Micro - Game Theory
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Transcript of Micro - Game Theory
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GAME THEORY
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Lectures 21 and 22
A. Madestam
What is game theory?
We focus on games where:
There are at least two rational players
Each player has more than one choice
The outcome depends on the strategies chosen by all
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players; there is strategic interaction
Example 1: Six people go to a restaurant
Each person pays his/her own meal a simple decision
problem
Before the meal, every person agrees to split the bill
evenly among them a game
Example 2: Prisoners Dilemma
John and Peter have beenarrested for possession of guns.However, the police suspects thatJohn and Peter also havecommitted a major robbery butthe police lacks the evidence toprove this
Ifno one confesses the robbery,they will both be jailed for 1years
Ifonly one confesses, he will bereleased and his partner ends upin jail for 20 years
If they both confess, they bothget 5 year
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Example 3: What to Do?
Philip Morris
No Ad Ad
ReynoldsNo Ad 50 , 50 20 , 60
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If you are advising Reynolds, what strategy do you
recommend?
4
, ,
Game theory has many applications
Economics
Politics
Business, etc, etc
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Examples
Prisoners dilemma
The entry or predation game
Battle of the sexes
Example 1: the prisoners dilemma revisited
Two robbers, X and Y, have been caught by the police and put
in separate interrogation rooms; the district attorney has
enough evidence to convict both for a lesser charge, but she
wants to convict them for a more serious crime, for which she
needs additional evidence
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The district attorney and her assistant go simultaneously (at
the same time) to each robber offering them a plea bargain a
reduced prison term in return for testifying against the other
robber (which will increase the latters prison term)
Each robber must choose between confess or keep quiet,
without knowing what the other is doing
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If only one of the robbers confess, he will be released, while
the non-confessing prisoner will go to jail for 20 yearsIf both confess, they will serve a 5 years sentence
If both keep quiet, they will both be convicted for a lesser
crime that carries a sentence of1 year
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Each player whishes to minimize the time he spends in jail,
hence we can represent the payoff as the negative of this
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Diagram to be drawn by YOU!
Example 2: the entry, or predation game
Firm X is considering entering a market that currently has a
single incumbent, firm Y. If X enters, the incumbent, Y, can
respond in one of two ways: it can either
i) accommodate the entrant, giving up some of its sales, i.e. it
can produce a low output level
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,
the market price, i.e. it can produce a high output level
IfXstays out of the market, it gets no profits, while Ygets 2 if
it produces a low output level and 3 otherwise
If Xenters the market and Yfights back and produces a high
output level, they both get -1; if Y instead accommodates the
entrant and produce a low output level, they both get 1
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The predation game
Y
(i) (ii) (iii) (iv)E
XNE
1 , 1
0 , 2
1 , 1
0 , 3
-1 , -1
0 , 2
-1 , -1
0 , 3
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i) L if E, L if NE
ii) L if E, H if NE
iii) H if E, L if NE
iv) H if E, H if NE
Example 3: the battle of the sexes
Pete and Maria are trying to coordinate on attending the ballet
or a (boxing) fight in the evening
They work at separate workplaces and cannot communicate -
yes, this was before the days of cell phones and for some
reason the fixed phone line is down
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ere w ey mee
Both Pete and Maria know the following:
Both would like to spend the evening together, if they
dont they both receive 0
But Pete prefers the ballet and receives 2 in this case,
while Maria receives 1 if she joins Pete
Maria prefers the fight and receives 2 in this case,
while Pete receives 1 if he joins Maria
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Diagram to be drawn by YOU (again)!
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Noncooperative game theory studies decision making in
situations where strategic behavior is important (e.g. shouldthe robbers confess or not?)
