Average change in margin: how to decide how many dice to roll in Lord of the Rings RISK

Average change in margin: how to decide how many dice to roll in Lord of the Rings RISK

Calculating the probability of particular outcomes in standard RISK only gets you so far and has been carefully modeled and calculated by Tan 1997, Blatt 2000, and Osborne 2003. There are five total possible outcomes in a given RISK battle: The defender can lose two armies, the attacker can lose two armies, the defender can lose one army, the attacker can lose one army, and both the defender and the attacker can each lose an army. Knowing the probabilities of each of these outcomes, however, doesn’t necessarily help you understand what attacking or defending strategy is optimal because of the difficulties of comparing the full complement of probabilities, which you would need to know if you were some kind of super-nerd, over-competitive prick, or, in all likelihood both. Maybe you don't have a lot of friends and love to spend your spare time concocting strategic ploys for stupid board games.

Furthermore, the previous work has centered on the standard version of RISK, ignoring the variants that have sprung up over the years. As an example, Lord of the Rings RISK, a variant designed primarily as a craven cash-in on the blockbuster trilogy included some fun new rule options, often included in a number of a number of the recently produced RISK variants. Two of these rule options: captains and strongholds, introduce a cool wrinkle into the game, by adding a bonus to the highest die roll of either the attacker or defender. It's a neat rule, but changes the probabilities calculated in the previous work. That’s where the average change in margin metric comes in handy. It's useful for helping the way-to-in-to-it RISK player kick the ass of his unenlightened opponent on an attack by attack basis (previous works focus primarily on the overall success of a given invasion). If this isn't a cry for help,then I don't know what is.

In any event, this metric is simply calculated by multiplying the probability of each outcome by the change in margin that such an outcome would net, and then adding all these terms together. This result is the average change in margin each time that there is a battle. If for example, you are attacking, and there is a 75% chance that the defender will lose one army and there is a 25% chance that the attacker will lose one army, you would make this calculation:

0.75 * (1) + 0.25 * (-1) = 0.50

Therefore, on this given skirmish, the attacker can expect to increase the margin of difference between his and the defender’s armies by 0.5 armies. You didn't read any of that last part did you? Your eyes just glazed over and you happily skimmed. Asshole.

This metric provides a clear way to compare who has the advantage in a given situation and which strategic setups have positive and negative expectations. The raw probabilities (included at the end of the document) were used to calculate the average change in margin for the defender. These expectations yield a number of basic insights, a few of which are a little surprising.

[Lots of charts]

1. For the most part, rolling as many dice as possible on offense and defense is a good idea. In normal RISK, this is an absolute truth. Ever notice how really insecure people cling to the concept of absolute truth? That was totally apropos nothing. Really. Fine, don't believe me.

2. The stronghold and leader system of LOTR RISK complicates matters just a little bit, however, and creates a few situations where you might want to roll less than the maximum number of dice. If the attacker has a captain, and is rolling three attack dice, it is slightly more advantageous to roll a single die on defense if you have no bonuses of your own. It's a minuscule advantage that will probably not really ever help you win any games, but, when all you have are minuscule advantages, it is something to cling to, like that memory of the one time where you felt like you had a really solid group of friends and you all got along so well and you can't remember what went wrong or why that era ended. You can't help but think it was your drinking, but there is no way to know these things for sure.

3. If you are attacking a defender with a captain, a stronghold, or both, you should always roll three dice if possible. If you have to choose between rolling two dice or one on offense in this situation, it’s slightly better to attack with only one die. When I say slightly better, this is completely relative, as attacking either 2v2 or 2v1 against a stronghold or a captain are two of the very worst moves expectation-wise that a player can make: You are practically guaranteed to lose an army. Which brings me to my next point:

4. Never attack in a 1v2 or 2v2 situation. Even if you have a captain and the defender doesn’t, you still don’t have a positive expectation. There has to be some massive pay-off for this scenario to ever be worth it. Sidenote: honestly the pay-off should be the lens through all of this strategy is considered. A thorough understanding of the principle of poker "pot odds" would help you evaluate an overall strategy of taking a particular territory or continent. This is obviously the most important part of RISK strategy, but because here we are celebrating the myopia of super- focused nerd minutiae we will, in the proud tradition of science, ignore it for now. Do I sound bitter? I was going for "arch," but I think I sound bitter.

5. If you are serious about defending a territory, a captain and a stronghold guarantee you a positive expectation in every possible skirmish. These things are quite good. You know, in a totally insignificant way that will never ultimately affect your happiness in more than the most trivial way imaginable.

6. The attacker has a positive expectation in a standard 3v2 match up. This finding was first found in Osborne 2003, but it bears repeating. You can be more aggressive than you might think. That paper has a lot of concrete info about the 3v2 standard matchup, and considering that this is most battles, it’s definitely worth reading. Particularly, if you have gotten this far and it's oh-so- clear that you are interested in learning and improving yourself and being a goddamn insufferable phony clinging to your Pyrrhic victories playing a goddamn board game that was adapted to include three hundred percent more motherfucking hobbits and little plastic cave trolls. Fuck you. Fuck you. Fuck you. No, I'm not crying. Fuck you.

[Raw Probabilities: lot more charts]

For those interested in reading up on the topic of mathematically modeled risk modeling, I would recommend following up on these works. Tan 1997 was the first major attempt at modeling the basic structure of a Risk turn while Blatt 2000 and Osborne refined the method in Tan. Vaccaro 2005 is less directly relevant, but shows a cool new tack for using evolutionary algorithms to develop Risk strategy. It's cool. You should read it. Or you know, go out into the world and make friends. I would probably read it, but that's me. Don't be me.

Blatt, Sharon. RISKy Business: An In-Depth Look at the Game RISK. Technical Report, Elon University, 2000.

Osborne, Jason. Markov chains for the RISK board game revisited. Mathematics Magazine 76 (2003) 129-135.

Tan, Baris. Markov Chains and the RISK Board Game. Mathematics Magazine 70 (Dec. 1997), 349-357.

Vaccaro, J.M. Planning an Endgame Move Set for the Game RISK: A Comparison of Search Algorithms. IEEE transactions on evolutionary computation. vol. 9, no. 6, pp. 641- 652, 2005.