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Risk Strategy Manual

Postby TheChymera on Mon Sep 29, 2014 3:47 pm

Hello again, about a month ago I wrote an article about calculating Risk dice odds(link), and put together a script to help do that.

Recently, I tried to combine the knowledge from that any my gameplay experience to formulate a number of strategy guidelines (for 1v1 games - since I haven't gotten to modelling players yet :D ). Some of the guidelines are quite obvious, some others will doubtlessly be met with a decent amount of disbelief. Nonetheless everything in the article is backed by quantitaive analysis - no bogus!

You can read it here!

I'd also be very happy for comments, and I's also gladly add you as a co-author if you have any contributions ;)
Last edited by TheChymera on Sun Oct 05, 2014 3:10 am, edited 1 time in total.
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Re: Risk Strategy Manual

Postby Dukasaur on Sat Oct 04, 2014 6:44 am

TheChymera wrote:Hello again, about a month ago I wrote an article about calculating Risk dice odds(link), and put together a script to help do that.

Recently, I tried to combine the knowledge from that any my gameplay experience to formulate a number of strategy guidelines (for 1v1 games - since I haven't gotten to modelling players yet :D ). Some of the guidelines are quite obvious, some others will doubtlessly be met with a decent amount of disbelief. Nonetheless everything in the article is backed by quantitaive analysis - no bogus!

You can read it here!

I'd also be very happy for comments, and I's also gladly add you as a co-author if you have any contributions ;)

It's nice. Somewhat basic, but I like the format and the clarity of the explanations.
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Re: Risk Strategy Manual

Postby Kaskavel on Sat Oct 04, 2014 7:45 am

Very good job. Some notices from my part, though not critical ones. The majority of the text is quite accurate but as an experienced 1 vs 1 player, I would like to add some details.
a) Reinforcing a region with 1 troop is not necessarily bad, especially if the action increases you to 2 or 4 troops. Basically, if the game is under a "find a 1 to hit" situation, it makes sence to cover up your 1s. Of course, if multiple 1s from both players are exposed, then indeed, you should reinforce into an attacking stack. Basically, it comes into contradiction with your later statement of covering up 1s.
b) The 4 vs 3 versus 5 vs 3 discussion is more complex than that. Indeed, in the first rounds, you should make multiple 4 vs 3 attacks if the objectives of the game are not concentrated to a singe objective. But on the other hand, keep in mind that a 5 vs 3 attack is superior to a 4 vs 3 attack, meaning that it is better to add two troops into a 3 vs 3 situation, than adding 2 troops into a 2 vs 3 situation. And both attacks are superior to a 5 vs 2 attack. Basically, you would love to attack with equal troops or 2n extra troops (8 vs 6, 5 vs 5, 4 vs 4, 6 vs 2 etc) but you also want to get rid of your 3s, so making 4s at the start is more important. The 3s are bad. They cannot attack (but have some wasted attacking potential you have not yet exploited) and they are the worst thing that may be attacked (except 1s of course). 2s defend at the maximum. So, in the first rounds, you do indeed want to make multiple 4s. Some of those will end up 2s (because of winning or losing, doesnt matter). In an open board, you do not touch those 2s unless there is a reason. Those of them that become 3s, must be boosted into 4s or bigger numbers again. When the number of 3s diminishes, it becomes more important to make attacks at +2n, rather than making 4s blindly
c) The concept of always attacking when having 4+troops is correct in general. There is an exception few players seem to understand. First, you do not want to release the stack behind. if you have 4 vs 3,3 in a bonus boarder, you may want to attack the boardering 3, but if you win first battle, reducing it to a 1, you may or may not want to continue. Second, making such a battle in a bonus boarder, offers information to your opponent. If you attack 6 vs 5 and end up 2 vs 5, opponent knows he needs to add 1 more troop next round to break the bonus (a good 6 vs 2 attack). If you end up winning and expanding your line with 2,2, opponent knows he must either switch to other bonuses or increase his deployment considerably. The meaning is that although attacking is preferable to defending, you hide information of this battle to your opponent, which may be of equal importance, even more. In hive where multiple bonuses are around to battle for, I do not attack 4 vs 2 from the boarders of cells I posess. If I win and split 2,2 opponent will ignore me and attack elsewhere. If I lose, he will attack me. By making this single battle I offer him some info. It is like I offer him the outcome of his first "attack", which helps him decide what he needs to deploy next round. It is like he deploys next round, knowing what the outcome of the first pair of dices will be (although on the other hand, this pair of dices were rolled at a slight disadvantage for him).
d) The major disagreement I have is with the 3 vs 1 attack. It is a complex situation, since sometimes you want to make the attack, sometimes you do not.
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Re: Risk Strategy Manual

