\item It is debatable whether the definition of subgame ought to include the
original game. Gibbon (1992, p. 122) does not, for example, and modellers
usually do not  in their conversation.

 \item Perfectness is not the only way to eliminate weak Nash equilibria like
{\it (Stay Out, Collude)}.  In Entry Deterrence I, $(Enter, Collude)$ is the
only iterated dominance equilibrium, because $Fight$ is weakly dominated for the
incumbent.

\item The distinction between perfect and non-perfect Nash equilibria is like
the distinction between {\bf closed loop} and {\bf open loop} trajectories in
dynamic programming.  Closed loop (or {\bf feedback}) trajectories can be
revised after they start, like perfect equilibrium strategies, while open loop
trajectories are completely prespecified (though they may depend on state
variables). In dynamic programming the distinction is not so important, because
prespecified strategies do not change the behavior of other players.  No threat,
for example, is going to alter the pull of the moon's gravity on a rocket.

  \item A subgame can be infinite in length, and infinite games can have non-
perfect equilibria. The infinitely repeated   {\it  Prisoner's Dilemma}  is an
example; here every subgame looks exactly like the original game  but begins at
a different point in time.

\item {\bf Sequential rationality in macroeconomics.} In macroeconomics the
requirement of {\bf dynamic consistency} or {\bf time consistency} is similar to
perfectness.  These terms are less precisely defined than perfectness, but they
usually require that strategies need only be best responses in subgames starting
from nodes on the equilibrium path, instead of all subgames. Under this
interpretation, time consistency is a less stringent condition than perfectness.

  $\;\;\;$ The Federal Reserve, for example, might like to induce inflation to
stimulate the economy, but the economy is stimulated only if the inflation is
unexpected. If the inflation is expected, its effects are purely bad.  Since
members of the public know that the Fed would like to fool them, they disbelieve
its claims that it will not generate inflation (see Kydland \& Prescott [1977]).
Likewise, the government would like to issue nominal debt, and promises lenders
that it will keep inflation low, but once the debt is issued, the government has
an incentive to inflate its real value to zero. One reason the US Federal
Reserve Board was established to be independent of Congress in the United States
was to diminish this problem.

 

\item Often,  irrationality--- behavior that is automatic rather than
strategic--- is an advantage.  The Doomsday Machine in  the movie {\it Dr
Strangelove} is one example. The Soviet Union decides that it cannot win a
rational arms race against the richer United States, so it creates a bomb which
automatically    blows up the entire world if anyone explodes a nuclear bomb.
The movie also illustrates a  crucial detail without which such irrationality is
worse than useless:    you have to tell the other side that you have the
Doomsday Machine.

$\;\;\;$ President Nixon reportedly told his aide H.R. Haldeman that he followed
a more complicated version of this strategy: ``I call it the Madman Theory, Bob.
I want the North Vietnamese to believe that I've reached the point where I might
do {\it anything} to stop the war.  We'll just slip the word to them that `for
God's sake, you know Nixon is obsessed about Communism. We can't restrain him
when he's angry--- and he has his hand on the nuclear button'--- and Ho Chi Minh
himself will be in Paris in two days begging for peace''(H. R. Haldeman \&
Joseph DiMona , {\it The Ends of Power}, 1978,  p. 83). The Gang of Four model
in section 6.4 tries to model a situation like that.

\item The ``lock-up agreement'' is an example of a credible threat: in a
takeover defense, the threat to destroy the firm is made legally binding.  See
Macey \& McChesney (1985) p. 33.

\item
 A famous paradox relating to sequential rationality is the ``Quiz on Friday
Paradox''. You are going to have a quiz next week, but I am going to surprise
you with my choice of day. If we reach Thursday without a quiz, however, you
will know the quiz must be on Friday and not be surprised, so the quiz must be
earlier. The same argument applies to looking ahead to Thursday on Wednesday,
however, and by iteration all days can be ruled out.  Philosophers  from Quine
(1953) to Schick (2003, ch. 5) have puzzled over this.



