Help My Game - RISK

calebb writes "RISK is a classic board game that's been around since 1959. This week, Science News posted an article titled 'Improving the Odds in RISK.' They mention that '...the chances of winning a battle are considerably more favorable for the attacker than was originally suspected.' Amazing! Risk is over 40 years old & nobody ever calculated the odds of winning a 5 vs. 5 battle!"

Comments

Interesting

[Taral]

So for equal battles, break-even is at 5 vs. 5, whereas one extra army always gets you 50%.

But there's no link to the actual paper. Anyone?

Re:Interesting

[theNote]

This article is bull.

Its says:

For instance, even when the number of attacking and defending armies is equal, the probability that the attacker ends up winning the territory is actually greater than 50 percent, provided that both sides have at least five armies each. The attacker also suffers fewer losses on average than the defender.

This means means you would be rolling 2 attacking and 2 defending dice, but ties go to the defender.
How could that possibly give the attacker an advantage?

The article also flips flops every few paragraphs as to wether it calculates equalness as being the total number of armies in a territory or the number of attcking vs. defending dice.

(This, btw, would make the largest possible number of attacking di 2)

Re:Interesting

[fehlschlag]

Back in the 80's, my cousin and I played Risk all the time, and with the newly acquired knowledge from Statistics class, calculated all the odds for attacking/defending by German rules (allowing up to 3 defender dice, whereas (I think) the US rules anly allowed 2 defenders max).

We found that even 2 defenders vs. 3 attackers gives a small greater than 50% advantage to the defender (mainly due to a tie being won by the defender). 3 defenders vs. 3 attackers was a significant advantage for the defender, statistically (not taking into account sticky dice, or beer spills on the board...).

Of course, if there are more armies involved in the 'war', the statistic advantage has more chance to take effect, so the results of the article are not so surprising.

With this defensive advantage on the dice, and other rules we had to further analyze due to lack of clarification in the skimpy rule book (such as the 'move armies not involved in the fight to an adjacent country'), our strategies evolved into very defensive ones, that made our games last up to 5 hours.

Re:Interesting

[PK_ERTW]

This means means you would be rolling 2 attacking and 2 defending dice

No it doesn't. It means you would be rolling three attacking dice and 2 defending dice. The extra attacker die is what gives the attacker an advantage.

PK

Re:Interesting

[Saige]

While IANAMathmatician, I would believe that the defender rolling only 1 die would actually be detrimental to their defense as opposed to rolling two.

By only rolling one die, whatever the result is on that die, it is guaranteed to be matched against the highest roll on the attacker's dice - and if they're rolling three dice to your one die, then that essentially gives them three chances to beat your roll. For example, if you roll a 5, then they have to get a 6 on one die to beat you - the odds of that are 91/216, or 42%

By rolling two defense dice, you decrease their chances of beating you, since they now have to win two matchups with the same three dice - they don't get three whole dice to beat each single die of yours. If you were to roll 2 fives, for example, they still have the 42% chance of winning one die roll, but what about the second? With your second five, they only have two chances to beat it with a six. The odds of that? 11/36, or 31% - a 11% improvement in the odds for you.

(For comparison, if you had a third defense die, the chances of a third five being beaten are only 16%, much better odds than the first and the second)

If you roll only one die at a time, the attacker gets the benefits - they get to focus more resources per defender, without any worries of greater loss.

Re:Interesting

[Golias]

In all my years of playing Risk, I've never seen an "isolationist" strategy win the game, except in those cases where they first conquer North America, and then somehow manage to hold it while Africa and South America are locked in dispute and everybody else is in a futile struggle to hold Europe or Asia.

If you don't have a major continent to yourself, the isolationist strategy will never win you the game. Europe and Asia have too many open borders to hold defensively without expanding. By the time you succeed at holding either one in a six-player game, you are probably already strong enough that you can expect to win the game. Australia has only one border, but rewards you with so few armies that it's just a matter of time before somebody (whoever eventually conquers asia) decided to wipe you out. South America is worse. No more armies that Australia, but with two border states to defend. Africa has three border states to guard, and whoever is playing in South America has to go through you to get to most of the rest of the world.

