 Hi everyone, this is AliceGal. In the previous video, we looked at our second normal form game, Dancing or Running, and we tried to apply dominant strategy equilibrium solution concept to the game, but unfortunately there is no dominant strategy equilibrium, so this solution concept does not give us a prediction of how the players would behave in this game. But based on our intuition, there are some clear outcomes that are reasonable outcomes, such as both going dancing and running. It turns out that we can formalize our intuition using a different concept called the Nash equilibrium. Let's take a look. The hero of her story is John Nash. He was an American mathematician. He made a lot of important contributions to game theory and various branches of mathematics. So he obtained his PhD from Princeton, and during that time, as part of his PhD thesis, he worked out the idea of the Nash equilibrium. And this idea, and with his other contributions to economics and mathematics, eventually won him the Nobel Prize in Economics. To give you a bit of a sense in terms of the history and how life was like at the time, and also about this historical figure, I want to show you John Nash's one-page paper on Nash equilibrium and also his PhD thesis. By working on the project, you should have had some experienced reading research papers and you know what they should look like. But back then, you can see this is John Nash's one-page paper on Nash equilibrium. And you can see this paper is still mostly, this one is printed, but a lot of the paper back then are type-setted by a typewriter. And this is the seminal paper on Nash equilibrium. I look at how short it is and look at the fact that there is one foot node and two citations at the end of the paper. And then that's it. Here is John Nash's PhD thesis. It's on non-cooperative games. That's one of the central topics in game theory. And now you can see that this is entirely written on typewriter, right? Physically typed out by John Nash. And let's scroll down and look at a few more details. So if we go to the body of the thesis, you can see all of these equations. So they were not able to write out a lot of mathematical notations using typewriters. So you need to type most of the text and then write in all the formulas. So this is the last page of the PhD thesis. And you can see the bibliography again. Only has two things where one of the two things is Nash's own paper. This was just something interesting to show you to help you get a sense of history. Another way you can get to know a bit more history about John Nash is by the film, A Beautiful Mind. But unfortunately, my personal opinion is that A Beautiful Mind is not a very good and accurate depiction of Nash's life. In contrast to, for example, the imitation game, I thought that was really good for telling stories about Alan Turing and the theory of everything. I thought that was a great film for telling stories about Stephen Hawking. But not about A Beautiful Mind. I thought A Beautiful Mind had a little bit too much about the stories on the mental illness that Nash had been through. And I didn't think it was very accurate. And also, there were some criticism about the fact that the concept of Nash's equilibrium was really not very accurately described in that film. Anyway, if you're curious, you should take a look. But I would take it as a grain of salt. So one of the main contributions by John Nash is the concept of Nash's equilibrium. And not only did he propose the concept of Nash's equilibrium, he also was able to prove that every finite game has at least one Nash's equilibrium. And this is a remarkable result, right? Because there is an endless number of games you can construct out there. And this is a really, really general statement. The statement means that if you give me a finite game, then I'm guaranteed to be able to describe, to derive a Nash equilibrium of the game. So it means potentially we can use the Nash equilibrium solution concept to predict how players behave in any finite game. That also makes Nash equilibrium a really applicable and useful solution concept, right? Because like we've seen with dominant strategy equilibrium, if we have a game where there's no dominant strategy, equilibrium, then this solution concept is not useful for that game, for making predictions of that game. All right, enough stories about histories. Let's look at the technical details about what is a Nash equilibrium. So the concept of Nash equilibrium is based on this idea of best response. Best response is defined as follows. First of all, we have a strategy profile. The strategy profile consists of a strategy for player i and then some set of strategies for all the other players. Now, when can we claim that the strategy for player i, sigma i here, is a best response to all of the other agent strategies? Well, we can claim this if the following inequality holds. So in this inequality, notice that on either side, we are fixing the other agent strategies. So for some fixed set of strategies for all of these other agents, then we are comparing sigma i with any other strategy for agent i, so sigma i prime. And if you look at the quantifier at the back, the quantifier is saying we are comparing sigma i with all other possible strategies for this agent. So sigma i prime can change where sigma i state is fixed. So the inequality says, well, sigma i gives agent i weekly higher utility than any other strategy for agent i. Notice the inequality here is greater than or equal to. So it's possible that there's a tie, but even in the event of a tie, sigma i is still weakly better than any other strategy. So in short, this definition says that given what the other agents are doing, sigma i is the best choice for me compared to any other strategy for me. So I am playing my best response if I am playing the best strategy for myself, given that the strategies for all other agents are fixed. Given the concept of best response is quite easy to define Nash equilibrium. Nash equilibrium simply says that a strategy profile is a Nash equilibrium if every agent is playing a best response to all other agent strategies. So as long as every agent strategy satisfies the definition of best response, then we're good. You can see that Nash equilibrium is trying to characterize a stable set of strategies. The set of strategy is stable because while fixing what all other agents are doing, then I don't want to change to another strategy because the current strategy I'm playing is already the best option, weekly best option, because it's greater than or equal to. If every agent thinks they're playing their best option, then nobody wants to switch to another strategy. So another way of characterizing Nash equilibrium is that no single agent wants to deviate to another strategy if the strategy profile is at a Nash equilibrium. All right, let's now apply the Nash equilibrium solution concept to our game. So our dancing we're running game, let's only consider pure strategy Nash equilibrium, which means we don't want to consider any mixed strategy, any distributions, we are only going to consider the player's actions. So given that which of the outcomes are possible Nash equilibrium of this game, is it just dancing, dancing, is it just running, running, or both dancing, dancing, and running, running, or maybe the game has more than two Nash equilibrium? So which one of these is the correct answer? Take some time, think about this yourself, and then keep watching for the answer. The correct answer is C. Both of the two outcomes dancing, dancing, and running, running are both Nash equilibrium of this game. So you can see that by using the Nash equilibrium concept perfectly captures our intuition, right? Our intuition tells us that these two are both plausible outcomes, and so does Nash equilibrium. Check out the separate video for more detailed explanation of why these two outcomes are both Nash equilibrium. That's everything for this video. Thank you very much for watching. I will see you in the next video. Bye for now.