The Game of Risk - Numberphile

one of the games I love playing with my kids is risk risk is a game about trying to take over the world in this case I've got two people playing two armies the blue Army and the white Army each time you will get more people that you can place on the countries um and the number of countries you have generate more uh armies for you to lay down if you control a continent then you also get more armies so if you actually control the whole of Europe like white does you generate um five extra pieces to be able to lay down on your next go um Asia which is a very big area generates seven um America you see blue controls the American side you actually generate five when you're laying out and trying to claim countries at the very beginning of the game which country should you go for which continents should you try and take over Brady you're from Australia so Australia is an interesting case because here the continent only has four different countries so it's very quick you can control that but it doesn't generate very much because australasia only generates two armies per turn although you can get it quite quickly doesn't generate much whilst Asia will take quite a long time to just fill those and you might be competing with another player so to control that whole continent you'll be rewarded with seven armies per turn one important thing here is to use a little bit of mathematics which is the mathematics of topology because actually it doesn't really matter what the map of the world looks like it's how the map is connected because you'll be able to try and Conquer another country on your turn if there's a border between a country you're on and a country your opponent's on so let's take Iceland so the white Army on their turn could try and take over the the blue Army on Greenland because there's a connection here but obviously I can only compete in this case against countries that I'm connected to so one of the important things is how are these continents and countries connected both America and Europe if you control those continents they both generate Five Armies if you control them but does one of these have an advantage to to occupy and I did a little new version of the map of the world and this is a a topological map which just shows the connections between the countries what's interesting is that actually Europe is connected to a lot of under continents so there are a lot of ways that Europe can be attacked so Europe is actually much more vulnerable than America which turns out actually to have um three from this side and actually there's another side because you can be attacked from from Oh Yes actually I've got this in because it's Asia can attack so kind of interestingly that's the beauty of a topology you don't have to worry about the layout it's just how they're connected so America turns out to be a better continent to go for because of the topological map map of the world which is actually difficult to see if you're just looking at the geographical map so I would suggest Asia is quite hard to occupy a lot of countries America I would say is the best place to go for I can see why that would Mak America easier to defend defend yes does it does it not make America less ideal for launching attack you 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 and defend but you've got to attack sometimes cuz you're going to try and uh take over the world but on any particular turn is it better to sit and soak up attacks or is it actually strategically better to go on the offensive well so how do you actually attack so let's suppose Greenland with three armies is going to attack Iceland with two and so you determine this with the throw of dice so Greenland gets to throw three dice for its three armies Iceland is defending with its two dice why don't you be Iceland okay uh and so uh I'm going to attack Iceland and so we both throw all of our dice so there's my three dice um of six six and a four so now you're going to defend so throw your two dice and so that's a five and a four now what you do is you peel off the top dice in each of our roles so my six beats your five so you lose an army for that one m and then we go down to the second and my six beats your four and so you lose a second Army and now I can invade but what happen suppose you got a six for example then the six would match my six and You' be safe and then this six would beat your five and I would just take one Army so you keep on playing until one of the countries is wiped out so the question is like well I've got three dice against your two but I have got to beat your dice so the question is you know okay so I've got to get more than the throws on your dice the Dynamics of this quite interesting and again this uses a a piece of mathematics which comes up again and again in playing games we see it in Snakes and Ladders we see it in uh Monopoly it's something called The markof Matrix which is the state of the armies now depends only on the state before that not you know three throws before so you can actually use a matrix which records the probability that given I've got three armies and you've got two um what's the probability we're going to go down to me with three and you with one me with three and you with zero me with two and you with two and so this Matrix records all the different possibilities of what the armies might look like after the throw of the dice but the first analysis of this turned out to be wrong because people didn't realize that there a dependence in the way the dice are being thrown very often when you're working out probability you know if I throw one dice it somehow is not going to affect the throw of the other dice so they're independent so the chance of getting two sixes well there one and a six chance that I get a six with one dice and one over six so the combined chance of me getting uh two sixes you multiply the probabilities together and so it's a one in a 36 chance of getting two sixes so that's clear so you think these things are completely independent but actually when you're comparing dice there be begins to be a dependence in working out the probabilities and this had been missed this dependence so um dependence in probability is very important and so the first analysis of the mathematics of risk said actually it's an advantage to defend that you're more likely to hold on to your country if you're defending but this misss that actually there's a dependence in the dice because if the second dice here is a five that means the top dice must have been a five or a six so the second dice actually influences what must have been in the top Dice and this was kind of missed if you got a one here the top dice can be very many things if it's a five the top dice only has two possibilities so this creates a dependence when you're comparing Dice and so when the mathematics was calculated with this dependence put inside the transition Matrix turns out now that you're an advantage in this game if you actually attack other countries this has actually changed the Dynamics of the game and and it's interesting that this kind of dependence of probabilities had been missed in this game because you're comparing guys they're not independent I get emailed about risk quite a lot and I've never played it and never known what the game is now that you've shown me I understand why I get lots of emails it Le seems very methy what what I think is um interesting is the um when this game was developed and I think it was developed by um a French filmmaker there would have been a lot of kind of beta testing because uh you want to work out well you know the topology of the map will be important you can find that one country will always be winning I think actually it's always uh you find austral Asia because of its isolation and the small number is never a good continent to sort of occupy the there are some dominant ones which are Asia Europe and America so it's it's really a sort of three-player game with those we're quite happy being safe down there don't yeah that's true yeah exactly but I think this subtlety and it you know took a uh actually a long time for the mathematics to really be teased out um shows that this is a beautifully balanced game because whether you're aggressive or defensive um it was really unclear which one was was going to be the optimal strategy but um now the mathematics seems to say that being offensive is the best way to win this game what's your playing style when you play Risk well now it's being influenced by the mathematics and that's I think what's intriguing is that you know because I love playing games and I love unpicking the mathematics that hides behind that game um it will often influence how I play so another game I love is Ticket to Ride which is about um uh laying down uh railway tracks across America America and and you get given challenges you know should you do a long one from New York to Los Angeles or or maybe a short one and again the Dynamics of that game rewards risk so of course a long trying to claim a long track um will require a lot more sort of gaming however the rewards if you can do that are such that you will probably win the game so again my analysis shows that I want to take the risks and not go for the short ones but go for the big long ones risk is one of many games well in fact 80 games covered in Marcus's latest book Around the World in 80 games find out more information from the link in the video description you can also support number file on patreon you see a list of some of our supporters on screen at the moment there's often little bonus things going on in fact hundreds and hundreds of our patrons recently received a unique prime number in the post just like this it's a great way to support our project again there'll there a link in the description if you're interested

Channel: Numberphile

Views: 926,443

Keywords: numberphile

Id: RdooKXXcWWc

Length: 10min 31sec (631 seconds)

Published: Tue Feb 06 2024