Expected losses on assaults

[ScottVal]

Hello-
I'm sure some one has calculated expected losses on assaults. For example, if you have 100 attacking armies versus 100 defending armies, and you press the "auto assault" button, what would the most likely end-result be?

I enjoy doing math calculations, so I started with the simple case of two armies versus one. What are the chances of the attacker winning one assault, and what are the chances of the defender winning. There are 36 different outcomes, with the attacker winning 15 and the defender winning 21.

Then I did the case of three armies versus one. This is obviously more complicated, but I figured it out as the attacker winning 125 and the defending winning 91. Let me know if you want to "see my work."

I haven't started the more useful case of four-plus armies versus two-plus. I thought I would post this topic and see what others have already come up with. This result can then be applied to answer the question about expected losses when a large group of armies assaults another group.

-Scott

[Timminz]

There's a new add-on that will tell you all you seek. Well, maybe. If you're more of a "right now" kinda guy, you could check out http://www.gamesbyemail.com/Games/Gambit/BattleOdds

[ScottVal]

That's nice, but it's not really what I'm looking for.

Say you have a territory with 90 armies in it, adjacent to an enemy territory with 100 armies in it.

Let's say you're the underdog in this war, and you think it might make sense to attack the 100 armies with your 90 armies. Even though you are outnumbered, the advantage of attacking might allow you to wear down the enemy. In other words, if you attack, he would lose more armies than you would lose.

To answer this question, I want to mathematically figure out the expected losses on either side when a group of 90 armies assults a group of 100 armies.

I started working it out on paper, and it is very complex. Has any one else figured this out?

I'm starting work on a computer program which will figure it out.
-Scott

[ScottVal]

OK, I have this to add to my last post:

According to the "BattleOdds" web page (http://www.gamesbyemail.com/Games/Gambit/BattleOdds), in a 3-dice versus 2-dice battle, the attacker wins 37%, defender wins 29%, and split 34%.

So, obviously in a battle between to large groups of armies it makes more sense to attack than defend. You have a 8% advantage when you attack. I think all good players know this instinctively.

So I guess I'm not going to write a computer program after all. I've more important things to do.

So, I guess the bottom line is that it might make sense to use a large army to attack an adjacent army which is even larger, in order to wear the larger army down.

Any other thoughts ya'll have on this are appreciated!
-Scott

[Timminz]

That's nice, but it's not really what I'm looking for.

I'm pretty sure, it's exactly what you're looking for.

[ScottVal]

You're right, it is what I'm looking for. Thanks. I tried to explain this in my last post. I even made a quote from the BattleOdds site.

Obviously this game is ultimately a game of resource management, competition and balance. If you fall behind, you'll lose.

If the game is close, i.e., the resources garnered by the sides are approximately equal, a tactic of using a large army to attack a territory with an even larger army might be useful, because it can cause the larger army to lose more armies than the smaller army. This is because of the advantage of attacking rather than defending.

What do you think?

-Scott

[Timminz]

What do you think?

I think you're catching on to this quite well. As long as you understand that the most probable event is not always the one that will occur, I think you'll do well here.

[BaldAdonis]

For example, if you have 100 attacking armies versus 100 defending armies, and you press the "auto assault" button, what would the most likely end-result be?

You're likely to lose 85 armies when you kill 100. In general, your expected loss attacking against n armies is 0.85n + 0.22*(-0.53)^n - 0.22

[ScottVal]

Hey, cool formula.

With large number of armies, the second and third terms become insignificant, leaving .85n.

I think another way of saying that is: You can expect a 85% casualty rate when pitting your large force against another large force.

Or, you can attack a force of 100 armies with a force of 89, and can expect to reduce the enemy to zero, with four of your armies left.

I think most people who have played Risk casually over the years with friends (over the board) have had the impression that you should have an overwhelming majority before attempting an attack. Like 3-1, or at least 2-1.

Obviously not true, unless (e.g.) you want to have a large force left over to continue impinging into the enemy position.

I'm playing a game now on the City Mogul map, and one of my opponents was getting a bit of a lead. So I attacked one of his tents which contained 138 armies with mine which contained 193 armies. It worked well, I wore him down to zero, with about 80 armies left over. I continued in and took a store and his house.

I set up a barrier to slow down his counter-attack, but he damaged me pretty bad the following turn. He also said he blundered; I think he meant he broke off his attack sooner than he wanted and pushed the wrong button or something. He could have hurt me a lot worse.

I have yet to win a game, but I think I'm getting closer.

Anyway, thanks for the cool equation. -Scott

[xelabale]

If you fall behind, you'll lose.

I don't agree with this though scott. The game is more than numbers. 1v1 yes of course. Any more than 2 players and there's tactics too. If A gets big, B and C may gang up on him. There's also distribution to take into account. If he has 100 armies but they can't attack me this go I may choose to dive in. I think it's relatively rare for the biggest guy to just steamroller the others - there's more to it than that. Having the most armies does help though!!