Situations in which strategic considerations are an essential
part of decision making are called games
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A decision maker is behaving strategically when she takes
into account what she thinks the other agents are going to do
The theory is labeled noncooperative because each decision
maker acts solely in her own self-interest; this does not mean
that cooperation is not a possible outcome of strategic behavior
A central feature of multi-agent interaction is the potential for
the presence ofstrategic interdependence
In multi-agent situations with strategic interaction, each agent
recognizes that the payoffs she receives (in utility or profits)
depends not only on her own actions but also on the actions of
other agents
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In her decision-making process, each agent should take into
account:
1) actions the other agents have already taken
2) actions she expects them to be taking at the same time
3) future actions they may (or may not) take as a result of her
current actions
Basic elements of a game
To describe a situation of strategic interdependence, we need
four basic elements:
i) Players: the decision makers in a game (who is involved?)
ii) Actions: the possible moves available to the players (what
can the do?
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iii) Strategies: the players plans of action at any stage of the
game (what are they planning to do?)
iv) Payoffs: the possible rewards enjoyed by the players (what
will they gain?)
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Strategy
A strategy is a complete plan, or decision rule, that specifies
how the player will act in
EVERY POSSIBLE DISTINGUISHABLE
CIRCUMSTANCE
in which she might be called upon to move
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Intuition: when a player specifies her strategy, it is as if she
had to write down an instruction book prior to play so that a
representative could act on her behalf merely by consulting
that book
Being a complete contingent plan, a strategy often specifies
actions for a player at circumstances that may not be reached
during the actual play of the game
Some definitions
Games in which all players move simultaneously (at the same
time) are known as simultaneous games
(e.g. prisonersdilemma; battle of the sexes)
Games in which players moves may precede one another are
known as sequential games
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(e.g. entry/predation game)
Games in which all players move knowing the earlier orsimultaneous moves of the other players are known as games
of perfect information
(e.g. entry/predation game)
Definitions contd
Games in which some players must move without knowing the
earlier or simultaneous moves of the other players are known
as games of imperfect information
(e.g. prisonersdilemma; battle of the sexes)
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Text to be added by YOU !
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Text to be added by YOU (and again)!
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Text to be added by YOU (and again)!
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Representation of games
Games can be represented in two forms, the normal form and
the extensive form
The normal form presents the game directly in terms of
strategies and their associated payoffs
When describin a ame in its normal form there is no need to
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,
keep track of the specific moves associated with each strategy
The prisoners dilemma in normal form
Y
C NC
C -5 , -5 0 , -20X
NC -20 , 0 -1 , -1
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The payoff matrix summarizes the payoffs associated with
each combination of strategies
Note that the normal form is practically useful only when there
are two players and the set of possible strategies is limited
The extensive formThe extensive form captures who moves when, what actions
each player can take, what players know when they move,
what the outcome is as a function of the actions taken by the
players, and the players payoff from each possible outcome
The extensive form relies on the conceptual apparatus known
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as a game ree
The circumstances in which agents are, or might be, called
upon to move are represented by decision nodes (little gray
squares in previous picture)
Each of the choices available at a particular decision node is
represented by a branch from the decision node itself
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The prisoners dilemma in extensive form
)5,5(
)20,0(
)0,20(
X
YC
NC
C NC
C
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: Decision node : Information set
The dashed oval around the two decision nodes for Y, known
as information set, is used to represent Ys inability to
distinguish between these two points at the time it makes her
decision; from Ys point of view, the entire information set is a
single decision node
)1,1( NC
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Diagram to be drawn by YOU (again)!
The dominant strategy equilibrium
A dominant strategy* is the best strategy regardless of what
any other player does
There is no reason for players to use anything other than their
dominant strategy, IF they have one (often dominant strategies
simply do not exist)
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Hence, when each player has a dominant strategy, the only
reasonable equilibrium outcome is for each player to use its
dominant strategy
A dominant strategy equilibrium is an outcome in a game in
which each player follows a dominant strategy
-------------------------------------------------
* The correct definition is actually strictly dominant
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Dominant strategies in the prisoners dilemma
Y
C NC
C -5 , -5 0 , -20X
NC -20 , 0 -1 , -1
Has Xa dominant strategy?
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Y
C NC
C -5 , -5 0 , -20X
NC -20 , 0 -1 , -1
Has Ya dominant strategy?
