 Hi folks, it's Matt again. So we've been looking at Nash equilibrium and understanding play and in settings where we make those kinds of predictions. And we've also been looking a little bit at dominance relationships. And now let's talk about strictly dominated strategies and iterative removal of those, which is another way to analyze a game. So when you're talking about game theory, there's many different ways that people can think about analyzing games in terms of stability, in terms of predicting what people are going to do, what logic can be applied. And this is another important way of looking at games, and it gives us some insights. So the idea when we start thinking about rationality in game theory that the basic premise here has been that players maximize their payoffs. So they're basically trying to maximize their payoffs. And again, it doesn't necessarily mean that they're just greedy payoffs could be that they are altruistic, public minded, etc. But the premise here is that there's something, some objective function that people have, and they tend to do things that'll give them higher payoffs rather than lower payoffs. Okay. So in terms of iteration on this logic, what we're going to be thinking about is what if all players know that other players maximize their payoffs and we have an idea of what the structure of the game is. Then what does that mean for the game? Can we deduce something about what should be played in the game? And what if all players know that all players know that players are rational in this sense? So you can take this, what if I know that you know that I know and so forth. You can take this at absurdum, but it's an important concept and understanding what it yields gives us some insight into games and gives us some predictability. Now, you know, going through very, very high levels of this are questionable, but nonetheless the logic here and the predictions that are made will give us some understanding of games that we can use in analyzing equilibrium and doing other things with it. So, you know, we can take this logic fully to its full logic conclusion. Okay. So in terms of strictly dominated strategies, that means a strategy which is, there's some other strategy which always does better than it. It can never be a best reply. So we'll make that clear in a second. So basically that means that if this is a strategy that never does well, there's something which does always better than this against any strategy of the other players, then basically it's never going to be played. So this is essentially a strategy we can just safely ignore if we think players are rational. They should never play a strictly dominated strategy. There's something else which does better in all circumstances for them. So we remove those from the game. And the idea of iteration is we take those out. Now we've got a simpler game. Now let's do the same thing, right? So there might be something which is now strictly dominated in this thing. So a player should never play this once we get to this reduced game. And then we take those out. And we get an even further reduced game and so forth. And we just keep iterating on that. It leaves us with some prediction. And then we think that the only thing that's logical if there's rational players and they understand that other players are rational and so forth, they're going to be left inside that sub game. Okay, so the running this process to its termination is called the iterated removal of strictly dominated strategies. So in terms of formal notation, what are we saying here? A strategy A sub i for some player i is strictly dominated by some other strategy of the same player, A prime i. If what's true, the payoff that the person gets from playing AI, the one that's strictly dominated is worse, strictly lower. Okay, it's important that this is a strict inequality. Then the payoff that they would get by playing A prime, no matter what the other players do. So this is a for all sign for every possible strategy of the other players. No matter what they do, this one AI is worse than the payoff from AI prime. Okay, so there's no circumstance in which it can do as well. It always does strictly worse. That means it's a strategy where you're just strictly better off playing A prime i. That's the concept of strict domination. Okay, so let's have an idea now of iterated removal of strictly dominated strategies. So here's a game, a three by three game. It's got a bunch of different payoffs in it. We look at it. We begin to think, okay, let's suppose you want to find the Nash equilibrium with this game. Well, it gets a little complicated because you have to if you're thinking about mixed strategies are pure and you have to consider all the possible combinations. One thing we can begin to do is look for strictly dominated strategies and just get rid of those. So for instance, in this game, what's true? If we look at this game, we notice that R is strictly dominated by C, right? So the column player gets a strictly lower payoff in every one of these entries than they get in every one of these entries. So you would be strictly better playing C than R, no matter what the other player does, whether the other player goes up, middle or down. Center always does better than R or even if the other player mixed. So whatever the other player does, you get a strictly higher payoff from the center than R. So we should just get rid of R altogether. And now we've got a simpler game, right? So the idea is, boom, we get rid of R altogether. Now