Economics 250

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Economics 250 Lecture 6 Game Theory
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Game Theory Basics Players Those who make
the decisions. Strategies The planned
decisions of the players. Payoffs The results
of different strategies.
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There are several ways to distinguish different
types of games in Game Theory 1. Timing of
Actions a. Simultaneous Move Games b.
Sequential Move Game 2. The Period of the
Game a. One Shot Games b. Repeated Games
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Our Sequence of Studying 1. Simultaneous Move
Games ltNormal Formgt A. One Shot B. Repeated
Infinite C. Repeated Finite-Certain D.
Repeated Finite-Uncertain 2. Sequential Move
Game One Shot ltExtended Formgt
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Normal Form Game A representation of a game
indicating 1. the players, 2. their
possible strategies, and 3. the payoffs
resulting from alternative combinations of
strategies.
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A typical normal form game
Player B Player B
Left Right
Player A Up 10, 20 15, 8
Down -10, 7 10, 10
Player A has strategies Up and Down Player B has
strategies Left and Right The four cells with
two numbers (a, b) tell the payoffs to Players
A and B respectively.
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Player B Player B
Left Right
Player A Up 10, 20 15, 8
Down -10, 7 10, 10
For example if Player A chooses Up and Player
B chooses Left Player A will receive a Payoff
of 10 Player B will receive a Payoff of
20 Player A should choose UP. Regardless of
what player B does, player A will end up with
the better result. In this game, UP is the
dominant strategy for Player
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Dominant Strategy A strategy that results in
the highest payoff to a player regardless
of the opponents actions.
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Player B Player B
Left Right
Player A Up 10, 20 15, 8
Down -10, 7 10, 10
Player B does not have a Dominant Strategy.
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Without a Dominant Strategy, Some players may
opt for the Secure Strategy if they are
completely risk averse. Secure (Risk Averse)
Strategy A strategy that guarantees the
highest payoff given the worst possible
scenarios.
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Player B Player B
Left Right
Player A Up 10, 20 15, 8
Down -10, 7 10, 10
Player Bs Risk Averse strategy is to choose
Right
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Secure strategies are often not the best
strategy if you are less than completely risk
averse.
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Player B Player B
Left Right
Player A Up 10, 20 15, 8
Down -10, 7 10, 10
What is the Risk Seeking strategy for player
B? Player B Will Go for the Highest Number,
20, and choose Left.
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Player B Player B
Left Right
Player A Up 10, 20 15, 8
Down -10, 7 10, 10
What is the Best strategy for player
B? Player B should realize that Player A
has a Dominant Strategy to choose Up.
Knowing this, Player B should choose Left.
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Player B Player B
Left Right
Player A Up 10, 20 15, 8
Down -10, 7 10, 10
If each player makes the move which gives them
the best Payoff, they will move Left and Up.
This is a Nash Equilibrium.
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Nash Equilibrium A condition Describing a set
of strategies in which no player can improve
her payoff by unilaterally changing her own
strategy, given the other players
strategy. In a cell containing a Nash
Equilibrium, neither of the players can gain by
making a different decision, holding the
other players move constant.
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Applications of One-Shot Games Pricing
Decision
Firm B Firm B
Low Price High Price
Firm A Low Price 0, 0 50, -10
High Price -10, 50 10, 10
So Oligopolists will be forced to charge a Low
Price because they cannot trust their
competitors. Note that any manager which was
honest and goes along with a collusion will
end up losing (getting -50). So managers
honest to their fellow colluders tend to be
weeded out.
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Advertising
Firm B Firm B
Advertise Dont Advertise
Firm A Advertise 4, 4 20, 1
Dont Advertise 1, 20 10, 10
So, again we see the case where companies are
forced into a behavior which is not to their
benefit.
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Monitoring Due to the Principal-Agent Problem,
Firms often have to monitor their employees,
suppliers, etc. Monitoring
Employee Employee
Work Shirk
Manager Monitor -1, 1 1, -1
Dont Monitor 1, -1 -1, 1
Where is the Nash Equilibrium in this case?
There is none. The best strategy in such a
case is to Randomize action to maximize the
expected benefit of the Payoff. Sometimes
called a Mixed Strategy
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In Calculating a Mixed Strategy, the expected
benefit will depend on 1. the probability of
the each choice for the other party and, 2.
the payoffs of each choice for your party.
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Bargaining Imagine that you are in management and
the union is asking for a pay raise. The
options of a 0 pay raise, a 50 pay raise,
and a 100 pay raise have been mentioned.
