Risk Legacy: Impact of Defensive Buffs

Over the weekend we played many games of Risk Legacy which is a Risk variant board game with some pretty wacky rules. The specifics of those rules are unimportant for the time being (and contain spoilers for anyone who might want to play in the future) but one of things that happens is some of the time the defender gets to modify their die rolls. These modifications are unambiguously good or bad (adding 1 to your roll can't be worse than not doing so, after all) but the question really is how big of an impact do they have. Sky had a few turns where it seemed like he couldn't hurt Pounder at all... Was this just a result of Pounder rolling high on small numbers of dice or did his defensive mod actually make the attack sketchy in the first place? I want to find out!

My first problem is I've never actually crunched the normal Risk attack numbers. I used to play Risk a fair bit way back when I was a kid and didn't really care about precise odds. 3 is bigger than 2, so attacking is good. Defender wins on a tie so that's good too. Those two good things OBVIOUSLY cancel each other out so it doesn't matter if you attack or defend, right? Right? The only thing that matters in that mindset is getting more guys. I've since been informed that the attacker actually has the advantage but I still have no idea how big an advantage that actually is. Therefore it is definitely time to crunch some numbers!

The numbers displayed below are how many troops the attacker expects to lose while killing 1000 defenders. So, not small numbers of dice at all, but it shows some trends. For reference, an ammo shortage subtracts 1 off of the defender's highest die (with a min of 1). A bunker adds 1 to the defender's highest die (with a max of 6). A fortification adds 1 to both defender's dice.

	Normal Risk	Ammo Shortage	Bunker	Fortification	Bunker+Fortification
3v2	853	510	1192	1893	2613
2v2	1566	1017	2260	3469	5000
3v1	516	211	1025	1025	2000
2v1	728	350	1400	1400	2724
1v1	1400	800	2600	2600	5000
1v2	2927	1541	6200	6200	14429

So it turns out attacking 1 on 2 into a bunker and a fort is a terrible, terrible idea. Who knew?

More seriously, the times when the attacker has an advantage are 3v2 normally and with an ammo shortage. 3v1 normally and with an ammo shortage. 2v1 normally and with an ammo shortage. And 1v1 with an ammo shortage. In every other situation the defender has the advantage. When you combine a bunker with a fort it's a huge undertaking to take it out. A bunker alone more than doubles your expected losses. A fort alone more than triples your expected losses. 

I think you'll want pretty good numbers, or a really good incentive, before attacking any of the defended positions. Being able to defend the spot yourself is a decent reason if you can win with enough troops remaining but that seems pretty unlikely if they had a decent force to start. Probably you'd be better off going someplace else. A good incentive would be a chance at straight up winning the game. The actual game is still played with small numbers of dice and maybe a slim shot is the best you're going to get. But I was more than willing to attack a defended position willy-nilly yesterday and that no longer looks like a terribly good idea. And I suspect if I look at how missiles impact things it'll look way worse for the attacker!