Gen 1 Question for Game Theory Experts

Hipmonlee

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Hi, I'm working on the prediction part of my RBY guide, and I think I am wrong about imperfect information games, so if anyone can correct me I'd appreciate it.

So I am using the example of Countering Snorlax with a Chansey.

So assuming Chansey has Counter and Lax has Body Slam and Earthquake we have a matrix, and slotting in some rough estimate payoffs we get a matrix that looks something along the lines of this:
Chansey CounterChansey switch to Exeggutor
Lax Body Slam-155
Lax Earthquake60

Ok, but in the scenario where Chansey doesnt have Counter and we have something like:
Chansey switch RhydonChansey switch Exeggutor
Lax Body Slam-25
Lax Earthquake40

So my understanding had been that if you knew that the Chansey has a 25% chance of having Counter you could combine these two together, and you would end up with something like this:
Chansey Counter or switch to RhydonChansey switch Exeggutor
Lax Body Slam(-15*.25) + (-2*.75) = -5.255
Lax Earthquake(6*.25) + (4*.75) = 4.50
And get Lax should Body Slam 31% of the time and Chansey should...
And once I got around to writing this out I realised at this point that this probably doesnt really actually make sense. Because the Chansey user knows whether or not Chansey has Counter.

I know that the correct way to solve this is to expand this out into a single matrix, but the game of Chansey selecting whether or not to run Counter is going to be far too complex to be of any practical benefit. So does anyone know how I could include a piece of knowledge like 25% of Chanseys will run counter into this kind of game?
 

Jorgen

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I took a crack at it:

No matter what mixed strategy Snorlax adopts, there exists a scenario where Chansey can use its extra information to benefit from it. From the payout tables provided, the equilibrium against Counter Chansey is to use Body Slam p = 3/13 of the time, whereas against non-Counter Chansey it's p = 4/11. Deviating from either of these permits a strictly dominant strategy for Chansey in some scenarios, and because 3/13 is not equal to 4/11, that means there's no mixed strategy from Snorlax that forces a mixed strategy from Chansey.

From there, we adopt a strategy that maximizes what value we can get, assuming Chansey can use its perfect knowledge to change its strategy to best address our own. We end up considering a piecewise function:

1) where Chansey is compelled to always predict EQ when p < 3/13
2) where Chansey ideally adopts a mixed strategy to address Snorlax IF it has Counter; otherwise it predicts EQ; when p = 3/13
3) where Chansey is compelled to always predict Slam IF it has Counter; otherwise it predicts EQ; when 3/13 < p < 4/11
4) where Chansey is compelled to always predict Slam IF it has Counter; otherwise it adopts a mixed strategy; when p = 4/11
5) where Chansey is compelled to always predict Slam when p > 4/11

In case 1, the expected return is 5p.
In case 2, the expected return is 15/13, or the maximum value of 5p from the previous case. We can now eliminate case 1 as a desirable strategy, as this is strictly better than anything case 1 can give us.
In case 3, the expected return is 6 - 21p in the case of Counter, and 5p in the case without. 6 - 21p is always less than 15/13 in this case, which in turn is less than 5p, which in turn would compel Chansey to always choose the "with Counter" subgame. We therefore eliminate this possibility to maintain our expected return at 15/13.
In case 4, the expected return is -18/11 in the case of Counter, and 20/11 (the maximum value of 5p from the previous case) without. Chansey will always choose to play the "with Counter" subgame, which gives us negative expected return, so we avoid this as well.
In case 5, the expected return is 6 - 21p in the case of Counter, and 4 - 6p without. Both cases yield worse results than the previous case, which we already discounted, so we eliminate this as well.

So case 2, where p = 3/13, is our best mixed strategy for Snorlax. In this case, Chansey adopts the Nash equilibrium mixed strategy if it has Counter (which is to use Counter 5/26 of the time), just in case Snorlax gets any funny ideas. Chansey uses the strictly dominant "always predict Earthquake" strategy if it doesn't have Counter.

Note that the rate of Counter Chansey usage was not required knowledge here. If you want Counter Chansey usage to be fixed, the reasoning about cases 3 and 4 in particular would need to be revisited (but, importantly, this only affects Snorlax's strategy, as Chansey's is conditioned upon whether or not it has Counter). Regardless, I think the key insight is that Snorlax cannot force Chansey to adopt a mixed strategy.
 

Hipmonlee

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Could you clarify what you mean by Chansey should 'choose the "with Counter" subgame'? I've been playing around with it this evening and I haven't got my head around that bit.
 

Jorgen

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It was my attempt to deal with Snorlax's incomplete information by framing this as some kind of dynamic game, one where Chansey makes the first move (either bring counter or don't), and then one where Chansey & Snorlax choose an option simultaneously. Naturally, there are more factors that play into whether Chansey spends a moveslot on Counter, so if you want to continue with the "25% of Chansey users will use Counter" line of reasoning, then your calculations for Snorlax's expected returns seem solid, it's just that Chansey now has fixed strategies of "always use Counter" when it does have counter, and "always predict EQ" when it doesn't. I think.
 

