Game Theory 101 (#16): Subgame Perfect Equilibrium

Game Theory 101: The Complete Textbook on Amazon: www.amazon.com...
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This lecture begins our adventure through sequential games, in which players take turns moving. Not all Nash equilibria are sensible in this context, so we introduce a new concept: subgame perfect equilibrium. A subgame perfect equilibrium requires all actions to be Nash equilibria in every subgame of the larger game. In essence, this requires all threats players make to be credible.
We consider a game between two firms deciding whether to enter a market and engage in a price war. Can a monopolist's threat to launch a price war convince a challenger to stay out of the market?

[@JassyB4592]

What annoys me is the fact that I am paying £9000 a year to go to university, and this guy, my now dear friend William (haha), has taught me a whole module with free videos and an amazing textbook. Thank you so much, without these videos I didnt stand a chance.

[@jevnoc365dd]

Same, this guy is my only hope now

[@lamortexotique]

Same here. I have been suffering from a nervous breakdown because I wasn't being able to follow my uni courses. If I pass this exam (and I have one chance of passing it), I owe it all to William Spaniel. I have never understood these concepts like this before. I really hope I can pass this exam.

[@AAABBB-eq8wd]

Exactly!

[@XtremeKillah101]

truth!!

[@ezekomaugoo5569]

Universities are only good at legitimatizing one's knowledge. They never guarantee one an absolute knowledge.

[@HeavenForbid7]

I saw some game theory videos of yours, you are so much better at explaining all of this than my microeconomics lecturer teaching at a prestigious university... they should hire you.

[@PunmasterSTP]

So I know it's been a full decade, but I came across your comment and was curious. How'd the rest of your game theory class go, and how have you been doing since then?

[@elvisboyaci8276]

"Some people just want to watch the world burn." -Firm 2

[@yannickingermany]

Hi William, I am an engineering student who borrowed the course; electronics marketing from another department. Your explanations are going to help me pass my exams. Thank you so much.

[@danielseda2450]

The conclusion here seems a bit strange. Suppose we had a similar hostage situation where the criminals were threatening to execute hostages if the police entered the building. Once the police enter the building, however, the criminals have nothing to gain by executing hostages, and have something to lose by facing higher criminal charges. This would mean that not executing hostages strictly dominates executing them in the subgame, rendering the threat non-sensible. By eliminating that strictly dominated strategy, the police have no incentive to negotiate with the criminals, and should be able to enter the building without risking hostages being executed. In the real world, however, we can't ignore that threat.

[@AnnihilateButtonGames]

I mean, if the criminals were really rational thinkers. They would likely not have committed such crimes.

[@absidd]

William Spaniel, you are a life saver. Thanks a lot.

[@mauritsbol4806]

Only correct strategy is for firm two to flip a die, or exploit gametheory inoptimal behavior. If firm one plays perfectly, and firm two play optimally, they will always opt out, but if you always allow firm 1 to enter the market, firm 2 is getting pushed around. If you just flip a coin, firm 1 now has a 1.5 gain from going to war, and 2 from opting out. Now you have to be willing to get your 'revenue' from 1 to 0.5 if firm 1 opts in, but this will be adjusted in game theory optimal.

[@DubsCat_]

Will you are a lifesaver for my Game Theory class, thank you for making all these videos

[@PunmasterSTP]

How'd your game theory class end up going?

[@Gametheory101]

Two. The full game is always considered a subgame. The other subgame is Firm 2's move.

[@susanxiao5726]

Thank you so much for the book + the website!!! I am so excited that I came up with these videos I learned a lot!

[@TheAwesomenesstw0]

GOAT video and video maker

[@niemand262]

A firm one in and out. These games are getting spicy. Great stuff.

[@PunmasterSTP]

It's getting Nashty up in here...

[@ARP2wefightforyou]

Firm 1 out should be 2, 4 because we see from Firm 2 accept that there are 1 + 3 = 4 points in the market and since Firm 2 has a monopoly on the market, it gets all of the point that are in the market.

[@jakeclifton3310]

They’re just arbitrary rewards to establish a hierarchy in terms of which result is most/least favorable.. it doesn’t matter what the magnitude in. We know that firm 1 out is best for firm 2, then the result where firm 1 enters and 2 accepts, then price war outcome

[@SuPeRev94]

This video was awesome, thank you SO MUCH for posting!

[@RandomGaming-the-OG]

Firm 2’s strategy is to burn out firm 1 cash then reraise the price after firm 1 exit. You do see that in real life.

[@bluedonkey180]

That’s assuming firm 1 has more money than firm 2

[@ddavis7561]

Thanks for the great example. Glad I watched

[@supersweetie92]

this is explained incredibly well! thankyou

[@Gametheory101]

Not sure what you mean. Firm 1 earns 3 by entering and 2 by staying out. So, it enters.

[@drhemlatacivilstudy]

Thankyou making all video

[@desolstudios]

Reading this going on my final paper ur the saver bro

[@Pa3kable]

Correct me if I'm wrong, but (In, accept) is a perfect equillibrium while (out, war) is just an "ordinary" Nash equillibrium.