 \end{itemize}

\noindent {\bf N4.3} {\bf An Example of Perfectness: Entry Deterrence I}
\begin{itemize}
  \item The Stackelberg equilibrium of a duopoly game (section
3.4) can be viewed as the perfect equilibrium of a Cournot game modified so that
one player moves first, a game similar to Entry Deterrence I. The player moving
first is the Stackelberg leader and the player moving second is the Stackelberg
follower.  The follower could threaten to produce a high output, but he will not
carry out his threat if the leader produces a high output first.

\item Perfectness is not so desirable a property of equilibrium in biological
games. The reason the order of moves matters is because the rational best reply
depends on the node at which the game has arrived. In many biological games the
players act by instinct and unthinking behavior is not unrealistic.

 \item Reinganum \& Stokey (1985) is a clear presentation of the implications of
perfectness and commitment illustrated with the example of natural resource
extraction.

 \end{itemize}

\newpage

\noindent {\bf Problems}

 \noindent {\bf   4.1.      Repeated Entry Deterrence } (easy) \\
Consider two repetitions without discounting of the game   { Entry Deterrence I
}  from Section 4.2. Assume that there is one entrant, who sequentially decides
whether to enter two markets that have the same incumbent. \begin{enumerate}

  \item[(a)]  Draw the extensive form of this game.

\item[(b)] What are the 16 elements of the strategy sets of the entrant?

    \item[(c)] What is the subgame perfect equilibrium?

 \item[(d)] What is one of the nonperfect Nash equilibria? \end{enumerate}

      \bigskip \noindent {\bf  4.2.  The Three-Way Duel } (medium)  (after
Shubik (1954)
\\ Three gangsters  armed with pistols, Al, Bob, and Curly,  are   in a room
with a suitcase containing 120 thousand dollars.  Al is the    least accurate,
with a 20 percent chance of  killing his target.  Bob  has  a  40 percent
probability.  Curly is slow but sure; he kills his target with 70 percent
probability.    For each, the value of his own life outweighs the value of any
amount of money.  Survivors split the money.

 \begin{enumerate}

  \item[(a)] Suppose      each  gangster has one bullet and  the order of
shooting  is  first Al, then  Bob,  then Curly.    Assume also that each
gangster must try to kill another gangster when his turn comes.  What  is an
equilibrium strategy profile and what is  the  probability that each of them
dies in that equilibrium? Hint: Do not try to draw a game tree.

 \item[(b)] Suppose    now that each gangster has the additional option of
shooting his gun at the ceiling, which may kill somebody upstairs but has no
direct effect on his payoff. Does the strategy profile that you found was an
equilibrium in part (a) remain an equilibrium?

\item[(c)] Replace the three gangsters with three companies, Apex, Brydox, and
Costco, which are competing    with slightly different products.  What story can
you tell about their  advertising strategies?

\item[(d)] In the United States, before the general election a candidate must
win the nomination of his party. It is often noted that candidates are reluctant
to be seen as the frontrunner in the race for the nomination of their party,
Democrat or Republican.  In the general election, however, no candidate ever
minds being seen to be ahead of his rival from the other party. Why?

\item[(e)] In the 1920's, several men vied for power in the Soviet Union after
Lenin died. First Stalin and Zinoviev combined against Trotsky. Then Stalin and
Bukharin combined against Zinoviev. Then Stalin turned on Bukharin.  Relate this
to Curly, Bob, and Al.

 \end{enumerate}

\bigskip \noindent {\bf   4.3.  Heresthetics in  Pliny and the Freedmens' Trial
} (easy)  (Pliny [105] ``To Aristo'',    Riker[1986, pp. 78-88])\\
 Afranius Dexter died mysteriously, perhaps dead by his own hand, perhaps killed
by  his  freedmen (servants  a step above slaves), or perhaps  killed by his
freedmen by his own orders.   The freedmen went on trial before the Roman
Senate. Assume that  45 percent of the senators favor acquittal, 35 percent
favor banishment, and 20 percent favor execution, and that the   preference
rankings in the three groups are $A \succ B \succ E$, $B \succ A \succ E$, and
$E \succ B \succ A$.  Also assume that each group has a leader and   votes as a
bloc.
  \begin{enumerate}
  \item[(a)] Modern legal procedure requires the court to decide guilt first and
then assign a penalty if the accused is found guilty. Draw a tree to represent
the sequence of events (this will not be a game tree, since it will represent
the actions of groups of players, not of individuals). What is the outcome in a
perfect equilibrium?