North America is the sweetest plum. Put up a massive force on the panama canal, and then leave South America alone. They will decide fighting you is not worth it and go attack Africa. Then split the rest of your force between holding your two northern borders, and collect 5 bonus armies per turn while picking off easy, strategically unimportant countries in Europe and Asia to get your cards. Once you are strong enough to devour South America with lots of armies left over, do so, and you will then be collecting 7 bonus armies, again with only three borders to hold. The game is pretty much over at that point. Just about the only way for the other 5 players to beat that strategy is if somebody "takes one for the team" and badly weakens themselves to make sure that the person taking North America does not succeed at holding it. Whoever steps up to do so will not win, either, so there is little incentive to do so, unless you really hate that guy. Otherwise, somebody holding Europe early in the game (which requires real ineptitude on the part of the other players) can rival North American power. I have seen the game won many times by a "Europe and Africa" empire... but most of the time, victory eventually goes to North America.

This is one of the reasons why my circle of friends eventually stopped playing Risk and moved on to other games, like Diplomacy.

Re:Interesting

[Andy_R]

Your analysis is correct if the defender rolls first, but that never happens in Risk. The throws are either simultanious, or attacker first, depending on which set of rules you got in your set. I the attacker rols first it can be better for the defender to only roll 1 die.

Consider what happens if the attacker rolls first and gets 2 or 3 sixes. The defender then only has a 1 in 6 chance of matching the 2nd 6 and therfore successfully defending with his second die.

He is better off not throwing it this time, since the attacker is likely to throw less than 2 sixes next time, which would give him better odds.

Wrong

[(trb001)]

An attacker with three or more attacking armies rolls three dice, one with two armies rolls two dice, and one with only one army rolls one die. A defender with two or more armies rolls two dice, and one with one army rolls one die.

Already, they're wrong...as an attacker, you have to have more armies than dice, i.e. you have to have 4 armies to roll 3 dice. The article already lost my confidence, every true Risk player knows this.

--trb

Re:Wrong

[ApharmdB]

Sounds to me like a misread statement. I take three or more attacking armies as three or more armies that are involved in the attack. The 4th army you refer to is not involved and is not considered an attacker. Begun, The Semantic War has.

Enhancing RISK

[AtariAmarok]

If you have multiple boards, you can add to the fun by trying "multiple world" Risk. Either place them side by side (so the Alaska connects to the Kamchata of the other board, and vice versa), or play such that the boards are "stacked" (one Egypt can attack another).

Re:Enhancing RISK

[person46]

We always player nuclear RISK. Instead of just being able to cash in your cards, you could elect to use them to nuke the country they represented. It's even better when the players aren't sober; one broken treaty and Armageddon ensues...

Re:Enhancing RISK

[dschuetz]

We always player nuclear RISK

The only game of Risk I've ever won (I'm lousy at these sorts of games) was a Nuclear variety. Rolling three sixes would slag a country such that nobody could go in, ever again. Rolling three threes killed everyone, and you had to wait six turns before you could go back in. Rolling three ones was a neutron bomb -- kills *all* of your opponent's armies in the war zone, but none of yours, and you can move right in.

We'd slagged right down through Eurasia, as well as Alaska, so we had a full-out east-vs-west battle in the one remaining non-slagged country: The middle east. We threw some "normal" nukes, built up and moved armies on both sides for six turns, threw another normal nuke, built up some more, etc., etc., until someone (me, luckily) threw a neutron bomb, and marched through the rest of the countries with little effort (since, naturally, all my opponent's armies were clustered conveniently nearby).

The craziest game of Risk I ever watched was a three-board, multi-day marathon session. I'm not sure any of the players went to class for at least three days.

Greetings Professor Falken

[Anonymous Coward]

I thought the WPOR told Professor Falken that "the only way to win was not to play". Now we are learning that the way to win is to attack, attack, attack.

Confimation

[Henry V .009]

I can confirm this. I recall one game with a friend years ago when we were both young teenagers. Early in the game we found that we were rather balanced as far as the map went. Since my friend had a triple set of armies, we simply built up our armies until we had used up all of the spare pieces. Both sides were perfectly balanced, but I attacked first.

