probability and gaming

[Don]

I was reading some gaming conference when Sid Meier was explaining probability. So let's say you got you got one side with a strength of 1, and the other side has a strength of 3. Civilization handles odds by (your strength/total strength) so the side with the strength of 1 has a 25% chance to beat the side of 3. So then someone asks what if I got a big army on both sides so it's 10 versus 30, and Sid Meier said well it's still 25% chance for the side with 10 to win, and the fans didn't like that, and then the commentary makes some observation about how people don't get probability. I thought it was funny at first, but then I realized the guy who doesn't get it is Sid Meier and the guy making the comment. If 10 versus 30 represents 10 times the troops one each side (which I assume it is, since this is Civilization), then the chance of the side with 30 losing is next to nothing.

Let's keep things simple and assume that there's a 25% chance 1 will take out the 3 without dying, and 75% chance represents all the way the side with the 3 can win (they might lose some guys versus the 1), then the side with the 10 has to get at least 7 of the 25% events in a row to have a 50/50 chance to be on equal ground after the first round (that'd leave them with a strength of 7 versus whatever an expected 6.75 strength left on the side of 30). I know it's more complicated than this to calculate but you basically have to come pretty close to repeating the original event X times when you multiply both sides by X.

I checked the 2008 election and Obama has 53% of the popular vote versus 46% for McCain, but this doesn't mean if you run the 2008 election 10 times you expect McCain to win about 4 times. An 8% deficit, when multiplied over millions of votes, becomes an insurmountable deficit.

It's interesting that it's Sid Meier who made this mistake since that is the foundation of his games. It also highlights why strategy games are hard to balance. Bigness absolutely trumps any element of chance, and while strategy should not be about luck, it's not exactly an interesting model when the bigger side has a 99.99% of winning and then you end up with just an empire building game. Note that bigness can factor in technology and other factors. 2 Battleships in any Civ game is a lot *bigger* than say, 10 Knights, due to their superior combat values. While it feels pretty stupid in Civ 1 when your Battleship loses to a Knight, I'm not sure having 99.999% a Battleship winning is much better. Yes it makes sense, but if the outcome is certain 99.999% of the time then you might as well call it a building game. Build more units than the other side and then you win the game. It might be true so far as strategy goes, but it doesn't make a very interesting game.

[Julius Seeker]

Civilization (as in real battles) do not work with absolutes. Many times in history armies of smaller size and perhaps even lower advantage (or relatively so) have succeeded in winning.

He sells his games short though since a lot of other factors need to be taken into account when calculating damage. I assume this is simple archer vs warrior with no modifiers (I believe they both have a hit rate of 1. Think of it this way, 1 warrior has a 25% chance to defeat 1 archer (no modifiers), 3 warriors have an equal chance, but 9 warriors have a 75% chance.

[Don]

I'm not sure how newer Civ handles battle with multiple units (the one he talked to is not any of the numbered series, seems like it's something on a handheld), but it's very different to say 1 warrior versus 1 archer, warrior has 25% chance of winning. You can verify that by running that scenario a few times with loading (assuming seed isn't saved, sometime they are so you can't cheese it). But now let's say you got 10 warriors versus 10 archers. Let's just simulate it as 10 concurrent 1vs1 matches and winners trickle up like a tournament (obviously if they're the same type of units then they chill out on the side next round). For the warriors to have a shot at this they essentially have to win 7 out of the 10 fights in the first round (because the 3 surviving Archers are still favored to beat the 7 Warriors that won their matches), and the chance of that happening is extremely slim. It's not quite 1/4^7 but it's definitely a lot less than 1/4.

At any rate modification to unit is just a different way of 'big'. Once the size gets big enough there is almost no element of chance involved. Maybe this is why the strategy games that focus more on the tactical portion tend to have a strict unit limit so you can't just get around the problem by sending an army that's twice as big.

[Julius Seeker]

Sounds like Civilization I (where an archer can take out a battle ship), since Civ 2 has HP. You are right about multiple battles; The function would include 10 battles initially:
First 10 with archers attacking, 2-3 for warriors, 7-8 for the archers. You now have 7.5 archers and 2.5 warriors. The next round takes you down to 0.625 warriors and 5.75 archers roughly, take these up to whole numbers: 1 warrior vs ~8 archers: and the probability of survival of attack from 8 archers is roughly 0.25^8=X on the archers second turn - that's somewhere around 1/100,000 chance of survival. Sid's off on his number crunching =)

[Don]

If I recall battleship has an attack of 18 and Phalanx has a defense of 2, so in Civ 1 there's a 10% chance the Phalanx can win (higher if it's behind city walls which it usually is).

Now in Civ 2 I think a Battleship has 40 HP and a firepower of 3 versus 10/1 for the Phlanax. But, it's not the same as just multiply its strength by 12. You'd get 216 versus 2, so the Battleship has a 99.1% chance to win. But I'm pretty sure in the actual game you will not see a Phalanx win 9 out of 1000 times. I don't even think a Phalanx can beat a Musketeer with these odds if it's not behind a city wall, even though you'll certainly find the higher tech units may end up with surprisingly few HPs left at some fights.

One of the thing I thought is weird about the Civ games and strategy games in general is that if you got 1 Warrior making a stand against 1 Archer you have a 25% chance of winning that stand. But if it's 10 Warriors versus 10 Archers you might as well just resign and save yourself the humiliation. A good strategy should minimize the element of chance, but it shouldn't be so good that there is no element of chance either.

[Julius Seeker]

I suspect Civ 5 will get it correct. They have done away with unit stacking and archers now have range. Though not too much has been revealed, importantly whether they have done away with armies and if this applies globally or to field armies only.

[Don]

Well to be fair Civilization has always been an empire building game rather than a tactical strategy game. The combat system is extremely simple that greatly favors attackers unless you're behind a city wall or similar defenses, so the optimal strategy is generally just build as many of the most powerful offensive units as you can and keep on attacking. I can't really imagine in a Civ game you can brilliantly turn around say a 20 Armor versus 30 Armor disadvantage and come back and win unless one side did something ridiculously stupid. It's both a strength and a weakness at the same time.

[SineSwiper]

Again, every time somebody talks about Civ, I start to miss Masters of Magic.

[Don]

MoM is more like a RPG than a strategy game. At the end it's pretty much having an army of superheroes that just stomp everything unless you ran into something crazy like 8 Great Drakes or 8 Great Wyrms, or if you've to deal with spells like Crack's Call. Some of the tougher nodes in Myrrar are almost like a boss battle in themselves, and on Impossible battling agianst Wizards that have like 8 Great Wyrms is pretty fun too (probably the worst possible combo because anyone who can summon them also has Crack's Call which magic immunity cannot stop). Of course, there's absolutely nothing wrong with a game like that. I think it'd be cool if you can play Impossible with no computer players, or at least computer players that don't attack you because you can't possibly beat them without resorting to load/save and cheesy combos early on.