The Game of Risk - Numberphile

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We've missed two huge strategic ideas here, both related to the number of players. The more players there are, the more difficult it will to conquer and hold a large continent. You're more likely to oppose multiple players.
If you take Australia, you're likely to be the first one with a continent, because it's small and remote, and there's only one player bordering you so you can devote 100% of your army to that front. No one ever has reason to go through your territory to get anywhere else.
Secondly, attacking does give you an advantage in that battle, but attacking is always a negative sum proposition, the best case is you lose no troops, but you are still dividing the same number of troops between more territories. So the real result is that the attacker will be better than the defender on average, but the 3rd-xth players will all be better than either the attacker of defender in the short term. Even if you attack and conquer a new territory, the bystander players will then have an army advantage to contest your new continent before you receive any benefits.
This is all without thinking about the politics of conquering territories and making yourself appear strong.
There is another unfortunate factor which is cards, which depending on your ruleset, possibly/probably render all other strategies irrelevant.

His kids: „Maybe today we can beat dad!“
Him: „So the mathematics of topology…“

Its funny that Marcus points out "Australasia is a weak position to hold because of its isolation." When I played with friends back in high school, we joked about "The Great Wall of Indonesia" strategy where someone would take Australasia and build a massive army in Indonesia that was super hard to get past, eventually Zerg rushing Asia if they were unchecked for long enough. Granted, if someone else took Asia you kinda get steamrolled, but it was such a nightmare to build up enough to finally breach the wall while also trying to fend of whoever had Europe. It almost always came down to someone taking Asia, trying to stay neutral in material with Australiasia, then once they took Europe they point to Indonesia and go "YOU IN PARTICULAR"

This video is beautifully shot. I also really enjoyed the stop motion!

Risk: the only reason any of us know what Kamchatka is.

One small correction (unless the rule has changed): the attacking player is not committed to seeing an attack all the way to a decisive conclusion; he can break off any time he wishes, which comes in handy if he sees that his attack is failing.

3:58 Greenland can only throw two dice because they have to keep one army back to occupy Greenland

Great video, but I can't leave without giving praise to your editor who put together those animations/stop motion. They were FANTASTIC. Clearly a ton of time went into them. And they helped visualize and understand the topic. Keep it up!

Math(s) is cool and all, but that stop motion cotton explosion was 🔥

In my experience, turtling is usually the best strategy. Hold a continent and keep conquering one country per turn to keep gaining cards. Doesn't matter if you hold that country or not, in fact, it could be better to lose it so you can re-conquer it the next turn. Build up armies from continent points, build up armies from trade-ins, and expand when the time is right. When you can take out an opponent in one turn, do it, gain their cards, and trade in for more armies

An important factor is the multiplayer dynamics. You grow an advantage over an opponent by attacking them a lot in your turn, but you'd be making *both* of you weaker against a 3rd player.
Ideally you want everyone else to destroy their armies fighting each-other and allow you to have large forces remaining to sweep the board.

My dad and I wrote a Monte Carlo simulator for Risk when I was a kid, with entire attack chain calculators and expected remaining units. It because pretty obvious pretty quickly that attacking was better at large unit counts, but the last couple rounds where defense wins dice ties had a pretty huge influence in non late game battles. Extended attack chains also quickly lose some of the advantage since you're losing a unit every time you move out of a country.
The other super interesting thing about Risk is that you can stop attacking whenever you want, so your odds of winning a battle go up and down during the battle and you can change strategy mid turn.

Another important factor in attacking is that you get cards for a sucessul attack which you then can exchange for additional armies

The cards are quite important as well. If you do not attack, you cannot win a new territory and your turn ends without you receiving a card. If you take a territory every turn, you are guaranteed a set of matching cards after at most 5 turns. The extra armies received for completing a set increase each time and absolutely will influence the end game.

there's another game called diplomacy which i think its topology is an interesting discussion. that game doesn't have any dice or any random elements, on top of that everyone makes a move at once, so the only advantage one has is the positional advantage of the topology. it would be interesting to know the chances of winning for each starting country and how would it affect decisions.

