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#This file was created by <hawk> Fri Mar 31 15:21:55 2000
#LyX 1.0 (C) 1995-1999 Matthias Ettrich and the LyX Team
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\layout Title
Game Theory II
\layout Date
January 21, 2000c
\layout Author
Richard E.
Hawkins
\layout Standard
Game theory can be quite useful in understanding economics.
It does not seek to explain the games we play for entertainment, but to
express decisions by multiple agents in the form of a game, modeling decisions
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as if
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individuals were players in a game.
\layout Section
Equilibrium in Game Theory
\layout Standard
\layout Section
he Prisoners' Dilemma
\layout Standard
Before considering games in general, let's consider the most popular game
of all, the prisoners' dilemma, as seen in Figure
\begin_inset LatexCommand \ref{pd1}
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.
This table may not have any immediate and obvious meaning, but it conveys
several types of information to those who can read it.
On the other hand, without a story to accompany it, it means absolutely
nothing.
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\begin_float fig
\layout Standard
\align center
\begin_inset Figure size 165 93
file pd1.ps
flags 9
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\layout Caption
\begin_inset LatexCommand \label{pd1}
\end_inset
The basic Prisoners' Dilemma game
\end_float
\layout Subsection
The Story
\layout Standard
Larry and Moe have successfully robbed the bank while well disguised.
District Attorney Curley knows this, but can't prove it--but he can prove
a minor assault charge for the two of them.
Larry and Moe, like all good crooks, have agreed not to rat on one another
if caught, but they've been put in separate cells and can't tell whetehr
theother is holding to the agreement.
\layout Standard
Curly makes an offer to the prisoners: Defect from your criminal enterprise
(thus the
\begin_inset Quotes eld
\end_inset
D
\begin_inset Quotes erd
\end_inset
in the chart), by ratting on the other, and get a better sentence.
If you confess, and the other doesn't, you can go free, but if you both
confess, it's two years.
If you don't confess and your partner does, it's five years.
If neither confesses, they will both be locked up for six months for assault.
\layout Subsection
Relating the Story to the Chart
\layout Standard
If you look at the numbers from the sentences, you will see that in Figure
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, the negative numbers are the sentence lengths in months.
In each pair of numbers, the first is the
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payoff
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\end_inset
to the player on the left, or what he receives, and the second is the payoff
for the player on the top.
Notice that the numbers are all negative--a jail sentence is worse than
nothing, and has negative value.
\layout Standard
To determine which pair of payoffs to use, look at each player's actions--if
Larry cooperates and Moe defects, chose the first row (C) and the second
column (D) for a payoff of (-60,0), meaning that since Larry kept his mouth
shut while Moe ratted him out, Larry does five years while Moe walks.
\layout Subsection
Predicting the Outcome of the Game
\layout Standard
Both players must choose their moves without knowing the other's move.
Nonetheless, if Larry and Moe are bright enough, we can predict what will
happen in this game.
Look at the game first from Larry's perspective.
If Moe cooperates, keeping his mouth shut, what is the best thing for Larry
to do? If Larry cooperates, he'll do six months, but he'll walk if he defects
by ratting out Moe.
Larry is clearly better off defecting if Moe cooperates.
Now consider what happens to Larry if Moe defects.
If Larry defects as well, he does two years--which increases to five if
he cooperates.
\layout Standard
No matter what Moe does, Larry's best result comes when he defects, and
Larry should defect.
Moe faces the same choices, and we conclude that Moe will also defect.
We predict that if both players in this game are rational, they will both
defect--in spite of the fact that there is a possible outcome in which
\emph on
both
\emph default
players are better off than the outcome we reach.
\layout Section
Some Key Topics in Game Theory
\layout Standard
Having seen a simple but quite common game, we have a framework to discuss
some of the problems that were involved.
Let's take a closer look at these.
\layout Subsection
Strategy
\layout Standard
A player's strategy is the set of choices that a player will make at each
possible point in the game, and with each possible information set (what
he could know at that point) that can occur.
In the simple prisoners' dilemma, there is only one such point, the beginning
of the game, with no knowledge other than the payoffs of the game--the
player moves blindly without knowing the other player's move, and thus
the strategy is simply C or D.
\layout Subsection
The Payoff Matrix
\layout Standard
This bit of knowledge leads to the next thing that a player must have: knowledge
the payoff matrix.
If the player doesn't know the payoffs in the game, he cannot move in an
intelligent manner.
While such a game
\emph on
could
\emph default
be played, it wouldn't provide information about how people make choices.
\layout Standard
Keep in mind that the payoffs must represent a
\emph on
complete
\emph default
account of the payoffs in the game.
In the prisoners' dilemma, the total payoff to the players actually includes
much more than just the jail sentence.
What happened to the money? If Larry rats out Moe, what will Moe do to
Larry when he gets out? What about his conscience? The payoff should include
all of these factors.
