GAME THEORY

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GAME THEORY

• PRINCIPLES OF MICROECONOMICS

Dr. Fidel Gonzalez Department of Economics and
Intl. Business Sam Houston State University
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OPPORTUNITY COST
MARGINAL ANALYSIS
Elasticity
Elasticity
SUPPLY
DEMAND
MARKET EQUILIBRIUM
CONSUMER SURPLUS, PRODUCER SURPLUS AND TOTAL
SURPLUS
MARKET EFFICIENCY
MARKET FAILURE
Pigouvian Taxes Quotas Coase Theorem Command and
control
TAXES
EXTERNALITIES
QUOTAS
PUBLIC GOODS
COMMON GOODS
ARTIFICIALLY SCARCE GOODS
GAME THEORY
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• Before we start with the definition of game
  theory, lets consider the following real life
  example
• The example we are going to consider is the
  infamous Central Park Jogger Case.
• On April 1989 a raped and beaten up woman was
  found in New Yorks central park.
• The victim was a young rich woman that worked in
  a very important investment bank in New York
  City. After the attack she was in coma and
  severely injured.
• The case was all over the news and the police
  was under pressure to find the attackers.
• Eventually the police found several teenagers
  who were suspected of committing the crime. Of
  these teenagers five of them were convicted to 10
  to 15 years.
• An important thing about the case is that the
  conviction of the five teenagers was based almost
  entirely on their own confessions of the crime.

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• Now, fast forward 13 years. In 2002, a prisoner
  unrelated to the five teenagers confess
  committing the crime and DNA evidence confirms
  that he did it.
• This means that the five teenagers were not
  guilty yet they confess committing the crime.
• Why? Why do innocent people confess?
• The short answer is that these five teenagers
  were placed in a strategic situation where it was
  optimal for them to confess.
• Game theory is going to help us explain this
  behavior and show us that in some cases is better
  to cooperate than to follow our own self-interest.

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Q What is game theory? A Game Theory is the
study of people in strategic situations. The
strategic situation is called a game and the
people in this strategic situation is called a
player. Q What is an strategic situation? A An
strategic situation is when the actions and
payoff of one player depend on the action of the
other player. A good example of an strategic
situation is a football game. The actions of the
defensive team depend on the actions of the
offensive team. If the defensive team believes
that the offensive team will run the ball then
the defensive team will adjust accordingly. But
the defensive team believes that the offensive
team will pass the ball then they will adjust in
a different way. This is also true for most
competitive sports where you play against an
opponent like soccer, tennis, basketball,
baseball, etc. However, this also applies to
economic situations as we will see later.
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• Lets see how the game theory works with another
  example
• A bank has been robbed and the police has two
  suspects Thelma and Louise.
• Thelma and Louise are on police custody on
  separate rooms when the police interrogate them.
• The police tell Thelma that is she confesses to
  the robbery but Louise does not, Thelma gets goes
  free and Louise gets 20 years in prison.
• However, if Thelma and Louise both confess then
  each of them get 10 years.
• In addition, if Thelma and Louise do not confess
  then each of them gets 3 years on some
  technicality.
• Finally if Louise confesses but Thelma does not
  then Louise goes free and Thelma gets 5 years in
  prison.
• Notice that this is an strategic situation and
  therefore it is a game because the payoffs (the
  number of years in prison) of one person depend
  on the action of the other person too. The
  players in this game are Thelma and Louise.
• The next step is to put the payoffs in a table,
  this is called to put the game in normal form.

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This is the table with the payoff or game in
normal form
Thelma
Louise
The table has Louises options to confess and not
confess and Thelmas option to confess and not
confess with the corresponding option. They
payoffs (or number of years in prison) for Louise
are on the left side of each cell and the payoff
(or number of years in prison) for Thelma are on
the right side of each cell. For example when
both confess each of them gets 10 years in prison
and thats what the first cell shows. When Thelma
confesses and Louise does not this corresponds
the bottom left cell and Thelma gets 0 and Louise
20. The next step is to figure out what will
Thelma and Louise do. But before we do that we
need to have some definitions.
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Strategy the action of a player Dominant
Strategy the strategy that is always best
regardless of the other players action. Nash
Equilibrium the outcome when each player
follows their dominant strategy. The Nash
Equilibrium is also known as the Non-cooperative
Equilibrium. It is known as non-cooperative
equilibrium because in this case each player goes
on his own, he does not try to cooperate or
coordinate with the other player. Lets use this
new concepts to the Thelma and Louise example. Q
What is the dominant strategy for Louise? To
answer this question we have to figure out what
is best for Louise all the time.
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Thelma
Louise