Postby Robespierre__ on Sat Oct 04, 2014 1:03 pm

Hey, Kask ...

I was always under the impression that (ignoring the offensive side), once you turn a 1 into a 2 for defense, that every additional troop has an equal contribution to the space not being successfully conquered. I had never heard/read that a 4 has a mathematical advantage over a 3 that is not equivalent to that of a 5 over a 4. Can you elaborate a little more?
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Re: Risk Strategy Manual

Postby Kaskavel on Sat Oct 04, 2014 6:51 pm

I am not certain I exactly understood what you ask, but I can probably imagine. Indeed, the 4 vs 3 does not have any advantage or disadvantage compared to the 5 vs 4, except for the fact that the second one has more "attacking advantage". In other words, the only advantage a 5 vs 4 has over a 4 vs 3 is of the same nature of the advantage a 400 vs 399 has, over a 4 vs 3. More troops in the battle, more advantage to the attacker. If there is any extra advantage or disadvantage between a 4 vs 3 and a 8 vs 7, it is probably random between random pairs and probably insignificant to the practical CC player.
That is not though what I casually mentioned in my point b, which is the point I guess created some confusion. What I wrote, is that it is favourable to attack with the difference of attackers and defenders being odd. Equal troops, +2 troops,+4 troops,+6 troops etc. This is indeed something that I am not aware any other player has ever suggested, it is my own invention and I cannot prove it, although I can try to explain it and see if it finds acceptance.
FACT The 2 is the best number in defence. A line of 2s is superior to any other distribution of defenders
FACT The 3 is the worst number in defence (after the horrible 1s of course)
ASSUMPTION It is not good to have 3s around the board. This is very close to a fact (for me, it is a fact) and I think we will agree. You do not want a bunch of 3s over the board, you want a bunch of 2s and some remaining stack. A 2,2,2,10 is better than a 3,3,3,7, both for defensive and for offensive reasons. You have bigger stack to attack and the 2s are very good in defence. If I have a 3,3,3,7 line, I will add one troop in all the 3s and the remaiming in the stack. Some of the 4s will win and split 2-2, some will lose and become 2s. All those 2s are now "fine". Those 4s that made a draw and became 3s again are problems that I will probably try to solve again next round, the same way, by making them 4s and attacking. A 3 is very bad, not only it is vunlerable as nothing else in defence, but it also contains an extra useless troop, that cannot be used for attacking purposes.

If we agree to the points above (and I think we should), here comes the idea

SUGGESTION It is better to assault with an attacking advantage (or disadvantage!) of 2n troops, rather than 2n+1 troops.

