Lecture 27: Sub Game Perfect Equilibrium- Need and Introduction - YouTube

Welcome to Mooc lecture on Strategy, An Introduction to Game Theory.

In this module, we are going to discuss a solution concept named sub game perfect equilibrium.

By the way, why do we need a new solution concept, let us go back to the technique that we learntto

solve an extensive form game.The name of the technique was backward induction.

So, just to recap, let us see, how do we do?

We start towards the end and wego back towards the beginning.

So, what happens if player 1given a chance, player 1would play,if player 1 moves in this

direction,if player 1 move in this direction player 1 gets 1,if player 1 moves inthis direction,he

gets 0.Notice here, there is no strategic interaction,as soon as the player 1decides

the game ends,sohe does not have to worry about anything.

So, of course, what should you do, heis a rational player,he is interested in maximizinghis

payoff, he does not care about the payoff that player 2 receives.If he cares,then we

have to build it in the model.

But,once we have return the payoff,then what he cares about is maximizing his payoff,he

does not careabout the payoff of other player.So,as 1 is greater than 0,he will move in this direction

not in this direction, sogame will proceed here.Now, player 2 at this point ifplayer

2 gets a chance to play,player 2 knows thatplayer 1 is rationaland he will move in this direction.

So, if player 1decides to move in this direction game will reach here and player2 will get

0.But,if player 2decides to move in this directions player 2 will get 2,of course,2 is greater

than 0, sogame will proceed in this direction as player 2 is not only rational,player 2knows

player 1is rational,so game will not proceed in this direction.Similarly, here we can see

player 2has to decidebetween two actions, one action is to move in this direction and

another is to move in this direction.

Moving in this direction gives player 2, 4, moving in this direction gives player 2,3of

course, 4 is greater than3, sogame will proceed in this direction not in this direction,coming

back to here.Why can we go back?Because, not only player 1is rational,player 1 knows that

player 2is rational,player 2 is rational and player 2 knows,player 1 is rationaland soon.

So, it is the implication of common knowledge, soplayer 1 sees that if he moves in this direction,game

will eventually reach to this point and his payoff will be 2.

And, if you proceeds in this direction game will reach here,and his payoff would be 3,

so of course, 3 is greater than 2, sogame will proceed in this direction not in this

direction.So, we know the outcome, let us say that, if we name this A and Band then

here C,then we can say the right in the beginning,player 1 will take action Band player 2 will take

action C. But,if you want to describe the strategy that

player 1 will take right in the beginning, we cannot simply say B, because B is not his

strategy,player 1 gets to play game at two nodes, one here and another time here.

So, his strategy should describe his action at both of these decision nodes.So, what should

he do?He should decide to play B here and let us again,let us give them name X and Y.

And of course, he should decides to play X here, soX strategy for player 1 is that he

should play B X.How about player 2?

Similarly, we have to see that player 2 decides to move here and at this node here.

So, hisstrategy should give act this particular action here and this particular action hereandthat

is how, we get the equilibrium.

But, let us lookat a particular limitation of this backward inductiontechnique.At this,

again there is no strategic interaction,player 2 either has to decideto play Xor to playY.Playing

Ygives him higher, soof course game will move in this direction,at this point also player

1 has two actionsA and B.Whichactionsshould he take?Taking action Ameans that player 2

will get 1and taking action B means he would get 1.

Of course, if player 2 takes actionA here, and game has reach to this point, the payoff

forplayer 1 could be 4and if player 2 takesaction B,payoff offor player 2 would be 0.But,notice

that hereplayer 2does not player about how much player 1 is getting,player 2 is interested

in only maximizing his payoff.

So, but he is indifferent, he can go in this direction, he can go in this direction,then

how should we,what should be do here,what should we take.

If, we think player 2 is going to take action A, then let us say, if we have action Pforplayer

1and Q for player 1.Of course, ifplayer 1thinks that player 2is going to play A, then he is

better off playing Q,because 4 is greater than 3 or 2.But,if player 1thinks that player

2is going to play B, then he is better off by playing P.So,how should here backward induction

plays?Itdoes not give as any answer and, we will see,because we have we may have multiple

equilibrium in this games, sothat is the reasoning.

Again, coming back to another limitation of backward induction,again let us say player

2 has two actionsX and Y.If,player 2 takesaction X player 2 gets 2, if player 2 takes action

Y,player 2 gets 3, player 2should take action Y, sofor all practical purposes, we can deletethis

part.Because,player 1 knows thatplayer 2 is rational, soif game reaches to this point,player

2 is going to play Y.How about here?Can we sayplayer 2 is going 2?

Let us say, this is Aand this is Band then, if this is thecase,we have to have Aand Bpresent

here.Wecannot havedifferent actionsat the different nodes of the same information’s

set.Again, you will have torefer if you are having difficulty in understanding this,then

you should look at a module called strategies.

So, player 2, can we say that player 2decides do take this actions A, we do notknow.Let

me tell you that, letme changethis payoff slightly, then it would become more interesting,

Let us make it 2.

So, we now we cannot say that player 2will take action Aor action B, because player 2does

notknow whether he,game has reach to this nodeor this node.If he thinks this node has

reached,he is better off by taking action A, because 1is greater than 0.