I just looked up my father's notes from 1959. He calculated the odds correctly, so I don't know where you got the idea that anyone calculated the dice rolls wrong "previously". 3:31

the best continents to go for can also depend on the amount of players playing the game. Playing with a larger amount of players can make going for Australia/Oceania a much better strategy as it can be difficult to establish continent control early on and the extra two troops for taking Australia early can be beneficial in asserting control over the game

This game and Axis and Allies were my favorites.
Unfortunately, nobody in my life was or is willing to endure a game.

The numbers matter, but so does understanding the psychological part of the game. In classic risk with fixed cards, the psychological element is of the most importance, where technical skill and numbers matter more in other maps and modes.
Excellent video!

This is fascinating. I few years ago I became curious about whether defenders REALLY had an advantage, when it felt attackers did better. I'm awful at stats, so I set up a Monte Carlo simulation that ran 3 attackers vs 2 defenders a million times and found that attackers would win 7.8% more than defenders (ie, for 1000 defender wins, there would be 1078 attacker wins). It was a really unexpected outcome.
One of the statements I made was that a monte carlo simulation was unnecessary, as you could figure this out using stats, so I'm glad to find the challenge wasn't quite as simple as I'd originally thought!
EDIT: Well, I never thought I'd revisit that project, but turns out my code from last time was wrong. For every 1000 defender wins, there will be around 1268 attacker wins. That's a significant advantage).

The “correct” Risk strategy depends very heavily on what strategy the other players will be using. In a tournament setting where most players don’t already know each other, but can be trusted to have a sophisticated understanding of game theory and the current “state of the art” of how Risk is “best” played, then an analysis like what is presented in this video makes sense. When playing with friends/family where much is already known about expected strategy used by the others, then any number of strategies that would appear sub-optimal might be actually be best. Also, there was no discussion of how many rounds have been played (thus how many cards are in other players hands) and if someone turned in cards early, the increasing value of turning in sets of cards definitely impacts decisions on relative levels of aggression for any particular round.

I didn't get the argument of dependency @6:58. Yes, if the SECOND dice is 5, then the TOP dice must be either a 5 or 6. But why should 5 be the SECOND dice and not simply the TOP dice itself? How would this result in one dice actually influence the result in the other dice unless you've previously and arbitrarily set that a 5 must have been the top dice?

I was in a Risk club in high school, we met three times a week, and it's amazing how the strategy outlined here (and a lot of variations mentioned in the comments) became apparent just through repetitions and an ever-growing sample size. I feel really lucky that the members were inclined to discuss exactly these sorts of analyses and were more interested in the theory than the competition lolol.

3:12 So difficult to draw the topological map? I only looked briefly and found 4 mistakes. How many do you find?

One thing about Europe though is that most of the spaces threatening it are Asia, and Asia has a real hard time stabilising, so Europe can be surprisingly solid despite its many connections. (There is also the fact that until later on, often people just want to break bonuses, and that just means counting how many people you are connected to, not all the individual spaces. Typically each person will be gunning for a complete territory and both Europe and N. America have 3 of those.)
Of course, this also means relying on Asia being a bit of a mess while building up elsewhere and thatll wreck you if Asia does successfully stabilise...
Whole thing can get real complicated even before you add in the social game.

At 3:55 this is not how the game works, the attackers must send forward every attacker upon winning a tile, they also must keep at least one unit in the tile attacking from. In this case the attacker (Greenland) has three units meaning they can attack with at most two, which would also be two dice.

As a strictly mathematical analysis, this is pretty solid, but it's worth noting the player dynamics are pretty key here. The game can become very political and interaction-heavy. Truces can effectively change the topology of the board (assuming they are honored). If one player looks particularly strong, others may gang up on them. Basically, the simply fact that there are always more than two players is a major factor in determining the outcome of the game.
That said, I like to play in South America. Fast to take over, and still relatively easy to defend (two borders instead of one). Unlike Australia, it also has better mobility -- Africa and North America are both solid defendable countries, and you can move into whichever is weaker, using the bonus you've gotten faster than anyone else. Sometimes you can even win by just staking down and raising armies: players attack each other instead of you, weakening them for an endgame where you cash some cards for massive payout and obliterate. It's a very flexible little continent.

