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\begin{LARGE}

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

 
 

\begin{center} {\bf Table  1:  Ranked Coordination  }

 \begin{tabular}{lllccc}
 &       &             &\multicolumn{3}{c}{\bf Jones}\\ &       &             &
$Large$  & & $Small$  \\ &   &  $Large$   &     {\bf 2,2} & $\leftarrow$  & $-1,
-1$ \\
& {\bf Smith} &     & $\uparrow$  & & $\downarrow$ \\ &  &       $Small$     &
$-1, -1$ & $\rightarrow$  & {\bf 1,1} \\ & & & &\\
 \end{tabular} \end{center}
\vspace{-24pt}
 {\it Payoffs  to:  (Smith, Jones). Arrows show how a player can increase his
payoff.   }

 
 \newpage



  \noindent {\it The {\bf strategic form} (or {\bf normal form}) consists of \\
1 All possible strategy profiles $s^1,s^2, \ldots, s^p.$ \\ 2 Payoff functions
mapping $s^i$ onto the payoff $n$-vector $\pi^i$, ($i =1,2,\ldots,p).$ }

 
 

\noindent
    {Follow-the-Leader I } 

Smith  has a strategy set of two strategies:  Small or Large. 

Jones has a strategy set of four different strategies:
 
 

 \begin{center} \noindent $\left\{ \begin{tabular}{l} ({\it L$|$L},  {\it
L$|$S}),\\ ({\it L$|$L}, {\it S$|$S}),\\ ({\it S$|$L}, {\it L$|$S}), \\ ({\it
S$|$L}, {\it S$|$S}) \\ \end{tabular} \right\}$ \end{center}

 Combining one strategy for each player, we get a strategy profile.  

That results in an action for each player, and a payoff.  

The normal form shows the strategies and payoffs, omitting the actions. 



\newpage


\begin{center}
{\bf Table  2: Normal Form for  Follow-the-Leader  I        }

 \begin{tabular}{ll | llll}
 &       &            \multicolumn{4}{c}{\bf Jones}\\
  & &  $ J_1$     &  $ J_2$ &  $ J_3$ & $ J_4$  \\ & & {\it L$|$L, L$|$S}    &
{\it L$|$L, S$|$S}   & {\it S$|$L, L$|$S} &{\it S$|$L, S$|$S}  \\
 & & & & &\\
\hline
  & & & & &\\
 &    $S_1: Large$   & {\bf   \fbox {2}, \begin{picture}(10,15) \put(0,-2)
{\dashbox{3}(8,12) {2}} \end{picture}  ($E_1$) }    &{\bf   \fbox{2},
\begin{picture}(10,15) \put(0,-2){\dashbox{3}(8,12) {2}} \end{picture}   ($E_2$)
}   &  \begin{picture}(10,15) \put(0,-2){\dashbox{3}(16,12) {$-1$}}
\end{picture},  $-1$ & $-1, -1$ \\ {\bf Smith} & &     &    & &   \\ & $S_2:
Small$  &  $-1, -1$ & $ 1, \begin{picture}(10,15)  \put(0,-2){\dashbox{3}(8,12)
{1}}\end{picture}  $ & \begin{picture}(10,15) \put(0,-2){\dashbox{3}(16,12)
{$-1$}} \end{picture},$ -1  $ & {\bf \fbox{1}, \begin{picture}(10,15) \put(0,-2)
{\dashbox{3}(8,12) {1}} \end{picture} ($E_3$)} \\
& & & &\\
 \end{tabular}
\end{center}
\vspace{-24pt}
 {\it Payoffs  to:  (Smith, Jones). Best-response payoffs are boxed (with
dashes, if weak) }

 

 


\begin{tabular}{ccc} {\bf Equilibrium} &  {\bf  Strategies} & {\bf Outcome}\\
$E_1$ &\{{\it   Large,  (L$|$L, L$|$S)}\} & Both pick $Large$    \\ $E_2$ &
\{{\it Large, (L$|$L, S$|$S)}\} & Both pick $Large$ \\ $E_3$ &       \{{\it
Small,(S$|$L, S$|$S)}\} & Both pick $Small$ \end{tabular}

 
\newpage 
 
 \noindent {\bf The Extensive Form  }

 

\noindent {\it A {\bf node} is a point in the game at which some player or
Nature takes an action, or the game ends.}

\noindent {\it A {\bf successor} to node X is a node that may occur later in the
game if X has been reached.

\noindent A {\bf predecessor} to  node X is a node that must be reached before X
can be reached.

\noindent A {\bf starting node} is a node with no predecessors.