we've got a simpler game. Okay, so let's iterate on that logic. So now there's no strict domination any longer for the column player because the column player is actually indifferent between left and center if the other player plays middle. But one thing we do notice here is that the middle strategy of the row player is now dominated, right? So the middle strategy does strictly worse than the up strategy for the row player, right? So three is better than one, two is better than one. No matter what happens, you're better off playing up than middle. So M is strictly dominated by U. In this case, we can get rid of M together. That collapse the game further. So now we're iterating. We've got a simpler game. Now we see that in this case, now once we've done this removal, now C is dominated by L, right? So the column player would always get better playing L in this game than, sorry, would always be better off playing C than L in this game. So the payoff is always higher playing C than playing L. So L is strictly dominated by C. We can get rid of L. Simplify the game further. You can see where this is going. Boom. We're down to a very simple game. Now if this is the game that's left, the row player is better off playing down than up. Boom. So what do we end up with? We end up with down and center being the only things that are left once we've done this full iteration. So we started with a fairly complicated game. We end up making a very simple prediction that the only thing that's left after iteratively eliminating strictly dominated strategies is down for the row player, center for the column player. That leads to a payoff of 4 and 2 for the two players. So in fact, given that a player, if we're looking for Nash equilibrium and there are things that have to be best replies, we know that they could never be playing a strictly dominated strategy. So we can rule those out. They can't actually be playing. You can convince yourself they can't be playing something that's strictly dominated in what's remaining and so forth. So the fact that we ended up with a unique prediction here actually tells us that this game has a unique Nash equilibrium and the only Nash equilibrium is for players to play down and center. So it actually in this case identifies a unique predicted play which coincides with the only Nash equilibrium of this game. So we've got the unique Nash equilibrium DC. So that worked very well in that game. Let's take a look at another game. I've slightly changed the payoffs of this matrix. Let's try again. In this case, R is still dominated. So in this case, R always leads to 0 for the column player, left or center give higher payoffs. So in this case, R is dominated by either L or C. We can get rid of R and then we can go through again. Now in this particular situation, there's something that's interesting. So now the column player is indifferent between the two. But when we're looking at the row player, we notice that the row player doesn't have any pure strategy domination relationships. So the player gets three here, four here compared to zero zero. So neither of these strictly dominate the other. They get one always by playing middle. So in this case, they sometimes do better than down. If they're playing middle, sometimes do better than up if they're playing middle. So there's no strict domination when we're looking just at pure strategies. But if players are willing to randomize, one thing to notice in this game is that let's suppose that you played a half on up and a half on down. What would your expected payoff be? So if the other player went left and you're playing half up half down, you get a payoff of 1.5. If you were doing this and the other player was playing C, you would get a half of zero and a half of four. You would get two. So there's when we look at playing a half half, then what would we end up with if we allow for that mixture, right? We put in a mixture. We would end up having, oops, how'd that happen? We would end up with 1.51 and a 2.1. So we end up here with something which strictly dominates middle. So playing a half on up and a half on down gives the row player a strictly higher payoff than they would get by playing middle. So in this case, M is dominated by the mixed strategy that selects U and D with equal probability. So in that case, we can still get rid of M. So down to a reduced game. Now this game doesn't really reduce any further. The column player is indifferent. The row player likes up better if the column player goes left, likes down better if the column player goes right. So what's going to happen in this game? Now you'd have to take a further analysis. And actually, if you want to go through and solve for the Nash equilibrium of this game, there's a lot of them, right? So there's in fact an infinite number of Nash equilibria given that the column player is fully indifferent in this game. So you can go through and analyze all the Nash equilibria if you want. But the iterative elimination of strictly dominated strategy still gave us a lot of predictive power in the sense that it collapsed the game down to a much simpler game. And then it's much easier to analyze what's left. Okay, so iterative removal of strictly dominated strategies. One nice thing about this is it preserves Nash equilibria. So you can use it if even if you were just wanting to analyze Nash equilibria, you can use it as sort of a pre-processing step, right? So before you try and compute Nash equilibria, get rid of all the strictly dominated strategies