[0:00]Hey internet, this is Jacob Clifford. A teacher on Facebook said their students are really struggling with game theory, so I made this video to help you out. Here it is, it's the definitive ultimate game theory breakdown video. This is a payoff matrix with two players, me and you. Each player can choose between one of two actions, choice A or choice B, they can't do both. The result is four possible outcomes. Now let's not include numbers yet, let's just keep it simple. Big idea number one, the final outcome where these two will end up is determined by both players. They can't just go get the outcome they want because they're interdependent. For example, if I really want outcome one but you choose choice B, outcome one is off the table. It's no longer possible. My only options now are outcomes three and four. And if you really want outcome four, but I choose A, then you can't get it. I chose A, so choice B for me is off the table. Notice, we ended up here at outcome three because we both made decisions. So I'll say it again, big idea number one, the final outcome is determined by the decisions of both players. Big idea number two, players prefer the outcome that's better for them. Remember, the goal is to make myself better off, not to hold back my opponent. For example, if choice A earns me $100 and choice B earns me $200, I'm going to choose choice B. It makes me better off. And it doesn't matter if my decision ends up making you way better off, I'm only going to focus on what's best for me. Also, it doesn't really matter what the choices are. Whether it's to increase prices or decrease prices or to advertise or not advertise. You're not like researching these options, it doesn't matter what they are. They could be reading the dictionary or watching paint dry. Whatever it is, you're going to choose the outcome that's best for you. Big idea number three, a player has a dominant strategy if one choice always results in the better outcome, no matter what the other player chooses. So for each player, their dominant strategy is either choice A or choice B, or they don't have a dominant strategy. Meaning sometimes they choose A, and sometimes it's better to choose B. Big idea number four, to figure out a player's dominant strategy, pretend the other player makes the first move. So if I want to find out my dominant strategy, I see what happens if you choose first. So if you choose A, I cover up B for you because that's off the table. Now I just choose the outcome that's best for me. For example, let's just say it's choice A. Now that doesn't mean that that's my dominant strategy, it just means that when you choose A, my best move is to also choose A. To see if I have a dominant strategy, I have to see what I would prefer if you choose B. So if you choose B, and choice A is off the table, now I can decide which outcome I prefer. If that best outcome for me is choice A, then choosing A is my dominant strategy. I'm going to do it every single time, regardless of what you do. But if my best choice was B, then I don't have a dominant strategy. Sometimes I prefer A, and other times I prefer B. Okay, now let's practice. I'm going to give you three different matrixes. The numbers in the boxes represent each player's daily profit and the numbers on the left represent the player on the left. For each one, pause the video and figure out each player's dominant strategy, then start the video back up again and I'll go over the answers.
[3:13]Buehler? Buehler? For example number one, if you're trying to find Ferris's dominant strategy, you let Cameron make the first move. If Cameron chooses A, then Ferris will choose B, and if Cameron chooses B, then Ferris will choose B again. So B is Ferris's dominant strategy. Now we do the same thing for Cameron and let Ferris choose first. If Ferris chooses A, then Cameron's going to choose A. If Ferris chooses B, then Cameron's going to choose A again. So choice A is Cameron's dominant strategy. Buehler? For example number two, if we're trying to find the dominant strategy for Bill, let Ted make the first move. If Ted prices high, then Bill's going to choose price high. If Ted prices low, then Bill's going to choose price high again. So pricing high is Bill's dominant strategy. There's no reason for Bill ever to price low, he's always going to price high. And to find the dominant strategy for Ted, pretend Bill chooses first. If Bill prices high, then Ted's going to price low. And if Bill goes low, then Ted's going to go high. Notice, sometimes Ted'll go low, sometimes he goes high, so Ted does not have a dominant strategy. How'd you do? Excellent! Okay, time for example number three. Are you ready? Oh, no. Oh, oh. No. Okay, for example three, we have two people deciding to advertise or not advertise. To figure out the dominant strategy for Shawn, we let Gus make the first move. If he advertises, then Shawn will also advertise. And if Gus chooses not to advertise, then Shawn will not advertise, so Shawn does not have a dominant strategy. And we do the same thing for Gus, let Shawn make the first move. If Shawn advertises, Gus will advertise. If Shawn does not advertise, then Gus will also not advertise. Since Gus sometimes advertises and sometimes doesn't advertise, he also does not have a dominant strategy. You don't even know what that means, do you? Leave me alone. Now notice, I made these on purpose so you can see the three different types. Either both players will have a dominant strategy, one will have a dominant strategy, or neither will have a dominant strategy. The numbers in the choices changed, but these questions are really just always the same. Whoa. Okay, if you're lost right now, stop the video, rewatch it because we're about to move forward and talk about Nash equilibrium. A Nash equilibrium is the final outcome where no player has an incentive to change their decision on their own, because switching would make them worse off. So here's the rule, there's always going to be at least one Nash equilibrium. If both players have a dominant strategy, finding the Nash equilibrium is easy, it's just the final outcome where they choose their dominant strategies. If one or both players don't have a dominant strategy, the Nash equilibrium is the final outcome where each player pursues what's best for them. So looking back at example one with Ferris and Cameron, this is the Nash equilibrium. Ferris is always going to choose B, and Cameron's always going to choose A, so they're going to end up right here. This is the final outcome. Buehler? Let's see if you can find the Nash equilibrium with examples two and three.
[6:11]Strange things are afoot at the Circle K. For example number two, we know that Bill is going to price high because that's his dominant strategy. He's never going to price low. Ted doesn't have a dominant strategy, but since he knows Bill's going to price high, he is going to price low, so right here is the Nash equilibrium. Okay, example number three is tricky because there's two Nash equilibria. In other words, there's two situations when switching would actually make you worse off. Neither player has a dominant strategy, but if Shawn decides to advertise, then Gus is also going to advertise. And if they end up there, there's no incentive for either one of them to switch because it makes them worse off. But it's the same thing if they both choose not to advertise. If they end up down here, neither player has an incentive to change on their own. Now you might be asking yourself if there's two Nash equilibria where they're actually going to end up. If they have this information, they're both going to end up advertising because that helps them both the most. But there are still two Nash equilibria. But the good news, this is kind of a unique situation. You won't see that many questions like this. Oh, sure. Okay, that's it. Let me know in the comments below if this video helped you out. If you need more help, take a look at the practice sheets in the ultimate review packet and the hundreds of questions in the ultimate exam slayer. Thanks for watching. Until next time.
