 Hi everyone this is Alice Gao. In this video, let's start looking at our second normal form game. By analyzing this game, I will introduce another solution concept called the Nash equilibrium. In this game Alice and Bob are choosing between signing up for one of two activities, dancing or running. They both prefer dancing over running. You can see from the utility matrix that their utility for dancing are both higher, but they also prefer signing up for the same activity over signing up for two different activities, because everything is more fun when you do it together. Now from the first game, you should already know my passion and interest in swing dancing. Now this game might not accurately reflect my attitude towards running. As a child, I really, really hate running and that's still the case right now. So in China, when you go from middle school, three years of middle school, two, three years of high school, there's an entrance exam. You have to get tested on the normal subjects like math, English, Mandarin, other things, but there is also a physical education exam. There's a PE exam with three subjects. One is doing crunches, which I was really good at. The other one is standing long jump. I was really terrible at it. But after a few months of practice, I can do it. And then the third one is running 800 meters. And then there is a time limit. So I think you can get perfect mark if it's a little bit less than four minutes. And you can pass if you can finish running 800 meters in almost five minutes. And I think I actually never really knew the perfect mark line very well, because I never got anywhere close to that. I was always barely passing and struggling with it. So ever since that, I hate any form of running whatsoever. Short distance, long distance. Yeah, just don't make me run. Anyway, enough about the rent about running. Let's come back to this game and let's figure out how Alice and Bob would like to play this game. As usual, first of all, let's think about what the players will do because our goal here is to predict how they would behave. So take some time and think about this yourself and then keep watching. I can't really ask you this question and show you the distribution of answers. But based on historical data, this is what I would predict. The the distribution we get will be a bimodal distribution. Most of you would say that Alice and Bob were both trues to go dancing. And some of you, a significant fraction of you would say that they would both go running. And both of you would be correct. These are two plausible, plausible outcomes of this game. And why is that? Well, because in some sense, these are the two stable outcomes of this game. And by stable here, I mean that if Alice sticks to dancing, then Bob does not want to go running instead. And similarly, if Bob sticks to running, then Alice does not want to go dancing instead, because deviating in that way will cause you to cause either player to get smaller lower utility. Now we've gotten this conclusion based on our intuition. Let's now see if solution concepts from game theory can help us to formalize this intuition. So far, the only solution concept we've talked about is the dominant strategy equilibrium. Let's try to apply the solution concept first and see what happens. So based on this game, dancing or running, is there a dominant strategy equilibrium of this game? If there is one, which of the four outcomes is a dominant strategy equilibrium? Think about this yourself, and then keep watching for the answer. The correct answer is the last one. This game has no dominant strategy equilibrium. And the short explanation is that first of all, the game is symmetric, so we can consider either player. If we consider Alice, then it turns out Alice does not have a dominant strategy. If Bob chooses a different action, then Alice would prefer a different action depending on Bob's choice. Therefore, there's no one action that's always the best for Alice. This means there's no dominant strategy. If there's no dominant strategy for a player, then there's no dominant strategy equilibrium. Unfortunately, this means that the dominant strategy equilibrium solution concept is not sufficient to capture our intuition about this game. Intuitively, we believe that dancing dancing or running running are both reasonable outcomes of this game. But based on dominant strategy equilibrium, it doesn't give us a prediction of how the players will behave. So we need to look at another solution concept and turns out Nash equilibrium would be a useful solution concept we can use to predict the player's behavior. In the next video, I'm going to introduce the concept of Nash equilibrium in more detail, and then use that to analyze this game. Thank you very much for watching. I will see you in the next video. Bye for now.