and iterate on that. Some games like the first one we looked at tend to be solvable using this technique. That's called dominant solvability if it actually collapses to a single point. You were able to solve that game just by using dominance arguments. Some games won't be, but it still could be useful to analyze these things. What about the order of removal? So, you know, we did things in a very particular order. So just noticing that the column player had won, you know, that the right play was strictly dominated and so forth. What if we started with a different player or would it make a difference? If we're dealing with strictly dominated strategies, then order doesn't matter. So no matter how you do this, whatever order you do it in, you'll end up with the same solution. You can spend some time trying to convince yourself of that. Think carefully about it. So that's something you'd have to prove. But in fact, order does not matter in eliminating strictly dominated strategies. So the logic is very tight in that respect. There's another type of domination which also makes some intuitive sense and people use in games. And that's to weaken the domination relationship. And instead of strict domination, we can consider weakly dominated strategies. What's the idea of weakly domination? It's very similar to what we had before, but instead of having the strict inequality hold everywhere, right? So instead of having this hold for all a minus i, it just has to hold sometimes and you just need a weak inequality for all strategies of the others. So the idea of a weakly dominated strategy is that it always does, a prime always does at least as well as a sometimes strictly better. So this is still a strategy you can say, okay, a prime really dominates a because it always does at least as well and sometimes does strictly as well. So if I'm uncertain at all, I might as well go with the one which always does as well and sometimes does strictly better. So weakly dominated strategies can be eliminated as well. You can go through, you can iterate, just go through games exactly like we did before, same kind of thing. But one thing that's true about weakly dominated strategies is that sometimes they could be best replies, right? So a strategy could be weakly dominated and still turn out to be a best reply. How could that happen? Let's suppose for instance, you know, we look at a very simple game where the role player can go up or down. If they go up, they get a payoff of one against left and right of the column player. And you know, here they get two, here they get three. So this would be a situation where down weakly dominates up, right? You always get at least as high a payoff and sometimes strictly higher. But nonetheless, it could be for instance that if left is the strategies actually chosen by the column player, then up is still a best reply, right? So for instance, if we put in payoffs here of one, one so that the column player is exactly indifferent between these two strategies, then this is actually a Nash equilibrium. And so eliminating that actually eliminates one of the Nash equilibrium of the game, right? So depending on what those payoffs are, we could end up eliminating a Nash equilibrium of the game. And you know, so this is a situation where, you know, we end up eliminating something which could be part of an equilibrium. What is true is at least one equilibrium is always preserved. What's unfortunate is that the order of removal can matter. So which order you remove things in can begin to matter. There are some games which it's useful to use in. So for instance, if you remember the Keynes beauty contest game that we talked about earlier where people were naming an injury between 0 and 100, think about trying to solve that via iterative elimination of weekly dominated strategies. What do you end up with? So it can still be a useful logic and that logic can help you in analyzing some games, but you do have to, it's not as tight as strict domination because there are situations where you might want to play a weekly dominated strategy. If you are sure that the other player was going to, you know, go in a certain direction. So for instance here, if we put in say 2-1, then if we eliminate the column players dominated, weekly dominated strategy first, right? So they left, weekly dominates right, we get rid of right, then what are we left with? We're left with a situation where the column player then, sorry, the row player is then indifferent between the two strategies, right? So if we sort of say, okay, look, this left dominates right, so we get rid of this, then we end up with a situation where up and left is still left. But if we removed the row players things first, we would remove up first and then we would end up with down left. So depending on how you go through this, you get different things that are left. So there are things that, you know, where the order matters and that is somewhat problematic. Okay, iterative strict and rationality, players maximize their payoffs, they don't play strictly dominated strategies, they don't play strictly dominated strategies given what remains. We iterate on that, Nash equilibria are a subset of what remains. So it's a nice, simple solution concept that helps us throw things out of the game and simplify what we're looking at. We can also ask whether or not we see such behavior in reality. Do people really act in ways that are consistent with eliminating strictly dominated strategies and moreover iterating on that process?