New rule: boardgames need to be reviewed by Numberphile _before_ Christmas. Thank you

The "stop motion" animation in this video is really fun

I wish that they had done an interview with an expert Risk player before this, because that was honestly pretty pedestrian. I figured that the attacking vs defending advantage for an individual battle was well known since the game came out. I remember doing brute force probability calculation problems of that in junior high, so I can't believe that the misconception lasted any significant amount of time.

I always go for South America first. It takes two stacks to defend but expanding to either North America or Africa only add one stack each.
It also helps to not get drawn into the Australia build up.

I'm sure I'm not alone when I say that as a child risk was truly my first step into the broader world of gaming... I still remember the day my grandmother bought it for me for my birthday.
Objectively there's lots of reasons to dislike risk, or things to complain about it, and I recognize them all as completely true, but the game will always hold a special place in my soul.

One very important factor of this game are cards, which you get for attacking countries, and if you defeat a player you get their cards. You can turn in cards to deploy a number of troops. There are two widely used risk variants, one where the card rewards are constant, and one where they increase linearly, reaching 50 after the 6th-or-so turn-in.
In the first variant, the game is very defensive and it's advantageous to hold a continent with a large troop bonus, like NA or Europe and the game not seldomly fizzles out to some kind of game-theoretic stalemate.
In the second variant, the game is super aggressive as eliminating one player can start a domino effect of turning in cards for a big army to defeat more players. It's not necessary to hold large continents at all since later in the game the bonuses are inconsequential compared to the massive card rewards. It's much more important to amass lots of troops early and get into the position to eliminate players who hold many cards.

The 1960 and 1970 boards and pieces were the Best!

The editing/stop motion in this video is so fun! I like it

The correct strategy is to play so that *you win every roll of the dice.*
This is accomplish by convincing the other players that their best move is to not attack you but to attack each other.

The stop shoot animation is 🔥

Perfect timing on this video. I had just suggested that when we get sick of Wingspan at our game nights, we could play Risk next. We hadn't had it at a game night since before COVID. 😂

In my experience starting in Australia was always the best tempo for launching a conquest of Asia, since whoever was trying to get Asia without a foothold for extra production was at Risk (heh) of losing their poorly defended flanks and thus their production. North America and South America are good starts too in my experience, but yeah it's usually pretty quickly resolves into a 3-player game (whoever came out on top in the Australia-Asia skirmishes and whoever claimed the Americas, and whoever got squeezed in the middle as the wild card, usually in Africa. That third player would rarely win, but they held a lot of power in forcing the other two to respect them or else end up getting flanked).

Risk is the reason my family doesn’t have family game night

To win the game you *must* attack on as many turns as possible. You get a card at the end of each turn during which you take a territory, and turning in sets of cards earns the player doing so progressively higher numbers of armies each time this is done. After a while the armies gained for doing this are ridiculously high, and in a long-running game this provides the majority of the armies raised during the game.

2:29 you missed a connection between southern europe and middle east region

Regardless of where the dice advantage is, attacking always weakens you, even if you lose no armies in the attack, because it forces you to split your armies across more territory. In a 3+ player game, the only player(s) advantaged by an attack is/are the one(s) not involved at all. I always found that in order to win, you have to appear strong enough for the other players to be wary of attacking you, but not dominant enough that other players must attack you anyway - at least until it's too late for them to do anything about it.

I *knew* there was a reason I loved that one scene from Katawa Shoujo where you play Risk with the student council president!