\layout Subsection
Normal Form Games
\layout Standard
This game is written in what is called
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\end_inset
normal form.
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\end_inset
Many games are simple enough to place into rows and columns, with each
possible strategy for the first player as the label of a row, and each
possible strategy for the second player labelling a column.
Games with only two players and a single simultaneous move are usually
explained more clearly in this form.
The extended form will be discussed later.
\layout Subsection
Non-Cooperative Games
\layout Standard
The prisoners' dilemma is a
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\end_inset
non-cooperative
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\end_inset
game.
This means that the players cannot make binding agreements with each other
in the game.
The players would be better off if they could make an agreement to cooperate,
seeing six month sentences rather than two year sentences.
In a cooperative game, such agreements are allowed.
\layout Section
The Terrorist Game
\layout Standard
Not all games have players moving simultaneously in ignorance of the moves
of the other players.
In the terrorist game, the terrorist has taken control of the plane and
has a bomb.
He demands a million dollars in return for not blowing up the plane.
This game is shown in Figure
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.
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\align center
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file tr1.ps
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\layout Caption
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\end_inset
The Terrorist Game
\end_float
The first move is for the airline, which can either pay (p) or not pay
(n).
The
\begin_inset Quotes eld
\end_inset
1
\begin_inset Quotes erd
\end_inset
at the top of the diagram is to show that it is player 1, the airline,
that moves at this point.
Each decision has a different
\begin_inset Quotes eld
\end_inset
node,
\begin_inset Quotes erd
\end_inset
indicated by the two black dots separated by a dotted line.
In this form, player 2, the terrorist,
\emph on
does know
\emph default
which move player 1 has chosen.
The information sets at the two nodes are therefore different--each contains
different information about how player 1 moved.
The second move is for the terrorist, who can explode the bomb (B) or not
explode it and live (L).
If the terrorist explodes the bomb, it costs the airline an additional
five million dollars, and the terrorist dies, for a payoff of -10.
\layout Standard
In deciding how to move, the airline should consider what the terrorist
\emph on
would
\emph default
do in each possible situation.
If the money has been paid, the terrorist presumably has no reason to blow
up the plane, takes the million for a payoff of 1, while the airline has
a payoff of -1 from paying the ransom.
On the other hand, if the airline does not pay, consider the terrorists
choices: die with a payoff of -10, or live with a payoff of -1 for going
to prison.
We therefore predict that the airline will not believe that the terrorist
will blow up the plane, and will refuse to pay, followed by the terrorist
choosing to live rather than carrying out the threat.
\layout Section
New Concepts from the Terrorist Game
\layout Subsection
Extensive Form Games
\layout Standard
The terrorist game has been shown in
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\end_inset
extensive form,
\begin_inset Quotes erd
\end_inset
in which the possible paths the game can follow are drawn out.
Much longer games are possible, and will be considered later.
\layout Subsection
Information Sets
\layout Standard
In this game, the terrorist could arrive at two different information sets,
depending upon the choice of the airline.
The information set for the terrorist in this case includes the move of
the prior case, but this is not always the case.
Suppose the airline is given a third choice, lying (l) that the ransom
has been paid, as in Figure
\begin_inset LatexCommand \ref{tr2}
\end_inset
.
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\layout Standard
\begin_inset Figure size 272 144
file tr2.ps
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\end_inset
\layout Caption
\begin_inset LatexCommand \label{tr2}
\end_inset
Terrorist Game where the airline can lie
\end_float
The terrorist has the same information set whether the airline pays or
lies, even though there are two decision nodes within the set.
The combined set is noted by the ellipse around the nodes.
\layout Standard
The prisoners' dilemma can also be written in extensive node as in Figure
\begin_inset LatexCommand \ref{pd2}
\end_inset
.
\begin_float fig
\layout Standard
\begin_inset Figure size 209 178
file pd2.ps
flags 9
\end_inset
\layout Caption
Prisoners' Dilemma in extensive form
\begin_inset LatexCommand \label{pd2}
\end_inset
\end_float
While the game was described as simultaneous, what if the prisoners' make
their decision in separate rooms a minute apart? In this case, the extensive
form may more accurately describe the situation faced by the second prisoner.
\layout Subsection
Credible Threats
\layout Standard
Perhaps the most important notion here is that of a credible threat--which
the terrorist's is not.
If the terrorist actually
\emph on
would
\emph default
blow up the plane if not paid, the airline would pay.
But the threat simply isn't credible, as the terrorist dies as well.
Later we will discuss the possiblity of the terrorist somehow irrevocably
committing himself to blowing up the plane in this circumstance
\emph on
before
\emph default
the airline's move, and how this would change the game.
We will also discuss experiments which show that people sometimes
\emph on
will
\emph default
take a loss in such a situation, even though we find it irrational.
\the_end