It took a lot of dice rolling, but I wiped out his entire force in one turn suffering only 1/3 losses or thereabouts.

I was interested, so I did a calculation of the odds. Yeah, they're stacked for the attacker.

Re:Confimation

[Andy_R]

This is because of a flaw in the rules. Nowhere in the rules (at least in any version I have seen) does it actually define when a player's turn ends!

If you assume that a player can attack as many times as they like before the, the first player in a 2 player game can always gain an overwhelming advantage before the other player gets to make a move.

Risk is only close to being a balanced game if you put some limit on the length of each turn.

Another bogus game article

[AtariAmarok]

"Already, they're wrong...as an attacker, you have to have more armies than dice,"

Indeed! Chalk this one up with the chess article that refers in passing to pawns moving in an "L" pattern, or the video game history article that talks about Pac-Man breaking bigger asteroids into little asteroids.

Next, Science News will publish an article about "Improving the Odds in Clue" that will tell us about good ol Colonel Ketchup in the mud room with the mastadon-leg as the weapon.

Most amusing RISK memory

[AtariAmarok]

It was in 1988, shortly after an event involving a certain famous hockey player switching teams.

The guy who had Western US and Alberta moved one army down from Alberta into Western US as a troop transfer.

Someone commented "There goes Wayne Gretsky".

No one calculated this?

[epine]

The analysis of Risk is a trivial problem. I once wrote a small program to do exactly the same analysis for Axis und Alies, a vastly superior game IMHO. Sometimes I played A&A for 24 hours straight and then couldn't sleep because of a pivotal scenario developing, usually a land grab in Indonesia by one side or another before a fateful invasion of Tokyo by the Americans.

Then I would get up and it would take another 24 hour day, several pots of coffee, and two trips to the beer store to play it out.

Axis and Alies has more degrees of freedom than Risk. The person losing units has the choice about which units to sacrifice. Not all units can attack other units. Submarines can't attack airplanes. (They laughed at me the first time I played for thinking I could do this, no one told me I couldn't, they let me buy four subs to defend against a fighter plane outpost. Later I was reading about submarine technology in WWII and I discovered that submarines often carried surface to air mortar shells--not terribly effective though).

Because of the problem of the exact order in which each player chooses to remove their own casualties, the complete tree explodes exponentially. However, in practice, the order of removal is automatic 90% of the time, and the cases where the removal order is debated tends to come at the end of a close battle where you are more concerned about what is left behind when the roles reverse than who actually wins.

In my program, the attack and the defense both submit their roster in a predetermined removal order. You could try different removal orders, but you couldn't make the removal contingent on battle outcomes.

My program calculated the exact probability of every terminal outcome. You don't even need Markov models, that's just a view of what the final math represents. The actual algorithm is like a fertilizer spreader that tosses little chunks of probability from one bucket to another until all the probability is sorted into buckets representating terminal outcomes.

It worked out to perhaps three pages of code. Memory requirements go up roughly on the cube of the number of armies involved IIRC.

I learned an incredible amount about the strategy of A&A from this program. Moral of the story: you can never have too many grunts. The strategic problem with grunts is they move so slowly. I learned to invest in waves of grunts (esp. for Germany) at the beginning of the game, get the waves moving outward, and once the fronts were established, replenish a fixed supply of tanks as these were consumed in battle, and grunts grunts grunts with the leftovers.

For my money, A&A is the best game I've ever played at pressuring the strategists to set up confrontations with the potential to break symmetry and channeling the game down unpredictable paths, forcing everyone to adapt their goals. Except for the Eastern front, where the game was a little too close to being historically accurate. For a five person game, being stuck with Russia was a chore in the early going.

No one has calculated Risk in 50 years? Phffff.

More likely, no one interested in that kind of analysis considers Risk much worth the bother.

The Markov analysis of Monopoly I saw a few years ago in SciAm was far more interesting to me. Verdict: Never underestimate "go directly to jail" as a form of rent control.

I had to laugh when I saw the comment that "go directly to Australia" was a fundamental Risk strategy.

