 Hello everyone, this is Alice Gao. In this video, I'm going to discuss the answer to the clicker question on slide 10 in lecture 23. In this question, we wanted to characterize pure strategy Nash equilibria of prisoner's dilemma. And the correct answer is that there is only one pure strategy Nash equilibrium, and that is defect, defect. So let's look at how we can derive this answer. This game is fairly small. There are only four outcomes. So one brute force way of answering this question is that we can go through all four outcomes and verify whether each outcome is a Nash equilibrium or not. So we can start from start with defect, defect. So start with this outcome to verify that it is a Nash equilibrium. We have to ask ourselves this question. Can either player improve their utility by deviating to another action? If you can find a player that prefers to deviate, that can improve their utility by deviating, then this outcome is not a Nash equilibrium. If no player wants to deviate, then this outcome is a Nash equilibrium. So for defect and defect, what happens? Well, if Alice chooses to deviate, then she will get a utility of minus 3 instead of minus 2. So she gets worse utility if she deviates. What about Bob? Well, if Bob wants to deviate, he gets a utility of minus 3 instead of minus 2. So Bob is worse off as well. So in this case, if both player are choosing to defect, then neither player wants to deviate by themselves. This is the argument we can use to say that defect, defect is indeed a Nash equilibrium of this game. So far, we verified that defect, defect is a Nash equilibrium. But to answer this question fully, we also have to verify that all three other outcomes are not Nash equilibrium. Let's do this. We can use our chain of best response argument that I've tried before. Let's suppose we start with co-operate, co-operate. If we start with both co-operating, then can either player improve their utility by deviating. Well, let's consider Alice. So if Alice chooses to defect instead of co-operate, she gets a utility of 0 instead of minus 1. So this defect is clearly better. If Alice wants to deviate, then both co-operating is not a Nash equilibrium. Similarly, we can verify that defect co-operate is not a Nash equilibrium. So we're looking at this outcome, this outcome right now, defect co-operate. Well, why is this not a Nash equilibrium? Because, well, although Alice is defecting, she is choosing her dominant strategy. Bob can do better, right? If Bob chooses to defect rather than co-operate, then he gets a utility of minus 2 instead of minus 3. So Bob wants to deviate, therefore this is not a Nash equilibrium. By the exact same argument, you can verify that co-operate defect is also not a Nash equilibrium because Alice wants to deviate to defect instead of co-operating. Therefore, we can conclude that there is pure strategy Nash equilibrium of this game and the equilibrium is that both players will choose to defect. There's actually a much simpler way of getting this answer. And the simpler way requires you to realize that dominant strategy equilibrium is a stronger concept than Nash equilibrium. So in fact, if an outcome is a dominant strategy equilibrium, then it definitely is a Nash equilibrium. And the reason for this is that, think about it, when we're characterizing dominant strategy equilibrium, we are finding a dominant strategy for each player. And the definition of dominant strategy is that, no matter what the other player does, I always want to take this action. Right? So if an action already satisfies this, if an action is already a dominant strategy, then it would definitely satisfy the requirements for Nash equilibrium as well. Because Nash equilibrium, the requirement is weaker. It says that given what the other player is doing, considering what the other players are doing, I want to play this action. I prefer to play this action. So in other words, if I prefer a particular action, regardless of what the other players are doing, then I definitely also prefer this action if I know considering what the other players are doing. So in the future, if you see an outcome that's a dominant strategy equilibrium, then you know right away that it must be a Nash equilibrium as well. That's everything for this video. Thank you for watching. I will see you in the next video. Bye for now.