For and against binary goals in strategy games (and against high score)

#61
Hopenager, I’m not convinced by the idea that “as the chance of victory gets further from 50% (in either direction), the chance that new actions will change the outcome of the match gets lower”. I think that causal leap is one a bit too wide for me to make. However, I am convinced by my own argument to a similar conclusion, so let’s not dwell on the point.

Ok Richy, it’s very useful to point out the distinction between actual WC and what the player believes WC to be. It is worth knowing that although the former is definite - there is one certain WC at any point in the game, since the system follows deterministic rules - the latter can be more or less indefinite, given WC is practicably incalculable for human players in deep strategy games.

The important point that follows from this distinction is that, adapting my argument for why WC should not deviate from 50%, it is expected WC and not actual WC that cannot deviate from 50%. So, in theory, WC can sit wherever it likes, as long as the player does not expect it to have deviated from 50%. (E.g. if the player is in the dark about WC and believes it to be wholly unknown, then they do not expect it to have deviated from 50%.) So long as this condition is met, decisions can be designed to be well-balanced for maximum ambiguity, which is what we’re after.

The questions that follow from this include: should the player be actively deceived about WC to preserve ambiguity? Is that even possible? Ignoring deception, how far can we hide the truth from the player? Is there still a limited imperative for actual WC to reside at 50%? All interesting questions I will have to deal with after a night’s sleep. Please get there before me and solve all those questions while I slumber, I won’t complain!
 
#62
@richy You're absolutely right that we can't calculate the winchance in any non-solved game. In fact, a "position evaluator", an algorithm that would compute the winchance given a gamestate, is impossible even in theory, because winchance depends not only on the gamestate, but also on the mindstates of the player(s).

However, the fact that we can't compute the exact value of winchance doesn't mean that it's useless as a concept ("depth" is another a value that can't be computed in a non-solved game but which is clearly not useless as a concept.) We can still develop hypotheses that provide guidelines about what types of winchance patterns are better or worse, and attempt to build games that conform to those guidelines (imperfectly, of course).

And even though winchance can't be computed, we can have an intuitive sense, as designers and as players, of what the chance of winning is for a given situation. Feeling that you have an advantage over your opponent, for instance, is equivalent to feeling that you have a higher chance of winning they do (and the degree of the feeling of advantage correleates with the degree of the deviation from 50%). And feeling that the game is already decided even though it isn't technically over (i.e. feeling that you are in a foregone conclusion), is equivalent to thinking that the winrate is near 0% or 100%. So even though winchance can't be calculated, it can certainly be approximated (and in fact getting better at a game can be thought of as a process of learning to better approximate winchance, since "good decisions" are just the decisions which lead to higher winchances)

Hopenager, I’m not convinced by the idea that “as the chance of victory gets further from 50% (in either direction), the chance that new actions will change the outcome of the match gets lower”. I think that causal leap is one a bit too wide for me to make. However, I am convinced by my own argument to a similar conclusion, so let’s not dwell on the point.
We might have to dwell, actually, because my reasoning leads to the opposite conclusions on actual vs believed winchance: what matters is that the actual winchance is at 50%, not merely that the player believes that it's at 50%. But before I get to that, I want to say that I doubt that it's possible to hide the truth of winchance from the player, at least for experienced players, given what I said above about getting better at a game really being a process of better approximating winchance.

What I said earlier, "as the chance of victory gets further from 50% (in either direction), the chance that new actions will change the outcome of the match gets lower”, wasn't a great way of explaining my actual position. My bad. A better way of phrasing this is in terms of of feedback efficiency: 50% is the point at which entropy is maximized with respect to performance, or in other words 50% is the point at which a win or loss gives you the most information about your performance. See this article for a deeper explanation. What this means, in practice, is that when the winchance is at 50%, you get feedback (from the game's end-state, at least) more efficiently than you do at any other winchance.

