 Hello everyone. This is Alice Gao. In this video, I'm going to talk about our first game, which is called home or dancing and Through this game, I'm going to introduce a solution concept called dominant strategy equilibrium The story of this game is as follows. I have a best friend from grad school My friend's name is not actually Bob, but it turns out using Bob here is a convenient choice My friend introduced me to swing dancing and they dragged me along to lessons and to social dances And eventually I became pretty good at it Well, okay, maybe I should be more modest and say I became not terrible at it So in this game, we are both deciding whether we should stay at home So in that case, we won't see each other or we should go swing dancing this evening In that case, we will see each other and both of us sort of prefer going dancing than staying at home Now the problem here is that for some reason our phones are dead. We don't have more than communications So we cannot communicate with each other We have to look at this game the rules of this game and each make our own decision But our happiness is sort of depends on depends jointly on both of our decisions What I'm showing you here is a two-person normal form game. Let me explain some components of this game So in the normal form game, we have a set of players in this case We have two players and we usually call the players Representing the roles as a role player. So here Alice is the role player and Bob is the column player Now each player has a set of actions in this case We have the simplest case where each place player only has two actions Right. The reason I call Alice the role player is because her two actions are represented by the two roles the top row is the actions staying at home where the bottom row is the action going dancing and For the column player Bob, his two actions are represented by the two columns So the left column represents the action of staying at home and then the right column represents the action of going swing dancing We often talk about an outcome of a game and this game has four different outcomes Each outcome consists of one action for each player So some examples of outcomes for example home home This is one possible outcome home dancing This is another possible outcome. We have four outcome corresponding to the four possible combination of actions for the two players We also have a payoff matrix this payoff matrix specifies how happy is each player for each outcome of the game, right? So for the top left outcome where both players choose to stay at home Then Alice's utility is zero and Bob's utility is also zero For the top right outcome where Alice stays at home and Bob goes dancing because both prefer dancing a little bit So Bob is a little happier. His utility is one where Alice's utility is zero for this outcome So for every outcome, we have a tuple of two numbers The first number denotes the role players utility and the second number denotes the column players utility That's it for the rules of a normal form game Now let's talk about how should the two players play this normal form game As you might remember from the story, the players cannot communicate with each other They each choose their actions separately and then their actions jointly determine the outcome of the game So here's how you can understand the game. You can think about the players choosing their actions at the same time they're not allowed to communicate with each other before or after and They cannot observe the other players actions before they choose their own actions So another name of these games is that it's called a Simultaneous move game because players move simultaneously to solve a normal form game, we are looking for a strategy profile and By a strategy profile, I'm really just saying that we need to choose one strategy for each player and The most general form of strategy is called a mixed strategy. So what do I mean by a mixed strategy? Well, let's introduce some notation So for each player will denote the player by index i and then that player will choose a mixed strategy called sigma i So in general a mixed strategy is a probability distribution Over the actions of the player. So a mixed strategy is saying that the player can choose to Play the actions randomly, but this randomness is based on a fixed distribution so for example a Player in this game can decide to stay at home with 80% chance and then going dancing with 20% chance This means that every time the player is playing this game. They need to Choose a random number and based on that random number from 0 to 1 based on the threshold Determine whether they stay at home or whether they go dancing and in the long term if they play this game enough times They will roughly end up staying at home 80% of the time and then going dancing 20% of the time Okay, so again in general a mixed strategy is a distribution probability distribution over the action it allows a player to play a game probabilistically rather than Deterministically Now there's a special case a special case of a mixed strategy is called a pure strategy For this special case it just says that the player will stick to one action So this is also a probability distribution except the distribution Plays all of its probability on one action. So the player is going to play one action for sure For the first few games before the matching quarters game. We will only Consider pure strategies for the players All right now that I've described what this game representation means Let's think about how we would play this game if you or Alice or Bob How would you play would you stay at home or would you go dancing? Think about this for a second and then keep watching. I Can't really ask you this question through the video and show you the distribution of the answers But here's my prediction. I predict most of you would choose to go dancing If you're either player so the most likely outcome is probably dancing dancing And this really makes a lot of sense to me because if you look at the payoff matrix Some numbers will stand out right these numbers two and two would stand out very much because they are the biggest Numbers in this payoff matrix if both players can get their biggest numbers then why not? That seems like a good choice to maximize their utility and make them happy, right? So this is indeed the right intuition and in the next video I'm going to formalize this intuition by Introducing the solution concept called dominant strategy equilibrium by using the solution concept We can predict that the two players will both goes choose to go dancing and the outcome of the game Would be dancing dancing That's everything for this video. Thank you for watching. I will see you in the next video. Bye for now