\item[(b)] Suppose that the acquittal bloc can pre-commit to  how they will vote
in the second round if guilt wins in the first round.   What will they do, and
what will happen? What would the execution bloc do if they could control  the
second-period vote of the acquittal bloc?

 \item[(c)] The normal Roman procedure began with a vote on execution versus no
execution, and then voted on the   alternatives in a second round if execution
failed to  gain a majority.  Draw a tree to represent this. What would   happen
in this case?

 \item[(d)] Pliny proposed that the Senators divide into three groups, depending
on whether they supported acquittal, banishment, or execution, and that the
outcome with the most votes  should win.  This proposal caused a roar of
protest. Why did he propose it?

\item[(e)]   Pliny did not get the result he wanted with his voting procedure.
Why not?

\item[(f)] Suppose that personal considerations made it most important to a
senator that he show his stand by his vote, even if he had to sacrifice his
preference  for a particular outcome. If there were a vote over whether to use
the traditional Roman procedure or Pliny's procedure, who would vote with Pliny,
and what would happen to the freedmen?

  \end{enumerate}



 \bigskip \noindent {\bf 4.4. Garbage Entry  } (medium) \\
 Mr. Turner is thinking of entering the garbage collection business in a certain
large city.  Currently, Cutright Enterprises has a monopoly, earning   40
million dollars   from the 40 routes the city offers up for bids.  Turner thinks
he  can  take away  as many routes as he wants from Cutright, at a profit of
1.5 million per route for him. He is worried, however, that Cutright might
resort to assassination, killing him to regain their lost routes.   He would be
willing to  be  assassinated for profit of 80 million dollars, and assassination
would cost Cutright  6  million dollars in expected legal costs and possible
prison sentences.

   How many  routes should Turner try to take away from Cutright?



\bigskip \noindent {\bf 4.5.  Voting Cycles   } (medium)  \\
  Uno, Duo, and Tres are  three people voting on whether the budget devoted to a
project  should be Increased, kept the Same, or  Reduced. Their payoffs from the
different outcomes, given in Table 2,  are not monotonic in  budget size. Uno
thinks the  project could be very profitable if its budget were increased, but
will fail otherwise. Duo mildly wants a smaller budget. Tres likes the budget as
it is now.

\textbf{Table 2:  Payoffs from Different Policies }\\

\begin{tabular}{l|lll} & Uno  &      Duo & Tres\\
 \hline Increase & 100 &2  &  4  \\ Same & 3 &      6 &  9 \\ Reduce & 9 &    8
&  1 \\ \end{tabular}



   Each of the three voters  writes down his first choice.   If a policy gets a
majority of the votes, it wins. Otherwise,  $Same$ is the chosen policy.

\begin{enumerate}

\item[(a)] Show that   ($Same, Same, Same$) is a  Nash equilibrium. Why does
this equilibrium seem unreasonable to us?

\item[(b)] Show that ($Increase, Same, Same$) is a Nash equilibrium.

\item[(c)]  Show that if each player has an independent  small probability
$\epsilon$ of ``trembling'' and choosing each possible wrong action by mistake,
($Same, Same, Same$) and ($Increase, Same, Same$)  are no longer equilibria.

\item[(d)]  Show  that   ($Reduce, Reduce, Same$) is a Nash equilibrium that
survives each player has an independent  small probability $\epsilon$ of
``trembling'' and choosing each possible wrong action by mistake.

