Nash Equilibriums // How to use Game Theory to render your opponents indifferent
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Nash Equilibria are one of the most powerful ideas in Game Theory, this was the subject of the movie Beautiful Mind. Given a normal form game, a mixed Nash Equilibrium is a choice of mixed strategy (i.e. probabilistic) for each player where no player can unilaterally improve their expected payoff. Incredibly, every such normal form game (with a finite number of players who can each choose from a finite number of pure strategies,) has a Nash Equilibrium! In this video I show you how to use pay-off matrices to computed the Expected Payoff aka Expectation Value for each action and use the Principle of Indifference to arrive at the Nash Equilibrium. We see two Nash Equilibrium examples, one from the football aka soccer world and one from economics with two firms in competition for a new market.
0:00 Intro to Nash Equilibriums
0:56 Football Example Intuitively
2:59 Mixed Strategies & the Indifference Principle
4:00 Expected Payoff
4:55 Definition of Nash Equilibrium
5:32 Nash Existence Theorem
6:06 Football Example Mathematically
10:14 Economics Example
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Пікірлер: 53
I love these game theory math videos. I think that there’s enough information about the topic to make an entire game theory college course.
@DrTrefor
2 жыл бұрын
That’s totally right! My plan is about ten vids for now, which corresponds to what I do in my math modelling course, about a three week unit.
@AvoniasStratigis
Жыл бұрын
@@DrTrefor While I was doing my undergrad in Physics, I took a course on Operations Research and came across Game Theory in the textbook I had. It was very math heavy and used Simplex method for introduction. I always wanted to learn Game Theory, at least get an overall intuition before I moved onto a math heavy treatment. These videos are really nice, gives a good working overview :) . Please do make a full fledged course!
You are very engaging. I finally understand what Nash Equilibria are!
you are a very good professor and explain concepts very good. thank you so very much
Found your channel when I was taking calc III last semester, I have stuck around since and have really been enjoying your game theory series! :D
@DrTrefor
2 жыл бұрын
Cool! Glad you have been enjoying:)
Great explanation !
gr8 video
great video
Your channel is awesome!
@DrTrefor
2 жыл бұрын
Thank you so much!
I absolutely love your content! Small point of feedback: your mic overpeaks sometimes, when you talk a bit louder, which makes the audio quality suffer. Investing in a better mic would have great returns!
@luukderuijter1332
2 жыл бұрын
I'd Suggest using a compressor or limiter, as he tends to almost yell when he gets excited
I got the Trefor discount on Brilliant! Thanks!
@DrTrefor
2 жыл бұрын
Awesome! Thank you!
❤️❤️❤️❤️
Too op
Why less views this is not fair
@coolgarrett17
2 жыл бұрын
Apparently youtube algorithms are a dominated strategy
Thank you Dr. Bazett. Can Game Theory be used to describe morality and ethics? If not, how can math best describe it?
@DrTrefor
2 жыл бұрын
Absolutely it can help inform this, my students did a class project related to e to ethics in medicine and game theory last year!
@luukderuijter1332
2 жыл бұрын
You can even expand this to evolution and population behavior as a whole
fat better than my lecturer. Thanks!
@DrTrefor
2 жыл бұрын
Sorry to hear that, but glad you enjoyed!
I need this if I return to goalkeeping.
@ricardobautista-garcia8492
2 жыл бұрын
Also, you are in Canada? Perhaps you mentioned this before but it is nice to hear. If only you were in NB.
@DrTrefor
2 жыл бұрын
I am! Victoria:)
@ricardobautista-garcia8492
2 жыл бұрын
@@DrTrefor Ah on the other side. Hopefully all is well over there.
💗
If a nash equilibrium always exists, then what is the nash equilibrium for the goalie example if you do NOT consider the mixed strategy?
@DrTrefor
2 жыл бұрын
There is no “pure” mash equilibrium here. The theorem is only for existence of mixed equilibria
@CristalMediumBlue
Жыл бұрын
@@DrTrefor In 5:40 you said that there is always at least one Nash equilibrium if the game has a finite amount of pure strategies. Did you mean "mixed" instead of "pure" strategies?
In your economics example, if you change the -50 to +50 then you get x=2 which is not a valid probability. Can you comment and perhaps address what constraints your require the reward matrix to satisfy?
@Joaking91
2 жыл бұрын
It means that you cant force indifference: no matter what you do, the competition will always benefit more by choosing to enter. It wins either 50 or 100 vs always 0 if they choose not to.
Is there a way to win with game theory? Not getting equilibrium but actually winning.
すげー
Is there a Discord channel, where we can all share and ask questions to each other? Just an idea.
@DrTrefor
2 жыл бұрын
Not yet but I’ve gotten a few requests for this so maybe I’ll do one at some point!
@aashsyed1277
2 жыл бұрын
@@DrTrefor ok!
Several classical games use this principle; notably poker. There's an old saying in poker; if you make a play 100% of the time you are playing 100% wrong.
Actually the best strategy is for the companies to collude, divide up the states and agree to not compete ... assuming they can avoid getting noticed by the FTC.
What about running to the ball fast, but hesitatng an instant . the goalie jumps , then kicking ?
this video is not in the game theory playlist!
@DrTrefor
2 жыл бұрын
What would I do without you?
@aashsyed1277
2 жыл бұрын
@@DrTrefor:)
Canadians can't play soccer. The ice is too slippery. ps. How does the Nash equilibrium thm consider the quality of execution in a real world game?
@DrTrefor
2 жыл бұрын
lol:D Typically what we do (and will see a bit of this later on) is to think of this probabilistically. For instance, we say something like that "when you kick to the opposite side of the goalie you score 75% of the time, and 25% of the time you are wide/hit the post/etc". These elements can then be incorporated into the model.
render them indifferent? great so they no longer care about the game :P
@DrTrefor
2 жыл бұрын
haha then we would definitely win!
Unfortunately sports require instinct and physical training not Nash Equilibriums
@unknowninfinium4353
2 жыл бұрын
Interesting point.
I like the video, but i could do without the yelling