\noindent An {\bf end node} or {\bf end point} is a node with no successors.}

\noindent {\it A {\bf branch} is one action in a player's action set at a
particular node.}

\noindent {\it A {\bf path} is a sequence of nodes and branches leading from the
starting node to an end node.}

  \newpage

\noindent {\it The {\bf extensive form} is a description of a game consisting of
\\ 1 A configuration of nodes and branches running without any closed loops
from a single starting node to its end nodes.\\
 2 An indication of which node belongs to which player.\\
3 The probabilities that Nature uses to choose different branches at its
nodes.\\ 4 The information sets into which each player's nodes are divided.\\ 5
The payoffs for each player at each end node.}

 

 \newpage
 



 \includegraphics[width=150mm]{fig02-01.jpg}

\begin{center} {\bf Figure  1:  {  Follow-the-Leader      I} in Extensive Form }
\end{center}

 



 \includegraphics[width=150mm]{fig02-02.jpg}

\begin{center} {\bf Figure 2:  Ranked Coordination in Extensive Form   }
\end{center}




\newpage

 \noindent {\bf The Time Line}

 



 \includegraphics[width=150mm]{fig02-03.jpg}

\begin{center} {\bf  Figure  3:    The Time Line for Stock Underpricing:  (a) A
Good Time Line; (b) A Bad Time Line}
 \end{center}

  {\bf decision time}  versus  {\bf real time }  

 
\newpage

\noindent {\it Player $i$'s {\bf information set} $\omega_{i}$ at any particular
point of the game is the set of different nodes in the game tree that he knows
might be the actual node, but between which he cannot distinguish by direct
observation.}

  

 \includegraphics[width=150mm]{fig02-04.jpg}

\begin{center} {\bf Figure  4:  Information Sets and Information Partitions.}
\end{center}





\newpage


 \includegraphics[width=100mm]{fig02-04.jpg}

\begin{center} {\bf Figure  4:  Information Sets and Information Partitions.}
\end{center}




 One node cannot belong to two different information sets of a single player.

 If
node $J_3$ belonged to information sets \{$J_2$,$J_3$\} and \{$J_3$,$J_4$\}
(unlike in Figure  4), then if the game reached $J_3$, Jones would not know
whether he was at a node in \{$J_2$, $J_3$\} or a node in \{$J_3$, $J_4$\}---
which would imply that they were really the same information set.
 

  If the nodes in one of Jones's information sets are nodes at which he moves,
his action set must be the same at each node, because he knows his own action
set (though his actions might differ later on in the game depending on whether
he advances from $J_3$ or $J_4$).

 
 
\newpage

\noindent {\it Player i's {\bf information partition} is a collection of his
information sets such that\\
 1 Each path is represented by one node in a single information set in the
partition, and\\ 2 The predecessors of all nodes in a single information set are
in one information set.}

 \includegraphics[width=100mm]{fig02-04.jpg}

\begin{center} {\bf Figure  4:  Information Sets and Information Partitions.}
\end{center}




  One of Smith's information partitions is
(\{$J_1$\},\{$J_2$\},\{$J_3$\},\{$J_4$\}).

  The definition rules out information
set \{$S_1$\} being in that partition, because the path going through $S_1$ and
$J_1$ would be represented by two nodes. 

 Instead, \{$S_1$\} is a separate
information partition, all by itself.

\newpage

 \includegraphics[width=100mm]{fig02-04.jpg}

\begin{center} {\bf Figure  4:  Information Sets and Information Partitions.}
\end{center}


  The information partition refers to a
stage of the game, not chronological time. The information partition
(\{$J_1$\},\{$J_2$\},\{$J_3$,$J_4$\}) includes nodes in both 1985 and 1986, but
they are all immediate successors of node $S_1$.

   Jones has the information partition (\{$J_1$\},\{$J_2$\},\{$J_3$,$J_4$\}).
There are two ways to see that his information is worse than Smith's.  First is
the fact that one of his information {\it sets}, \{$J_3$,$J_4$\}, contains {\it
more} elements than Smith's, and second, that one of his information {\it
partitions}, (\{$J_1$\},\{$J_2$\},\{$J_3$,$J_4$\}), contains {\it fewer}
elements.

\newpage



 \includegraphics[width=100mm]{fig02-table02-03.jpg}



\begin{center}
 {\bf Table  3: Information Partitions  }
 \end{center}

Partition  II  is {\bf coarser}, and  partition I is {\bf finer}. 

 Partition II
is thus a    coarsening of partition I, and partition I is a refinement of
partition II.  

 The ultimate refinement is for each information set to be a {\bf
singleton}, containing one node, as in the case of partition I. 


   A finer information partition is  the formal definition for ``better
information.'' 

 





\newpage


  Coarse information
can have a number of advantages. 

(a) It may  permit  a player to engage in trade
because other players do not fear his superior information. 


 (b) It may give  a
player a stronger strategic position  because he usually has a strong position
and  is better off not knowing that in a particular realization of the game his
position is weak.

 (c)  Poor information may permit players to insure each other.

\newpage

 (c)  Poor information may permit players to insure each other.

 

 Suppose Smith and Jones, both risk averse,  work for the same  employer, and
both   know that one of them chosen randomly will be   fired at the end of the
year  while the other   will be promoted. The one who is fired will end with a
wealth of 0  and the one who is  promoted will  end with   100.

 The two workers
will agree to insure each other by pooling   their wealth: they  will agree that
whoever is promoted will pay 50 to whoever is fired.  Each would then end up
with a guaranteed utility  of U(50).

 If   a helpful outsider offers to tell them
who will be fired before they make their  insurance agreement,  they should
cover their ears and refuse to listen.