• To figure out the dominant strategy for Louise
  she has to figure out what is her best strategy
  of Thelma if 1) confess and 2) does not confess.
• If Thelma confess (the blue cells)
• In this case Louise has to choose between
  confessing which will give her 10 years or not
  confessing which will give her 20 years. So, if
  Thelma confess the best strategy for Louise is to
  confess and get only 10 years.
• 2) If Thelma does not confess (red cells)
• In this case Louise has to choose between
  confessing which will give her 0 years or not
  confessing which will give her 5 years. So, if
  Thelma does not confess the best strategy for
  Louise is to confess and get only 0 years.

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Thus, from the previous slide we can conclude
that is always best for Louise to confess.
Therefore, Louises dominant strategy is to
confess. Now, lets figure out the dominant
strategy for Thelma.
Thelma
Louise

• To figure out the dominant strategy for Thelma
  she has to figure out what is her best strategy
  of Louise if 1) confess and 2) does not confess.
• If Louise confess (the blue cells)
• In this case Thelma has to choose between
  confessing which will give her 10 years or not
  confessing which will give her 20 years. So, if
  Louise confess the best strategy for Thelma is to
  confess and get only 10 years.
• 2) If Louise does not confess (red cells)
• In this case Thelma has to choose between
  confessing which will give her 0 years or not
  confessing which will give her 5 years. So, if
  Louise does not confess the best strategy for
  Thelma is to confess and get only 0 years.

