 Hi everyone, this is Alice Gao. In this video, I'm going to give a detailed explanation of the answer to the clicker question on slide 30 in lecture 22. In this question, we looked at the dancing or running game, and we want to characterize the pure strategy Nash equilibria of this game. Turns out there are two pure strategy Nash equilibria. One is dancing, dancing, and the other one is running, running. So I'm going to talk about the general approaches we can use to derive a Nash equilibria, a pure strategy Nash equilibria of any normal form game. One approach I'd like to follow is that we can use the best responses and follow a chain of best responses until we reach a stable point, a stable outcome of the game. So this is what I mean. Let's consider our game and start with an arbitrary outcome. We have four outcomes. So let's randomly pick one, say let's start with dancing and running. And once we start with the outcome, we can pick either player and try to verify, is that player currently playing a best response to the other players? If the player is not playing a best response, then we'll switch that player to playing another strategy that's a best response to the other player strategies. So we're basically picking a player and trying to improve their strategies to be a best response to the other players. And we'll keep doing this until every player is playing a best response to the other player strategies, and that's a Nash equilibrium. So what happens in this case? Let's consider Bob, for example. Now, in this case, we're fixing Alice to be going dancing and consider Bob. Bob's current strategy is running, which gives him a utility of zero. Another strategy for Bob is going dancing, and that gives him a utility of two. So clearly dancing is better than running, which means in the current strategy profile, Bob is not playing his best response. Now, because of this, we are going to switch Bob's strategy from running to dancing. That means we'll switch from the profile dancing, running to the profile dancing, dancing. Okay. So now we have this new profile, and we just improved Bob's strategy so that Bob's strategy is a best response to Alice's strategy. Now let's see if we can improve Alice's strategy in a similar way. So currently, we are at this left strategy profile, and let's fix what Bob is doing. So let's suppose Bob is sticking to going dancing, and can Alice improve her strategy? Well, in the current strategy profile, Alice is going dancing, and she gets a utility of two. If she switches to going running, she gets a utility of zero. So in fact, Alice is currently playing a best response to Bob's strategy, because she is getting the highest utility that she could possibly get. Switching running actually causes her to get a lower utility. So she does not want to do that. In fact, we have reached a stable point, because both Alice and Bob are playing best responses to each other's strategy. Therefore, this is an ash equilibrium. By using a similar reasoning, you can also derive that running-running is an ash equilibrium as well. You can start from dancing-running, or you can start from running-dancing, and either way, if you choose the correct player, then you will end up with running-running. And turns out that's a stable outcome as well. This is one approach we can use to find all the pure strategy-nash equilibria of a normal form game. I've also written down the thought process for another approach. So for the second approach, we are looking at all possible strategies for each player, and then we're trying to figure out the best responses for all of them. So since this game is symmetric, I only need to consider one of the two players. I'm going to consider changing Bob's strategy and determining what's Alice's best response with respect to Bob's strategy. So here's my reasoning. If Bob goes dancing, then I'm going to determine Alice's best response. And then if Bob goes running, I'm also going to determine Alice's best response. It turns out if Bob goes dancing, then we're looking at this yellow column that I highlighted. And Alice prefers going dancing as well. And if Bob goes running, then Alice prefers going running as well. So since the game is symmetric, Bob's best responses are the same as Alice's, right? Which means you can think about this game as a coordination game. And in any kind of coordination game, it's best if players coordinate on doing the same thing, taking the same action, right? This is the case here. Dancing is the best response to dancing and running is the best response to running. So this leads us to our conclusion, which are there are two stable outcomes and both of them are Nash equilibria of this game. The difference between the two approaches is that the first approach is somewhat ad hoc, right? You randomly start with an outcome and then you randomly choose a player to try to improve their strategy. So you likely will be able to find at least one Nash, one pure strategy, Nash equilibrium, but it's not guaranteed that by following the first strategy, you can find all the pure strategy Nash equilibria. But for the second, the second approach is more systematic. You end up analyzing all the possibilities, all the possible best responses. And given the analysis, you should be able to find all the pure strategy Nash equilibria of a game. That's everything for this video. Thank you very much for watching. I will see you in the next video. Bye for now.