If,he thinks this node has reached, then he is better off bytaking action B, but the problem

is here, game is off in prefect informationgame is off imperfect information.

So, we cannot, the player 2does not know, where the game has reached.

So, we cannot decide,we cannot reduce, we cannot take out one of the branches and say

this is what player 2 would do, sobackward induction would not work here.

We should also, but you may say that why are you taking about a new solution concept, as

we already have asolution concept known as Nash equilibrium.

So, for that I wantto remind you,that this was the entrant that we discussed earlier.What

did we see?That, if we usebackward induction,fortunately backward induction works here and what did

we see that entrant inters,here we have entrant and incumbent accommodates, this is the outcome we get.

But,when we use the solution concept Nash equilibrium,we get two different possibilities.

E and A is of course, one of the possibilities, but we also get O comma F as another possibility,

this is really upset,because itdoes notsatisfysequential rationality, because game is ofcommonknowledge,entrant

would know if game reaches to this point player 2has no option, but to accommodate.

So, saying O comma Fit may be rational at this point, but it is not rational, when game

reaches to this point.

So, in that sense O comma F is an observed outcome recommended by Nashequilibrium.Can

we have atechnique to get rid of thisobsolete outcome?Of course, we have that is backward

induction, but we saw thatbackward induction does not work in some of the cases.

So, what do we really need,backward induction is a good concept for extensive form game,

but we need some modification in backward induction to obtain a solutions in more in

all of the extensive form games.

So, let us define something called sub game.What do you mean by a sub game?Again, let usgo

back before I define sub game.

Let us, look at this entry game, because game is moving in a linear fashion,first entrant

gets todecide, whether to inter or remain outof the gameand then,incumbent gets to reside.When

at look at the game at this point, one can say that a new game is beginning here,whatever

happen in the past that incumbent takes as given and now, he has, hedoes not need to

consider,what has happened in this part.Only he needs to consider, what is going to happen

from now on wards, so that is how we get the concept of sub game.

We can say, this is the new game which is beginning at this point,when this node has

reached and this is a sub game.

So, every moment you know as we are moving in the game,we do not we of course,we have

to consider the past, but only in the sense that we have to take it as given.If game has

reachedto this particular node, then nothing can be done for the past nodesthat can only

be taken as given, so that is why we define sub game.

So,what do we have a sub game,what we say thatsub game begins at some decision node.Say

AI am just saying one of the, I am giving thatname A, which hasa single terminformation set.

So, very, very clear asub game begins at the single term in formation set.Why not aninformationset,

which has more than one element?

Let uslook at it, here game cannot a sub game cannot begin at this information said, why

because play at,let say here we have player 1here, we have player 2,player 2does not know

whether this has reached or this has reached.

So, he does not have aclear cut picture form the past, sothat we cannot say again begins

here, but incase in case, we have situation like this, where clear to here,as clarity

that this node has rest if it have reached.Then, he does not needs to consider,what ishappening

here or what is happening here,he will consider a new beginning of a new game from this on word.

So, that is what I am saying that sub gamebegins only at the information sets, which are single

term,second requirement is that it includes all the successor arenote.Again, it would

be,let us say,let us take another example,this is the case,here we have player 1,here we

have player 2here we have player 3.What we are saying?That, if we say that sub gameis

beginning here, then all the successor nodes should belong to this sub gamewhy, because

notice we are deciding now,game is at this is stage.

So, we do not know whether game will proceed in this direction are in this direction or

in this direction.

So, we cannot consider,let say this,this,this and not this, sowhat we say that once we a

node is included in the sub game all the successor node of this node would be included in the

sub game.And, I have already said I just want to emphasize that if it includes one node

from the information said, then it includes all the nodes of that information said.

Notice, that whole game is also a sub game of itself, because whole game would satisfy

all the criteria.

So, we define again for the shack that we have something called propersub gameit is

the part of the game tree whoseroot is I think I have written it incorrect here,proper sub game

is all the sub game of the game excluding the whole game itself.

And of course, hereeverything is that we have notion of moment,if game is moving in this

direction it should be clear.

So, for example, how many sub games,do we have in this game one of course,beginning

write at the initial node, because whole game issub game of itself.

So, all the whole game is not proper sub gameof itself, but it a sub game of itself,another

is beginning here.

So, we can say, if we start drawing this is one sub gamethis is another sub gamethis is

another sub gameand then, the whole game is sub game of itself.

So, how many we get 1 2 3 4 this game has 4sub games 3 proper sub gameshere.

Another example, how many sub games do we havein this game,you should pass and think

before you say I suggest that you write it in your notebook, that how many sub game do

we have in this game of course,onesub game isstarting here.And, another the whole game

is also sub game of itself, but can we say a of sub game is starting here,no we cannot

say,why because the third criteria says that if a node is included in the sub gamethe node

belonging to information’sall the nods belonging to the those information said present in this

sub game should also be in the sub game.

For example, let me also, let me say it again that, if let us say this node is present in

the sub gamethis node also belongs to the same information said the information to the

information said that this node belongs to, but if we say this is the sub game,we are

excluding this, so this cannot be a sub game.

In fact, this gamehasonly two sub games, so that is it about the sub game, now we are

going to learn the solution concepts.