With 3, you can only roll 2 dice on attack

This video is missing two key rules - the first is that the amount of troops you get to place each turn is influenced by the amount of territories you control. It's something like, for every 3 or 4 territories, you get an extra troop spawn. The second is the use of cards which you earn through playing and can discard, which again bolster the troops you can place when using said cards. I played an online version of this game some years ago and all of the highest rated players/games with the highest rated players would be about grabbing what you could at the start - sometimes that involved contesting a continent, but that weakens both you and your opponent so usually people avoided it unless already in a strong starting position - and then you build up usually just one stack and grab as many territory as you can on your turn with a little bit of defensive measures to ensure you might keep a continent bonus or in bottlenecks/chokes to prevent territory loss. This prevents the other players from getting both continent and territory troop bonuses whilst adding it to yourself. Eventually, it just comes down to probability and "luck" for who has the largest stack and the strategy is all about who you are weakening on your turn - even at the start - and that continent bonuses are luxuries you should not prioritise keeping but to prioritise removing. The reason that Australia is bad is the same as in the video - it's far from everything else - but I think the way to get to that conclusion is different. In a casual game of Risk, Australia is probably going to win the most games because players will be more cautious with their armies and the land they own.

I would love to see Numberphile videos covering some more recent board games!

Back in the mid/late 1980s, I wrote a program to brute-force compute whether 3-dice attacking was better than 2-dice defending. It ran every permutation and then sorted the dice from low to high and compared. In AppleBasic, this took minutes to run. Eventually, it showed attacking was better, though I had a hidden bug for a few weeks that claimed defending was better. Doh!

I was taught to start in Australia, but more recently I have been leaning towards South America. It's almost as easy to defend, so gives you an early game boost, but it's a lot easier to move on to a new continent from there.

The +1 strategy is my tried and true method. Hold either Australia or South America and then a "buffer" country. For Australia, this means holding Siam. For SA, this means holding Central America and North Africa.
Then you can (1) slowly stack armies on these "buffer" countries to provide adequate defense/prepare for assaults and/or (2) attack a border country once per turn to earn a card.
After a few turns, assuming no opponent breaks through your "buffer", your "buffer" countries will have enough armies due to the +2 armies compared to opponents, plus any armies you get from cards (4/6/8/10, or the double multiplier). You do this until you can creep into claiming other continents, knock out another player (and claim their cards), and/or launch longer assaults. If/when done right, when you math out the attacks, and if/when the dice favor you, you can knock out two players during your turn, claim 2-8 cards, and claim nearly half the board.

Maybe we were doing it wrong but we always tried to dash for Australia in the beginning because once you had it you could just park a big army in the one country and leave the others empty. Conquering the other continents is always a longer term campaign in which the 2 extra armies every term come in very handy

Your topological map is colorblind friendly! Such a simple solution that's accessible for everyone! Thank you!

My dad is a stubborn Australia guy, I used to be a South America guy, but these days I feel like North America is the best reinforcements to entry points ratio, and combined with diplomacy and cards has always worked out well for me.

Afghanistan in game migrated about 1000km north-west. this mountainous land suddenly gained a coastline almost looks like they mixed it up with Kasakhstan.

More episodes about more boardgames would be great!
Did you know you can make a complete Turing Machine in Magic the Gathering?

Nitpicking. 1. The attacking territory always leaves one troop behind, so only two of the blue from Greenland would attack Iceland. 2. In terms of overall strategy it makes sense to attack on your turn because conquering a territory gives you a territory card and making sets of these will give you bonus troops when you turn them in. You don;'t get a card if you don't conquer and you don't conquer if you don't attack..

Love the stop-motion animations! They're really fun :)

I never had friends to play Risk, so there was no Risk involved. It was pure DOMINATION.

Sp you're basically saying that among millions of people plaing Risk through the years, not a single one of them was clever enough to put together an Excel sheet to check all those dice throws possibilities and find out offense wins overall?

9:15 In Ticket To Ride, if you manage to get a long route completed, you're very likely also the player with the longest route which nets you another 10 points, so going for at least one long route is always worth it.

Love the stop motion animations! I also love Ticket to Ride

I played this game so much as a kid 🥲 the nostalgia i feel !!

It started with him messing up the rules of combat, and then claiming that everyone thought the defender has advantage (which if you’ve played the game more than once you’d instantly recognise as ridiculous), but saying that taking Australia is the worst choice convinced me he’s not actually played the game.
In the couple hundred games I’ve played (physically and online), the Australia Turtle strat won the vast majority of the games it got used in, even when other players tried to unite against it.

