Section 2d Game theory

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Section 2d Game theory

• Game theory is a way of thinking about situations
  where there is interaction between individuals or
  institutions. The parties may have conflicting
  interests. The theory is a structure to help us
  sort out our thinking in these situations.

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Strategy
In a game of tic-tac-toe your next move will be
influenced by my move and my move will be
influenced by your move. A strategy is a plan
for acting that responds to the reactions of
others. Game theory deals with situations of
strategy. To characterize a game we must specify
3 things 1) the players, 2) the strategies of
each player, and 3) the payoffs to each player
for each strategy.
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Prisoners dilemma
A classic game is the prisoners dilemma. Here
is an example, slightly different from the
book. Two career criminals are picked up for a
recently committed crime (which they may or may
not have committed) due to the circumstances of
their recent activity. The two are taken to the
police station and put in separate cells where
they can not communicate with each other.
Each is offered this deal.
Confess and you get 1/2 year if the other does
not confess, but you get 5 years if the other
also confesses. If you dont confess and the
other does confess you get 7 years, but if the
other does not confess we can get you for 1 year
in jail.
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Strategic form of game
The information about the criminals on the
previous page is typically put into a payoff
matrix. One player is put in the rows of the
matrix or table and one is put in the columns.
In each cell we see the payoff for each player,
given that outcome. A negative number represents
a lose and a positive number represents a gain.
The first number in each cell is the payoff to
the row player and the second is the column
player payoff. Lets put the information about
our example into a table.
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Our example
column player confess keep
quiet row confess -5, -5 - 0.5,
-7 player keep quiet -7, -0.5 -1,
-1 The assumption here is that each player would
like to have the highest number possible - in
this case that means serve the lowest jail
sentence for themselves. Each player has to
decide what to do in isolation. We will start
with the row player.
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Row player
The row player sees her options as the first
number in each pair. Now she will say If the
column player confesses the best I can do is to
confess - I get 5 years instead of 7. If the
column player does not confess the best I can do
is confess - I get a half year instead of 1
year. So the row player sees that no matter what
the column player does it is best for her to
confess. The row player is said to have a
dominant strategy - here confess.
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Column player
The column player sees his options as the
second number in each pair. Now he will say If
the row player confesses the best I can do is to
confess - I get 5 years instead of 7. If the row
player does not confess the best I can do is
confess - I get a half year instead of 1 year. So
the column player sees that no matter what the
row player does it is best for him to confess.
The column player is said to have a dominant
strategy - here confess.
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Nash equilibrium
In this game each player will confess because
that is the best each can do. So each will get
5 years. Would there be any reason for either
to change? If each thought the other would not
change, then there is no reason to change.
Changing would just make the player worse off.
This idea of no reason to change is called an
equilibrium and here it is called a Nash
equilibrium in honor of the guy who thought about
this idea a lot. Some games have no Nash Equil.,
some have more than 1.
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Pareto-efficient
Again I ask the question, would either want to
change from the confession? If each could be sure
the other would not confess, they both would be
better off not confessing. So the both
confessing solution is not pareto-efficient
because both can be better off with a move. The
problem here is could both trust the other to not
confess? If the row player confesses, what would
keep the column player from keeping quiet? The
row player here will not risk cooperating with
the column player, and vice versa.
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The super bowl
Say that I have a special bowl in my possession.
I can use it as a bowl, but also in exactly one
year the bowl will unravel into 100 single dollar
bills and the bowl will be gone. Would you pay me
105 today for the bowl? How about 95.45, or
90 or 80? Would I accept all your offers?
Well if the interest rate, r, is 10 then
every dollar today can be made into 1.10 by next
year --- 1 1(.1). In general P dollars today
can be thought of as F in one year, where F P(1
r).
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The super bowl
You would not pay 105 because next year, with
the bowl, you would only have 100. But if you
invested your 105 you would have 115.5. If you
paid 90.91 for the bowl you would have 100 with
the bowl, but also 100 without the bowl.
90.91(1 .1) 100 (if you round to nearest
penny). So you would not pay more than 90.91 for
the bowl because you would be worse off at the
end of the year. Would I accept 80? If I took
the 80 and invested at the interest rate I would
have only 88. I would lose out if I accepted
this trade. I would take exception to the trade.
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Present value of a future amount
The equation F P(1 r) means that if you have
P dollars today and you can earn interest rate r,
then you will have F at the end of one year. You
can change the equation to P F / (1 r) and
say that if you want F dollars at the end of one
year and you can earn interest rate r then you
have to start with P dollars today. If the time
frame is longer than one year then you have to
take the term (1 r) and raise it to the power
n, where n is the time frame.
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Basic formulas
If F and P are n years apart, then the
relationship between F and P can be stated F
P(1 r)n or P F / (1 r)n . Sometimes will
may see a stream of payment at various points in
the future. The present value is simply the sum
of the present value of each part.
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The theory of asset pricing
Assets like our labor, or maybe an apartment
building, can give us a future stream of
payments. What are we entitled to if someone
damages our asset and lowers the future stream of
payments? Typically we say the net present value
of the damage to the future stream.