week 10 the basics of game theory
TRANSCRIPT
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8/20/2019 Week 10 the Basics of Game Theory
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The basics of Game
Theory
Understanding strategicbehaviour
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The basics of Game Theory
As we saw last week, oligopolies are aproblem for classical theory The best strategy for a rm depends on what
the other rm decides to do
Unless some assumption is made, the solutioncan’t be found...
Game theory is the study of the strategicbehaviour of agentsot !ust useful in economics, but also in
international relations, games of money, etc.
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The basics of Game Theory
The prisoner’s dilemma
ash e"uilibrium and welfare
#i$ed strategy e"uilibria
%etaliation
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The prisoner’s dilemma
The prisoner’s dilemma is the &historical' game thatfounded game theory as a specic area of study( This is because the solution to this game is sub)
optimal from the point of view of the players. This
means that there is a solution that makes bothplayers better o*, but the rationality of the agentsdoes not lead to it .
The prisoner’s dilemma shows quite elegantly
how difcult it is to get agents to cooperate,even when this cooperation is benecial to allagents.
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8/20/2019 Week 10 the Basics of Game Theory
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The prisoner’s dilemma
They o*er the criminals a &deal'... 0f one of them &spills the beans' on his
colleague, he gets a reduced sentence 12months3, and the other guys gets a e$tended
one 14 years3Payo Matri
+st criminal
5onfess 6eny
nd criminal5onfess
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-
4
-.4
6eny-.4
4
+
+
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8/20/2019 Week 10 the Basics of Game Theory
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The prisoner’s dilemma
The prisoner’s dilemma applied to a duopoly
Two rms competing on a market can(5ompete 1This leads, for e$ample, to the 5ournot
solution35ollude and share monopoly prots 1cartel3.
7rot in a cartel 8 prot in a duopoly.
0f collusion is not illegal, then it is clearly theoptimal situation from the point of view of thesetwo rms. 9ut is it the e"uilibrium the market endsup in /
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The prisoner’s dilemma
players ( rms 1A and 93 producing the same good
1Airbus:9oeing ts well;;3
strategies (7roduce at the duopoly level7roduce at the cartel level 1which is lower3
Given players and strategies, there are< possible market congurations These are listed in the payof matrix
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The prisoner’s dilemma
=et’s put some numbers on the di*erent possible prots(
>or the 5artel case( ?ach rm earns a share of the monopoly prots(
@c +- >or the duopoly competition case (
?ach rm earns duopoly prots, which are lower(
@d
>or the &cheating' case( The rm producing at duopoly level captures the market
share of the other rm, and makes very high prots (
@t +4
The other rm is penalised and earns minimum prots (
@m -
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The prisoner’s dilemma
PayoMatri
>irm 9
Bd Bc
>irm A
Bd
-+4
Bc+4
-
+-
+-
For firm A:
Qd if firm B chooses Qd
Qd if firm B chooses Qc
Note: the game is symmetric, so the dominant
strategy is to produce the duopoly quantity.
What is the best strategyfor each firm?
For firm B:
Qd if firm A chooses Qd
Qd if firm A chooses Qc
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The basics of Game Theory
The prisoner’s dilemma
ash e"uilibrium and welfare
#i$ed strategy e"uilibria
%etaliation
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ash e"uilibrium and welfare
!enition o" a #ash equilibrium$
A situation where no player can improve his outcome byunilaterally changing his strategy
%entral properties$ The ash e"uilibrium is generally stable ?very game has at least one ash e"uilibrium(
?ither in pure strategies ( 7layers only play a singlestrategy in e"uilibrium
Cr in mi$ed strategies ( 7layers play a combination ofseveral strategies with a $ed probability
The proo of this result is the main contribution of Dohnash 1and the reason why it is called a Nash e"uilibrium3
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ash e"uilibrium and welfare
Let’s go back to the Duopoly eample:
Payo
Matri
>irm 9
Bd Bc
>irm A
Bd
-
+4
Bc+4-
+-+-
Is the “Qd-Qd” equilibrium
a Nash equilibrium ?
Can firm A or B improe
their outcome b! shiftin"
alone to the cartel quantit!
Qc ?
“Qd-Qd” is indeed a Nash
equilibrium
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ash e"uilibrium and welfare
Payo
Matri
>irm 9
Bd Bc
>irm A
Bd
-
+4
Bc+4-
+-+-
#o the dominant strate"!
is to produce “Qd”
But the “Qd-Qd”
equilibrium is not socially
optimal
$ith a small number of
a"ents% indiidualrationalit! does not
necessar! lead to a social
optimum
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The basics of Game Theory
The prisoner’s dilemma
ash e"uilibrium and welfare
#i$ed strategy e"uilibria
%etaliation
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#i$ed strategy e"uilibria
A pure)strategy ash e"uilibrium does not e$istfor all gamesE
?$ample of a penalty shoot)out(
players( a goal)keeper and a striker strategies ( shoot : dive to the left or the rightFe assume that the players are talented( The
striker never misses and the goalkeeper alwaysintercepts if they choose the correct side.
This is not re"uired for the game, but it simpliesthings a bit;
Fhat is the payo* matri$/
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#i$ed strategy e"uilibria
For the stri&er:' if the &eeper "oes (
( if the &eeper "oes 'PayoMatri
Goalkeeper
= %
triker
=+
-
-
+
%-
+
+
-
For the "oal&eeper:
( if the stri&er shoots (' if the stri&er shoots '
No pure!strategy Nash equilibrium "
$hateer the outcome%
one of the pla!ers can
increase his sucess b!chan"in" strate"!
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#i$ed strategy e"uilibria
PayoMatri
Goalkeeper
= %
triker
=+
-
-
+
%-
+
+
-
)here is% ho*eer% a mi+edstrate"! equilibrium
#trate"! for both pla!ers:
,o ( and ' ./ of the time 01out of t*o% randomly 2
)hat *a! :o 3ach outcome has a
probabilit! of .45o )he stri&er scores
one out of t*o% the other is
stopped b! the
"oal&eeper
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#i$ed strategy e"uilibria
=et’s check that this is actually a ash e"uilibrium(
The goalkeeper plays = and % 4-H of the time. 5an thestriker increase his score by changing his strategy/
The striker decides to play 2-H left and
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The basics of Game Theory
The prisoner’s dilemma
ash e"uilibrium and welfare
#i$ed strategy e"uilibria
%etaliation
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%etaliation
>inally, the stability of the e"uilibrium alsodepends on whether the game is repeatedor not. The very concept of a mi$ed strategy
e"uilibrium depends on the repetition of thegame through time.
?ven for a pure strategy e"uilibrium, the abilityto replay the game can inLuence the outcome
7layers can retaliate, and thus inLuence thedecisions of other players
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%etaliation
Actually, this depends on whether the gameis repeated a $ed number of times orindenitely 1open)ended3...
=et’s say that our rms decide to play thegame 4 times 14 years3Fhat is the best strategy on year 4 /
Fhat about year
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%etaliation
=ets imagine now that our rms have anopen)ended agreement. The threat of retaliation can bring the social optimum
The optimal retaliation strategy is also thesimplest one( &tit for tat'%obert A$elrod( !ust choose what your opponent
did last period( cooperate if he cooperated, cheatif he cheated.
9ut the threat needs to be credible i.e. theopponent needs to believe that it wille*ectively be carried out.