Nash Equilibrium

economicsdetect...
Nash equilibrium is a solution concept in game theory. It is named after its creator, John Nash, who invented the idea in his dissertation in 1950. He later won the Nobel Prize and was portrayed by Bruce Willis in the movie "A Beautiful Mind"...or something.
Anyways, a set of strategies are in Nash equilibrium when no player has an incentive to change his strategy given every other player's strategy. The equilibrium concepts looked at in my previous videos, the dominant strategy equilibrium, and equilibria found by deleting strictly dominated strategies, were both forms of Nash equilibria.
Now, let's look at a simple game for an example of a Nash equilibrium. Let's say you and I are walking in opposite directions down a hallway. I can go to my left or to my right, and you can go to your left or to your right. If we go in opposite directions, we'll walk into each other. Let's say that corresponds to a payoff of negative one for both of us. If we both go left, or if we both go right, we won't crash into each other. Let's say that corresponds to a payoff of one. Maybe I'll tip my hat at you and say "good day" as we pass.
Or at least I would. If I had a hat. If you have a spare hat and want to mail it to me, send me an email at garrett.m.petersen at gmail.com, and I'll give you my mailing address. Fedoras are preferred over baseball caps.
In any case, let's try to find the Nash equilibria in this hallway game. If you play right, then I want to go right, so I'll circle that. If you play left, then I want to go left. Similarly, if I play right, you want to go right, and if I play left, you want to go left. What I've done here is circle all the best-responses for each player. By the definition of a Nash equilibrium, if we play strategies where I am playing my best-response to you while you play your best-response to me, that's a Nash equilibrium. So you see, we have two Nash equilibria in this game.
Now let's look at another game. We're going to do the same thing we did before, and circle the best-responses to each possible strategy by the other player. Note that if two strategies are tied for the highest payoff, we consider them both to be best-responses.
Top is a best-response to left, and so is bottom. Bottom is a best response to right. Left and right are both best-responses to top, and left is a best-response to bottom. So top left and bottom left are both Nash equilibria.
Now I'm going to leave this more complicated game on the screen, and I want you to pause and find the Nash equilibria...Have you found them? There are the best-responses, and you can see that there are two Nash equilibria, one at top left and one at bottom centre. Remember, some things are best-responses for one player, but still aren't a Nash equilibrium because they aren't best-responses for the other player.

[@fuego5532]

I DONT UNDERSTAND IM GOING TO CRY SOMEONE HELP ME

[@omarkhan5223]

How am I the only one who got the joke about Bruce Willis, he showed Gladiator with Russel Crowe so he obviously made that mistake on purpose

[@collinsmusumba4460]

seriously, gladiator = beautiful mind and Bruce Willis = Russel Crowe I don't get the Math

[@widiopranoto1]

Well, not only you.

[@bcnicholas123]

this is what like half of my whole econ final is, so thanks for the video

[@mitran.writes]

Russel Crowe not Bruce Willis.

[@RobRandolph80]

I'm confused as to why the bottom right combination in the last example isn't a nash equilibrium.

[@RobRandolph80]

Nevermind, I got it.

[@wunnanyeinhtoon2977]

what about middle centre (1,1) in the last example? why we can't choose it like top left (1,1)?

[@yasasp5781]

The reason is because if "I" were to pick middle, "YOUR" best response its to pick left. This is because you are only comparing the value on the second row (as these are the possibilities if "I" pick middle) and "left" has the best outcome at 2 (compared to 1 and -1). Note you shouldnt just be looking for matching numbers because the numbers are only significant relative to your other options given what the other person is doing. you are choosing your best RESPONSE to what that person does. Hope this helps.

[@AditeeArt]

Hey, thank you for these videos they're really helpful and your animation is just brilliant. The only thing I would suggest is slow down a bit when you show how to decide what's the best response for each player, i.e. maybe you can block out the right side of the normal form when you want to show player 1's options when player 2 chooses left. On that note-since your animation is so lovely, would you mind telling me what program you use to add/edit them in? Would be super useful for my own videos (nail art... you're perhaps not my target audience ;) )

[@GarrettPetersen]

I used Keynote. Thanks for watching!

[@stuckupcurlyguy]

If clothing choices were represented on a huge grid, fedoras would be -5's all round, m8.

[@manutemanute]

Dude great jokes!!! Literally blew my headache away!

[@AHMEDTALAATHASSAN]

the movie (Beautiful Mind) the actor was (Russel Crow)

[@aryaman61]

Thank you, I appreciate it! You saved my day.

[@91099Babar]

??? How did you calculate the Best Response in it ???

[@witchinghour_om4090]

If I play right, he will play Bottom; and make Bottom/Right a 3rd Nash equilibrium?

[@LapineTingTing]

You videos are awesome!!! Wish you can make more! >_

[@MeoLaKid]

Thank you so much! You've helped a lot!

[@TheRepain]

Still don't understand...

[@anap4852]

Thank you so much for your videos! Now everything seems clear! However, could I ask you to explain us weakly dominant equilibrium?My best regards and thank you so much for your videos!

[@GarrettPetersen]

A strictly dominant equilibrium is one in which all players have strictly dominant strategies, meaning that the payoffs are strictly higher under the dominant strategy than under all other strategies. A weakly dominant strategy is one where the payoffs are at least as good as the alternatives, and better in at least one instance. If all players have weakly dominant strategies, there is a weakly dominant equilibrium.

[@TheWitness1001]

Oh Yea, that Aussie actor: Bruce Willis!!!

[@jasminesisa]

Thank you so much ! That's so informative !! (Y)

[@pankajbansal8787]

A Beautiful mind or something. haha. Bruce Willis wishes he acted.

[@rznirvana]

Bruce Willis??

[@eroselee9019]

the movie wasn't gladiator, it was A Beautiful mind

[@stroyaworskon5111]

It was a joke, Beautiful Mind.

[@pratikrane17]

both take opposite direction not go in opposite direction. You had me confused.

[@germanaquila2666]

Coming soon...

[@msctbeats]

Bruce Willis...?

[@issailio3568]

thanks for the videos but why did you stop ???

[@GarrettPetersen]

Video is a very time-consuming medium. It took me hours to make a video that you could watch in minutes. I am now working on producing a podcast, so I will be able to deliver more content with less work. Thanks for watching!

[@issailio3568]

thanks for answering ...good luck dude

[@nobad6134]

+The Economics Detective Any luck on the podcast. We need more content please.

[@GarrettPetersen]

I have produced 15 episodes of the podcast. They are available on iTunes and Stitcher, and also here: economicsdetective.com/

[@TheCandyHole]

*tips fedora* m'economist

[@Openreality]

Russell Crowe.........not Bruce Willis

[@Warrantofficer13]

Russell Crowe not Bruce Willis!!!!!

[@djSpinege]

this video needs a redo

[@pradgadagkar8557]

Bruce Willis? Are u all a pack of jokers?

[@helyatremblay613]

You really should pay more respect to Nash by doing your research. The actor who portrayed Nash was not Bruce Willis. The title of the movie did not contain the words "or something"