 
\newpage





 \noindent {\bf Common Knowledge}

 

\noindent
  {\it Information is {\bf common knowledge} if it is known to all the
players, if each player knows that all the players know it, if each player knows
that all the players know that all the players know it, and so forth ad
infinitum.}

 

  

Models are set up so that the extensive form is assumed to be  common 
knowledge. 

  The Blue-Eyed Islander Puzzle. An island  starts with  2 blue-eyed people and 48 brown-eyed, but the people do not know these numbers.  If a person ever decides  his eyes are blue, he must leave the island at dawn the next day. There are no mirrors and people may not talk about eye color, but they see each others' faces. 

What will happen? -- nobody leaves. 

Now an outsider comes to the island and says, "At least one of you has blue eyes". The next dawn, nobody leaves, but on the second dawn, both blue-eyed people leave. 

 The reason: Both blue-eyed people realize there are either 1 or 2 blue-eyed people. When nobody leaves  on the first dawn, each realizes that there must be 2-- and he is one of them. 


\newpage

 

 \noindent
 Information Categories: 

Perfect: each information set is a singleton

Certain: Nature makes no moves

Symmetric: No player has information different from any other

 Complete: Nature does not move first, or her initial move is public information. 



 
 \newpage

\noindent {\it In a game of {\bf perfect information} each information set is a
singleton.} {\it Otherwise the game is one of {\bf imperfect information}}.

  The strongest informational requirements are met by a game of perfect
information, because in such a game each player always knows exactly where he is
in the game tree. No moves are simultaneous, and all players observe Nature's
moves. {Ranked  Coordination} is a game of imperfect information because of its
simultaneous moves, but Follow-the-Leader         I     is a game of perfect
information.  Any game of incomplete or asymmetric information is also a game of
imperfect information.

\newpage


  \noindent
 {\it A game of {\bf certainty} has no moves by Nature after any player moves.
Otherwise the game is one of {\bf uncertainty.}}

 
 


 \includegraphics[width=100mm]{fig02-05.jpg}




  \begin{center} {\bf Figure 5:  { Follow-the-Leader      II} } \end{center}

    {\bf von Neumann-Morgenstern utility functions } are necessary when there is either uncertainty or random (mixed) strategies. 

 

Maximizing their expected utilities, the players would behave exactly the same
as in  Follow-the-Leader  I.  

The players can differ
in how they map money to utility-- introducing risk aversion.  It could be that (0,0) represents (\$0,
\$5,000), (10,10) represents (\$100,000, \$100,000), and (2,2), the expected
utility, could here represent a non-risky (\$3,000, \$7,000).


\newpage

\noindent {\it In a game of {\bf symmetric information}, a player's information
set at \\ 1 any node where he chooses an action, or \\ 2 an end node\\ contains
at least the same elements as the information sets of every other player.
Otherwise the game is one of {\bf asymmetric information.}}

 
  The one point at
which information may differ is when the player {\it not} moving has superior
information because he knows what his own move {\it was}; for example, if the
two players move simultaneously.  Such information does not help the informed
player, since by definition it cannot affect his move.

\newpage

 
  \noindent {\it In a game of {\bf incomplete information}, Nature moves
first and is unobserved by at least one of the players. Otherwise the game is
one of {\bf complete information.}}

 A game with incomplete information also has imperfect information, because some
player's information set includes more than one node. Two kinds of games have
complete but imperfect information: games with simultaneous moves, and games
where, late in the game, Nature makes moves not immediately revealed to all
players.

  Many games of incomplete information are games of asymmetric information, but
the two concepts are not equivalent. If there is no initial move by Nature, but
Smith takes a move unobserved by Jones, and Smith moves again later in the game,
the game has asymmetric but complete information.  The principal-agent games  of
Chapter 7 are again examples: the agent knows how hard he worked, but his
principal never learns, not even at the end nodes.  A game can also have
incomplete but symmetric information: let Nature, unobserved by either player,
move first and choose the payoffs for $(Confess, Confess)$ in  the  Prisoner's
Dilemma to be either $(-6,-6) $ or $(-100,-100)$.

Harris \& Holmstrom (1982):  Nature assigns different abilities to workers, but workers don't know them.


\newpage

\noindent {\bf Poker Examples of Information Classification}

\noindent In the game of poker, the players make bets on who will have the best
hand of cards at the end, where a ranking of hands has been pre-established. How
would the following rules for behavior before betting be classified?  (Answers
are in note N2.3)

\noindent \begin{enumerate} \item All cards are dealt face up.\\ \item   All
cards are dealt face down, and a player cannot look even at his own cards before
he bets.\\
\item    All cards are dealt face down, and a player can look at his own
cards.\\ \item   All cards are dealt face up, but each player then scoops up his
hand and secretly discards one card.\\ \item  All cards are dealt face up, the
players bet, and then each player receives one more card face up.\\ \item   All
cards are dealt face down, but then each player scoops up his cards without
looking at them and holds them against his forehead so all the {\it other}
players can see them (Indian poker). \end{enumerate}

\newpage

 \noindent
 {\bf  2.4 The Harsanyi Transformation and  Bayesian Games  }

\bigskip \noindent
  {\bf The Harsanyi Transformation: Follow-the-Leader         III}

\noindent The term ``incomplete information'' is used in two quite different
senses in the literature, usually without explicit definition.  The definition
in Section 2.3 is what economists commonly {\it use}, but if asked to {\it
define} the term, they might come up with the following, older, definition.

