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\noindent 19 September 2009   Eric Rasmusen, Erasmuse@indiana.edu. Http://www.rasmusen.org

\begin{LARGE} {  Follow the Leader I}, which has three pure strategy Nash equilibria  of which only one is reasonable.

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

\includegraphics[width=200mm]{fig04-01.jpg}

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

 \newpage

 . 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.}

(Note: pedantic people will call this a ``proper 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.}

 This is an application of  BACKWARDS INDUCTION or SEQUENTIAL RATIONALITY.

\newpage

\noindent {\bf Reasons why we use perfect equilibrium}

(1) sequential rationality

(2) robustness

 On (2): Suppose there is small probability $\epsilon$ of a ``tremble'': a player might pick the wrong move by mistake.

Non-perfect Nash equilibria are all weakly dominated (why?) and so would then disappear.

\newpage

The tremble approach is NOT equivalent to sequential rationality.

  Nash equilibria, all weak:\\ {\it (Out, Down)},  {\it (Out, Up)},  and {\it (In, Up)}.

\includegraphics[width=200mm]{fig04-02.jpg}

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

Think of the basic Bertrand Game too. The only Nash equilibrium is in weakly dominated strategies--- picking price to equal marginal cost. The Tremble idea rules this out.

\newpage

  \noindent {\bf  Predatory Pricing}

 McGee (1958):  a price war would hurt the incumbent more than collusion with the entrant.

\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.   }

\newpage

\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}

 Two Nash equilibria :\\ ({\it Enter}, {\it Collude}) and ({\it Stay Out}, {\it Fight }).  Perfectness rules out threats that are not credible.  (Schelling idea)

\includegraphics[width=200mm]{fig04-03.jpg}

 



 \newpage

 \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}

\newpage

\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=200mm]{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.

  \newpage

 \noindent {\bf Introducing Risk Aversion}

Add a final move by Nature to decide who wins.

    $\gamma x$  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 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.

\newpage

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

Now change the order of moves. The plaintiff pays 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. 

 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.

\newpage

If
 \begin{equation}
   \label{e4.1} -c -p  + \gamma x + d  \geq 0 \end{equation} 
then the  perfect  equilibrium is :

\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.

   \newpage

 \noindent {\bf The Open-Set Problem   }

    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.

  Shouldn't  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? 

Why  would the plaintiff risk holdi  out for 60 when he might be rejected and   receive  0 at trial, when he could offer 59 and give the  defendant a strong incentive to accept?

(1) No other   equilibrium exists besides   $s=60$.  

(2)  The  objection's premise is false because  the plaintiff bears no risk whatsoever in offering $s=60$ 

(3) The problem is an artifact of using a model with a continuous strategy space 

\newpage

(3) The problem is an artifact of using a model with a continuous strategy space 

  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 $s \leq 60$.  

 
 \newpage

\noindent
 {\bf The Ultimatum Game}

 1. Smith  proposes how to share \$10. He    offers Jones share $x$. 

 2.  Jones accepts or rejects. 

If Jones accepts, the payoffs are $(10-x)$ for Smith and $x$ for Jones. If Jones rejects, the payoffs are 0 for both players. 

There are many non-perfect Nash equilibria. The unique perfect equilibrium is ($x=0$, Jones accepts any offer of $x \geq 0$). 

  But experiments show that Jones would not follow this strategy. Why? 



 \newpage

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

     How would we model the idea that the plaintiff dislikes the defendant? 

 
  {  Nuisance Suits III}: let $\gamma= 0.1$, $c=3$, $p=  14$, $d=50$, and $x=100$, and   the plaintiff receives additional utility of  0.1 times the defendant's disutility.

Let   the settlement $s$  be in the middle of the settlement range. 

The payoffs  conditional on suit being brought are $$  \pi_{plaintiff}(Defendant \; accepts) =    s - c + 0.1 s = 1.1s -3 $$
  $$ \label{e4.0a} \begin{array} {ll} \pi_{plaintiff}(Go\;to \;trial)  & = \gamma x - c -p + 0.1 (d + \gamma x)\\ & \\ &  = 10 -  3 - 14  + 6 = -1.\\ \end{array} $$
$$
\pi_{plaintiff}(give\; up)  = -3. 
$$
 
   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. 

\newpage

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   $s= 30.91$.  

\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.

 \newpage

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

 Suppose we think Pareto-dominant equilibria are what will be played out. 

That idea has further implications in dynamic games. 


 

 {\bf Coalition-proof Nash equilibrium}: no coalition of players could form a self-enforcing agreement to deviate from it.

Combining this with sequential rationality: 
  no coalition  would deviate in future subgame--- {\bf renegotiation proofness} (most common name), {\bf recoordination}  {\bf pareto perfection}  

\newpage
 

\includegraphics[width=200mm]{fig04-05.jpg}

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

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).

 
  
 \end{LARGE}

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