 In the celebrated example of prisoner's dilemma, we have this general scheme where the two prisoners can either cooperate or not, defect, as it's called. If they both cooperate, they get some payoff A, and if they both defect, they get a different payoff D, where A is greater than D. However, if they miscoordinate and one of them cooperates and the other defects, then the cooperator gets the lowest possible payoff and the defector gets the largest possible payoff. And that's true symmetrically here as well. And this a very well known example that has a rather counterintuitive paradoxical properties. Most games are not as conceptually confusing. Here's an example that's conceptually very clear, and these are games of pure competition. The situation here is limited to two players, where one player's payoff is exactly the complement of another player's payoff. So they always sum to some constant C. Often that constant is zero, and we call them, for that reason, zero sum games, as opposed to constant sum games. And since they do sum to zero or to a constant, we only need to remember one number, the payoff to one of the players, and we can infer the payoff to the other player from that. Here's the most simple version of it. These are games of matching pennies. So you and I each need to pick either heads or tail for the coin. If we pick the same side, either heads or tailed, I win, which means that I get a payoff of one and you have minus one. If we miscoordinate, and so I pick heads and you tailed, or the other way around, then you win. A very straightforward game of pure competition. Here's another very well-known similar game with three actions from both of us, and that's the game of rock, papers, and scissors, also known as Rochambeau. And so if we pick the same action, then it's a draw. And otherwise, there are rules for who wins. For example, if I pick rock and you paper, then you win. If I picked rock and you scissors, then I win, and so on. Again, the payoffs in both cases sum to zero. This parenthetically, this very simple children game actually has an annual competition that carries a non-trivial prize of $10,000. And it's actually a sobering thought that when we look at this trivial game then, perhaps chuck a little bit, if we actually participate in this competition, we'd actually think hard about how to play it. Here's the other extreme of a game of pure coordination or pure cooperation. In this case, all agents have exactly the same interest. In other words, their payoffs for every action vector that they take is the same. And so the utility for player i is always the same of utility for player j for every action, action vector that they choose. And so, again, here too, we only need to write in each cell of the matrix only one number because it's common to all the players. It drives home the unfortunate term non-corporary game theory that describes this dominant strand of game theory we're discussing for now. The name would suggest that these are games that describe situations that are inherently conflictual, but as we see, they apply also to games in which the interest of the players coincide. So here's a game that describes a pure cooperative situation. You and I walk toward each other on the sidewalk. We can each decide whether to go to our respective left or our respective right. And if we pick the same side, then all is good. We avoid a collision. If we don't, then we do collide and that's equally bad for both of us. Of course, in general, games will be neither purely cooperative nor purely conflictual. And here's a game that exemplifies that. This is a game that's called Battle of the Sexes. Imagine a husband and a wife who want to go out to a movie. There are two movies they could choose from. Let's say Battle of Armageddon and Flower Child, the one a violent war movie and the other is a romantic comedy. Above all, they want to go together to the movie. If they go to different movies, then they are equally unhappy. So they want to go to the same movie, but they have conflicting preferences. The wife clearly would prefer to go to Battle of Armageddon and husband, romantic as he would like to go to Flower Child. So both cooperation and competition in this game.