\noindent {\bf Old definition}\\
  {\it In a game of {\bf complete information}, all players know the rules of
the game.  Otherwise the game is one of {\bf incomplete information.}}

   The old definition is not meaningful, since the game itself is ill  defined
if it does not specify exactly what the players' information sets are.  Until
1967, game theorists spoke of games of incomplete information to say that they
could not be analyzed.  Then John Harsanyi pointed out that any game that had
incomplete information under the old definition could be remodelled as a game of
complete but imperfect information without changing its essentials, simply by
adding an initial move in which Nature chooses between different sets of rules.
In the transformed game, all players know the new meta-rules, including the fact
that Nature has made an initial move unobserved by them. Harsanyi's suggestion
trivialized the definition of incomplete information, and people began using the
term to refer to the transformed game instead.  Under the old definition, a game
of incomplete information was transformed into a game of complete information.
Under the new definition, the original game is ill defined, and the transformed
version is a game of incomplete information.

  Follow-the-Leader         III       serves to illustrate the Harsanyi
transformation.  Suppose that Jones does not know the game's payoffs precisely.
He does have some idea of the payoffs, and we represent his beliefs by a
subjective probability distribution. He places a 70 percent probability on the
game being  game (A)   in Figure 6   (which is the same as     Follow-the-Leader
I), a 10 percent chance on game (B), and a 20 percent on game (C). In reality
the game has a particular set of payoffs, and Smith knows what they are.  This
is a game of incomplete information (Jones does not know the payoffs),
asymmetric information (when Smith moves, Smith knows something Jones does not),
and certainty  (Nature does not move after the players do.)





 \includegraphics[width=150mm]{fig02-06.jpg}


\begin{center} {\bf Figure   6:    Follow-the-Leader   III: Original }
\end{center}

   The game cannot be analyzed in the form shown in Figure 6.  The natural way
to approach such a game is to use the Harsanyi transformation. We can remodel
the game to look like Figure 7, in which Nature makes the first move and chooses
the payoffs of game (A), (B), or (C), in accordance with Jones's subjective
probabilities. Smith observes Nature's move, but Jones does not. Figure 7
depicts the same game as Figure 6, but now we can analyze it. Both Smith and
Jones know the rules of the game, and the difference between them is that Smith
has observed Nature's move. Whether Nature actually makes the moves with the
indicated probabilities or Jones just imagines them is irrelevant, so long as
Jones's initial beliefs or fantasies are common knowledge.

\includegraphics[width=150mm]{fig02-07.jpg}



\begin{center} {\bf Figure 7:   Follow-the-Leader  III: After the Harsanyi
Transformation } \end{center}

   Often what Nature chooses at the start of a game is the strategy set,
information partition, and payoff function of one of the players. We say that
the player can be any of several ``types,'' a term to which we will return in
later chapters. When Nature moves, especially if she affects the strategy sets
and payoffs of both players, it is often said that Nature has chosen a
particular ``state of the world.'' In Figure 7 Nature chooses the state of the
world to be (A), (B), or (C).

\noindent {\it A player's {\bf type} is the strategy set, information partition,
and payoff function which Nature chooses for him at the start of a game of
incomplete information.}

\noindent {\it A {\bf state of the world} is a move by  Nature.}

As I have already said,  it is good modelling practice to assume that the
structure of the game is common knowledge, so that though Nature's choice of
Smith's type may really just represent Jones's opinions about Smith's possible
type, Smith knows what Jones's possible opinions are and Jones knows that they
are just opinions. The players may have different beliefs, but that is modelled
as the effect of their observing different moves by Nature. All players begin
the game with the same beliefs about the probabilities of the moves Nature will
make--- the same priors, to use a term that will shortly be introduced.   This
modelling assumption is known as the {\bf Harsanyi doctrine}.   If the modeller
is following it,  his model can never reach a situation where two players
possess exactly the same information but disagree as to the probability of some
past or future move of Nature. A model cannot, for example, begin by  saying
that Germany believes  its probability of winning a war against France is 0.8
and France believes it is 0.4, so they are both willing to go to war. Rather, he
must assume that beliefs begin the same but diverge because of private
information. Both players initially think that the probability of a German
victory is 0.4 but that if General Schmidt is a genius   the probability rises
to 0.8, and then Germany discovers that Schmidt is indeed a genius. If it is
France that has the initiative to declare war,   France's mistaken beliefs may
lead to a conflict that would be avoidable if Germany could credibly reveal its
private information about Schmidt's genius.

 An implication  of the Harsanyi doctrine is that players are  at least slightly
open-minded about their opinions. If Germany indicates that it is willing to go
to war, France must consider the possibility that Germany has discovered
Schmidt's genius and update the probability that Germany will win (keeping in
mind that Germany might be bluffing).  Our next topic is how a player updates
his beliefs upon receiving new information, whether it be by direct observation
of Nature or by observing the moves of another player who might be better
informed.

