Preventing mini-maxing (Was: Re: PBM design and formulae - a fresh start

From: markiel@callisto.pas.rochester.edu (Andrew Markiel)

Date: Wed, 07 Jul 1993 08:03:11 +0000

Here I want to counter the arguement that one should hide the formulas
to a game to prevent mini-maxing (i.e., a person playing via numerical
optimization of a set of known parameters). This isn't inherently
wrong, but lots of people don't like to play this way, so it's reasonable
to design a game trying to prevent this.

Now, I claim that hiding the formulas is a poor way to this. The
reason is that, if the formulas are deterministic, then they are
certain to produce the same output with the same input parameters;
thus whenever I get a result from the game, that gives me a perfect
data point. While I may not be able to derive the general formula,
whenever the same circumstances come up again I will know the exact
result and thus I can optimize based on that exact result.
The more data points I get, the better I can optimize. Furthermore,
this is inherently unfair, in the sense that an expert player has
knowledge that is *in principle* unknowable by a beginner: namely
not what might happen in a given circumstance X, but what is
*certain* to happen in any given circumstance X that the experienced
player has seen before.

I think there are four reaasonable ways to solve this problem of
unfairness:

1) Release all the formulas, so everyone has the same (exact) knowledge.
This certainly makes things fair. Of course, the realists in the
audience will immediately point out that this is completely unrealistic.
However, my point is that the lack of realism is implicit in the
choice of deterministic equations: in real life, seeing a previous
result still does not allow me to predict perfectly a similar future
result, because the result depends on many many "micro-parameters",
little things that change from situation to situation but have an
effect on the result. An explicit formula, however, does not have
this feature, and so mini-maxing is built into the game system.

2) Make the formulas depend on all of these "micro-parameters", to
reproduce the uncertainty of reality. In essence, with each result
I still get an exact data point, but the parameter space is so vast
that it is statistically unlikely to ever encounter that exact
situation again, so my perfect knowledge is not directly applicable
to future events. Obviously, this is the holy grail of simulations,
and I agree that if you can do this, and still have the results of
the formulas be intuitive, then the formulas should not be released.
However, I would point out that, the more parameters you choose to
consider, the more complicated the formula becomes, which means
(a) the probablility of bugs becomes high (which can't be detected
by the players since they can't check the formulas), and (b) it
it becomes much harder to make your formula fit reality: obviously
it's easier to model any desired situation with a 3-parameter
equation than a 15-parameter equation, as you have to check that
all 15 of those parameters affect the formula in an intuitive way.
In other words, this is usually too hard to be a desirable
design alternative.

3) Abstract the "micro-parameters" into a random factor. Thus, you
insert uncertainty into the equation to make it represent the
uncertainty of reality. But in this case, even if I know the
formulas, I can't generate exact results, because there's still
one piece I don't know, the random factor. Thus I claim that in
this case you can release the formulas without allowing people
to mini-max the equations. (If you wish to argue that they can
do an approximate mini-maxing over the probability distribution,
I counter that they can achieve equally useful results via
analysis of their results from the hidden formulas, and both are
of marginal utility).

Also note that in this case, the formula is realistic in the sense
that it allows you to calculate a probable value without allowing
you to calculate an exact or certain value. I'll also point out
that this option is still not completely realistic, since the
random factor will presubably be artificially constrained to
some set of values, which introduces an artificial behavior
unlike reality (where, in theory at least, any result could
happen but very radical results are extremely unlikely to
occur). This is a minor point, but, depending on how artificially
the random factor is constrained, it may be appropriate to tell
the players the range of possible values. This, of course, again
tells them only what might happen, without giving perfect knowledge.

4) Keep the equation deterministic, but depend on one or more
parameters that are not known exactly by the player (hidden
information). Even exact equations cannot generate an exact
result without exact initial conditions. However, from the
players point of view, these hidden parameters are just random
variables (with a probability distribution determined by the
probability of the parameter taking any particular value). Thus
from the player's point of view, this is exactly the same as in
(3), and the same arguments apply: thus in this situation as well
knowing the exact formulas does not allow you to do precise
mini-maxing, because you don't have all of the information you need.

Thus, I argue that, except in the special case where the formula
has so many parameters that circumstances are essentially never
the same, mini-maxing is not an issue in deciding to reveal the
formulas because it is either (a) inherent in the choice of
completely deterministic equations, or (b) not useful due to
imperfect knowledge, even knowing the "exact" equations.

-Andrew
markiel@callisto.pas.rochester.edu