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# N5 - Game Theory, Oligopoly.pdf

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University of Toronto Mississauga

Economics

ECO204Y5

Kathleen Wong

Winter

Description

Elements of Game Theory
- players
o decision makers in a game
o e.g. individuals or firms
- strategies
o possible actions in a game
o e.g. event , output choices
- payoffs
o rewards or returns to players for each combination of
strategies chosen by players
o note: depends on both players’ decisions
players cannot determine their own outcomes
o e.g. utility, profits
- Nash Equilibrium (NE):
o the outcome or equilibrium in game
o made up of and defined as a set of strategies (s *, s *)
1 2
o occurs when both players are playing their best response
(BR), given that their opponent is playing their BR
o a NE is an “intersection of BR”
s1* = BR (1 2)
s2* = BR (2 1)
- assumptions:
o all players always play their BR
o all players are rational
always choose strategies with max payoffs
o all players have perfect info
both players know:
• the strategies available to their opponents
• the payoffs associated with each of their
opponent’s strategies
Game Theory, Oligopoly
1 Example 1: normal form game with pure strategy
P2
L R
P1 T 3, 5 7, 1
B 6, 2 8, 4
- example 1 is a 2-by-2 game
o each players has 2 strategies to choose from
- player 1 is the row player
o strategies are listed as rows
o choice: top (T), bottom (B)
- player 2 is the column player
o strategies are listed as columns
o choice: left (L), right (R)
- within each cell, payoffs are specified in the following order:
payoff = (row player, column player)
- because of this, BR strategies for players should also be listed
in this order: NE = (1*, s2*)
o NE strategies listed in the reverse order are incorrect
o points will be deducted
- goal: determine which strategy you should play (choose)
o what is your BR?
- strategy: condition your choice of strategy on what your
opponent does
o for each of your opponent’s options, decide which
strategy gives you the highest payoff
o ask yourself, “if my opponent chooses this option, which
of my strategy options gives me the highest payoff?”
o recall 1* = BR (1 2), s 2 = BR (2 1)
o in other words, “which strategy is my BR?”
- starting with P1:
Game Theory, Oligopoly
2 o if P2 chooses L, what is P1’s BR?
o if P2 chooses R, what is P1’s BR?
- now onto P2:
o if P1 chooses T, what is P2’s BR?
o if P1 chooses B, what is P2’s BR?
- outcome: the choices I make (the circledayoffs) are a BR
because they’re based on what my opponent does
o when both players are playing their BR, this is an
equilibrium
o why?
o this is the cell where:
both players are playing their best response
neither player as any incentive to change strategy
conditionalon their opponent also playing their BR
“intersectionf best responses”
- answer: NE = (B , R)
- note: a NE is defined as a set of strategies
o describes how each player is going to act
o your answer must specify the strategythe players choose
o it is incorrect to list the payoffs
- practice: use the BR method to find all pure-strategy NE
1) P2
L R
P1 U 3, 100,
D 0, 031,
2) P2
L R
P1 T 2, 120,
B 1, 203,
Game Theory, Oligopoly
3 3) P2
L C R
U 2, 01, 24,
P1 M 3, 41, 32,
D 1, 30, 03,
4) P2
L C R
U 5, 62, 12,
P1 M 8, 94, 53,
D 0, 43, 22,
Types of games: differ along different dimensions
- types of strategies
o purestrategies (what you just practiced)
o mixed strategies (discussed later)
- timing: how players make decisions, reveal their strategy
o simultaneously(above)
o sequentiallylater)
- examples above are pure-strategy, normal form games
- normal form games:
o purestrategies
o players reveal chosen strategies simultaneously
Example 2:
Firm 2
0 6 5 4
Firm 1 45 8100, 8100 6750,0
60 9000, 6750 7200,0
- recall that NEs take into account what your opponentoing
Game Theory, Oligopoly - case by case comparison
- as Firm 1:
o if Firm 2 chooses 45 , what is Firm 1’s BR?
Firm 1 should choose 60
π 160) > π 145)
o if Firm 2 chooses 60 , what is Firm 1’s BR?
Firm 1 should choose 60
π (60) > π (45)
1 1
- as Firm 2:
o if Firm 1 chooses 45 , what is Firm 2’s BR?
Firm 2 should choose 60
since π 2(60) > π2(45)