\bigskip \noindent {\bf Updating Beliefs with Bayes's Rule}

\noindent When we classify a game's information structure we do not try to
decide what a player can deduce from the other players' moves. Player Jones
might deduce, upon seeing Smith choose {\it Large,} that Nature has chosen state
(A), but we do not draw Jones's information set in Figure 7 to take this into
account. In drawing the game tree we want to illustrate only the exogenous
elements of the game, uncontaminated by the equilibrium concept. But to find the
equilibrium we do need to think about how beliefs change over the course of the
game.

  One part of the rules of the game is the collection of {\bf prior beliefs} (or
{\bf priors}) held by the different players, beliefs that they update in the
course of the game. A player holds prior beliefs concerning the types of the
other players, and as he sees them take actions he updates his beliefs under the
assumption that they are following equilibrium behavior.

  The term {\bf bayesian equilibrium} is used to refer to a Nash equilibrium in
which players update their beliefs according to Bayes's Rule. Since Bayes's Rule
is the natural and standard way to handle imperfect information, the adjective,
``bayesian,'' is really optional. But the two-step procedure of checking a Nash
equilibrium has now become a three-step procedure:

\noindent 1 Propose a strategy profile.\\
 2 See what beliefs the strategy profile generates when players update their
beliefs in response to each others' moves.\\ 3 Check that given those beliefs
together with the strategies of the other players each player is choosing a best
response for himself.

  The rules of the game specify each player's initial beliefs, and Bayes's Rule
is the rational way to update beliefs.  Suppose, for example, that Jones starts
with a particular prior belief, $Prob(Nature \; chose \;(A))$. In { Follow-the-
Leader III}, this equals 0.7. He then observes  Smith's move --- $Large$,
perhaps. Seeing  $Large$ should make Jones  update to the {\bf posterior}
belief, $Prob ( Nature  \; chose \; (A)) |Smith \; chose\; Large)$, where the
symbol ``$|$'' denotes ``conditional upon'' or ``given  that.''

 Bayes's Rule shows how to revise the prior belief in the light of new
information such as Smith's move. It uses two pieces of information, the
likelihood of seeing  Smith choose $Large$ given that   Nature chose state of
the world (A),   $Prob (  Large|  (A) )$, and the likelihood of seeing Smith
choose $Large$   given that   Nature  did not choose state (A),   $Prob (
Large|  (B)\; or\; (C) )$.     From these numbers, Jones  can calculate $Prob
(Smith \; chooses\; Large )$, the {\bf marginal likelihood} of seeing $Large$ as
the result of one or another of   the possible states of the world that Nature
might choose. \begin{equation}\label{e2.1} \begin{array}{ll}
Prob (Smith \;
chooses\; Large ) &= Prob (  Large|  A  ) Prob(A) + Prob (  Large| B  ) Prob(B)
\\ &  + Prob (  Large| C  ) Prob(C).\\ \end{array} \end{equation}

 To find his posterior, $Prob ( Nature  \; chose \;(A)) |Smith \; chose\; Large)
$,  Jones   uses the  likelihood and his priors.   The  joint probability of
both seeing Smith choose $Large$  and   Nature  having  chosen (A) is
\begin{equation}\label{e2.2} Prob(Large,A) = Prob(A|Large)Prob(Large) =
Prob(Large|A)Prob(A). \end{equation}

  Since what Jones is trying to calculate is $Prob(A|Large)$, rewrite the last
part of  (\ref{e2.2}) as follows:
 \begin{equation}\label{e2.3}
 Prob(A|Large) = \frac{ Prob(Large|A)Prob(A)} {Prob (Large)}.
 \end{equation}
Jones needs to calculate his new belief --- his posterior --- using $Prob(Large)
$, which he calculates from his original knowledge using (\ref{e2.1}).
Substituting the expression for $Prob(Large)$ from (\ref{e2.1}) into equation
(\ref{e2.3}) gives the final result, a version of Bayes's Rule.
   
 \begin{equation}\label{e2.4} Prob(A|Large) = \frac{Prob(Large|A)
Prob(A)} {Prob(Large|A)Prob(A) + Prob(Large|B )Prob(B)+ Prob(Large|C)Prob(C)}.
\end{equation}
 

 More generally, for Nature's move $x$ and  the observed  data,
  \begin{equation}\label{e2.5}
  Prob(x|data) = \frac{Prob( data|x)Prob(x)} {Prob(data)}
  \end{equation}

  Equation (\ref{e2.6}) is a verbal form of Bayes's Rule,  which is useful for
remembering the terminology,\footnote{The name ``marginal likelihood'' may seem
strange to economists, since it is an
unconditional likelihood and when economists use ``marginal'' they mean ``an
increment conditional on starting from a particular level''.   The statisticians
defined marginal likelihood this way
because they start with $Prob (a, b)$, and then derive   $Prob (b)$.
  That is like going to the margin  of a graph in $(a,b)$-space, the $b$-axis,
and asking how probable the value of  $b$  is integrating over all possible
$a$'s.} summarized in Table  5.
 