I am not sure exactly what you're reasoning is for thinking that 50% is ideal if it isn't this. You've said that 50% is the point at which there is "maximum ambiguity", but what exactly do you mean by ambiguity? And why is ambiguity, however you define it, a good property?
 

Myko

New member
#63
@richy50% is the point at which entropy is maximized with respect to performance, or in other words 50% is the point at which a win or loss gives you the most information about your performance. See this article for a deeper explanation. What this means, in practice, is that when the winchance is at 50%, you get feedback (from the game's end-state, at least) more efficiently than you do at any other winchance.
To be more precise, this should read "when the initial winchance is at 50%", right? I don't think it infers that that a randomly sampled winchance during the match being 50% tells us anything about the end-game feedback.
 
#64
No, it doesn't just apply to the initial winchance. What would make the argument apply only to the first moment of the game, and not to any other moments? The argument in the article I linked was initially intended to make a point about the initial winchance, but in fact the same argument proves the more general point that I'm making about winchance at any particular time during the match.
 
#65
  • I(p) is anti monotonic, as p increases I(p) decreases and as p decreases I(p) increases.
  • I(0) is undefined.
  • I(p) is always greater than or equal to 0.
  • I(1) = 0, events that are guaranteed to occur communicate no information.
  • I(p1 p2) = I(p1) + I(p2), information due to independent events is additive.
(Almost) flawless! Except the last one, not a clue what’s going on there...

It turns out that the following function has all these properties,

I(p) = log(1/p) = -log(p)

We will use this function in the next section as our measurement of the information content of an event with probability p.
Not so much. I’m afraid this is affirming the consequent - just because the function you’ve offered matches the properties you’ve described of information, doesn’t mean that that function describes information in results of decisions. Maybe, you could find another function with the same properties! The problem is, those properties you’ve given are accurate, but not a complete description of information: not everything that has those properties is necessarily information. In order for a proof, you have to use nothing other than the properties you’ve described.

In any case, I think we are still concerned about expected chance of success rather than actual chance of success. Something isn’t informative when it contradicts your expectations (as you point to yourself in the article). So if something contradicts your expectations, even if it was in fact the likeliest result, that is informative, and vice versa - if you totally expected something to happen, that is uninformative, even if it was actually very unlikely.

So your proof doesn’t hold, and even if it does, we’re still talking about expected results, not actual.
 
#66
Not my article btw, I just linked to it.

Using -log(p) for the information content of an event is standard information theory, afaik (I'm no information theory expert, so let me know if you more than I do and I'm gettting something wrong). See this wikipedia article. And in fact that last property, I(p1 p2) = I(p1) + I(p2), is actually quite important, because, to quote the wikipedia article, "The class of function f(⋅) having the property such that f( x ⋅ y ) = f( x ) + f( y ) is the logarithm function of any base" (and, from what I understand, logarithms are the only continuous real-valued functions that satisfy that functional equation).

The proof in the article, as far as I understand, assumes that the "player" knows the true probability. In other words it assumes that both the expected and actual probabilities are 50%, not just one or the other. Anyways, this is probably a moot point. I doubt it's possible to conceal the true winchance from a player over many matches if the player is seriously trying to improve at the game, so you can't permanently maintain a difference between expected and actual.
 
#67
Ok, since I’m not an expert either, I’ll surrender to authority on the premise that only logarithmic functions can describe additive functions. However, I’m still not convinced that information, as we are meaning the word here, is additive; I will need a proof of that. I’m not able to surrender that to authority, because we may be using the word “information” to mean something different to the sort of “information” that is additive.

Assuming that the player knows the win-chance isn’t very helpful, since more often that not, the player has only a very nebulous idea of what their win-chance is, though you’re right that in the long term, they generally won’t be wrong.

It’s more helpful to examine the concepts used without adding any new assumptions. The article does clearly state that “information” is a consequence of surprise. Surprise is independent of the actual probability, and entirely dependent upon the expected probability.

(Of course, why surprise should necessarily lead to “information” in the sense of enhancing the player’s decision-making skill is another long logical leap...)