As management you know that there is 100
available. You will get whatever is left
over Bargaining
Union Bid Union Bid Union Bid
0 50 100
Management Bid 0 0,0 0,50 0,100
50 50,0 50,50 0,0
100 100, 0 0, 0 0, 0
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Union Bid Union Bid Union Bid
0 50 100
Management Bid 0 0,0 0,50 0,100
50 50,0 50,50 0,0
100 100, 0 0, 0 0, 0
Where are the Nash Equilibria? (Diagonal from
low left to upper right) You are able to
eliminate some strategies of your opponent
because these strategies are not in the
opponents best interest.
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Experimental Economists have discovered that
often players in games do tend to look for a
fair outcome. This is called the Focal Point
of the game.
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Infinitely Repeated Games Infinitely Repeated
Game A game that is played over and over
again forever and in which players receive
payoffs during each play of the game. You
expect a payoff in future periods, but you will
discount future payoffs by the discount rate.
Trigger Strategies a strategy that is
contingent on the past plays of players in a
game.
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Consider this duopoly situation again
Firm B Firm B
Low Price High Price
Firm A Low Price 0, 0 50, -10
High Price -10, 50 10, 10
In the one shot game, the Nash Equilibrium is
for each firm to cheat and charge a low price.
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Firm B Firm B
Low Price High Price
Firm A Low Price 0, 0 50, -10
High Price -10, 50 10, 10
Infinitely Repeating Game Each firm agrees to
charge a high price and threatens to use the
trigger strategy. In the repeated game case,
you have to consider all future outcomes If
you keep colluding you get 10 today and
10 forever after. If you stop colluding you
get 50 today and 0 forever after.
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Player A will cheat Player B if the Benefit
of Cheating gt the Cost of Cheating Ex Collud
e 10 10 10 10 Cheat 50 0 0 0 So Which is
Greater? 50/1r or 10/r (Assuming money
comes at end of period)
The Cost of Cheating will be great (and You will
not Cheat) if either 1. Interest Rate is Low
(The Future Matters), or 2. There is only a
Small Extra Profit to Gain by Cheating (Relative
to Cooperating)
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Product Quality
Firm Firm
Low-Quality Product High-Quality Product
Consumer Dont Buy 0,0 0,-10
Buy -10,10 1,1
What is the One-Shot Nash Equilibrium here?
Upper, Left.
In a repeated game, Reputation Matters.
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Important Behavioral Results 1. More likely
to Cheat Customers when a. you will Never
See Them Again (one-shot game), b. the
Interest Rate is Very High, or c. the Nash
Equilibrium (Trigger or Cheating) Payoff is
Much Greater than the High Quality
(Colluding) Payoff 2. If you do give Bad
Quality even once, you can Lose that Customer
Forever. 3. The need for Brand Loyalty is
Important Enough that Legal Interference and
protection is Not Always Necessary
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Finitely Repeating Game With a Certain Ending
Date Assume Each firm has to make this
decision over and over. Both firms know
the game will be over in some future period.
For example Each firm knows that the
product they sale will be obsolete in 7 years
due to a new law, but until then they are a
duopoly.
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Collusion cannot work in a Finite Repeated Game
because 1. There is no credible trigger
strategy in the last year. 2. Both firms will
not collude in the final play of the game. 3.
The period before the last now becomes the final
period. 4. There is no credible trigger
strategy for that year. 5. This reasoning
continues, backwards, bringing about no
collusion.
This is sometimes called the End of Period
Problem
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Finitely Repeating Game With an Uncertain
Ending Date For Example Two firms know a
secret process. Eventually others will learn
the process, but until then they are a
duopoly. If you are not sure when the period
will end you must discount the future earning
by the chance of market shut down. So when
you are developing strategies, consider 1. How
much is the benefit to cheating. 2. How much do
you value future payments. 3. What is the
probability the market is going to shut down.
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The main points As the probability of shut
down increases you count the future less and
you are less likely to collude (more likely to
cheat). As the probability of shut down
decreases you count the future more and you
are more likely to collude (less likely to cheat).
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Summary of Cheating and Colluding Possibilities
so Far Simultaneous Move Games ltNormal
Formgt One Shot Collusion Cannot Occur
Will Cheat, No Trust Repeated Infinite
Collusion Can Occur if Interest Rate is
Low Enough and No Big Profit from
Cheating Repeated Finite-Certain Collusion
Cannot Occur Race to End, No
Trust Repeated Finite-Uncertain Collusion
Can Occur if Interest Rates are Low
and No Big Profit from Cheating and
Chance of Shutdown is Low