Hipmonlee

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Ok, I built a script to play around with this. And from simple experimentation it seems like the Snorlax user should choose whether to play the Counter game or the non Counter game based on the likelihood that the enemy Chansey has Counter. But once it has decided which game to play it just uses the probabilities based on that game. And the tipping point appears to be just under 20%.

So in summary, if you agree with my numbers, which you probably shouldnt, then the best strategy for Quake Lax against Counter Chansey is to assume the enemy Chansey has Counter if you think that the likelihood of it having Counter is higher than 20%. And if you think the likelihood of it having Counter is less than 20% then you should assume it does not have Counter.

I think the next step for me is to figure out how to do the maths to determine what that 20% cutoff is without just trial and erroring it.

I've attached the script if anyone is interested.
 

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Jorgen

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I think that roughly-20% cutoff is, in exact terms, 5/26. Or roughly 0.1923.

Which is exactly the same rate that (I thought) Chansey should click on Counter, assuming it has Counter and everyone already knows it. I'm not sure if this points to a fundamental link between those two rates or a fundamental misunderstanding on my part.
 

Hipmonlee

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Ok I think I have it. You're right.

Ultimately it comes down to the fact that for Chansey having Counter dominates not having Counter. So ultimately the Chansey needs to Counter enough to force Lax to play the Counter game.

So in my script, I assume that Chansey only plays the Counter game when Lax plays the Counter game. But when Lax plays the Counter game it doesnt matter what Chansey does. This is misleading since the Chansey user doesnt actually know the probability that Lax will Body Slam. I think Chansey should actually Counter often enough that the likelihood of it having Counter * the likelihood of it using it comes to 5/26.

Having said that this seems to suggest there is no reason for Chansey to have Counter more than 5/26. That would also imply that Chansey should always Counter when it has it against Lax (assuming you use Counter on your Chan the correct amount of times. But! This doesnt really make sense, because then Lax could know whether you have Counter or not based on whether or not you used it the first time (assuming it survived). I guess these things will affect your payoff matrix, and come into the decision of how often you want to have Counter. I guess for those reasons it does make sense to have Counter more often than you intend to use it.
 

Jorgen

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Yes, we simplify a ton in this example, which we treat as a one-shot game with a fixed payoff matrix. In reality, Chansey could, for instance, wait for a game state that offers a particularly juicy (expected) payoff, or wait for a moment when the Lax player assigns an appropriately low probability of Chansey having (the yet-unrevealed) Counter.

Also, as a reminder for somebody reading this, because it's easy to go too deep into these toy examples: these numbers and payoff matrices are extremely fake. 5/26 is probably within an order of magnitude of the correct Counter rate, but hardly precise.
 

Hipmonlee

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Also its important to note that there isnt one correct counter rate. Battle state will affect this massively.

Having said that, in general for most of the sort of standard lines where Lax finds itself against a paralysed Chansey for the first time, I think we have come up with numbers that feel about right, from my experience anyway (bearing in mind that Lax wont always have Earthquake either).
 

Bughouse

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5/26 is probably within an order of magnitude of the correct Counter rate, but hardly precise.
I haven't read this thread thoroughly at all, but isn't saying 5/26 (or ~20%) is within an order of magnitude effectively narrowing the range of possible rates from 2% to 100% (since 200% doesn't really mean anything more useful than 100%). Which is to say, it has no useful predictive power at all.
 

Isa

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I haven't read this thread thoroughly at all, but isn't saying 5/26 (or ~20%) is within an order of magnitude effectively narrowing the range of possible rates from 2% to 100% (since 200% doesn't really mean anything more useful than 100%). Which is to say, it has no useful predictive power at all.
thank you for this insight into semantics. do you wish to contribute to the topic at hand as well or are you just here to nitpick?
 

Bughouse

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...no I'm asking a legitimate question about whether this thread has come to a useful conclusion that could be used in games, or if it's essentially determined that there really is no dominant strategy, even in one pretty particular defined battle state.

I'm literally just asking if the proposed answer to the question is actually a useful one. It's no one's fault if the answer isn't useful. It might still be the right answer.
 

Jorgen

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I'll admit I didn't put much thought into "order of magnitude". I'll be unambiguous and say we have no reason to believe these numbers, per se, are pertinent to playing RBY. However, the process is relevant, particularly for navigating those so-called "50-50"s that pop up every now and again.

That said, If you wanted to talk seriously about orders of magnitude in this context, it might make more sense to talk about the odds rather than the probability itself. So in this case, we're talking within an order of magnitude of 5/21 (use counter / don't use counter), which would be between 5/210 (5/215 rate) and 50/21 (50/71 rate), which is an obscene spread. If you work in base 2 because you love computer, it's more like 5/42 (5/47 rate) to 10/21 (10/31 rate), which is more reasonable.
 

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