\begin{equation}\label{e2.6}
 ( Posterior \;for\; Nature's \; \;Move) = \frac{ (Likelihood\;of\; Player's \;
\;Move) \cdot (Prior \ for\; Nature's \; \;Move)} {(Marginal\; likelihood\; of
\; Player's \; \;Move)}.
 \end{equation}
 
   Bayes's Rule is not purely mechanical. It is the only way to rationally
update beliefs. The derivation is worth understanding, because Bayes's Rule is
hard to memorize  but easy to rederive.

\begin{center} {\bf Table  5:  Bayesian Terminology }

 \begin{tabular}{ll}
 \hline
 & \\
{\bf Name} & {\bf Meaning} \\
 & \\
\hline
 & \\  Likelihood & $Prob(data|event)$\\
 Marginal likelihood & $Prob(data) $\\
Conditional Likelihood & $Prob(data$ X$| data$ Y, $event)$\\
Prior & $Prob( event)$\\
Posterior & $Prob( event|data)$\\
 & \\
 \hline
  \end{tabular}
 \end{center}

 \bigskip

 \noindent
 {\bf Updating Beliefs in  Follow-the-Leader      III}

\noindent
 Let us now return to the numbers in  Follow-the-Leader      III    to use
the belief-updating rule that was just derived. Jones has a prior belief that
the probability of event   ``Nature picks  state (A)''  is 0.7 and he needs to
update that belief on seeing the data  ``Smith picks $Large$''. His prior is
$Prob(A) = 0.7$, and we wish to calculate $Prob (A|Large)$.

  To use Bayes's Rule from equation (\ref{e2.4}), we need the values of $Prob
(Large|A)$, $Prob (Large|B)$, and $Prob (Large|C)$. These values depend on what
Smith does in equilibrium, so    Jones's beliefs cannot be calculated
independently of the equilibrium. This is the reason for the three-step
procedure suggested above, for   what the modeller must do is propose an
equilibrium and then use it to calculate the beliefs. Afterwards, he must check
that the equilibrium strategies are indeed the best responses given the beliefs
they generate.

   A candidate for equilibrium in {Follow-the-Leader III} is for Smith to choose
$Large$ if the state is (A) or (B) and $Small$ if it is (C),  and for Jones to
respond to $Large$ with $Large$ and to $Small$ with $Small.$ This can be
abbreviated as $(L| A , L| B , S| C ; L|L, S|S)$.    Let us test that this is an
equilibrium, starting with the calculation of $Prob(A|Large)$.

If Jones observes $Large,$ he can rule out state (C), but he does not know
whether the state is (A) or (B). Bayes's Rule  tells  him that the posterior
probability of state (A) is
 \begin{equation} \label{e2.7}
  \begin{array}{ll}
Prob(A|Large)  & = \frac{(1)(0.7)}{(1)(0.7) + (1)(0.1) + (0)(0.2)}\\ & \\ &  =
0.875. \end{array} \end{equation} The posterior probability of  state (B)  must
then be  $1-0.875 = 0.125$, which could also be calculated from Bayes's Rule, as
follows: \begin{equation} \label{e2.8} \begin{array}{ll}
(B|Large) &=
\frac{(1)(0.1)}{(1)(0.7) + (1)(0.1) + (0)(0.2)}\\ & \\ & = 0.125.
 \end{array}
\end{equation}

 Figure 8  shows a  graphic intuition for Bayes's Rule.  The first line shows
the total probability, 1, which is the sum of the prior probabilities of states
(A), (B), and (C). The second line shows the probabilities, summing to 0.8,
which remain after $Large$ is observed and state (C) is ruled out.  The third
line shows that state   (A)  represents an amount 0.7 of that probability, a
fraction of 0.875. The  fourth line  shows that state   (B) represents an amount
0.1 of that probability, a fraction of 0.125.

\includegraphics[width=150mm]{fig02-08.jpg}



\begin{center}
 {\bf Figure 8:  Bayes's Rule} \end{center}

   Jones must use Smith's strategy in the proposed equilibrium to find numbers
for $Prob(Large|A)$, $Prob(Large|B)$, and $Prob(Large|C)$. As always in Nash
equilibrium, the modeller assumes that the players know which equilibrium
strategies are being played out, even though they do not know which particular
actions are being chosen.

 Given that Jones believes that the state is (A) with probability 0.875 and
state (B) with probability 0.125, his best response is $Large$, even though he
knows that if the state were actually  (B) the better response would be $Small$.
Given that he observes $Large$, Jones's expected payoff from $Small$ is $-0.625$
( $= 0.875[-1] + 0.125 [2]$), but from $Large$ it is $ 1.875$ ( $= 0.875[2] +
0.125 [1]$).  The strategy profile  $(L| A , L| B , S| C ; L|L, S|S)$ is a
bayesian equilibrium.

 A similar calculation can be done for $Prob (A|Small)$.  Using Bayes's Rule,
equation (\ref{e2.4}) becomes
 \begin{equation} \label{e2.9} Prob (A|Small) =
\frac{(0)(0.7)}{(0)(0.7) + (0)(0.1) + (1)(0.2)} = 0.
 \end{equation}
 Given that
he believes the state is (C), Jones's best response to $Small$ is $Small$, which
agrees with our proposed equilibrium.

