 Hi everyone, this is Alice Gao. In the next few videos, I'm going to discuss a game called Prisoner's Dilemma. Prisoner's Dilemma is one of the most well-known games in game theory. And the interesting thing about this game is that if the two prisoners, if the two people, are able to communicate, then they can achieve the best outcome for both of them. But if they're not able to communicate, and if they act purely out of self-interest, that means they act to pursue their own goals to maximize their own utility, then it turns out that they are going to achieve the worst outcome of all possibilities. There are many versions of this game out there, let's look at one version right now. In this version, Alice and Bob have been caught by police, and each of them has been given a chance to testify against the other, so offer the deal. And originally, of course, they don't want to testify against each other, and if they don't say anything, you will see that they end up with a pretty good outcome. But now the police decided to separate them and interrogate them separately. So they won't be able to communicate, and they have to independently make this decision of whether to testify against the other person or not. I've written the two actions as cooperate and defect. So here cooperating means cooperating with the other prisoner, which means not testify, and defect means testify against the other person. So there are three types of outcomes they can achieve in this game. If both of them refuse to testify, this corresponds to the outcome where they both cooperate. So I'm highlighting this outcome right here. If they both cooperate, then both of them will be convicted for minor charge because neither of them is talking, and there's not enough evidence. So each of them will serve one year in prison. So here the numbers are referring to the number of years served in prison, and your utility is kind of inverse of the number of years you serve in prison. Now in the second case, one person testifies against the other person, the other person does not. So in this case, if one person testifies, then the person who defects, the person who testifies is going to go free. So this is part of the deal. And then the other one will be convicted of a serious charge because now all the evidence is against the other person. And in this case, one person goes for free, so their utility is zero. And the other one has to serve three years in prison, so minus three right there. And finally, if both people, both Alice and Bob choose to testify against the other person, so both of them defect. In this case, well, they will both be convicted of a major charge and serve two years in prison. So you can see that the final outcome is better than if you don't say anything, and the other person testifies against you, right? But it's still worse than if both of them shut up and don't say anything. So this is one instance of the prisoner's dilemma game. In fact, you can change the reward, the utility numbers, but as long as the utilities follow a specific relationship, then the outcome, the result of the analysis will still be the same. We're going to look at three questions. So so far, we've looked at a couple of solution concepts. Dominant strategy equilibrium, we've looked at Nash equilibrium. We've talked about Pareto optimal outcomes. So I have three questions, one for each of these concepts. And in this video, I'm only going to discuss the answers. Please watch the separate video for a longer discussion of how I derive the answers. Here's the first question regarding dominant strategy equilibrium. Does this game have a dominant strategy equilibrium? If it does, which one is it? Take some time, think about this yourself, and then keep watching for the answer. The correct answer here is D. There is a dominant strategy equilibrium and the equilibrium is both players will choose to defect against the other one. So both players will choose to testify against the other person, and both of them ended up with a major charge and must serve two years in prison. Let's look at the second question. The second question is regarding pure strategy Nash equilibrium. So if we consider the four possible combinations of actions, how many of these four outcomes are pure strategy Nash equilibrium? Take some time, think about this question yourself, and then keep watching for the answer. The correct answer is B. There is one pure strategy Nash equilibrium and again is defect and defect. So both people testify against each other and they both need to serve two years in prison. Finally, let's apply the concept of Pareto optimality on prisoners dilemma. So this question says there are four outcomes, how many of these four outcomes are Pareto optimal? This is the trickiest question of all three questions on prisoners dilemma. So really go back and review the definition of Pareto optimum, then carefully think about the definition and apply it here and try to get the answers yourself before you keep watching for the answer. The correct answer is D. Out of these four outcomes, there are three outcomes that are Pareto optimum. And interestingly, the only outcome that's not Pareto optimum is the one that's both a dominant strategy Nash equilibrium and the pure strategy Nash equilibrium. So beside the only Nash equilibrium, all three other outcomes are Pareto optimal. If you didn't get this right, make sure you watch the separate video for detailed explanation of verifying why these three outcomes are Pareto optimum. So after thinking about these three questions, let's come back and think about this game a little bit. If we were to rationally and look at this game, we can all agree that cooperate, cooperate, both cooperating will be the best outcome for both players and both defecting will be one of the worst outcomes, right? Both of them will serve two years in prison where cooperating, if they both cooperate, then they only serve one year in prison. But it turns out there's one dominant strategy Nash equilibrium and there's one Nash equilibrium, pure strategy Nash equilibrium. And both of these are defect, defect. So the worst outcome is, turns out to be the stable one turns out to be predicted by two different solution concept. And even more interestingly, the only Nash equilibrium and the only dominant strategy equilibrium is the only outcome that's not Pareto optimal, right? So that's the worst outcome in multiple senses yet. It is the only stable one. There's in fact quite a bit of research on how we can motivate people to achieve the cooperating outcome when people play this kind of game in practice. So it turns out if there's a lot of economic experiments on this, turns out if the two players are only going to play this game once or a very small number of times, then it's very difficult to sustain or motivate cooperation. So people say, Oh, I'm not, I'm really not going to see you ever again. So if we play this game, might as well defect against you and try to maximize my own utility, right? But if it turns out I'm playing with someone who I'm going to repeatedly interact for many, many times. So in a repeated game scenario, we observe a lot more cooperation between the players because they can sort of build a trust against each other. And that trust is sufficient to sustain some cooperative behavior. If you're interested, you can look into some literature about person's dilemma. That's everything for this video. Thank you very much for watching. I will see you in the next video. Bye for now.