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Games on auto pilot

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Infernal
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Joined: 12/31/1969
Games on auto pilot

Ok here is an example from an actual game. I will explain the problems and how they solved them.

The units:
Horse Archer beat Infantry and Heavy Cavalry but are beaten by Archers
Infantry beat Heavy Cavalry adn are beaten by Horse Archers and Archers
Heavy Cavalry beat Archers but are beaten by Horse Archers and Infantry
Archers beat Horse Archers and Infantry and are beaten by Heavy Cavalry

As it stands in this game Infantry are next to usless. There would be no situation where you had other units and needed Infantry.

The designer fixed this by createing a system where, during combat a unit had to either give ground or take ground. He then decided to allow the special ability of infantry to be that they were then only one that could hold ground in combat, providing a defensive units to maintian control of territories. This gives it an ability that is useful in certain circumstances where the other units would not work.

This nesessitated changes to the mechanics and rules to impliment.

A game where this unit dominance system exists (without the fix) is the computer game "Age of Empires". The designers included the infantry unit but the resource mecanics meant that holding ground was not needed. The infantry unit was not used, even when the designers halved (and that is a big change) the cost of infantry it was still not worth using.

So the examples that I listed were to demonstrate that having certain paterns of dominance of choices over others can healp break the "game on autopilot" syndrome, by giving players a choice that will effect their position in meaningful ways (give them an interesting chioce). Of course this is not the only way to do it, and not the only aspect to consider. It is only 1 tool in the toolbox of game design.

Zomulgustar
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Joined: 07/31/2008
Games on auto pilot

Actually, I find analyzing elementary games to be very interesting...best to take the opportunity to completely analyze those few games which are amenable to such analysis.

To keep things simple, let's move things to a symmetric situation where fog-of-war makes unit selection effectively secret/simultaneous, not unlike RPS itself. While it's true that the equilibrium of

0 1 0 -1
-1 0 1 0
0 -1 0 1
1 0 -1 0

is unstable, is it necessarily the case that ANY even number of options can only generate unstable equilibria? I'll admit that a counterexample matrix doesn't spring readily to mind, but I don't see any immediately obvious reason why it should be a universal property, and I'd be very curious to learn of any proof to this effect.

Besides, in the context of something as simple as a matrix game, balancing these exactly isn't something that generates a significant development cost, and in a more realistically complex game many other factors can make the effective payoffs have uncertain values, allowing a little wiggle room for what would otherwise be dominated strategies to have their moments in the sun. So practically speaking, building your game units around a genuine but unstable equilibrium doesn't doom your game as thoroughly as the chapter you refer to might imply, especially since it encourages a wide variety of strategies as viable options: everything in the planes (A=C) and (B=D) are valid choices in the above example, and anything close to them loses only slightly...IMHO of course.

I'm not sure where you're getting the forced stalemate/runaway leader concern from, but from what I can tell it derives from problems with the structure of the rest of the game, rather than a problem with the equilibrium itself. With perfect understanding of the game, Chess is almost certainly either a forced stalemate or a runaway win for the first player to move...but people still play since the search for that perfect strategy is itself interesting. If you feel it's not acceptable to have your game's balance working on the 'security through obscurity' principle, then you have to add an element of chance or hidden information.

As for the non-zero sum issue Infernal pointed out before, I MEANT to say paper gets only one point vs. ROCK, i.e. the matrix

0 2 -1
-2 0 1
1 -1 0

with a single equilibrium at (1:1:2)...but since Sid Meier did Civ (Will Wright did SimCity), I suppose we're even. ^_^

Infernal
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Joined: 12/31/1969
Games on auto pilot

Quote:
As for the non-zero sum issue Infernal pointed out before, I MEANT to say paper gets only one point vs. ROCK, i.e. the matrix

0 2 -1
-2 0 1
1 -1 0

with a single equilibrium at (1:1:2)...but since Sid Meier did Civ (Will Wright did SimCity), I suppose we're even. ^_^
That is how I read it the first time, but as I was re-reading it I picked up on the error. At least you got mine the first time around :P .

Quote:
I'm not sure where you're getting the forced stalemate/runaway leader concern from, but from what I can tell it derives from problems with the structure of the rest of the game, rather than a problem with the equilibrium itself.

I just had a look at my "simple game" in my example:

Quote:

Archers beat Warriors, tie with Sorcerers, lose to Barbarians
Warriors beat Sorcerers, tie with Barbarians, lose to Archers
Sorcerers beat Barbarians, tie with Archers, lose to Warriors
Barbarians beat Archers, tie with Warriors, lose to Sorcerers

Taken as a purely thoretical excersize (where each player puts out one unit at atime and the loser removes their unit and ties are left on the board), the players have a non-symmetrical starting posiotion (it could be because one player goes first), and if there are no dice or other outside effects, and all other things being equal, then one player would have, from the start, a runnaway leader effect.

The system its self is gives the run away leader problem, not the payoff matrix.

again: my bad. :oops:

I'll explain (as this also has to do with games on autopilot and interesting choices).

In the game because ties and winners are left on the board it would be possable to use these units again in another combat. If I get just 2 units (either Archer and Scorcerer, or Warrior and Barbarian) then I can use 1 of these units to beat any unit you throw at me, my position as leader is unassailable and the best you can do is to throw everything at me to keep my position where it is, other wise I can just keep puting units out with a runnaway leader situation.

This could be fixed by, for example, randomness in the outcome of battles (but it still wouldn't effect it much over the long term), or have units that can only be out for a limited time (eg. each battle causes acumulated damage which kills the unit after so many battles).

Even then these fixes are not going to completely fix the problem, but it would give the loosing player some chance to catch up.

So although the root of the problem lies in the matrix, it is not the matrix itself that is the direct cause of the runnaway (positive feedback) leader.

Zomulgustar
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Joined: 07/31/2008
My brain hurts...

It turns out there IS a reason why even-n nxn matrices don't work, but it involves words like 'eigenvalue'. The proof doesn't actually show that the equilibria are unstable, per se, but that there is no unique mixed equilibrium.. I figure I probably had better check if anyone actually cares before attempting to translate from Nerd into Geek.

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