 Smith's best responses are much simpler. Given that Jones will imitate his
action, Smith does best by following his equilibrium strategy of   ($L| A , L|B,
S| C$).

  The calculations are relatively simple because Smith uses a nonrandom strategy
in equilibrium, so, for instance, $Prob(Small|A) =0$ in equation (\ref{e2.9}).
Consider what happens if Smith uses a random strategy of picking $Large$ with
probability 0.2 in state (A), 0.6 in state (B), and 0.3 in state (C) (we will
analyze such ``mixed'' strategies in Chapter 3).  The equivalent of equation
(\ref{e2.7}) is
 \begin{equation} \label{e2.10}
Prob(A|Large) = \frac{(0.2)(0.7)}
{(0.2)(0.7) + (0.6)(0.1) + (0.3)(0.2)} = 0.54  \;\;(rounded). \end{equation} If
he sees $Large$, Jones's best guess is still that Nature chose state (A), even
though in  state (A) Smith has the smallest probability of choosing $Large$,
but Jones's subjective posterior  probability, $Pr(A|Large)$,  has fallen to
0.54  from   his  prior of $Pr(A)=0.7$.

The last two lines of Figure 8 illustrate  this case. The second-to-last line
shows the total probability  of $Large$, which is formed from the probabilities
in all three states and sums to  0.26 (=$0.14 + 0.06 + 0.06$). The last line
shows the component of that probability arising from state (A), which is the
amount 0.14 and fraction 0.54 (rounded).

\newpage

\noindent
 {\bf Regression to the Mean }

 
  The teacher does not know the individual student's ability, but
does know that the average student will score 70 out of 100. If a student scores
40, what should the teacher's estimate of his ability be?

It should not be 40. A score of 30 points below the average score could be the
result of two things: (1) the student's ability is  below average, or (2) the
student was in a bad mood the day of the test. Only if mood is completely
unimportant should the teacher use 40 as his estimate. More likely, both ability
and luck matter to some extent, so the teacher's best guess is that the student
has an ability below average  but was also unlucky. The best estimate lies
somewhere between 40 and 70, reflecting the  influence of both ability and luck.
Of the students who score 40 on the test, more than half can be expected to
score above 40 on the next test. Since the scores of these poorly performing
students tend to float up towards the mean of 70, this phenomenon is called
``regression to the mean.''  Similarly, students who score 90 on the first test
will tend to score less well on the second test.
 

 In bayesian terms, the teacher in this example has a prior mean of 70, and is
trying to form a posterior estimate using the prior and one piece of data, the
score on the first test. For typical distributions, the posterior mean will lie
between the prior mean and the data point, so the posterior mean will be between
40  and 70.

 

 Suppose that the firm will not spend \$100,000 on an
investment with a present value of \$105,000.  This is easily explained if the
\$105,000   is an estimate and the \$100,000   is cash. If the average value of
a new project of this kind is less than \$100,000 --- as is likely to be the
case  since profitable projects are not easy to find --- the best estimate of
the value will lie between the measured value of  \$105,000 and  that average
value, unless the staffer who came up with the \$105,000 figure has already
adjusted his estimate. Regressing the \$105,000 to the mean may regress it past
\$100,000. Put a bit differently, if the prior mean is, let us say, \$80,000,
and the data point is \$105,000, the posterior may well be less than \$100,000.
Regression to the mean  is an alternative to
strategic behavior in explaining certain odd phenomena.  
  

\newpage

  The ``Two-Armed Bandit''
model of Rothschild (1974).

 In each of a sequence of periods, a person
chooses to play slot machine A or slot machine B. Slot machine A pays
out \$1 with known probability 0.5 in exchange for the person putting in
\$0.25 and pulling its arm.

 Slot machine B pays out \$1 with an unknown
probability which has a prior probability density centered on 0.5.
The optimal strategy is to begin by playing machine B, since
not only does it have the same expected payout per period, but also
playing it improves the player's information, whereas playing machine A
leaves his information unchanged. The player will switch to machine A
if machine B pays out \$0 often enough relative to the number of times
it pays out \$1, where ``often enough'' depends on the particular prior
beliefs he has. If the first 1,000 plays all result in a payout of \$1,
he will keep playing machine B, but if the next 9,000 plays all result
in a payout of \$0, he should become very sure that machine B's payout
rate is less than 0.5 and he should switch to machine A. But he will
never switch back. Once he is playing machine A, he is learning
nothing new as a result of his wins and losses, and even if he gets a
payout of \$0 ten thousand times in a row, that gives him no reason to
change machines. As a result, it can happen that even if machine B
actually is better, a player following the ex ante optimal strategy
can end up playing machine A an infinite number of times.

\newpage



 {\bf  Cascade model}.   Bikchandani, Hirshleifer \& Welch (1992).

 A
sequence of people must decide whether to Adopt at cost 0.5
or Reject a project worth either 0 or 1 with equal prior
probabilities, having observed the decisions of people ahead
of them.

Each has an  independent private signal that takes the
value High with probability $p>0.5$ if the project's value is 1
and with probability $(1-p)$ if it is 0, and
otherwise takes the value Low.