RISK 2210 AD

[trevorrowe]

I liked risk, but it was to much of the same. Castle risk tried to fix this, but IMHO Risk 2210 AD takes the cake.

I don't know how many of you have played this game, its a refreshing approach to risk.

New Features include:

-New pieces, mech-like units.
-5 new units (generals) that add attacking and defense bonuses (namely they roll 8-sided)
-No more changing in cards for the 'next big bonus'. Extra units awarded for how many territories you control, plus complete continent bonuses. Very well balanced, the bonus scales, so it remains very fair.
-Territories are chosen round robin style at the beginning of the game, so you wont get shafted with bad luck.
-Sea colonies and moon colonies to expand into, these are vaccant at game start.
-general cards (strategy cards that enchance generals' powers).
-Random territories nuked at the beginning of the game (significantly alters the map every game, new choke points, battle strategies, etc).
-Energy is gained through combat and territories that allows you to hire generals, purchase general cards, and more.
-Turn order is bid for each round, with a 5 round limit (faster games). Energy is used for bidding, so strategies exist around the saving of energy for turn order (imaging going last one round and then first the next!)
-more, more more, and much more!

No, I don't work for Hasbro(current game owner?), but I do really like this game. There are options for playing classic risk too. I have found a few copies at WoTC, but have heard rumors that they are out of print know, don't know. Expect a 30+ price tag, there are a lot of pieces.

Risk is Terrible

[Apreche]

The game of risk is not that terrible, but it does suck in comparison to real board games. If you've ever played the likes of Puerto Rico or El Grande you will realize the lack of strategy in risk. American board games are too much based on luck. Sure, you can calculate the odds and make better decisions, but european board games have almost no luck involved in deciding the winner.

After a game of risk the winner can not safely say they are strategically superior to the losers. In a game of Puerto Rico there is no doubt who is a better person.

So true

[Schezar]

American board games are generally garbage. I was always amazed my how my friends could enjoy a game like LIFE, where the outcome is 100% random, or Trouble, where the outcome is also 100% random (yea, you can make decisions, but there is always an obvious "best" decision, so unless you're a complete moron the game is random). Why even play the games, when you could just flip a coin and declare a winner?

Axis and Allies was a decent game, except that it's unbalanced. All players being of equal competance, the Allies WILL win. Period. If they don't, then the Axis made a fatal and/or stupid decision early on. It's not a game of strategy or skill, it's just a game of avoiding obvious errors until America lands in Normandy.

Puerto Rico, El Grande, Settlers of Catan, Entdecker, Tikal, Mexica, Java, Carcassone... Those are real games. Tactics and strategy, deterministic outcomes, and real competition. You're comparing brain power instead of comparing dice rolls.

Re:So true

[Quill_28]

Auction my friend auction.

Yes, the games is tipped to the Allies.

You simply auction off the Axis, whoever bids the lowest get the Axis the that much extra money.

Also, Russia can't attack first turn.

Game is much more even when this happens.

Re:Paws in chess

[fred_sanford]

Computer Science graduates speak no English well. nor wright [sic].

Diplomacy

[ggambett]

Doesn't anyone play Diplomacy? It's much more interesting. No dice involved, no randomness at all.

Re:The explanation is in the culture

[Schezar]

"The fun for my friends and I as far as board games go also comes mostly from the social interaction, and almost not at all from the game."

Then why even play the game? Why not just hang out and socially interact? You don't need an excuse to get together with your friends.

Risk Probability Calculator

[puusism]

I made a small web application to calculate battle odds for the board game Risk, because there were questions in our game group whether to attack or defend in certain situations. I thought I would share the address, if anyone is interested to see how various battle situations could turn up. The calculator is in the following web address:

http://db.cs.helsinki.fi/t/ipuustin/webrisk/webris k.jsp

Use of the program should be pretty straightforward: user chooses the number of attackers and defenders, checks the rules version and presses the button. The result diagram shows horizontally all possible end-states (the remaining forces in the winner's army) and vertically their probabilities.

The algorithm is exact, meaning that the result is not an approximation and thus does not vary in several battles with the same parameters. The program works in time O(n*m), where n is the number of attackers and m is the number of defenders. The program is made with Java.

All comments are welcome!