[@jakeclifton3310]

Yes except they’re both pure strategy NE... it’s just that (In, Accept) will be perfect in subgame, and (out, war) is not... because firm 2 would not profit from choosing war over accept

[@alam5055]

I don't understand something. Why does the option of Firm 1 - out and Firm 2 - war even exist in the payoff matrix? Such option is impossible, since Firm 2 can only choose war if the Firm1 enters the market. So in the payoff matrix, the square for (out, war) shouldn't even be included and have assigned payoffs. Why is it included then?

[@EuroUser1]

This model ignores three important points: 1. The benefits of bankrupting your competitor and getting back your monopoly. 2. The costs of failing to follow-through on a threat. Future threats will sound far less believable. 3. The not-so-rational impulse towards revenge. Revenge is so useful - for strong animals - that our brain always wants to have it. Pretty much like we always desire more sugar, even if we're already overweighted.

[@dackss3520]

I mean, this is a model (and a simple one at that) so it's not supposed to accurately depict every facet of corporate decisions however: 1. The price war option for firm 2 IS trying to bankrupt its competitor. Firm 2 is trying to benefit from being in the market for longer by outpricing the newcomer firm 1. Firm 2 however, have to suffer from smaller profit margins or perhaps even a loss in order to outprice form 1 so it is a risk for firm 2. This model just chooses to show this price war as a lose-lose for both firms since firm 2 looses money and firm 1 struggles to enter the market. 2. Following through on threats isn't something that would apply to this game because it is perfect information meaning both players (or firms) know what each other can do and their payoffs. If this game had imperfect information and firm 2 didn't know what type firm 1 was (I'm assuming firm 2 is the one that has unknown information because I'm not really sure which game is making "threats") then firm 1 could benefit from threats and following through on them. 3. Game theory assumes that all players are rational and able to keep track of all the information available to them. It is true that there are not-so-rational impulses but in the perfect vacuum that is game theory, it is assumed that players will ignore their impulses and chose the optimal strategy.

[@phatnoir1986]

I don't understand, wouldn't it always be more profitable for Firm 1 to opt out? I understand Firm 2's incredible threat, but I don't see why knowing it's incredible should make you want to accept less profit. Thanks for any light you could shed on this.

[@Gametheory101]

It won't. I don't know of anyone else who says that a full game is not considered a subgame, but the point is trivial.

[@subhroacharjya8528]

Bro one thing i didn't understood is why why will be even consider out-war strategy as equilirium. If firm 1 chooses out strategy then that's it, firm2 will move only when firm1 has choosen in strategy. Microeconomics is not even my subject i am going through this tutorial because it seems fun😅, so please explain in latent terms.

[@mylesallen8055]

Thanks William! Super helpful mate.

[@Gametheory101]

You could think of the 2 meaning that they get to keep their investment.

[@Gametheory101]

That's correct.

[@StRaWbeRrYsasha]

So how would you interpret a real life scenario such as the UK supermarkets entering a price war? or is that totally different?

[@tzhou3301]

compared to the video, pay-offs are different. that means 1. their cost functions are not identical. 2. their service are not 3. sometimes in a short period of time they don't care about their profits but more about market share. 4. all in all this model is merely an abstract from the real complicated case with relatively unimportant information omitted so that we can see the real picture. we can't count on such a simple model to solve a lot of questions. however, what we can extrapolate from this way of thinking is what that matters.

[@StRaWbeRrYsasha]

I understand, thanks for answering x

[@god8831]

In the real world though, a monopoly will benefit in the long term from declaring a price war, since the new firm will be forced out of the market after a while. The monopoly firm can however absorb the losses until that time and then continue in thier merry way afterwards.

[@asdfasdfuhf]

Hence the best move is to accept the war

[@syildirimbuup]

This is wrong because if you look at the payoffs (3,1) it tells us the new firm is more efficient in production of the good. If firm 1 declares price war, in the long run they will be the ones who are forced out of the market by the new firm.

[@zethayn]

Thank you very much!

[@user-km1gq5nm2b]

william is the greatest man ever

[@cuscuzboi]

Thank you, great content!

[@phatnoir1986]

Ah, I was looking at the wrong numbers. Thanks for the quick reply! I understand now.

[@SuitedBunny]

Can you do Nash Bargaining Theory and Rubinstein's Bargaining Model?

[@maxcash4707]

Mr. Spaniel you da man!

[@Gametheory101]

Should be the same thing.

[@xitec75]

The problem I see here is that humans act irrational or one company has enough money to do a 0,0 game until the other one runs out of money.

[@GhostGlitch.]

Boris Köster in some form of long term game you would be correct, and that could be the more profitable solution. Or if the goal was simply to do better than the other guy, which is often the sort of strategy monopolys use, it would be calculated into the profits. But in a short term get as many points for yourself as possible sort of game war is never a good option.

