 We will now speak about max min strategies. These make a particular sense in the context of zero sum games, but actually are applicable quite to all games. What is a max min strategy? It's simply put a player's strategy that maximizes their payoff, assuming the other player is out to get them. We will concentrate primarily on the true player case here again, because when we get to zero sum games, those really make only sense in the case of two players. But keep in mind that one could define this more generally when we speak about max min strategy. So the max min strategy is a strategy that maximizes my worst case outcome, and my maximum value or safety level is that payoff that's guaranteed by the max min strategy. And here it is defined formally. The maximum strategy for player i is the strategy s1 that maximizes the minimum that the other player, remember the minus i is the player other than i, would hold a player i down to. And the maximum value is defined similarly to be the value of that maximum strategy. Now, why would we want to think about the maximum strategy? One can think of it either as simply a sort of a certain cautionary, maybe the other people will make some mistakes and they won't act in their best interest. Maybe I'm not sure exactly what their payoffs are. There are a lot of interpretations, or you can simply be paranoid about them and think that they're out to get you. And you know the saying, even the paranoid have enemies. That's the max min strategy. And just to confuse things, we'll also speak about the minimax strategy. The minimax strategy is strategy against, if you wish, the other player in the two-player game, is the strategy that minimizes their payoff on the assumption that they're trying to maximize it. And so here is the formal definition. The minimax strategy for player i is playing against the other guy, which we know by minus i, is the strategy that minimizes the maximum payoff as attempted by the other guy of the payoff to the other guy. And the minimax value is simply the value of that minimax strategy, the value to player one. Now, why would player one want to harm the other guy? Well, you could just be out to get him. That's a possibility. Or they could be playing a zero-sum game. And in a zero-sum game, hurting the other guy is tantamount to improving your own payoff. And so in the setting of zero-sum games, max min and minimax strategies make a lot of sense. And in fact, in a very famous theorem, due to John von Neumann, it proved that in a zero-sum game, by definition, we consider only two player such games, any Nashicler BM, the player receives a payoff that is equal to both his max min value and his min max value. And that means that we'll call it the value of the game, the value for player one is called the value of the game. And that means that the set of maximum strategies are really the same as set of the min-max strategies. That is, trying to improve your worst case situation is the same as trying to minimize the other guy's best case situation. And any maximum strategy profile or min-max strategy profile, because they're the same, constitute a Nashicler BM. And furthermore, those are all the Nashicler BM that exist. And so the payoff in all Nashicler BM is the same, mainly the value of the game. One way to get a concrete feel for it is graphically, and here's the game of matching pennies. This is a game where each of us chooses heads and tails some probability. And if it comes up either, if the result of our random optimization, I end up choosing head and you end up playing tail, you win, and vice versa, if I chose tail and you head. But if we both chose a head or both chose tail, I win. And so here are the payoffs. You see here the strategy spaces. This is player two is kind of increasing the probability of playing heads, and this is player one. And on this dimension, you have the value of the game of the payoff to player one. And the only Nashicler BM is for both to randomize 50-50. It's just right here. It's kind of uniquely looked by slicing the three-dimensional structure in this way. And you sort of see that it's got to be an equilibrium in the sense that player one could be moving along this curve. But if, as he does it, his payoffs would only drop. And so he's trying to maximize the value. He wouldn't do it. And conversely, player two can only traverse along this. But if he does that, the payoffs would only increase. And he's trying to minimize the value. So you get a stable point, which for obvious reasons is called a saddle point. So although there are general-purpose procedures for finding a Nashicler BM in particular in two-by-two games, we can use the max-min definition to find it directly in zero-sum games. And let's see how it happens in the game we'll call the penalty kick game. So in this game, we have a penalty kicker and a goalie. It's a zero-sum game. And the goal of the kicker is to score a goal. And the goal of the goalie is to prevent it. And let's assume that they each have two strategies. Kick to the left and kick to the right for the kicker. And jump to the left and jump to the right for the goalie. The payoffs will be the, each of those will determine a probability of the