Hi, in this lecture what I want to do is just describe the basic Colonel Blotto game, and then give you a sense of some of the main results from the game. So here's how the game works. Colonel Blotto again is this game of strategic mismatch. I've got a bunch of troops, you've got a bunch of troops. You've got to decide where to allocate them across fronts. Whoever's got the most troops on the front is going to win. So how do we formalize that? Do it as follows, we construct a game, the game has the following assumptions. There's two players, each has T troops, so we're going to start out by assuming we've got the same number of troops. There's N fronts, so these are the places that we can put our troops. And we're going to assume that T is large relative to N, so you see these two greater than signs, that means T is a lot bigger than N. So if we have ten fronts, we may have thousands of troops. Now the action I'm going to take, my strategy in this game, is going to be to allocate my troops across those fronts. So how many troops do I put on each of those fronts? And we determine the winner of the game by just sort of counting up who wins each front. So it's the number of fronts, won. So if I have more troops on a front, I win. If you have more troops on a front, you win. That's it, Really simple game. It turns out, it's sort of a complicated game, It's complicated for the following reason, It's zero sum. So zero sum games mean that there's one person gets in this case, plus one, the winner. The other person gets minus one. So when I add those up they get zero. So there's someone who wins, someone who loses. If you're in a game where there's someone who wins, someone who loses, it's naturally gonna be really competitive. Because suppose you're at some sort of equilibrium. In that equilibrium, one player's gonna be winning, one player's gonna be losing. Which means the player who's losing is gonna try and do something different so that they don't lose. If you talk to successful business people, one thing that you' [laugh] frequently hear from them is they'll say, stay out of zero sum games. The reason you wanna stay out of zero sum games is pretty straight forward. You've gotta work really, really hard, and at best, you're gonna break even. Because somebody's gonna win, somebody's gonna lose. You'd rather be in a game that sort of positive sum where everybody can win. So zero sum games by their nature, tend to pretty darn competitive. We'll see that's gonna definitely be the case of Blotto. So again, here's the example. Let's suppose we got T's 100. So each person's got 100 troops and there's three fronts. So I'll be player one. I'm player one. I say, well, the obvious thing to do is to evenly allocate my troops across the three fronts. So I'll do 33, 33, 33. I've got one left over so I'll put that on front one. Now you're sitting there, you're thinking okay, I'm player two. You're player two and you're thinking, okay what should I do. Well, a lot of us thought that Scott did the obvious thing, which was to put a third, a third, a third. So what I'm going to do is, I'm going to sacrifice one front. I'm gonna sacrifice this third front and only put twenty troops there And I'm going to put forty troops on each of these other two fronts. And by putting forty troops on each of these other two fronts, I'm gonna win. Each of those fronts. That's what you were thinking. So you're gonna win those fronts. Whereas I'm only gonna win front number three. So the point is you're gonna win two, I'll win one so you're the winner of the game. Player two wins. So if you think about Colonel Blotto, what you see is equally allocating your troops doesn't make a lot of sense because somebody can beat it. So you need a better strategy. But here's the problem [laugh]. Any strategy can be defeated. Let's see why, let's go back. So here was the first case. I did 34, 33, 33. You did 40, 40, twenty, so you beat me. So we know that 34, 33, 33 can get beat. What about 40, 40, twenty? So suppose now it's, we're sitting, we're gonna play again. And I know, b oy, you beat me with 40, 40, twenty. And I think that you think that, well, you're just gonna keep the same strategies that worked before. Well, what I could do, is I could do, I could put 30 here so I win there, and 50 here, and then put twenty there. So I do twenty, 50, 30. So I beat you. Well how could beat my twenty, 50, 30? Well you could say, well hmm, why don't I just go 50, zero, 50 and you'd beat me? So you can see is any position can be defeated and there is no real obvious beat strategy to play. Here's another insight about Blotto. You don't need all of your troops to win. So let's go and look at this example again. You're at 40, 40, twenty. I could beat you with 62 troops. I could have 41, 21, and zero. So even though we both have 100 troops, I wouldn't need a full 100 troops to beat you. That's another sort of interesting insight [laugh]. So the thing about this Blotto game was, it's fairly complicated. And it's all about these strategic mismatches, and where you allocate your troops. Anything can be defeated, and you can beat something without even using all your troops. So what happens in this, This game, Is it an equilibrium? Is it a cycle, are we going to alternate? Is it going to be complex? Or is it going to be completely random. Who wins? Well, what you get in this is actually a mixture of these two things. It's gonna be sort of an equilibrium and random. What do I mean by that? What there's gonna be is there's gonna be equilibrium where we choose strategies randomly. And so therefore, the winner is gonna be random. So what you can think of is this, Here's the fronts, Here's front number one, here's front number two, here's front number three. If I put all my troops on front one, then I'd be taking a strategy here. Now, that's a horrible strategy, because that can be beat by just about anything. Right here in the center would mean I'm doing 33, 33, and let's say 34. So I'm sort of evenly allocating my troops. What you can show in Colonel Blotto is if I make a little. Like hexagon here . That you should just sort of randomly pick a strategy in here. If you randomly pick a strategy in here, there's nothing the other person can do to beat you. What they can do is basically have an expected value of zero. So what's gonna happen is if you play this game, both players will randomly choose strategies in that space, and they're equally likely to win. So what we're gonna get is we're gonna get equilibrium. Equilibrium is gonna be with both randomly [inaudible] strategies, but the winner is gonna be random. So if we think about Blotto, then, we can ask, is winning at Blotto skill or is it luck? Because that's something we've talked about [laugh] in the past. Well, depends on how you think of it. If you think of the number of troops as your ability, Then if you got more ability, you should have a better chance of winning because you have more troops to move around. So, if we're playing lotto with different numbers of troops and troops just represent our ability then there's, there's definitely some skill involved. Now it's also true that allocating those troops is strategic. If I could figure out how you're going to allocate your troops for some reason, then my increased in strategic ability over you, my advantage, may enable me to beat you sometimes. But if you're playing a mixed strategy, If you're playing a equilibrium strategy, then there isn't any strategic ability. We can just cross this out and it's all just going to come down to luck. So against really smart players, Blotto may end up equal numbers of troops, Blotto's probably luck. We have maybe one player who is smarter than the other, or one player with more troops than the other. Then Blotto starts becoming more skilled. But again, the interesting thing about Blotto. Anything can be beaten. You don't need all your troops, and it really comes down to, Where is that other person gonna put their troops. So, what you want to do is not be understood. You want to be confusing to the other person so you want to random off. So, it's an interesting g ame. Alright so, that's the basic Blotto. Where we're gonna go next is we're going to think a little bit more deeply about this idea of one side having an advantage and see what that means for the nature of competition. We're also gonna talk about why Blotto has become interesting again. Why there's been a renewed interest in Blotto. Remember initially Blotto was about moving troops on battle fields, and that sorta fell out of favor. We're gonna see why it's generally. Insight about strategic mismatch has a lot to say about competition generally, okay, thanks.