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Therefore, the dominant strategy for Thelma is
also to confess. The Nash Equilibrium (or the
equilibrium in which each player follows it own
path and does not cooperate with the other
player) is for both players to confess. Notice
that this is not the optimal solution. The
optimal solution will be for Thelma and Louise to
cooperate and agree to not confess, that way each
of them get only 5 years instead of 10 years.
However, they will not cooperate. They will not
do this because the payoff are set up in such a
way and they will always end up confessing. The
most remarkable thing is that we do not know if
they are guilty or not. They will end up
confessing regardless. This is known as the
Prisoners Dilemma. The prisoners dilemma
shows how the player follows an strategy that is
not optimal but it is dominant. QHow can they
achieve an equilibrium where both of them do not
confess? A By having an enforceable agreement
that they will never confess regardless. This
enforceable agreement is hard to have in real
life and that is why confessions in many cases do
not say much.
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Q Why does this matter? A Remember that we said
before that self-interest and competition is all
that society needs to find an efficient
allocation of resources. Game theory has shown
to us that this is not necessary true. That in
cases of strategic situations doing what is best
for one player is not necessary what is best for
the group and the for the player. Therefore, we
have found another situation where the market
fails to allocate resource efficiently. This
implies that in some cases is better to cooperate
than to follow our self-interest.
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Another Example Consider the following problem
of two fraternities ß? and p? They are throwing
a big party to celebrate the end of the semester.
They will invite a lot of people and are not
charging to enter the party. Each fraternity
should buy one half of the beers consumed in the
party. They calculate that they need about 10
beer kegs for the whole party. Thus, ß? should
buy 5 kegs and p? should buy 5 kegs. Each kegs
costs 8 dollars. The key is that each fraternity
arrives to the party around 8 pm with the beer
and at that time all the liquor stores are closed
and there is no way to buy beer. Each fraternity
has to buy the kegs way before the party and they
have not way to know what the other fraternity
did until they arrive to the party.
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Q What kind of good is the beer ONCE the beer is
already at the party? A When the beer is at the
party, beer is a common good. It is
non-excludable and rival. It is non-excludable
because once the beer is at the party everyone
inside the party is allowed to drink beer.
Fraternities can not exclude people from drinking
beer. It is rival because every beer drank by one
guest can not be consumed by another guest. As
we studied before common goods suffer from over
consumption. The cost for each party attendant of
an extra beer is zero. In other words, the
marginal cost of a beer is zero. Each party
attendant will consumer beer until the marginal
benefit of an extra beer is equal to the marginal
cost. In this particular case, MB of beer MC
of beer MB of beer 0
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This is the problem for each fraternity 1) If
ß? buys beers but p? does not (only 5 kegs of
beer bought) ß? pays 5x8 40 ß? gets a
benefit from drinking beer of 30, net benefit
for ß? is30-40 -10 p? gets a benefit from
drinking beer of 30, net benefit for p? is 30
-0 30 (remember the beer is non-excludable and
rival so everyone at the party gets to enjoy the
beer) Note that in this case p? free rides on ß?
. 2) If ß? and p? buy beer (10 kegs of beer
bought) ß? pays 5x8 40 and p? pays
5x840 Each fraternity gets a benefit from
drinking beer of 60 (double than before
because there is the double amount of beer). Net
benefit for each fraternity is 60 40 20 3)
If p? buys beers but ß? does not (only 5 kegs of
beer bought) p? pays 5x8 40 p? gets a
benefit from drinking beer of 30, net benefit
for p? is30-40 -10 ß? gets a benefit from
drinking beer of 30, net benefit for ß? is 30
-0 30 (in this case, ß? free rides on p?) 4)
Nobody buys beer ß? and p? pay zero ß? and p?
get a benefit of zero (without beer the party is
boring and nobody has a good time).
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This is the pay off matrix for each fraternity
Fraternity p?
Fraternity ß?
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Fraternity p?
Fraternity ß?
Fraternity ß?s dominant strategy is to not buy
beer, why? If fraternity p? buys beer, then
fraternity ß? can buy beer and get a net benefit
of 20 or dont buy beer and get a net benefit of
30. ß? will choose not to buy beer. If
fraternity p? does not buy beer, then fraternity
ß? can buy beer and get a net benefit of -10 or
dont buy beer and get a net benefit of 0. ß?
will choose not to buy beer. Regardless of
whatever fraternity p? chooses to do, fraternity
ß? is always better off by not buying beer.
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Fraternity p?
Fraternity ß?
Fraternity p?s dominant strategy is to not buy
beer, why? If fraternity ß? buys beer, then
fraternity p? can buy beer and get a net benefit
of 20 or dont buy beer and get a net benefit of
30. p? will choose not to buy beer. If
fraternity ß? does not buy beer, then fraternity
p? can buy beer and get a net benefit of -10 or
dont buy beer and get a net benefit of 0. p?
will choose not to buy beer. Regardless of
whatever fraternity ß? chooses to do, fraternity
p? is always better off by not buying beer.
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Fraternity p?
Fraternity ß?
The Nash Equilibrium (or non-cooperative
equilibrium) is for both fraternities not to buy
beer. This is the worst outcome for both
fraternities. Both of them will be better off if
both of them buy beer for the party (this is the
Prisoners Dilemma), each fraternity gets 20
instead of zero. Each fraternity doing what is
best for them does not provide the best solution
for them. In this case, self interest is not
enough to provide an efficient outcome. Remember
that we are at an inefficient point if we can
make someone better off without making someone
else worse off. In this case, we can make both of
them better off, so not buying beer is an
inefficient allocation of resources.
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Extra problem covered in class

• Two airlines Continental and Southwest
• They offer a flight to Cabo. They can charge a
  high price or a low price. If they charge a high
  price but the other line charges a low prices,
  then the high price airlines has low profits
  because the other airlines gets more customers.
  The following table shows their profits for the
  different strategies.
• Obtain
• The dominant strategy
• The Nash Equilibrium
• Is the Nash Equilibrium optimal? What will be the
  optimal?

Southwest
Continental
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Southwest
Continental

• Answers
• The dominant strategy the dominant strategy for
  both is to charge the low price.
• The Nash Equilibrium they will both charge the
  low price and get a profit of 1500 each.
• Is the Nash Equilibrium optimal? What will be the
  optimal? No, the Nash Eq. is not optimal. The
  optimal solution for them is to charge the high
  price. However, to do this they need to agree on
  charging the high price, that is they have to
  cooperate. Cooperation is difficult because each
  airlines has the incentive to deviate (to charge
  the low price) when the other airline is charging
  the high price.