Week 10 the Basics of Game Theory

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 prisoner’s dilemma

ash e"uilibrium and welfare

#i$ed strategy e"uilibria

%etaliation


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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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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

-

-

4

-.4

6eny-.4

4

+

+


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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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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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=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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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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%etaliation


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!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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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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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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%etaliation


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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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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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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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=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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%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.

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Week 10 the Basics of Game Theory