o if Firm 1 chooses 60 , what is Firm 2’s BR?
Firm 2 should choose 60
π 260) > π 245)
- conclusions:
o regardless of what firm 2 does, firm 1 earns the largest
payoff by choosing q 1 = 60
therefore, q 1= 60 is a dominant strategy for firm 1
o regardless of what firm 1 does, firm 2 earns the largest
payoff by choosing q = 60
2
therefore, q 2= 60 is a dominant strategy for firm 2
- dominant strategy always give the player the highest payoff
o relative to other optional strategies
o regardless of what his opponent does
o think about how this differs from earlier examples
- NE = ( , )
- although this is the NE, could both players do better ?
o answer: Yes!
o by choosing to produce q i= 45, both firms could gain
larger profits
Game Theory, Oligopoly
5 o as a result, the NE need not be the optimal outcome
Example 3: Prisoners’ Dilemma
P2
Mum Confess
P1 Mum 1, 1 20,
Confess
0, 20 10,10
NE = ( , )
- highlight:
o could have been better off by choosing to collude to “keep
mum”
o the NE is not thetimal outcome
o why don’t we observe prisoners colluding as an
equilibrium?
each has the incentive to deviate
defies the definition of equilibrium
- is there a dominant strategy for P1 and P2?
o yes: each player will always choose to “confessr
“keep mum ”
o reason: it leads to less punishment, regardless of what the
other prisoner does
Custody Battle (Example 4)
- in a custody battle over their children, divorcing parents must
decide whether or not to hire a lawyer, which is costly
Dad
Lawyer No Lawyer
Mom Lawyer ½¼, ½ ¾,
No Lawyer ¼½, ¾ ½,
Game Theory, Oligopoly
6 - what is the NE and does either Mom/Dad have a dominant
strategy?
- NE = ( , )
- dominant strategy:
o another case where NE is not necessarily optimal
o could have achieved same outcome w/o the expense of
hiring a lawyer
Building on your understanding of dominant strategies:
- second way to solve for NE: Iterated Elimination of Strictly
Dominated Strategies (IESDS )
- based on assumption that rational players never play strategies
that are strictly dominated
- back to custody battle example:
o never choose “no lawyer ” as an option
so we say “no lawyer” is dominated by “lawyer”
alternatively, “lawyer” dominates “no lawyer”
- how does this help?
o if we know strategy options are strictly dominated, we can
eliminate them from the game
o example: mom knows dad will never play “no lawyer”,
dad knows mom will never play “no lawyer”
- note: why “strictly dominated”?
o only eliminate strategies where payoffs are strictly greater
than other strategies
o if two strategies have the same payoff, you cannot
eliminate the strategy
- indicate dominated strategies by crossing them out
(eliminating them)
Game Theory, Oligopoly
7 Dad
Lawyer No Lawyer
Mom Lawyer ½, ½ ¾,
No Lawyer ½, ¾ ½,
- since Mom will never choose “no lawyer”, cross out
lawyer” to eliminate it as a possible strategy
- since Dad knowshat Mom will never choose “no lawyer”, he
only considers the outcome of Mom choosing “lawyer”
o he will never play “no lawyer” too, so eliminate by
drawing a line through it
Example 5:
P2
L C R
P1 U 3 2, 2 3, 1 1,
M 1 1, 3 1, 0 3,
D 7 0, 4 0, 5 2,
Drawback of IESDS:
- does not guarantee unique solution
- may leave multiple remaining outcomes
- not allmaining outcomes are NE
Example 6
P2
L M N O
P1 A 4, 2 7, 43, 14,
B 6,7 6, 64, 57,
C 3,112, 81, 64,
D 3,7 4, 99, 72,
Game Theory, Oligopoly
8 Eliminations:
1.
2.
3.
4.
- 2 possible strategies remain for each player:
o player 1:
o player 2:
- 4 possible outcomes remain:
- arrange strategies, payoffs into a 2-by-2 in normal form:
P2
P1
- NE =
- now use the “best response” method to solve for NE:
P2
L M N O
P1 A
4, 12 7, 2 3, 4 5, 14
B 6, 7 6, 3 4, 6 7, 5
C 3, 7 2, 11 1, 8 4, 6
D 3, 7 4, 5 9, 9 12, 7
- take-away point: outcomes should be the same regardless of
the method used to solve for NE
Game Theory, Oligopoly
9 - drawbacks to iterative procedure:
1) each step requires strong assumptions about rationality
2) won’t always get precise predictions compared to BR method
- equilibrium in dominant strategies: I’m doing the best I can,
regardless of what you do
o doesn’t take into account what opponent does
o excluding information when making decision
- equilibrium in best response strategies: I’m doing the best I
