More Boring Risk Commentary

I mentioned the other day that I played a game of Risk with some rule variations and got some abnormal patterns, so I'm going to expand on that. Actually it was only one rule variation. There is a list of rule variations for "advanced" play towards the back of the instructions, and sometimes I play with those. In this case, it was one which altered the value of the sets of cards. If you aren't aware, in Risk, you get one card at the end of each turn on which you have taken at least one territory. Each card has either a soldier, a cannon, or a mounted soldier, in addition to a territory. You get a set of cards if you have three of a kind or one of each kind, and a set can be turned in at the start of a turn for a number of armies that increases as sets are turned in (by anyone). There are also two wild cards, which can be used as any of the three types, and all the cards but the wild card have a territory on them as well. The territory is largely irrelevant except when used as a selection method for initial territories. The only other effect the territories have is that if you have one of the territories on a card in a set that you turn in, you get an additional two armies on that territory.

The value of the sets turned in usually goes 4, 6, 8, 10, 12, 15, 20, and so on by 5's. The variant rule reduces the number of armies to 4, 5, 6, and so on.

Since the cards are awarded to players who take a territory, they can be seen as an incentive to be aggressive, and we would expect players that are aggressive enough to get sets to do better than those who aren't. Any good RISK player can probably attest to the trend of sets becoming massively important, especially later on in the game, as the value of the set surpasses the value of any territory or even continent rather quickly. To point this out, let's look at the other methods of gaining armies, via territories and via continents.

A player is awarded armies every turn for the number of territories he (forgive my gendered pronouns) has, t. The number of armies awarded is t/3, which is rounded down to the nearest whole number. However, even if he should be awarded fewer armies by this rule, he automatically receives 3 armies at the start of each turn. An interesting result of this minimum adjustment to the rule is that the value of territories drops significantly for players with few territories, that is, players who are probably not doing well.

It is difficult to state exactly this army-value of a territory precisely because the armies aren't awarded until the beginning of the turn, so if a player takes a territory and subsequently loses it before his next turn, it was of no (actually negative) army-value to do so. Also, since territories make up continents, we would have to factor in the value of continents. Overall, this is leading us to an equation for finding the expected value of moves in a given situation, which we could then hopefully use to generalize to good overall strategies.

Regardless, it is safe to say that the value of any given territory (as long as it does not lead to taking a continent) is very little. There are only 42 territories total, so even if you have all the territories, at which point you would have already won, you would only receive 14 armies the next round based on territories alone. Let's say for example that you control half the territories on the board, 21. At this point, you are probably nearly assured victory in any game with more than two players. However, the territories alone only yield 7 armies. Considering that even an opponent controlling only one territory would receive 3 armies, it is fairly clear that territories themselves are not particularly valuable in terms of armies. We draw from this that a strategy of merely taking territories quickly is probably not a good one, which we could expound on further, but is pretty clear.

So, anyway, let's expound. For reasons I will maybe get to later, experienced Risk players often shy away from Asia early in the game despite its high army-value, which often leads to many Asian territories being left with the minimum one army early in the game, so we have at least one real-game scenario in which there are numerous easily taken territories around. If it is our plan take these easily gained territories for the purpose of amassing armies, then we could look at the expected value. Assuming that we have enough armies that we can roll three dice to a defender's one, we have a 95/144 (I have computed this but maybe it is wrong) chance of winning on the first roll in each territory we take.

In a six-player game, every player will start with seven territories. Seven territories yields only 2 armies by the t/3 rule, so we would be awarded three armies at the start of each turn. Since 7=1 (mod 3), we need to take two more territories to step up a rank in territories. However, even if we do so, we will only have 9 territories, which would yield again 3 armies, so we would actually have to take 2+3=5 territories to see any gain in armies. Taking five territories, even if they only have one army defending them, we only have (95/144)^5 likelihood of doing that without losing any armies. I'm not going to do the full calculations here, but it is pretty clear, then, that we are likely to lose at least one army in the process of taking these five territories. So, overall, the expected value of taking these territories is negative. Does this mean that we are better off just sitting and not attacking? No, but that analysis will be much more complicated. That's it for now