Hi, we're talking about Colonel Blotto game, and in this lecture, I want to do two things. I want to talk about what it means to have a troop advantage, why a troop advantage is so useful. And then, I want to extend the Blotto game to instead of thinking of just two people playing Blotto, I want to think about a lot of people Blotto. So, there's a bunch of people in reach, playing each other, sort of one on one. You see what that means in general about competition, if it's really going on as a Blotto game. So, let's talk about troop advantage first. Now, it seems fairly clear that if I have more troops than you; I'm going to be at big advantage, so let's suppose there's three fronts, one two three. And I have 180 troops, and you've only got 100 troops. So, if I go 60, 60, 60 and you only got a hundred troops, there is not much you can do. If you go 50, 50, 0, you'll lose all three of them. So, the only thing you can do to possibly win one would be to say, well, I'm going to put 70 troops on this front but then you only got 30 left over. So, if you do 70, 30, 0, you are still only going to win one. So, what you see is if I got more troops, I'm at a big advantage. Well, let's think of that a little more deeply there. So, here I've done it with only three fronts. What if I increase the number of fronts? Is it, is it, having more troops still as useful? Well, here's a result you can prove that's sort of interesting. As the number of fronts increases, so as I add more fronts; if I want to guarantee victory, if I've got the one that's got more troops, I need relatively larger allocation of troops. Another way to put this is this. The advantage of having more troops decreases as we add more fronts. Let's see why that's the case through some examples. So, let's suppose that there's three fronts. I've got 150 troops. I'm player A. And player B has 96 troops. So, player A can just say, I'm just going to go 50, 50, 50 across the three fronts, and there's nothing player B can do because player B, at best, can do 48, 48, 0 and can't possibly win. But suppose it's the case that we now have five fronts, so increase the number of fronts from three to five. Well, now player A can do 30, 30, 30, 30, 30, if they do an even allocation across the, those three things. The player B can do 32, 32, 32, 0, 0, and that means player B is going to win, all three of these fronts and win the game. So, what we see, which is interesting is that, when there's only three fronts, a 150 to 96 advantage is enough to guarantee victory, but when there's five fronts, it's not enough. Hence this claim, as the number of fronts increases, the country needs a larger relative resource advantage to guarantee victory. Now, there's another insight that comes from this. So if there's, if we've got three fronts, and I'm going to lose. But with five fronts, I can win, if I'm player B. Then, it seems like if I'm at a resource disadvantage, what I want to do is I want to create more fronts. And in fact, that's a novel inside that comes from our model, which is really interesting. So, we take this very simple Colonel Blotto model, and what we see is if you're the weaker player, so if I have less ability, what do I want do? I want to add dimensions. I want to add fronts. How would that play out in practice? Well, it'd mean I'd add new dimensions to the competition. So, for the sporting event, I may add trick plays. If I'm competing in the business world, I may add a new dimension to the product. So if I'm lower, my product's lower quality than your product, I may try and add color or packaging, or some other dimension that I can add to it, in order to make it so when people do the comparison, mine looks better. If I'm a terrorist organization, what I want to do I want to attack in places that you'd never expect it. So, if you're the weaker player, you want to add new dimensions, and, in fact, people who study war, people who study insurgents, this is one of the key insights that they have. They say, you always find the weaker opponents are adding new dimensions. Or sort of attacking in place that weren't expected beforehand. And we see why that's true from Blotto. Because if you got very few fronts, the stronger player has a big advantage. If you got lots of fronts, the stronger player has less of an advantage. Okay, let's move on to the next thing, which is Multiple Player Blotto. So here, I've got a thing with a whole bunch of people. They're all allocating their troops, whatever, you know, whatever these troops are, their resources, across these fronts, and they're going to play each other pair-wise. So, let's suppose you've got the follow three players, player one, player two, player three, and what they've got is we've got five fronts, and we've got 20 troops. So, they've allocated their 20 troops across these five fronts, and let's see what happens. So, when player one plays player two, player two wins on these three fronts. So, player two is the winner. So, two beats one. When player two plays player three, player three wins on these fronts. So, three beats two. So, now we have the standard thing. Two beats one, three beats two. What do we think happens, when three plays one? Now, we'd expect, three should win, but, in fact, the opposite holds. Player one, wins on these three fronts, so player one wins, so we basically have one beats three, who beats two, who beats one. So, we have one of these cycles. So, what we get is that in Colonel Blotto for playing Multiplayer Blotto, we should expect to see quite a few cycles. And that's something that, again, you might not expect if you had, remember the skill luck model we did before, you might not expect to see cycles like this. The Blotto cycles would be something you'd expect to see almost all the time. In fact, and we can think of this, Blotto, if played correctly with a whole bunch of people, really just becomes a really sophisticated version of Rock, Paper, Scissors. You want to choose something that's like rock. Somebody else is going to choose paper. Somebody else is going to choose scissors. And this is going to be random who wins. Okay. So, this will be done in two weeks extension to Blotto one, changing the amount of troops and what we see is the weaker player wants to add more fronts. And the advantage of being stronger really depends on there being, you know, not as many fronts for over the troops. And then, we've seen if we go to a Multiplayer Blotto game, that we're likely to get cycles where one player beats two, two beats three, and then three can beat one or something like that, so we get these interesting cycles. We don't get sort of a consistent winner. So what Blotto does, if we have a situation, a competitive situation that looks like Blotto, we have some understanding of what the structure of winners should look like, and that's different than [INAUDIBLE] what we've seen in our other models. And that's where we'll go next. All right. Thank you.