SUPPORT OF THE SUGGESTION The suggestion cannot be proved mathematically, or it is very difficult to do so. There are 2 reasons I support the idea.
ARGUMENT 1 An "odd" assault has more chances to end up in a finishing 3 vs 1 attack, which may make the difference. A 8 vs 6 attack can end up in a 3 vs 1 attack more easily than a 7 vs 6 attack. All is needed is a draw in last dice (or an odd number of loses in n vs 1), while the other one needs an even number of losses in n vs 1. I have not checked the math, someone could do for me if he likes, what chances do a 8 vs 6, a 7 vs 6 and a 9 vs 6 end in a 3 vs 1 (with attacker stopping attacking if he is left with 3 troops of course, not attacking 3 vs 3)? If the board supports making 3 vs 1 attacks and/or the battle involved is important to be won, then this 3 vs 1 attack, an "extra cheating" in fact, may make a difference.
ARGUMENT 2 Argument 1 may in fact be wrong, it may be an illusion of the mind. It may be wrong, I have never done the math. But this is the more important one. An "odd" assault cannot end up 3 vs 2. Ending the battle at 3 vs 2 is a bad bad senario. Defender is left with a wonderful 2 and you are left with a problematic 3 (as mentioned in ASSUMPTION above). Of course, one might reverse the argument and suggest that an "odd" assault cannot also end up in a 2 vs 3, which is the reversed situation. True, but a 2 vs 3 is not a significant success for the attacker for the reason that it is now defender's turn to play and he can "fix" the mistake by immediately taking his turn and either adding up many troops to the 3 to attack or adding a single troop and attack (possibly 66% solving the problem) or, if his position is defending in nature (a bonus boarder or something), add a single troop and stay with a 4 in defence (which is the next best thing to a 2 that exists). The 3 vs 2 senario is bad and the 2 vs 3 is not so good as it may seem. In "odd" battles the outcome may be 3 vs 1, which is good, 2 vs 2, which is also good leaving decisions of continuing the assault or ignoring the now "stalemated boarder" for next round and the 3 vs 3 which is bad, but not as bad as the 3 vs 2 one, because in the 3 vs 2 senario, the defender may as well ignore your problematic 3, while now, in the 3 vs 3 senario, he is forced to take some action to prevent his 3 from getting attacked.
EXAMPLE you have to make a 10 vs 7 and a 8 vs 6 attacks and you have one extra troop to place. You do not care winning the battles, just to make optimal troop exploitation in numbers. I believe you have to increase the 10 vs 7 into a 11 vs 7 "odding" both assaults. The alternative is to "even" both attacks. People tend to choose the option that equalizes the difference of the assaults, in the above example most people would bring both attacks at +3, rather than a +2 and +4...it just somehow seems more natural to do so, but I think it is wrong if it evens both differences. Be careful here, the argument is NOT that 11 vs 7 and 8 vs 6 are a mathematically superior pair to 10 vs 7 and 9 vs 6. It may be or it may be not, there should be some minor difference if someone does the math, randomly favoring the "odding" or the "evening" depending the example. The math will suggest the option that maximizes the chance to finally battle against 1s in both assaults (in cases like the above where we have superiority in numbers, you see, the example I brought is not in my favor I think) or the option that maximizes the chance to face one 1 (in cases we are below in troops overall) The argument is not that. The argument is that the odd assaults will leave the attacker in a much better situation "after the round" with no chances of getting stuck with a 3 vs 2 inferior outcome

CONCLUSION Since the argument takes into account the events of the following round, it is very difficult to prove it correct or wrong. I think many CC players have such a behavior from experience already, mainly by feeling that a 5 vs 2 attack is a "bad attack". I have heard and seen some good players avoiding those attacks, while having no problem making 5 vs 3 attacks, although they do not seem ready to explain the reasons of their behavior.