\item[(e)]  Part (d) showed that if Uno and Duo are expected to choose $Reduce$,
then Tres would choose $Same$ if he could hope they might tremble-- not
$Increase$. Suppose, instead, that Tres votes first, and publicly. Construct a
subgame perfect  equilibrium in which Tres chooses $Increase$. You need not
worry about trembles now.

\item[(f)]   Consider the following voting procedure.  First, the three voters
vote between $Increase$ and $Same$. In the second round, they   vote between the
winning policy and $Reduce$. If, at that point, $Increase$ is not the winning
policy, the third vote is between $Increase$ and whatever policy won in the
second round.

What will happen? (watch out for the trick in this question!)

\item[(g)]  Speculate about what would happen if the payoffs are in terms of
dollar willingness to pay by each   player  and the players could make binding
agreements to buy and sell votes. What, if anything,  can you say about which
policy would win, and what votes would be bought at what price?

\end{enumerate}

 %---------------------------------------------------------------

 \newpage

\begin{center}
  {\bf  US Air for Sale:  A Classroom Game for  Chapter 4}
  \end{center}


        On October 2, 1995, USAir, at the time  the nation's fifth largest
airline, announced that it had approached United Airlines and American Airlines
about a possible buyout.  United and American are the two largest U.S. airlines,
and both are interested in growth in the highly competitive airline market.
United and American must  now consider    what to do.

  Financial analysts for both United and American have made projections for all
the possible scenarios, on the request of the strategic  bidding consultants.
They say that United and American start in equally strong positions, but that
can change depending on who wins the auction.

  The three possible outcomes for an airline   are:


 \noindent
 {\it Neither firm bids enough for US Air to accept.} Both United and American
will maintain their strong positions, and they can expect future profits of \$50
billion each.

 \noindent
 {\it Our airline wins.}  The winning airline will become the dominant firm in
the market, and  can expect future profits of \$80 billion. The purchase cost of
US Air, however, must be subtracted from this in order to calculate net profit.
Therefore, the payoff for the firm with the winning bid is
\begin{equation} Payoff_{Winner}= 80 - B_{winning}. \end{equation}
  The minimum price US Air will accept is \$10 billion.

 \noindent
 {\it Our airline loses.} The losing firm may have trouble competing with the
larger network of the winner because of the greater variety of flights a large
airline can offer. The more, however, that the winner pays for US Air, the
better off is the loser, because the cash flow  the winner uses  to pay for the
acquisition is no longer available for other investments in new equipment and
lenders will be more reluctant to lend to the   newly enlarged firm.    The
analysts suggest that a good approximation to the payoff of the losing firm will
be
\begin{equation} Payoff_{Loser}= 30 + 0.25*B_{winning}. \end{equation}

   In particular, if the winner pays just \$10 billion for US Air, the loser's
profit will be  \$32.5 billion.

   \noindent
  We will look at two kinds of auction rules: ascending, and first-price.

   \noindent
{\bf  First-Price  Auction Rules:}    US Air will solicit  bids in writing from
anyone who cares to bid.  Once all bids are received, they will be announced
publicly.   If there is just one bid, it wins if it is \$10 billion or above. If
there are no bids, US Air will continue as an independent firm.   If there are
two bids, and  at least one is  \$10 billion or above,  the winner has purchased
US Air, ties being broken   by the flip of a coin.

 \noindent
{\bf  Ascending Auction Rules:}    US Air will solicit initial bids in writing
from United and American, choosing the first bidder randomly and announcing
its bid  publicly before asking  the other for a written bid. 
   If there are no written bids of at least 10 billion dollars, US Air
will continue as an independent firm.   If there is at least  one bid of 10
billion dollars or above,  US Air  will allow   counteroffers, and an  ascending
auction commences.  The winner has purchased US Air.

 Students playing the game will be put into groups of three representing either
United or American. Groups will be paired up to play out the auction under the
ascending or first-price rules. First, play all the pairs  by the first-price
rules, then by the  ascending rules.



\end{small}

\end{document}





