 Hello everyone, this is AliceGal. In this video, I'm going to start talking about Meek strategy Nash equilibrium. This is the final solution concept that I'm going to talk about. And we'll look at the game called matching quarters. This is the game of matching quarters. It's supposed to be called matching pennies. That's the most well-known name of this game. But because we're in Canada and we don't have pennies anymore, so the smallest currency we have is a quarter, and we're going to match quarters instead. So Alice and Bob are playing this game, and they each have a quarter, and they will simultaneously show one side of their quarter. Alice wants the size of the two quarters to match, whereas Bob wants the size of the two quarters to mismatch. So from the payoff matrix, the utility matrix, you can see that if the two quarters have the same sites, so either both has or both tails, then Alice is happy, Bob is not as happy, whereas if the two quarters, the two sides are different, then Bob is happy, Alice is not as happy. Now given this game, now we have a few different concepts we can use to analyze it and try to predict what the players are going to do. Let's use the concept of pure strategy Nash equilibrium. So let's think about this question. How many of the four outcomes of this game are pure strategy Nash equilibrium? Think about this yourself, and then keep watching for the answer. The correct answer is A0. Turns out none of the four outcomes is a pure strategy Nash equilibrium. That may or may not be surprising for you. Now you might wonder if you remember when I introduced John Nash and the Nash equilibrium, I said that this is a really useful concept because every finite game has a Nash equilibrium. So what's happening here? We don't have one. How is that possible? Didn't John Nash prove this? Turns out he did prove this, but his exact statement was that every finite game has a mixed strategy Nash equilibrium, and a pure strategy Nash equilibrium is a special case of a mixed strategy Nash equilibrium. So we have a game here where we don't have any pure strategy Nash equilibrium, but we definitely have at least one mixed strategy Nash equilibrium. Next, let me show you how we can derive this mixed strategy Nash equilibrium. I'm going to show you the derivation. So the derivation answers the first question I have on the slide is if we're given a game, how do we calculate the mixed strategy Nash equilibrium? But after deriving the mixed strategy Nash equilibrium, we are also going to think about the second question. So the second question is more looking at the pure Nash equilibrium, reflecting it and trying to interpret it. So the second question says if a player is mixing between two actions, how should we understand this? Why is the player really doing this? Let's take a look at how we can derive the mixed strategy Nash equilibrium of this matching quarters game. So first of all, each player is playing a mixed strategy and recall that a mixed strategy is a probability distribution over all the actions for the players. In this case, there are two actions per player. So Alice needs to play heads with some probability and tail with the rest of the probability and Bob also needs to play heads with some probability and tails with the rest of the probability. So in order to derive the mixed strategy Nash equilibrium, we need to specify, we need to derive these mixed strategies. So let's assume that Alice plays heads with probability P because there are two actions, so we only need one probability to define the distribution. And similarly, let's assume Bob plays heads with probability Q. So I've taken the game matrix and highlighted the corresponding probabilities. So later on we can use these to calculate the expected utility of the players. Now, what's the trick? How should we derive these probabilities? It turns out the rule is quite simple. The rule says one player, this is a two-player game. So one player is going to choose their probability, their mixing probability such that the other player is indifferent between their actions. So in other words, for our game Alice chooses a value for P such that Bob is indifferent between his two actions. And similarly, Bob chooses a value for Q such that Alice is indifferent between her two actions. Now here indifferent means that the expected utility of playing either action is the same for that player. So Bob being indifferent between his actions just means both of his actions gives him the same expected utility. So he sort of doesn't care about which one he plays. Now, given this general principle of deriving the mixed strategy Nash equilibrium, let's now look at the calculations. So first of our Alice is going to choose P to make Bob indifferent between his actions. So if Bob is indifferent, we need to calculate his expected utility for each action and equate them together that will allow us to solve for P. The expected utility for Bob if he plays heads, if Bob plays heads, we are in this left column. We are in this column right here. And if Bob plays heads, then we're looking at these two numbers, right, zero and one. So if Bob plays heads, then with probability P Alice plays heads. So Bob got the utility of zero, right? So with probability P, Bob gets the utility of zero. And then with probability one minus P Alice plays tails and Bob gets utility of one. So Bob's expected utility is the sum of the two, which is one minus P. Similarly, if Bob plays tails, then we are looking at the right column. Let me highlight this. And then we're looking at these two numbers, right? If Alice plays heads with probability P, then Bob gets utility of one. If Alice plays tails with probability one minus P, Bob gets utility of zero. So Bob's expected utility is P. We equate these two. That gives us an equation involving P and we can solve for P, which is 0.5. Okay, this means in order to make Bob indifferent between his two actions, Alice should choose P equals 0.5. Basically means she should play heads and tails randomly. And the calculation for Bob is very similar, but let me still go through them. So Bob wants to make Alice indifferent. So if Alice plays heads, then we're looking at the top row. And in particular, we are looking at, since this is Alice, right? So we're looking at the first two numbers. Now if Bob plays heads, that's with probability Q, Bob plays heads and Alice gets a one. With probability one minus Q, Bob plays tails and Alice gets a zero. So that's the expected utility in this case is Q. And then for the case when Alice plays tails, in that case, we are looking at the bottom row. And we're looking at the first two numbers as well. So with probability Q, Bob plays heads and Alice gets zero. With probability one minus Q, Bob plays tails and Alice gets one. So Alice's expected utility is one minus Q. Again, equate these two together, you get 0.5. So overall, what's happening? Well, there is one mixed strategy Nash equilibrium. And at this mixed strategy Nash equilibrium, Alice plays heads with probability 0.5 and Bob plays heads with probability 0.5 as well. So to summarize, how do we derive a mixed strategy Nash equilibrium of a game? The general principle is that each player is going to choose their mixing probability to make other players in different between their actions. Now, given this derivation, I hope you get a sense of how to answer the second question as well. So the second question says, if a player is mixing between two actions, if they are playing both actions, each with some probability, what does this mean? What does this tell us about the two actions and the expected utilities for the players? So it turns out when a player is mixing between two actions, it means that the two actions have the same expected utility for the player. It means that the player is indifferent between the two actions. So what's happening at this mixed strategy equilibrium is that each player is choosing their mixing probability to make the other player indifferent between their two actions. And then once the other player is indifferent between their actions, it really doesn't matter how much they play each action because both actions are the same. Once they're indifferent, they can mix with any mixing probability, except they have to choose their mixing probability to make the first player indifferent between their actions as well. Okay, so there's an interesting dynamics going on where each player is choosing their mixing probability to make the other player indifferent. But once they're indifferent between the two actions, really the mixing probability doesn't matter to that player anymore. Right, that's everything for this video. Thank you very much for watching. I will see you in the next video. Bye for now.