As Agreed 30
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GameTheoryI.pdf
Game Theory: The Basics
BUS291 Microeconomics A
Session 10
Introduction
• Some fundamental terminology
• Some basic outcomes
• Look at a couple of straight forward
applications
• Credible threats and pre-commitments
A Game
• A situation in which strategic behaviour is an important part of decision making
• Strategic behaviour involves taking into account what a rival might do
• Our focus is non-cooperative games, each decision maker acts in their own self- interest
– This does not rule out cooperation when it is in the players’ interests
Key Terms
• Players are the decision makers in a game
• A strategy is a player’s plan of action
• Actions are particular things done according to a player’s strategy
• Payoffs are the rewards accruing to a player at the end of the game
• A game tree is a way to represent the elements of game such as each player’s alternatives and when it is their turn to move
Basic idea of game theory
Put yourselves ‘in the shoes’ of the other person
Assume that they are putting themselves in your shoes
And therefore they are seeing what you see when in their shoes
Which includes what you see when you are in their shoes …
This is called “common knowledge of rationality”
Common knowledge of rationality
A key assumption underlying basic game theory analysis.
Each player knows the other players are rational
Each player also knows that the other players know other players
are rational
… and so on
“I believe that everyone will act rationally, given their beliefs,
which includes the belief that I will act rationally.”
Leads to a key insight: look ahead, but reason back, before you
decide!
Movie Examples
• In the ‘Princess Bride’ the ‘Battle of wits’
clip
http://www.youtube.com/watch?v=U_eZm
EiyTo0
• And the ‘Doomsday weapon’ scene in ‘Dr
Strangelove’
http://www.youtube.com/watch?v=2yfXgu3
7iyI
• And almost all of ‘The Battle of Red Cliff’
Nobel Laureate 1994 John Nash
An Illustration of a Game
• Set out in text then as a game tree – Let there be two firms A and B each with just
two options, to produce a high or a low level of output
– If both firms produce a high level of output each earns $4 000
– If both firms produce a low level of output each earns $3 000
– If one produces a high level of output and the other a low level of output the former earns $6 000 and the latter $ 1 000
– Let A move first
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
Illustration: Some Observations
• Firm A’s strategic options are limited to either High or Low
• A strategy for firm B has to specify what it will do at both its decision nodes
– Eg if A produces High, I (B) will produce Low and if A produces Low I will produce High
• A decision rule is a strategy that specifies what action will be taken conditional on what happens earlier in the game
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
A Dominant Strategy
• A strategy that works at least as well as
any other, no matter what the other player
does
• In our example firm B has a dominant
strategy: produce High
• A player will use a dominant strategy if
they have one
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
Dominant Strategy Equilibrium
• As it turns out in our illustration firm A also
has a dominant strategy High
• Both firms will play their dominant
strategies producing a dominant strategy
equilibrium (High, High)
A Dominant Strategy Equilibrium is also
a Nash Equilibrium
• Neither firm can gain by unilaterally
changing what it is doing in light of what
the other firm is doing
• For a Nash equilibrium each players’
