 Hi everyone. This is Alice Gao. In a previous video, I introduced the homework dancing game and I discussed the intuition, which is that both players Alice and Bob seem to be happier if they choose to go dancing. In this video, I'm going to show you how to formalize this intuition using a solution concept called the dominant strategy equilibrium. To describe the concept dominant strategy equilibrium, let me introduce some notation first. First of all, some notations to denote strategies. We are going to use the index i to represent a player, so the strategy of player i would be denoted as sigma sub i. And then when we're talking about an equilibrium concept, it's often useful to represent the strategy of all the players except i. So we're going to use sigma sub minus i to represent the set of strategies of all the players except player i. Okay, then we need some terminologies for representing utilities. So when we're talking about utilities, all of the players' utilities are determined jointly by all of their strategies. So first of all, we need a set of strategies for all the players. This is called the strategy profile and we'll use sigma without any subscript to represent the strategy profile. So this is a set of strategies one for each player. Then we'll use big O sub i to denote the utility of agent i under the strategy profile. So for the strategy profile sigma, u sub i of sigma is the utility of agent i. If I break down sigma into more components, then you can write sigma as sigma i and sigma minus i. So the strategy of player i plus the strategy of all the other players. Now we're ready to talk about the solution concept. It's called dominant strategy equilibrium. First of all, we need to define what does it mean for one strategy to dominate another strategy for a player. So here let's consider a particular player, say player i, and let's compare two strategies sigma i and sigma i prime. So when can we say that sigma i dominates sigma i prime? We are going to compare the player's utility ui for the two strategies for sigma i and for sigma i prime with respect to some set of strategies for all the other players. So with respect to sigma minus i for the other players. Now I have two equations here. So the first equation says that for any set of strategies for the other players, the current player prefers sigma i to sigma i prime. So in other words, no matter what the other players do, I always prefer sigma i than sigma i prime. But notice here that the prefer here is a weak prefer. It's greater than or equal to that means in some situations, I might be the two might be the same to me. I might get the same utility for both of them. But in other situations, I might get a higher utility for playing sigma i than sigma i prime. So again, the first inequality says regardless of the other player's strategies, I weakly prefer sigma i to sigma i prime, i being the player i. Now the second inequality is very similar, except there are two differences. One is that this is a stricke inequality rather than a greater than or equal to. And the other difference is that the quantifier is not existential rather than universal. So the second equality can roughly be translated as this. There exists one set of strategies for the other players, such that I strictly prefer playing sigma i to playing sigma i prime. Now if I were to summarize both inequalities, that is for any set of strategies for the other players. So no matter what the other players do, I weakly prefer sigma i to sigma i prime. And for at least one situation, so for one set of strategies for the other players, I strictly prefer sigma i to sigma i prime. If both of these conditions are satisfied, then we say that for player i, strategy sigma i dominates strategy sigma i prime. So the first concept explains the relationship between two strategies for a player. So we can compare two strategies and figure out whether one dominates the other. Then next we will say that a strategy is a dominant strategy for a player if it dominates all other strategies. So essentially it is the best strategy for that player in some sense. It's possible that a player does not have a dominant strategy. Sometimes some strategies are not comparable using this concept, but in other cases a player does have a dominant strategy. So this strategy has to be better in the sense of the dominating relationship than any other strategy. Then finally, now we're ready to define a dominant strategy equilibrium. There exists a dominant strategy equilibrium if every player has a dominant strategy. So if every player has a dominant strategy, then we can take this set of dominant strategies for all the players, combine them together. That strategy profile is called a dominant strategy equilibrium. This is the concept of the dominant strategy equilibrium. Let's now apply the solution concept to the home or dancing game. Here's a question. Which of the following statements is correct about this game? Option A, home home is the only dominant strategy equilibrium. Option B, dancing dancing is the only dominant strategy equilibrium. Option C, one of the other two outcomes is the only dominant strategy equilibrium. Or maybe the game has more than one dominant strategy equilibrium. Maybe it has no dominant strategy equilibrium. Think about this yourself. Try to determine whether each player has a dominant strategy. And if each player has a dominant strategy, then you can put them together and that would be a dominant strategy equilibrium. Take a few minutes and then keep watching for the answer. The correct answer is B. The only dominant strategy equilibrium is dancing dancing. So if we are making a prediction based on the dominant strategy equilibrium concept, then we'll predict that both players will choose to go dancing. I've included the detailed explanation in a separate video. That's everything for this video. Thank you very much for watching. I will see you in the next video. Bye for now.