kicker scoring a goal. And we'll have that probability being the payoff, the value of the game, that is the payoff to player one. And therefore, minus the payoff to player two, namely the goalie. And so here they are. So for example, if the kicker kicks left and the goalie guesses correctly and jumped left also, then the goalie has not too bad a chance of stopping the shot, namely probability 0.4. If he jumps to the wrong side, his probability of stopping it is much lower, it's 0.2. Similarly, if the kicker decides to kick to the right, if the goalie guesses wrong, his probability is low, even lower than if he gets wrong in the other case. And if he guesses right, his probability is higher, although not quite as high as that had he guessed right in the left case. So he's better at stopping shots when the kicker kicks to the left. So how does the kicker maximize this minimum? That's what we're after. So here is the expression, right? We want the maximum value of the following. So they each have some mixed strategy of playing left and right with some probability. So S1L is a probability that the kicker kicks to the left, and S2L is a probability that the goalie jumps to the left. And as we saw, in that case, the value is 0.6. And similarly, the value of 0.8 is if we end up in this situation, 0.9 and 0.7 in these situations. So this is the expression that we somehow need to compute. So what is the minimum that the kicker should keep in mind? So the kicker says, I'm going to pick, I'm going to play my strategy S1, whatever it is. When I play it, player two is going to play S2 so as to minimize my payoff. So let me write down this entire expression. And this is simply copying it over, replacing S expressions such as S1R by 1 minus S1L. That's all it's doing. And the same for S2R. And so we have this is our expression. And player one is saying to himself, what would player two do if I were to play my strategy S1? And this is simply rearranging the terms, nothing else going on here, as a function of S2L. Because player one says, I'm going to pick S1. What would S2 be best response, namely its minimum? And so this is arranging it as a function of S2 strategy. And now all that remained is to look for the minimum of the strategy. And the minimum is taken by taking the first derivative with respect to S2, holding S1 as a constant. And so we have this expression. And then we solve for S1. And we get that S1 of L is a half. And therefore S1 of R is a half as well. And so by this maximum calculation, we see that player, the kicker, figures out that in equilibrium, they better randomize half, half between left and right. What does the goalie kind of figure out? Well, he's trying to minimize the kicker's maximum. That's the one way to look at it. He's doing the min-max strategy. And so here is the min-max strategy for player 2, just writing it down. And as before, we'll simply rewrite S1 of R as 1 minus S1 of L and so on for S2 of R. And we'll rearrange the term. This time is a function of S1 of L because player 2 is saying player 1. Player 2 says player 1 is whatever I choose, namely S2, player 1 will want to best respond, namely to maximize. So let me write it down as a function of S1's choice and now figure out what the maximum would be. Well, here's the maximum. And when you solve for L2 now, you get that the randomization in equilibrium for S2 for player 2 is a quarter and three quarters. So this illustrates how we can use the maximum theorem to actually compute the natural equilibrium in zero sum gains, at least in 2 by 2 games. In general, we can use the min-max theorem to compute the equilibrium of zero sum gain. And we do it by simply laying out a linear program that captures the game. And here it is. So U1 star is going to be the value of the game. That is, the payoff to player 1 in equilibrium. And so we're going to specify from player's 2 point of view. We could have done it the other way around also. So what player 2 is saying is simply says, for each of the actions of player 1, each action that player 1 might consider, I want to find a mixed strategy S2. So here's my mixed strategy S2. It will look at all my pure strategies K and make sure that the probability that is probability distribution over those, say sum to 1 and they're non-negative. So what I'd like to do is that the best response to my strategy by player 1 for any of these actions will never exceed this value of the game because I'm trying to minimize it. So I'm going to find the lowest U that has a property that player 1 doesn't have a profitable deviation by any of his pure strategies. So when I look at the payoff for player 2, when I play A2K and he plays A1J, that J that I'm considering right now, and I multiply the probability of in my mixed strategy playing A2K, I don't want that other player 1 to have a profitable deviation. So it's got to be that his expected payoff will be no greater than the value of U1 star. So clearly, this is a correct formulation of the game and it is a linear program. As we know, linear programs are efficiently solvable, in theory, by an interior method that is provably polynomial in practice by procedures that are worst case exponential but in practice work well.