equilibrium strategy must be the best
response to the other player’s equilibrium
strategy
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
A Dominant Strategy Equilibrium Also Meets
the Credibility Condition
• Each firm’s move at each decision node must be in its self-interest
• In our example this is easy to see for firm A
• Once firm A has chosen High it is certainly in B’s interests to choose High also
• The key question is if a threat by B to choose High if A chooses Low is credible
High
Low
High
Low
High
Low
B
B
(4, 4)
(6, 1)
(1, 6)
(3, 3)
A
Another Illustration
• Our first illustration works out so neatly
because there is a dominant strategy
equilibrium
• This is just a reflection of the particular
pattern of pay-offs
• Let us consider another game in which B
has a dominant strategy but A does not
(1, 1)
(6, 2)
(2, 6)
(5, 5)
High
Low
High
Low
High
Low
B
B
A
High
Low
High
Low
High
Low
B
B
(1, 1)
(6, 2)
(2, 6)
(5, 5)
A
High
Low
High
Low
High
Low
B
B
(1, 1)
(6, 2)
(2, 6)
(5, 5)
A
High
Low
High
Low
High
Low
B
B
(1, 1)
(6, 2)
(2, 6)
(5, 5)
A
High
Low
High
Low
High
Low
B
B
(1, 1)
(6, 2)
(2, 6)
(5, 5)
A
Observations
• There are two Nash equilibria
• It is rational to expect that B will make the best response to what A does
• Backward induction
• This eliminates non-credible threats
• A set of strategies that satisfies both the Nash and the credibility conditions is called a perfect equilibrium
An Application: Oligopoly with
Entry • Let us apply a similar payoff pattern to a
two firm entry game
• The two firms are firm J, an incumbent,
and G, a potential entrant
• “High” and “Low” again refer to levels of
output
Enter
Stay Out
High
Low
High
Low
J
J
(-2, 5)
(6, 6)
(0, 12)
(0, 8)
G
High
Low
High
Low
J
J
(-2, 5)
(6, 6)
(0, 12)
(0, 8)
G
Enter
Stay Out
High
Low
High
Low
J
J
(-2, 5)
(6, 6)
(0, 12)
(0, 8)
G
Enter
Stay Out
High
Low
High
Low
J
J
(-2, 5)
(6, 6)
(0, 12)
(0, 8)
G
Enter
Stay Out
Credible Threats and
Commitment • A player can irreversibly alter its payoffs in
advance so that it will be in that player’s
self-interest to carry out a threatened or
promised action if the need arises
• Suppose in our entry game the incumbent
invests in a large plant with low marginal
cost, so that a high level of output is its
profit-maximising response to entry
(-2, 4)
(6, 3)
(0, 11)
(0, 5)
Enter
Stay Out
High
Low
High
Low
J
J
G
High
Low
High
Low
J
J
(-2, 4)
(6, 3)
(0, 11)
(0, 5)
G
Enter
Stay Out
High
Low
High
Low
J
J
(-2, 4)
(6, 3)
(0, 11)
(0, 5)
G
Enter
Stay Out
High
Low
High
Low
J
J
(-2, 4)
(6, 3)
(0, 11)
(0, 5)
G
Enter
Stay Out
An Expanded Game Tree
• The decision of the incumbent to enter into
a pre-commitment can be seen as a
preliminary step in an expanded game
• In our illustration this is done by putting the
two games together
Enter
Stay Out
Enter
Stay Out
High
Low
High
Low
High
Low
High
Low
J
J
J
J
(-2, 5)
(6, 6)
(0, 12)
(0, 8)
(-2, 4)
(6, 3)
(0, 11)
(0, 5)
Small Plant
Large Plant
J
G
G
Enter
Stay Out
Enter
Stay Out
High
Low
High
Low
High
Low
High
Low
J
J
J
J
(-2, 5)
(6, 6)
(0, 12)
(0, 8)
(-2, 4)
(6, 3)
(0, 11)
(0, 5)
Small Plant
Large Plant
J
G
G
Enter
Stay Out
Enter
Stay Out
High
Low
High
Low
High
Low
High
Low
J
J
J
J
(-2, 5)
(6, 6)
(0, 12)
(0, 8)
(-2, 4)
(6, 3)
(0, 11)
(0, 5)
Small Plant
Large Plant
J
G
G
Enter
Stay Out
Enter
Stay Out
High
Low
High
Low
High
Low
High
Low
J
J
J
J
(-2, 5)
(6, 6)
(0, 12)
(0, 8)
(-2, 4)
(6, 3)
(0, 11)
(0, 5)
Small Plant
Large Plant
J
G
G
Conclusion
• Game theory and key terminology
introduced
• Sequential move games our focus
• A nice illustration in the form of an entry
game
• Pre-commitments as a way of influencing
outcomes