NB: playing Cis dominant for both players! Hence, {C,C} is a
dominant strategy equilibrium for the prisoners dilemma
The Nash equilibrium
The Nash equilibrium is the most widely used solution
concept in applications of game theory to economics
(http://www.princeton.edu/mudd/news/faq/topics/Non-
Cooperative_Games_Nash.pdf)
Consider a game with two players, Xand Y; a pair of strategies
form a Nash equilibrium if:
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actually played by Y
ANDii) the strategy played by Y is optimal given the strategy
actually played by X
In general, in a Nash equilibrium, each players strategy choice
is her best response to the strategies actually played by her
rivals
In simultaneous games of imperfect information, playerscannot directly observe the rivals moves (e.g. prisoners
dilemma; battle of the sexes)
Hence, each player forms ideas/conjectures (really guesses)
about what the rivals will do, and reacts consequently by
choosing her best response to the conjectured rivals strategies
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In a Nash equilibrium, these conjectures turn out to be
correct: each players strategy reveal itself to be the best
response to the rivals actual moves
In sum: players do not have incentives to unilaterally deviate
from the equilibrium once the rivals move become observable
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Y
The Nash equilibrium in the prisoners dilemma
Y
C NC
C -5 , -5 0 , -20XNC -20 , 0 -1 , -1
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C NC
C -5 , -5 0 , -20X
NC -20 , 0 -1 , -1
Hence, {C,C} is not only a dominant strategy equilibrium, but
also a Nash equilibrium for the prisoners dilemma
NB: all dominant strategy equilibria are Nash equilibria (by
definition), while the converse is false
Y
C NC
C -5 , -5 0 , -20X
NC -20 , 0 -1 , -1
Y
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In this case, the conjectures are not mutually correct! Hence,
{C,C} is the unique Nash equilibrium for this game
C NC
C -5 , -5 0 , -20
XNC -20 , 0 -1 , -1
IfPete goes to the ballet, Marias best response is to go to the
Nash equilibria in the battle of the sexes
MB F
B 2 , 1 0 , 0P
F 0 , 0 1 , 2
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a e as we ; ar a goes o e a e , e e s es
response is to go to the ballet as well (2 > 0)
IfPete goes to the fight,Marias best response is to go to the
fight as well (2 > 0); ifMaria goes to the fight, Petes best
response is to go to the fight as well (1 > 0)
Hence, { B , B } and { F , F } are both equally plausible Nash
equilibria for this game
NB: there may be several Nash equilibria in a given game
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The battle of the sexes is a coordination game
Two equilibria exist
Pete and Marie prefer different equilibria
How to achieve the most desirable outcome for you?
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- sequential moves: the bossier one in the relationship
chooses the equilibrium by leaving the office a bit earlier.
Who do you think leaves early?
- strategic moves: can you commit? Suppose Pete goes to
the ballet early but knows that Marie will go to the fight
regardless of what he does, will he stay to watch the Swan
Lake?
Nash equilibria in the Predation Game
Y
(i) (ii) (iii) (iv)
E 1 , 1 1 , 1 -1 , -1 -1 , -1X
NE 0 , 2 0 , 3 0 , 2 0 , 3
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IfYplays (i), Xplays E; ifXplays E, Yplays (i) or (ii)
IfYplays (ii), Xplays E; ifXplays E, Yplays (i) or (ii)
IfYplays (iii), Xplays NE; ifXplays NE, Yplays (ii) or (iv)
IfYplays (iv), Xplays NE; ifXplays NE, Yplay (ii) or (iv)
NB: { E , (i) }, { E , (ii) }, and { NE , (iv) } are Nash equilibria
Consider the three Nash equilibria:
1 E L ifE L ifNE outcome: 1 1
Y
(L,L) (L,H) (H,L) (H,H)
E 1 , 1 1 , 1 -1 , -1 -1 , -1X
NE 0 , 2 0 , 3 0 , 2 0 , 3
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,
2) { E ; L ifE , HifNE } ; outcome: (1,1)
3) { NE ; HifE , HifNE } ; outcome: (0,3)
NB: the first two equilibria generate the SAME outcome,
because Ys strategies differ only at the decision node that is
NOT reached during the actual play of the game
Does this really make sense?!