can, based on what you do
o takes into account what opponent does, case-by-case
situation
o making use of all information when making decision
- practice:
- first use Iterated Elimination of Strictly Dominated Strategies
to find remaining strategies
- then use the BR method to find all NE
1) P2
Red Yellow Blue
7, 2 8, 5 1, 1
Orange 3, 1 6, 3 4, 7
P1 Green 9, 18 7, 14 8, 3
Purple
P2
Red Yellow Blue
Orange 7, 2 8, 5 1, 1
P1 Green 3, 1 6, 3 4, 7
Purple 9, 18 7, 14 8, 3
2) P2
L M N O
A 3, 0 12, 8 5, 2 6, 6
P1 B 9, 6 3, 12 8, 16 13, 11
C 11, 1 0, 3 10, 0 9, 12
D 7, 13 1, 4 3, 13 10, 8
P2
L M N O
A 3, 0 12, 8 5, 2 6, 6
P1 B 9, 6 3, 12 8, 16 13, 11
C 11, 1 0, 3 10, 0 9, 12
D 7, 13 1, 4 3, 13 10, 8
Game Theory, Oligopoly
10 3) P2
I J K L
5, 6 0, 6 4, 1 9, 1
E 8, 8 2, 3 2, 0 4, 3
P1 F 9, 5 10, 0 0, 0 2, 7
G 1, 2 8, 7 3, 8 4, 5
H
P2
I J K L
5, 6 0, 6 4, 1 9, 1
E 8, 8 2, 3 2, 0 4, 3
P1 F 9, 5 10, 0 0, 0 2, 7
G 1, 2 8, 7 3, 8 4, 5
H
Normal Form Games Summary:
- model interactions between players, who reveal their strategies
simultaneously
o two ways of solving: best response, IESDS
- the resulting equilibrium of these games is a single strategy
outcomes per NE
- also called “pure strategies”
o 100% of the time, pick strategy A
2 Game: Mixed Strategy NE
- instead of having pure strategies, mixed strategy NE provide
players the option to behave probabilistically
o reveal their decisions simultaneously
o goal: find NE
- twist: assign probabilities to the possible strategies
- for example:
o 25 % of the time, pick strategy A
o 75 % of the time, pick strategy B
Game Theory, Oligopoly
11 Example 7: Battle of the Sexes
t a P
Opera Game
Chris Opera 2, 10, p
Game 0,20 1, 1 – p
s 1 – s
- definitions:
p = prob that Chris chooses opera
1 – p = prob that Chris chooses game
s = prob that Pat chooses opera
1 – s = prob that Pat chooses game
Each player will:
- choose strategy that maximizes utility
- incorp probabilities into decision by calculating expected
utility (EU)
- recall: BR are conditional on what your opponent does
s* = BR (s *)
1 1 2
s2 = BR (2 *1
- for each strategy option available to you, look at the payoffs
assoc with that option
- in a 2-by-2 game, you and your opponent each have 2
strategies to choose from
o each of your strategies can have two potential
depending on your opponent’s choice
- calculate the EU (an average) of each strategy by multiplying
the probability that your opponent plays strategy by the
payoffs you get
o note: the prob with which your opponent plays a strategy
is unknown to you
o leave the probability as an unknown variable o in the end, we will have a game-plan for whatever
probability p or s can take on
o since probabilities range from 0 to 1 (or 0% to 100%), we
will be specifying a game-plan for all ranges ofp or s
- recall the earlier example:
o if 0 < p < 25%, pick strategy A
o if p = 25%, indiff btw A or B
o if 25% < p < 100% of the time, pick strategy B
- for player 1:
EU 1(event A) = (payoff to P1of event A) (prob P 2hoose event A)
+ (payoff to P of event A) (prob P choose event B)
1 2
- depending on what Pat chooses , Chris can earn two different
payoffs:
EU C(opera) = (Chris’ payoff for Opera) (prob Pat chooses Opera)
+ (Chris’ payoff for Opera) (prob Pat chooses Game)
- for Chris:
o EU C(Opera) =
o EU C(Game) =
- for Pat:
o EU P(Opera) =
o EU (Game) =
P
- goal: for each individual, calculate a range of probabilitiesor
which the individual would:
o choose event A
o is indiff btwthe two options
o choose event B
- requires that you solve for p and s
- for Chris: choose opera (p = 1) if EUC(opera) > EU Cgame)
Game Theory, Oligopoly
13 - choose game (p = 0) if EUC(game) > EU (oCera):
- indiff btw game and opera if EU (gCme) = EU (operC):
- for Pat: choose opera (s = 1) if EU Popera) > EU (gPme):
- choose game (s = 0) if EUP(game) > EU (oPera):
Game Theory,14ligopoly - indiff btw game and opera if EU (gaPe) = EU (operaP:
- we use this info to help label the axis
- recall the definitions of p and s
o reps the probability Chris, Pat will choose each strategy
- if p = 0, (1 – p) = 1
o Chris chooses game with 100% probability, certainty
- if p = 1

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