Nash Equilibrium

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.

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.

[fuego]

I DONT UNDERSTAND IM GOING TO CRY SOMEONE HELP ME

[omarkhan]

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

[chris]

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

[stuckupcurlyguy]

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

[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 ;) )

[wunnanyeinhtoon]

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

[RobRandolph]

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

[aryaman]

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

[AHMEDTALAATHASSAN]

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

[manutemanute]

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

[witchinghour_om]

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

[TheWitness]

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

[MeoLaKid]

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

[anap]

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!

[Babar]

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

[LapineTingTing]

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

[eroselee]

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

[pankajbansal]

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

[pratikrane]

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

[jasminesisa]

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