 



 

\newpage
 

\begin{center} {\bf The Png Settlement Game } \end{center}
 {\bf Players}\\
The
plaintiff and
the defendant.

\noindent
{\bf The Order of Play} \\ 0 Nature chooses the defendant to be Liable
for injury to the plaintiff with probability  $q=0.13$ and Blameless otherwise.
The defendant observes this but the plaintiff does not.  \\ 1  The plaintiff
decides to Sue or just to Grumble.  \\
 2   The defendant  Offers a settlement amount of  $S=0.15$ to the plaintiff, or
Resist, setting $S=0$.  \\ 3a   If the     defendant offered $S=0.15$, the
plaintiff agrees to Settle or he Refuses and goes to trial.  \\ 3b   If the
defendant  offered $S=0$,   the plaintiff  Drops the case, for legal costs of
$P=0$ and $D=0$ for himself and the defendant,  or chooses to Try it, creating
legal costs of $P=0.1$ and $D=0.2$ \\ 4  If the case goes to trial, the
plaintiff wins damages of $W=1$ if the defendant is Liable and  $W=0$ if the
defendant is Blameless. If the case is dropped, $W=0$.

\noindent {\bf Payoffs}\\
The plaintiff's    payoff is  $S+W -P$. The
defendant's payoff is $-S-W -D$.

 \newpage

\includegraphics[width=150mm]{fig02-09.jpg}


 \begin{center} {\bf Figure 9:  The Game Tree for The  Png Settlement Game }
\end{center}

This model assumes  that the settlement amount, $S=0.15$, and the amounts spent
on legal fees are exogenous.  Except in the infinitely long games without end
nodes that will appear in Chapter 5, an extensive form should incorporate all
costs and benefits into the payoffs at the end nodes, even if costs are incurred
along the way.  If the court required a \$100 filing fee (which it does not in
this game, although a fee will be required in the similar game of Nuisance Suits
in Section 4.3), it would be    subtracted from the  plaintiff's payoffs at
every end node  except those resulting from his choice of  {\it Grumble}.  Such
consolidation makes it easier to analyze the game and would not affect the
equilibrium strategies unless payments along the way revealed information, in
which case what matters is the information, not the fact that payoffs change.

 We assume that if the case reaches the court, justice is done. In addition to
his legal fees $D$, the defendant pays damages $W=1$ only if he is liable.  We
also assume that the players are risk neutral, so they only care about the
expected dollars they will receive, not the variance.  Without this assumption
we would have to translate the dollar payoffs into utility, but the game tree
would be unaffected.

  This is a game of certain, asymmetric, imperfect, and incomplete information.
We have assumed that the defendant knows whether he is liable, but we could
modify the game by assuming that he has no better idea than the plaintiff of
whether the evidence is sufficient to prove him so.  The game would become one
of symmetric information and we could reasonably simplify the extensive form by
eliminating the initial move by Nature and setting the payoffs equal to the
expected values.  We cannot perform this simplification in the original game,
because the fact that the defendant, and only the defendant, knows whether he is
liable strongly affects the behavior of both players.

  Let us now find the equilibrium. Using dominance we can rule out one of the
plaintiff's strategies immediately --- {\it Grumble} --- which is dominated by
({\it Sue, Settle, Drop}).

    Whether a strategy profile is a Nash equilibrium depends on the parameters
of the model---$S, W, P, D$ and $q$, which are the settlement amount, the
damages, the court costs for the plaintiff and defendant, and the probability
the defendant is liable. Depending on the parameter values, three outcomes are
possible: settlement (if the settlement amount is low), trial (if expected
damages are high and the plaintiff's court costs are low), and the plaintiff
dropping the action (if expected damages minus court costs are negative).
Here, I have inserted the parameter values   $S = 0.15, D = 0.2, W = 1, q =
0.13$, and $P = 0.1$.  Two Nash equilibria exist for this set of parameter
values, both weak.

 One equilibrium is  the  strategy profile \{({\it Sue, Settle, Try}), ({\it
Offer, Offer})\}.  The plaintiff sues, the defendant offers to settle (whether
liable or not), and the plaintiff agrees to settle. Both players know that if
the defendant did not offer to settle, the plaintiff would go to court and try
the case.  Such {\bf out-of-equilibrium} behavior is specified by the
equilibrium, because the threat of trial is what induces the defendant to offer
to settle, even though trials never occur in equilibrium.  This    is a Nash
equilibrium because  given that the plaintiff chooses ({\it Sue, Settle, Try}),
the defendant can do no better than ({\it Offer, Offer}), settling for a payoff
of $-0.15$ whether he is liable or not; and, given that the defendant chooses
({\it Offer, Offer}), the plaintiff can do no better than the payoff of $0.15$
from ({\it Sue, Settle, Try}).

The other equilibrium is  \{({\it Sue, Refuse, Try}), ({\it Resist, Resist})\}.
The plaintiff sues, the defendant  resists and makes no settlement offer, the
plaintiff would refuse any offer that was made, and goes to trial.  Since
  he foresees   the plaintiff will refuse a  settlement offer of $S=0.15$,
   the defendant is willing to resist,  because his action makes no difference.


 
 

 
 
 

 
 

\end{LARGE}

\end{document}



