The Game of Risk - Numberphile

One of the

game

s I love to play with my children is

risk

. Risk is a

game

about trying to conquer the world. In this case, I have two people playing with two armies, the blue army and the white army, each time you will get more people. you can place in countries and the number of countries you have generates more armies for you to establish. If you control a continent, you'll also get more armies, so if you actually control all of Europe as the target, you'll generate five. additional pieces so you can place them in your next step. um Asia, which is a very large area, generates seven. um America, you see, blue controls the American side.

You actually generate five when you're designing and trying to claim countries from the start. of the game, which country should you go to, which continents should you try to take over Brady, you are from Australia, so Australia is an interesting case because here the continent only has four different countries, so it is very fast, you can control it, but it's not like that. It doesn't generate much because Australasia only generates two armies per turn, although you can get it pretty quickly, it doesn't generate much, while Asia will take a long time to fill them up and you may have to compete with another player to control that entire continent.

You will be rewarded with seven armies per turn. One important thing here is to use a little math, which is the math of topology, because it doesn't really matter what the world map looks like, but rather how the map is connected. because you will be able to try to conquer another country on your turn if there is a border between a country you are in and a country your opponent is in, so let's take Iceland so that the white army, on its turn, can try to take over the country. Blue army in Greenland because there is a connection here, but obviously I can only compete in this case against countries that I am connected to, so one of the important things is how these continents and countries are connected, both in America and in Europe, yes You control those continents.

Both spawn Five Armies if you control them, but does one of these have the occupy advantage? I made a small new version of the world map and this is a topological map that only shows the connections between countries. The interesting thing is that Europe is actually connected to many subcontinents, so there are many ways Europe can be attacked, so Europe is actually much more vulnerable than the United States, which turns out it actually has three of this side and actually there's another side because you can be attacked from Oh yeah, I actually have this because Asia can attack in such an interesting way, that's the beauty of a topology, you don't have to worry about the design, it's just how they are connected, so America turns out to be It would be a better continent because of the topological map of the world, which is actually hard to see if you just look at the geographical map, so it would suggest that Asia is quite difficult to occupy many countries.

America, I would say. It is the best place to go. I can see why that would make America easier to defend defend yeah, doesn't it? Doesn't that make the United States less ideal for launching an attack? You picked up a really interesting question about

risk

for many years, it was thought that It was easier to win this game if you just sit back and defend, but sometimes you have to attack because you're going to try to take over the world, but on any particular turn, Is it better to sit and absorb the attacks or not? Actually, strategically it is better to go on the offensive, so how do you actually attack?

Suppose that Greenland with three armies is going to attack Iceland with two and then you determine this by rolling dice so that Greenland can roll three dice for its three armies. Iceland is defending with its two dice, why aren't you Iceland? Okay, uh, so, uh, I'm going to attack Iceland, so we both roll all our dice, so there's my three dice, um six, six and four, so now you're going to defend, so roll your two dice and that's a five and a four. Now what you do is remove the top dice in each of our roles so that my six beats your five and so you lose an army for that m and then we go down to the second and my six beats your four, so you lose a second army and now I can invade, but what happens?

Suppose you have a six, for example, then the six would equal my six and you'll be safe and then this. six would beat your five and I would just take one army to keep you playing until one of the countries is eliminated, so the question is, well, I have three dice against your two, but I have to beat your dice so that the question It is, you know? Okay, so I have to get more than your dice rolls. The dynamics of this are quite interesting and again, this uses a part of mathematics that appears again and again in games, we see it in Snakes and Ladders.

Look at it in Monopoly, it's something called The markof Matrix, which is the state of armies, now it depends only on the previous state, you don't know three throws before, so you can use a matrix that records the probability that given has three . armies and you have two um what is the probability that we go down to me with three and you with one I with three and you with zero I with two and you with two and then this Matrix records all the different possibilities of what the armies after the roll of the dice, but the first analysis of this turned out to be wrong because people did not realize that there is a dependence on the way the dice are rolled very often when calculating probability.

You know, if I roll one die, it will somehow not affect the roll of the other dice, so they are independent, so the probability of rolling two sixes is one and six chances of me rolling a six with one die and one more . six, then the combined chance of you getting two sixes is multiplied together, so it's a one in 36 chance of getting two sixes, so it's clear, so you think these things are completely independent, but in reality, When you compare dice, there begins to be a dependence when calculating the probabilities and this dependence had been overlooked, so the dependence on the probability is very important and that is why the first analysis of the mathematics of risk said that in reality It is an advantage to defend that you are more likely to keep your country passing if you are defending, but this overlooks that there is actually a dependency on the dice because if the second die here is a five, that means that the top die must have been a five or a six, so the second die actually influences what should I've been into top dice and this was overlooked if you rolled a one here, top dice can be many things if it's a five, the top die only has two possibilities, so this creates a dependency when you compare dice etc.

When the math was calculated with this dependency put inside the transition matrix, it turns out that you now have an advantage in this game if you actually attack other countries, this has actually changed the dynamics of the game and it is interesting that this type of dependency of Odds were lost in this game because you're comparing guys, they're not independent. I get a lot of emails about risk and I've never played it and never knew what the game is. Now that you've shown it to me, I understand why. I receive a lot of emails. It seems very confusing.

What I think is interesting is that when this game was developed and I think it was developed by a French filmmaker, there would have been a lot of kinds of beta testing because, uh, you. I want to solve well, you know the topology of the map will be important. You may find that one country will always win. I think it's actually always Southern Asia because of its isolation and the small number is never a good continent to classify. of occupation, there are some dominant ones which are Asia, Europe and America, so it's really a kind of three player game that we're pretty happy to be safe down there, no, yes, that's true, yes, exactly , but I think this subtlety and you know, it actually took a long time for the math to really figure out, shows that this is a wonderfully balanced game because whether you're aggressive or defensive, it wasn't really clear what the strategy was going to be. optimal. but now the math seems to say that being offensive is the best way to win this game.

What is your playing style when you play? Risk, well, it's now being influenced by mathematics and I think the intriguing thing is that you know because I love playing. and I love discovering the mathematics behind that game. It will often influence how I play, so another game that I love is Ticket to Ride, which is about, um, setting up railroad tracks all over the United States, United States, and they give you challenges that you know you should. you do a long one from New York to Los Angeles or maybe a short one and again the dynamics of that game risk the rewards, so of course trying to claim a long track will require a lot more type of gameplay, however the rewards do. what you can do is such that you will probably win the game, so again my analysis shows that I want to take risks and not go for the short ones but for the big ones.

Risk is one of many games, well, in fact, 80 games. Covered in Marcus' latest book Around the World in 80 Games learn more at the link in the video description. You can also support the number archive on Patreon. You see a list of some of our followers on the screen right now. There are often little extra things going on. In fact, hundreds and hundreds of our backers recently received a unique prime number in a post like this. It's a great way to support our project again. There will be a link in the description if you are interested.