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To get the intuition, consider the predation game in extensive
form:
)1,1(
)1,1(
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X
Y H
EL
H
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,
)2,0(YNE
L
Suppose that Xhas already chosen to enter (E): what would Ys
optimal response be?
Y would clearly choose to accommodate the entrant and
produce a low output level (L), since 1 > -1!
)1,1(
)1,1(
)3,0(
)2,0(
X
Y
Y
H
NE
E
L
L
H
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what would Ys optimal response be?
This time, Y would clearly choose to produce a high outputlevel (H), since 3 > 2!
Once X has made its move, Y finds it optimal to play L if X
played E, and HifXplayed NE
Hence, the strategy L ifE , H ifNE is the credible strategy
for Y, and Xhas to take this into account!
From Xs point of view, the sequential game reduces to the
following:
E)1,1(
Y
)(L
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NE )3,0(Y
)(H
If X takes into account that L if E , H if NE is the only
credible strategy for Y, X will evidently choose to enter (E),
because 1 > 0
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The procedure used, which involves solving first for optimal
behavior at the end of the game and then determining what
the optimal behavior is earlier in the game given the
anticipation of this later behavior, is known as backwardinduction
Using this procedure, we realized that:
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; ,
is the only Nash equilibrium in the predation game that seems
credible what do I mean with credibility?
The Subgame Perfect Nash equilibrium
In a sequential game, sometimes not all Nash equilibria are
equally plausible: some of them may be based on non-credible
threats
Consider the three Nash equilibria in our predation game:
when Xplays E, actions at the decision node that is unreached
by play of the equilibrium strategies do not affect Ys payoff
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(i.e. the first two Nash equilibria generate the same outcome)
Therefore, Y can plan to do anything at this decision node:
given Xs strategy of choosing E, Ys payoff remains
maximized
To rule out non-credible outcomes, we introduce the principleof sequential rationality: a players strategy should specify
optimal actions at every point in the game tree
In other words, given that a player finds herself at some point
in the tree, her strategy should prescribe play that is optimal
from that point on given her opponents strategies
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What is the link between sequential rationality and credibility:
under sequential rationality, each time an agent is
called upon to make a move (i.e. at each of the
agents decision nodes), it is in the agents self-
interest to carry out the action called for by its
strategy (this means in terms of our example)
strategies that respect the principle of sequential
rationality are credible
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A Nash equilibrium that satisfies the principle of sequential
rationality, i.e. specifies only credible strategies, is known as a
(subgame) perfect Nash equilibrium
Hence { E ; L if E , H if NE } is the only subgame perfect
Nash equilibrium in the predation game
The other two Nash equilibria
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{ E ; L ifE , L ifNE } , { NE ; HifE , HifNE }
are clearly based on non-credible threats: Y would NEVER
play L ifNE or HifE!
Summing up
)1,1( Y HIn this
subgame, Y
IfYplays L ifE,HifNE,
XplaysE
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X
NE )3,0(
)2,0(Y L
H
In this
subgame, Y
playsH
,
{E;L ifE, HifNE} is the (subgame) perfect Nash equilibrium
Note that the perfect Nash equilibrium generates an outcome,
(1,1), that is not the best possible outcome for Y
The Nash equilibrium { NE ; H ifE , H ifNE } generates a
much better outcome from Ys point of view: (0,3)
This Nash equilibrium, however, is based on a non-credible
threat, since Ywould never play HifXdecides to enter
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a comm s o p ay no ma er w a oes
X
NE
E)1,1(
Y
)(H
)3,0(
Y
)(H
IfXtakes the threat
seriously, it playsNE,
and Ygets its max.
payoff
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Extensive form or normal form?
To find the (subgame) perfect Nash equilibrium of sequential
games we should focus on their extensive form representation
and solve them using backward induction
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To find all Nash equilibria of simultaneous or sequential games
we should focus on their normal form representation