 Hello everyone. This is Alice Gao. In this video, I'm going to discuss the answers to two clicker questions. One on slide 19 and the other one on slide 20 in lecture 23. Both clicker questions are about the mixed strategy Nash equilibrium of the dancing or concert game. So this is another coordination game except that the two players have different utilities for coordinating on the two on the two actions and we are going to derive the mixed strategy Nash equilibrium. So the correct answer here is that the role player prefers taking the top action with two third probability and then the column player prefers taking the left action with one third probability. Let's take a look at the derivation. So we're deriving a mixed strategy Nash equilibrium, which means we have two probability distributions. One over the actions of the role player Alice and the other one over the actions of the column player Bob because there are two actions. So we need one probability to specify each distribution. So let's start by assuming that Alice goes dancing so chooses the top action with probability P and then Bob chooses the left action with probability Q. I've also labeled the utility matrix with these probabilities so we can use those for our calculations later. Now as I explained in the main lecture video when we're deriving a mixed strategy Nash equilibrium because each player is is mixing between the two actions. That means the two actions must give the player the same expected utility. This is the main principle you need to remember and this requirement is going to help us to derive conditions on these probabilities. So we have two separate conditions. One is that the two actions give Bob the same expected utility and the other one is that the two actions give Alice the same expected utility. The first condition is going to help us derive P and the second condition is going to help us derive Q. Okay let's look at the first condition. So the first condition Bob is in different between the two actions so let's calculate Bob's expected utility for playing each action. The top one is the first one is dancing. So if Bob is going dancing let's highlight what's happening here. We are looking at the left column and then we're looking at Bob's utility which are 1 and 0. So if Alice also goes dancing Bob gets a utility of 1. If Alice goes to a concert Bob gets a utility of 0. So now what are the probability for these to happen? Well Alice goes dancing with probability P. In that case Bob gets a utility of 1. Alice goes to a concert with probability 1 minus P. Bob gets a utility of 0. So the total is P and for concert is very similar so we'll highlight in this case we're looking at the right column and if Alice so let's take a look. With probability P Alice goes dancing in that case Bob gets a utility of 0 with probability 1 minus P Alice goes to a concert and Bob gets a utility of 2. So the total of that is 2 minus 2P. Equate these two together we get that P is equal to 2 third. Okay you can see that all we have to do is look in the utility matrix find the right numbers calculate the expected utility equate them and then we can derive the probability. The process for the second one is very similar let's go through that. So for the second one what's Alice's expected utility if she goes dancing? Well in that case we are looking at the top row right this is the top row where she goes dancing. In that case with probability Q Bob goes dancing as well and Alice gets a utility of 2 with probability 1 minus Q Alice goes to a concert and sorry Bob goes to a concert and Alice gets a utility of 0 so the total is 2Q. What about if Alice goes to a concert in this case we are looking at the bottom row so for the bottom row well with probability Q Bob goes to goes dancing and Alice gets utility of 0 with probability 1 minus Q Bob goes to a concert and Alice gets a utility of 1 so the total is 1 minus Q. Again equate them together we get that Q is equal to 1 third. So previously when I explained the requirements I said that Alice is going to choose P to make Bob indifferent between the two actions and Bob is going to choose Q to make Alice indifferent between the two actions. Now you might find this counter intuitive you might think when Alice is choosing P why does she have to think for think about Bob or when Bob is choosing Q why does he need to think about Alice? So if you find this counter intuitive you can simply drop the choosing P and choosing Q part all you need to remember is the two things I've the two sentences I've written on the on the page here so remember that each player needs to be indifferent between the two actions okay so as long as you use this to calculate to write out the expressions and equate them then it just ends up being that the condition on Bob right the first the the first one I've highlighted here the condition on Bob help us to derive the mixing probability for Alice and then the condition on Alice helped us to derive the mixing probability for Bob right that's all it's happening all right that's everything for this video thank you very much for watching I will see you in the next video bye for now