[@jakeclifton3310]

^i agree with what he said.. in the long run it might be better for form 2 to choose war if they can bully the other out and eventually obtain market control again.. or at the very least they signal to others not to enter market in future or we will go to war

[@ulvumustafayev1823]

Hi.Thank you for this excellent presentation.I have watched all the videos and find them super useful.Unfortunately, there 2 exercises about game theory that I can't solve.Pleaseee could you help me? Pleasee

[@yendar9262]

2:59, if 2's choice is irrelevant, why is it choosing War and not Accept? In other words: why are the NEs (In, Accept) and (Out, War) instead of (In, Accept) and (Out, Accept)?

[@kristielektra]

Same question

[@saksham_agarwal]

Hi, Can this game be converted into a mixed strategy Nash equilibrium where credibility of an action (say, probab of war by Firm1 is 'w')? That way, we may be able to chart out the possibility of firm1 going from (2,2) to (3,1) depending upon the weakness of 'w'. I am asking this because there are two Nash equilibriums and one is unstable (the (2,2) one, since there the full game doesn't play out as firm1 never got the chance to act)

[@Warrantofficer13]

Why would firm 1 get 2 from going out? It is not in the market therefore payoff would be O

[@MrBboyrox]

Just to clarify that Firm 1 (In) and Firm 2 (Accept) is the SubGame Perfect NE. Firm 1 (Out) would be an NE, right?

[@denisnevsky3734]

No, because since going to war is strictly dominated move in the turn tree (Not in the payoff playoff square) it is removed, therefore the only outcomes for firm 1 to consider are Out(2) or Accept(3) and since Accept is the better outcome, Out isn't considered a NE.

[@nicoleluo6692]

How to find SPNE in strategies?

[@adipurplishchic]

gibbons book says that a full game is not considered a subgame. if i do consider it a subgame, it shouldnt really change anything?

[@christiangeraudn.b.2837]

Does each firm know the expected payoff of their choices?!

[@yasemintoprak7542]

How many subgames does this game have? Two?

[@Wmabc123def]

hey, do you know some videos /sites where I can learn more about Moore-Repullo mechanisms

[@BrainForce95]

Can someone tell me maybe why Firm1-> out, and Firm 2->war is an equilibrium ? or how you can get this equilibrium by doing a backward induction

[@kristielektra]

@@datfinancial but strategy "accept" is the dominant strategy for firm 2. How strategy "war" can be part of equilibrium?

[@alam5055]

​@@datfinancialhere's the part I don't understand: "Given that Firm1 decides to stay out, Firm2 can't do any better by choosing Accept. So it decides to go for war". The problem is, if Firm1 stays out, Firm2 can't decide to go to war. This options is only available if Firm1 enters the market.​ So the scenario "Firm1-> out, and Firm 2->war" is impossible and can't happen. So even if you convert it into simultaneus game, that option shouldn't even be included in the payoff matrix.

[@drhemlatacivilstudy]

Nice

[@DataQuestLive]

whats sad is that, my proffessor teaches way different topics on game theory then you. But he didnt go over the basics as well. so thanks!

[@ngyx57]

So how many subgame perfect nash equilibria are there in this entire game? One?

[@Gametheory101]

+ngyx57 One.

[@shalinigupta3515]

@@Gametheory101 but you said two earlier?

[@treeorwh2]

Hi William SpAniel!

[@williamchan1613]

This concept completely ignored the profit of being a monopoly of the market, which is not true in the real world. However, what I have to agree is that this youtuber (William) explained all the concepts in the game theory much more clearly and effectively than professors in my school. And I am a student in Durham University, which the university suppose to have the best teachers to teach us. AND!!! THE PROFESSOR USED 55 MINS TO EXPLAIN A CONCEPT WHICH WILLIAM FINISHED EXPLAINING CLEARlY IN 8 MINS!!! WHAT A SHAME.

[@adipurplishchic]

incredible threats? Gibbons book calls it non-credible threats. And I have been reading about subgame perfect outcome. How is that different form this?

[@therealist3012]

Adity Das Gupta A subgame perfect outcome is the actual payoff, while an equilibrium is always a set of strategies. So, a sub game perfect Nash equilibrium is a set of strategies which create a Nash equilibrium in every subgame, no matter whether it is played or not. Better late than never ;)

[@PunmasterSTP]

I really wish that I had found this video 9 years ago.

[@StinkinDoctor]

What kind of dog do you have?

[@enditend2]

don't get the difference between a subgame equilibrium and equilibrium

[@OlliePage]

+enditend 2 an equilibrium is a threat that doesn't have to be credible. A subgame (perfect) equilibrium is a threat that has to be credible

[@scottwitoff8932]

Can this game be used to explain the benefits of capitalism?

[@carsonkearl]

the payout system of the game was poorly chosen, but other than that great video

[@sixtysixx3933]

it was deliberately poorly chosen, so that it could explain why it was a perfect subgame, which rarely happens. But when it does, the counterpart at least knows what to choose.

[@henriquewaksman8217]

brabo demais ta louco

[@macdavidroberts7136]

appreciate the video but I cannot follow along at all

[@carloschongfon5815]

I paid 70 grand last year at davis for what I could get free