In my understanding risk is most and foremost a game about manipulating other players into doing things in your favour and less about the technical things. Because of the secret missions everyone has, people tend to believe others are only playing against them, so games get very emotional quite often. There are also missions which want you to beat a specific player only. The best manipulator on the table will most likely win.

The problem with Risk is that you always have somebody (or multiple people) who do things against their own best interest, and go to war with someone in a way which only hurts both players.

I've used America for decades to win at Risk. It's easy to defend and you can attack multiple continents from there. I always got a kick out of people that used the Australia strategy.

Okay, cool. Now do a mathematical breakdown of "Escape From Colditz".
I loved playing both games as a kid. I preferred the old style Risk army pieces to the modern ones.

what dependency was forgotten? 06:48, why does dice 1 influence 2? I am sure it was not explained yet in the video ...

And to think that I only came here to watch "301 views"

Great video! Please make another one about Ticket To Ride!

Defense has about a 1/3 advantage in roll outcome but the offense has 50% more dice. If it's one on one it always makes sense to attack as long as you have at least 4. This is why even 2 player risk has the "allied powers" as the real strategy is in balancing multiple opponents.

One big advantage of Australia is that it has no backdoors, in addition to netting you a continent bonus. You can amass an army on the one tile facing the world, likely outpacing anyone in Asia who has to defend multiple fronts. Then, you can safely attack with your entire army when it suits you, and safely back down if you get unlucky dice rolls.
I do agree on the notion that North America gives the best bonus compared to territories to defend (assuming the ones behind your line of defense only requires 1 army each. It does however have entrance from 3 different continents, which makes it likely that 3 different players will want to breach your defense on all sides, and hard to hold from the start.
If two players are sitting and amassing armies (turtling), the one with the best army gain will obviously be at an advantage. Australia could easily get islandlocked by whoever is in South-East Asia if that player does well.

The best strategy is Australia first. There is only 1 way in, small amount to control, and you can use your other spread out forces to bunker down and just prevent someone else from taking a continent until your Australia forces can slowly expand.
Another reason why America is so bad to take is that Risk isn’t a perfect world, you’ll lose spots and gain them back the next turn or two. So when Europe takes Greenland, you have to spread your troops across 3 different areas to defend from further invasion. While in South America it is normally only 1 or 2 areas you have to defend past the entry point so you can spend less troops to defend if an area falls

I always lose a friendship playing Risk.
Totally worth it.

Thanks!

You cannot earn Risk Cards without attacking. So a strictly defensive policy will lead to inevitable defeat.

I loved Risk as a kid, but sold my copy after discovering more modern similar games, that are usually more strategic, less luck dependent and waaay faster. I recommend Kemet, Cyclades, Rising Sun or Root.

There is a great, more advanced online version called "Warzone" (previously known as "Warlight"). Has a very nice community that is very active even though the game is over a decade old.

How much work went into the stop motion animation at 7:34? It's very pretty.

my man explains risk so beautifully that I was actually gonna think that it’s a good board game (it isn’t). It’s a classic for sure, it’s very light complexitywise, it’s easy to teach but it’s so dated, so dated that there are many other and better board games with the same mechanics(territory control is a mechanic).

I fondly remember playing this when I was younger. I’d forgotten about it. I think I’ll buy it to play with my kids.

In the version I played (might be homerule) the defender can choose to defend with one or two dice. This made the game hugely defender favoured because you can choose how many units do you risk, depending on the chance of beating the attackers roll.

I think everyone else has hit on a lot of the key decisions of risk, but I wanted to summarize a few I have found to be helpful.
1- you should always conquer at least 1 territory per turn. The card bonuses are huge.
2- Australia is better than South America in the early came because there is only one entry point. South America is better for mid game, because you can attack through North America (which is relatively easy to defend)
3- North America is better than Europe because it has fewer areas to defend while giving the same bonus.
4- you should almost always reinforce your troops so that areas facing the enemy have the most troops, and rear areas have a single troop. Not 100% sure on the math, but I suspect having more troops in the attacking country(border country) is better than having (2 ) troops on every interior country. Goes back to attackers having the advantage.
5- you can defend Asia easier by placing your troops in Eastern Europe. You can defend from one country instead of 2(Afghanistan/irkurts?). It acts as a natural choke point .
6- depending on how your allocate troops at the beginning, North America and South America is where you should focus. You can control the borders of these two continents from only three territories (Brazil, Greenland and Alaska). 7 bonus troops for only 3 entry points is one of(if not the best) in the game.
7- I’ve found Africa to be the hardest to control outside of Asia. It tends to get attacked from all sides, and does not have enough bonus troops to make it worth the trouble. I only ever go for it after already securing NA and SA.

