 Hello everyone, this is Alice Gao. In this video, let's look at another example where we'll practice calculating a mixed strategy Nash equilibrium. This game is another well-known normal form game in game theory. It used to be called the battle of sexes but I really don't like that name. So the name comes from the fact that we have two players and the players again would prefer to sign up for an activity together except they each prefer one of the activity more than the other one. So in this case, Alice prefers dancing more over going to a concert where Bob prefers going to a concert more than dancing. This is again channeling my best friend from grad school. It's actually a she and she her research is actually on computer music. So using she now works on using deep learning to generate music, which is really cool. And so she would often like to go to these concerts for contemporary music, very interesting music, but at the same time I prefer to go to go dancing rather than go to one of these concerts. So from the utilities, you can see that if both of us go dancing, we're both happier but I am much more happier than Bob. Whereas if we both go to a concert, we both get positive utility but Bob is happier than I am. Given this game, I'm going to ask you three questions. The first question already went over the main idea. So I'll just talk briefly about it. Then the next two questions are basically asking you to derive the mixed strategy Nash equilibrium and each question is asking you about the mixing probability. So the first question first, first question is about why would a player be willing to mix between two actions? And the question says, consider any two player normal form game and fixed Bob strategy. Suppose Alice is willing to mix between playing hats and playing tails with some probability, then which of the following statements is true? I already discussed this in the previous video, so let me just give you the answer and the brief explanation here. If Alice is willing to mix between two actions, she's willing to play either action with positive probability, that must mean that both actions give her the same expected utility. Think about it. If one action gives her strictly better expected utility than the other one, then she would not play the worst action at all. Why would she? She should always play the better action. So if the two actions give her different expected utility, she would always play the better one. It's only in the case when the two actions give her the same expected utility that she's willing to mix between the two. So the confusing point here is that the mixing probabilities has nothing to do with expected utility. She can be mixing with any probability, and that always means that the expected utility for the two actions are the same. So it's not the case that if she, for example, plays heads with probability 90% versus if she's playing heads with probability 30%, those two cases, there's no difference. For both of those two cases, the expected utility for the two actions must be the same. All right, for the next two questions, I'm simply asking you to derive the mixed strategy Nash equilibrium. For the first question, I'm asking, at the mixed strategy Nash equilibrium, with what probability does Alice go dancing? So for the top action, with what probability is Alice going to play this action? And then for the last question, I'm asking the strategy for Bob. So with what probability does Bob go dancing? So with what probability is Bob going to play the left action? Try calculating the mixed strategy equilibrium yourself, and then keep watching for the answer. Here are the correct answers. I've labeled the answers around this utility matrix. For Alice, she goes dancing with probability two-third and goes to a concert with probability one-third. And for Bob, he goes to dancing with probability one-third and then goes to a concert with probability two-thirds. And this is the mixed strategy Nash equilibrium. Notice something interesting here is that each player is going to their preferred option, taking their preferred action with a higher probability, right? So Alice is going dancing more, whereas Bob is going to a concert more. For a detailed explanation of how I derive the answer, please watch the separate short video. That's everything for this video. Thank you very much for watching. I will see you in the next video. Bye for now.