If that is true, then, although at the start of the game it is good to make many 4 vs 3 attacks to get rid of our 3s, it is not necessarily a good things to build up 4 vs 3s attacks later on from a 2 vs 3 situation. If we bother increasing a 2 vs 3 into a 4 vs 3, then we probably need to further increase it into a 5 vs 3 as well.
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Re: Risk Strategy Manual

Postby Kaskavel on Sat Oct 04, 2014 7:30 pm

A more practical example. You have a 4 vs 2 attack somewhere in order to break a small cell in hive (for example) and an extra troop to deploy somewhere. What do you gain by adding 1 more troop there, making it a 5 vs 2?
In case you lose 0-2, nothing
In case you win 2-0, nothing, except perhaps the possibility to leave a 1 behind and attack again, if the situation allows something like that
In case draw comes 1-1, you will now attack 4-1 instead of 3-1, which is something like a 1/12 troop gain.
No big deal. Especially if you are not interested in attacking again, like the hive example above. Better to use that 1 troop elsewhere. The 4-2 is superior to 5-2 in troop exploitation terms. But shouldnt this mean that the 5-3 is superior to the 6-3 as well? And the 6-4 superior to the 7-4?
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Re: Risk Strategy Manual

Postby TheChymera on Sun Oct 05, 2014 2:58 am

Hey there! many thanks for the positive feedback and for chipping in with your own ideas! Let me address a few of those:

Kask's:
(a) I indeed support reinforcing with one troop to get 4 troops on attacking territories (see my PR2). However, I am yet to find a normative method with which to compare the loss in attack potential which you get from 3-troop territories (vs 2-troop territories) with the respective gain in defense potential. In support for 3-troop territories I can only indicate that according to the compound odds table the chances to conquer a 2-troop territory with 4 troops on the attacking territory are above 50% while for 3 troops on the defending territory the chances drop below 50% (though I do not necessarily recommend you attach too much significance to the 50% figure). Not least of all 3-troop territories allow you greater flexibility, since you only need to invest 1 troop in the placement phase to get them attack-ready. A point you also address at the end of (b).

(c) I actually meant to write about this - but for some treason I thought I had addressed this elsewhere. I added that under AR3


Robespierre__, Kask:

Regarding the advantage gained by adding one more troop to the defense: it depends on both the number of troops you already have on the defending territory AND on the number of troops you could be potentially attacked by. Again, this paper provides a really nice figure (Figure 1) where you can see a sequence of sigmoid curves. Notice how they get flatter the larger the attacker number (+1 on defence counts less) and how the maximal addition value for +1 to teh defence is given by the last troop before the inversion point (where the second derivative is zero - these points are all connected by a line). So practically the advantage of 3 over 4 is not only unequal to that of 4 over 5 (and cetrainly to that of 400 over 399), but also contingent on the prospective number of attackers. Note that this is NOT the added advantage-for the attacker effect of large numbers I am talking about - though it is a consequence of it.

Kask:

"A 2,2,2,10 is better than a 3,3,3,7" - the former is better for attack, which is why I would agree that it is better, in principle. However, I was still unable to figure out any mathematical way to show which of the two is better for defense, and also to check if for instance 7,3,3,3 might also be better than 3,3,3,7. I guess this could be done via a markov chain, where winning over the next number of the tuple comes at a cost of -1. I have a strong suspicion that the best defense configuration is 10,2,2,2 - but I have no idea how to actually calculate it. Do you?
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Re: Risk Strategy Manual

Postby Kaskavel on Sun Oct 05, 2014 11:59 am

TheChymera wrote:Hey there! many thanks for the positive feedback and for chipping in with your own ideas! Let me address a few of those:

Kask's:
(a) I indeed support reinforcing with one troop to get 4 troops on attacking territories (see my PR2). However, I am yet to find a normative method with which to compare the loss in attack potential which you get from 3-troop territories (vs 2-troop territories) with the respective gain in defense potential. In support for 3-troop territories I can only indicate that according to the compound odds table the chances to conquer a 2-troop territory with 4 troops on the attacking territory are above 50% while for 3 troops on the defending territory the chances drop below 50% (though I do not necessarily recommend you attach too much significance to the 50% figure). Not least of all 3-troop territories allow you greater flexibility, since you only need to invest 1 troop in the placement phase to get them attack-ready. A point you also address at the end of (b).

