Some Background Info
A couple of links, first.
These posts are basically about RPS, and indirectly hint at game theory.
First off, RPS. Now, I’d never argue that RPS is a particularly good game. What it does do, however, is completely remove the analytical and execution phases of decisions, leaving only the decision itself - outguessing your opponent. And it can be done, if you play more than one game.
Roshambot - go ahead, try to beat it. Don’t say I didn’t warn you.
What’s interesting is that game theory can pretty much explain what Sirlin was trying to say, only a bit more succinctly.
Pai and Akira
| Pai/Akira | Throw | Big Move |
| Counterthrow | .5,-.5 | -.5,.5 |
| Block | -.8, .8 | .8, -.8 |
So, what we see here is pretty close to Sirlin’s description. If Pai counterthrows successfully, she gets to put some hurt on Akira. If Pai tries a counterthrow and Akira gets his big hit off, Pai takes some damage, but not as much as if Akira got the throw out.
This is good balance. The moves don’t have to be equal, but what’s important is that there are no strategies that are dominated - for each character, each move is useful in some situations. Complete balance wouldn’t necessarily hurt - but it’s not necessary.
Dominated Strategies - Rock, Paper, Scissors, Bomb
Let’s look at a bad example of balance: Rock, Paper, Scissors, Bomb. It’s just like RPS, except that there’s a new move, the Bomb. The Bomb blows up Paper and Rock, but gets its wick cut by Scissors. Bombs blow each other up for a tie. So, the decision matrix looks like this:
| Rock | Paper | Scissors | Bomb | |
| Rock | 0 | -1,1 | 1,-1 | -1,1 |
| Paper | 1,-1 | 0 | -1,1 | -1,1 |
| Scissors | -1,1 | 1,-1 | 0 | 1,-1 |
| Bomb | 1,-1 | 1,-1 | -1,1 | 0 |
So, what impact does the Bomb have? The important thing to see here is that in every situation, the Bomb does as well, or better than, Paper. So in this case, the strategy of Paper is dominated by the strategy of Bomb.
That’s bad. Why have an option in the game that doesn’t do anything?
But the Bomb isn’t unbalanced in comparison to rock or scissors. Rock still outperforms the Bomb against Scissors, and Scissors outperforms the Bomb against both Scissors and the Bomb.
In fact, there’s an interesting fact about both a "normal" RPS matrix and a RPSB matrix, minus the Paper (since nobody would play that). Here’s RPS:
| Rock | Paper | Scissors | |
| Rock | 0 | -1,1 | 1,-1 |
| Paper | 1,-1 | 0 | -1,1 |
| Scissors | -1,1 | 1,-1 | 0 |
And here’s RBS:
| Rock | Bomb | Scissors | |
| Rock | 0 | -1,1 | 1,-1 |
| Bomb | 1,-1 | 0 | -1,1 |
| Scissors | -1,1 | 1,-1 | 0 |
Look familiar? They should… they’re exactly the same! The only thing that adding the bomb did to the strategy of the game is to make Paper obsolete - once you ignore Paper, the game has the exact same strategies it did before. This is a good example of adding complexity (the player has to know never to play paper) without adding any depth (once the player knows that, it’s the exact same game).
Equilibrium
Another thing to look out for is equilibrium. Equilibrium just indicates a point of balance, that creates a general state of lack of motion.
The most common type is a Nash Equilibrium. This is a situation where neither side can make a gain by changing their strategy, unless the other side makes a change as well.
Consider:
| Hide | Run | |
| Hide | 0,0 | 1,-1 |
| Run | 1,-1 | 1,1 |
This represents a shootout. If both sides hide, they don’t get anything. If both sides run, they both get away.
But, if one side runs while one side hides, they get shot while running away. Their opponents then get away and take out the dastardly other team.
So what happens if both teams start by hiding?
Well, if either team runs, the other team shoots them. So neither team can move, even though neither team gains anything by hiding. This is a Nash Equilibrium.
Avoid these.
Smash Brothers
Some people like to play the Smash Brothers games without items. Personally, I think it’s missing the point. The items are there for a reason.
Here’s an example - some levels in tournaments are banned, because players can get in a defensive position that is basically unassaultable.
This leads to a simple decision matrix, almost exactly like the shootout one above. Neither player can gain anything by moving, but the first player to move puts himself in a weaker position than they were in.
Well, let’s go back to an earlier post - Opportunity Cost. What’s the opportunity cost of sitting in an unassaultable spot? The fact that you’re in exactly one spot. And, if items are turned on, there’s lots of items that can break an "unassaultable" position quite easily. And if you’re in one spot, you don’t get them.
Maybe those Nintendo folks aren’t as dumb as they look, eh?
Game AI Roundup Week #20 2008: 7 Stories, 2 Quotes, 2 Videos — AiGameDev.com | 18-May-08 at 2:39 pm | Permalink
[…] Balance is not Equality […]
Chris Paff | 07-May-09 at 2:33 am | Permalink
As an avid Super Smash Bros. player, I’d just to nitpick a bit. :p
When playing Smash purely for fun, items of course are either used or not used depending on personal preference, and stages are similarly banned or not banned based on personal preference.
However, it becomes troublesome for tournament play:
The items-off method, as you pointed out, does have the equilibrium problem on some stages, which unfortunately means that those stages must be banned. However, the items-on method has its own issues:
1. Items, when available, are typically much more powerful than the characters’ built-in attacks, which means that they dominate the built-in attacks. Furthermore, the number of ways to use an item are typically much fewer than the number of ways to use built-in attacks, which means that the items actually dominate more options than they add, reducing the depth of the game. Of course, items are only available part of the time during the match, but during those times the depth is reduced, which is generally frowned upon. (I suppose the overall complexity with items is higher, however.)
2. Items spawn at random times, at random locations, and have random effects. This means that the outcomes of matches between players of similar ability will frequently be determined by randomness from the item spawns. In tournaments, randomized outcomes are of course frowned upon heavily.
So really, what this means is that, for tournaments, both the items-off approach and the items-on approach have significant flaws, but generally players consider the items-off approach to be better overall.