I've played quite a bit of Risk in my day, mostly in the form of Risk II on PC, which is a great way to crank out a lot of games quickly (and with as many players as you want). Risk II also has additional optional territories and connections that can change up the game, as well as Same Time mode, which only allows for attacking one space into enemy territory at each border, which dramatically changes how the game is played.

First: the dice count that can be used for a country with three armies is only supposed to be 2. Because you have to leave a single army behind, so you can only attack with 2.
Second: some of the math should also take into account the likelihood of a rival randomly appearing in a continent that you are trying to take and the likelihood that a rival will appear on a neighboring border.
Third: besides math and probability (in 3 person or higher player games) there is more importance that should be placed on the social/diplomatic component of the game. Alliances form and players play the game to not just take advantage of the math when progressing but the need to diminish enemies where the math is more in their favor at any given time. If anyone played this game just using probability to winning and defending against engagements (in a 3 or more player game) they will most likely not win. The players that incorporate the diplomatic aspect of this game will have a much greater chance of winning.

Turtling in Australia early in the game has always been the best strategy for players. You build your army up and let the other players take each other out. You then go on the offensive when someone is about to take over Asia and use the stockpile of troops you gathered while turtling. It's important to keep a significant force on the border of Australia as an anchor in case things don't go well on the attack.

I feel as if I've been waiting for this video since I subscribed five years ago.... Love it

Risk is fun because you get to explain the law of large numbers every time your friends say you're just lucky

An important part which was missed is that most common rule sets include the use of the cards. When you gain at least one territory in a turn, you gain one card. When you have a set of cards you can trade them for more armies (the number of armies depends on the rule set, if it's a progressive or set based system). You also get one bonus army for each territory you control for which you also hold the card for. So cards can be hugely influential. The attacker might mathematically be more likely to win a fight to the death (at least in the 3 vs 2 die scenario), but the extra reward of gaining at least one territory makes attacking extra attractive.

I love playing risk but I always feel like the games i play go on for too long

Risk is an interesting game. Playing the online version is so much different than the board game version. Its a lot faster online. Computers calculate if you win or loose. And you can raise extremely large armies if the game is mellow for a half hour. My favorite way to play online is the captial conquest mode. Each player chooses a territory to make their capitol. The capitol has an extra die and its super hard to take. I also like playing with the fog of war and blizzard effect. Fog of war blocks off the part of the map you aren't in. Blizzard chooses a few territories to make completely inaccessible. It can really change the game. Imagine if western africa was blocked off. Then you would have to go all the way around the map to get to south america

this guy does not know Risk. Along with other mistakes mentioned, at 4:53, you don't play until a country is wiped out, the attacker can do that OR they can stop attacking whenever they want.

In Brazil, the game is called WAR. It was 'cloned' by a local company called Grow that, to be honest, made some pretty nice board games.

I wish you showed the Markov Matrix and the probabilities

Risk is so fun. My roommate back in the day had it on the PS2, which obviated the need to set up boards and roll dice and whatnot. Damn we played a ton of games of that. Usually 4 or 5 players. There's a lot of dynamics to it, especially when you have random placing.

0:27 those were the days, but not anymore.

When I was in high school, I wrote a BASIC program to similate risk dice throws and found out the attacker had a slight advantage.

I am Indian in class 8 but I can understand these videos I think I am lucky that I able to understand thanks NUMBERPHILE and happy birthday to GH HARDY in advance ( 7 Feb )

The rules basically force you to attack on each turn, because if you don't conquer something, you don't get a card.
Also, getting Australia first is the best strategy. End of story.