(c) I actually meant to write about this - but for some treason I thought I had addressed this elsewhere. I added that under AR3


Robespierre__, Kask:

Regarding the advantage gained by adding one more troop to the defense: it depends on both the number of troops you already have on the defending territory AND on the number of troops you could be potentially attacked by. Again, this paper provides a really nice figure (Figure 1) where you can see a sequence of sigmoid curves. Notice how they get flatter the larger the attacker number (+1 on defence counts less) and how the maximal addition value for +1 to teh defence is given by the last troop before the inversion point (where the second derivative is zero - these points are all connected by a line). So practically the advantage of 3 over 4 is not only unequal to that of 4 over 5 (and cetrainly to that of 400 over 399), but also contingent on the prospective number of attackers. Note that this is NOT the added advantage-for the attacker effect of large numbers I am talking about - though it is a consequence of it.

Kask:

"A 2,2,2,10 is better than a 3,3,3,7" - the former is better for attack, which is why I would agree that it is better, in principle. However, I was still unable to figure out any mathematical way to show which of the two is better for defense, and also to check if for instance 7,3,3,3 might also be better than 3,3,3,7. I guess this could be done via a markov chain, where winning over the next number of the tuple comes at a cost of -1. I have a strong suspicion that the best defense configuration is 10,2,2,2 - but I have no idea how to actually calculate it. Do you?



a) Indeed. Comparing your odds with the potential of defence next round is problematic. Of course chances of 4 vs 2 are better of chances of 4 vs 3, I miss your point here. And indeed, 3s only need 1 troop to attack, so we need to add this troop in first opportunity to get rid of what I think is a problematic 3. To exploit this unused energy of the extra troop (from 2 to 3) and transform this 3 into a 2.

b) I need some time to read the article, but I want to be first sure we are talking about the same thing. Our question is if an attack 6 vs 5 and an attack 18 vs 17 have any difference EXCEPT from the increased attacking advantage of the second case.

c) You can put 1000 troops to defeat both chains and calculate the expected losses. They will be greater in the case of 2s. Also, while our initial discussion did not involve an order (when I said 3,3,3,7 I didnt mean them in a specific order, but rather in random places across the board, we are talking about 1 vs 1 RISK), the answer is that the 2,2,2,10 is the more tough to beat. This is irrelevant to our subject, but becomes significant important in escalating sweeps in multiplayer risk
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Re: Risk Strategy Manual

Postby Kaskavel on Sun Oct 05, 2014 12:06 pm

An irritating problem is that most statistics calculate odds for attacker's winning provided he attacks to the end. I am mostly interested in statistics in 2 specific cases
1. Attacker stops when he is left with 3 troops
2. Attacker stops when he is left with 3 troops, unless opponent has a single defender (meaning he executes the 3 vs 1 attack)
and perhaps some cases when 2 vs 1 and 3 vs 2 attacks are also welcome (there is a bonus to break or someth
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Re: Risk Strategy Manual

Postby TheChymera on Sun Oct 12, 2014 8:48 am

@Kask:

c) and how would you calculate the expected losses? it's not only a matter of adding up the attrition from my table (see the bottom of the page), it's also a matter of determining the probability that x% of the attacks will be "3v2" (my notation!), "3v1", or "2v1", or "2v2" respectively - and scaling the attrition by that. The usual way to address this is a Markov chain analysis. I think you could also do it with a Monte Carlo simulation, but that seems non-trivial to write in this case. I also think that the sweep question is also highly relevant for 2-player RISK (especially with increasing cards).

1, 2) Yes, I also found that these compound odds tables lack precisely the information I need. I went ahead and wrote a script which either calculates the odds or simulates single attacks. I believe one could (and it was my original intent to) use this to create more flexible compound odds estimators (for exactly cases like the ones you mentioned). If you're good at coding (or interested in learning based on this interesting question) I'd be very happy if you can make additions to the script to accommodate for the added functionality.
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