Skill, or glorified guesswork?

[MrIndigo]

The mathematics of prediction: [Avinash Q. Dixit and Barry J. Nalebluff; Thinking Stategically]

Let's say you have a tennis player receiving and his opponent is serving. If he correctly guesses his opponent will serve to his forehand, he will return it at 90% success. If he correctly guesses his opponent will serve to his backhand, he will return at 60%. If he predicts forehand, but his opponent serves backhand, he will return at only 20%. It is slightly better if he guesses backhand and opponent serves forehand, with a 30% return rate.

What is the best strategy for server and returner?

The equilibrium point is found by graphing the following:

On the x-axis, you have "percentage of times the server aims to forehand", ranging from 0 to 100. On the y-axis, you have "percentage of successful returns". Plot the line "anticipate backhand" between the endpoints (0, 60) and (100, 30). Plot the line "anticipate forehand" between the points (0, 20) and (100, 90).

Now, the SERVER wants to find the x-value where his opponent cannot change strategy to get a better result; i.e. the highest minimum value of these two curves. This occurs at the crossing point of the two curves.

Similarly, the RECEIVER wants to find the y-value where his opponent doesn't reduce the line by changing his percentage of forehand/backhand. The crossing point is the place this occurs.

The crossing point is the value (40, 48) in this scenario.


This same logic can be applied to the equilibrium point of prediction. You have to weigh up what the success value is for making 100% prediction one way or 100% prediction the other way, graph the lines of values for your opponents' different options, and find the equilibirum point. This tells you what your optimum percentage spread for making the decision will be.

Note, you have to be random. If you got 33%, you shouldn't alternate between Choice 1-Choice 2-Choice 2, Choice 1-Choice 2-Choice 2, etc. You should roll a die before each decision, and if you get 1 or 2, make Choice 1. If you get 3-6, you make Choice 2.

 

[mtr]

This thread is getting really intense...

Let's try another simple endgame scenario: but this time, there's an element of recursion, which complicates the math.

Aggro player has Infernape on the field @ 25% health. Stall player has Celebi @ 70% and Blissey (Aromatherapy/Softboiled)@ 65% (so a crit makes no difference to ease calculation). Infernape has Life Orb, Celebi and Blisey have Leftovers. A Sandstorm is raging.

So Infernape has Fire Blast (the miss chance complicates things) and Close Combat, and will be dead in 2 attacks (Shoddy Battle doesn't count Life Orb recoil until after the turn). It basically has to match the correct attack on the first turn to win (unless it misses), and otherwise it loses. What is the best strategy for the aggro and stall players?

-If Infernape uses Fire Blast and the defender uses Celebi, the aggro player has an 85% chance of winning, and a 15% chance of being forced to repeat this cycle, as he is then @18.75% from sandstorm, which still allows him 2 attacks (each successful attack takes 16.25%, counting sandstorm).
-If Infernape uses Close Combat and the defender uses Celebi, the aggro player loses.
-If Infernape uses Fire Blast and the defender uses Blissey, the aggro player has a 15% chance of losing and an 85% chance of repeating the cycle (if Fire Blast misses, the cycle repeats, but only once more).
-If Infernape uses Close Combat and the defender uses Blissey, the aggro player wins.

So let's assume that Fire Blast misses either Celebi or Blissey(a 15% chance). The attacker still has 2 attacks, and must repeat the cycle. But this time, it's somewhat simpler:
-If Infernape uses Fire Blast and defender uses Celebi, aggro player has 85% chance to win, 15% chance to lose (if he misses this attack, the residual damage only allows him one more attack, meaning that he can't kill 2 people)
-If Infernape uses Close Combat and defender uses Celebi, aggro player loses.
-If Infernape uses Fire Blast and defender uses Blissey, aggro player loses (even with a miss, the sandstorm damage ensures he can only fire off another attack).
-If Infernape uses Close Combat and the defender uses Blissey, the aggro player wins.

So what are the best strategies for the aggro and stall players?

I'll write up an attempted solution later. I might need some help with this, so thanks. Remember that Fire Blast misses, which is the whole point of this. Assume that players do not have a conservative or liberal playstyle.

 

[mtr]

My work:

In these tables, r signals a chance of recursion. The first probability is the aggro player's win chance or a chance of a recursion, the second one is the stall player's win chance or the chance of a recursion.

The first time around.
              Cele        Bliss
FB         .85,.15r      .15r,.85
CC          0,1           1,0

The second time around.
              Cele        Bliss
FB         .85,.15        0,1
CC          0,1           1,0

I'm modeling this after Hipmonlee's example. First, we'll do the aggro player. So we have f(x) = expected value of the staller's switch and b = probability of using Fire Blast. This becomes rather complex due to the recursion.

The expected value of Celebi is equal to the sum of the value of the CC and the value of FB, but as of now, the value of FB has no definition in terms of p, so we might have to do algebra. But to make this simple, we will break it down into a Celebi-->Celebi and a Celebi-->Blissey.

f(Cele-->Cele) = 1*(1-p) + .15(1*(1-p)+.15*p)
= 1-p + .15*(1-.85p)
= 1-p + .15-.128p
= 1.15 - 1.128p

f(Cele-->Bliss) = 1*(1-p) + .15(1*p+0*(1-p))
= 1-p + .15p
= 1-.85p

So IIRC, we add the two together

f(Cele) = f(Cele-->Cele) + f(Cele-->Bliss) = 2.15 - 1.978p

to get out function for choosing Celebi. Now we do the same for Blissey. Remember that we have to take into account the chance of the aggro player picking Close Combat as well, but for Blissey, return on Close Combat is zero.

f(Bliss-->Bliss) = 0*(1-p) + .85(p) + .15(0*(1-p)+p)
= p

f(Bliss-->Cele) = 0*(1-p) + .85(p) + .15(1*(1-p) + .15p)
= .85p + (.15-.15p+.0225p)
= .15 + .7225p

So f(Bliss) = f(Bliss-->Bliss) + f(Bliss-->Cele) = .15 + 1.723p

Now we set the two equal to one another:
f(Bliss) = f(Cele)
2.15 - 1.978p = .15 + 1.723p
2.00 = 3.70p
p = 54.05%

So assuming optimal play, the Infernape should use Fire Blast 54.05% of the time.

I'll do the stall player later. As for now, I'm going to let this stand as my 500th post, halfway to the 1K.

 

[Hipmonlee]

I got a slightly different result to you, and I believe it is because you are assuming the chance of fireblasting will be the same the first turn and the second. Which is not the case.

By my calculations the probability of fireblasting on the second turn would be 54.05% but on the first turn it would be 54.41%. Though it surprises me that my probability for fireblasting on the second turn would be the same as yours for both turns..

What I did is I calculated the chance of the aggro player winning on the second turn, by subtracting 54.41% from 1 (IE the expected return of the opponent switching to blissey, which of course should be the same as the expected return of switching to celebi). Then I multiplied that by .15 and added that to .85 to find the value of fireblast for the aggro player.

I got

The first time around.
            Cele          Bliss
FB          .9189,.0811   .0811,.9189
CC          0,1           1,0

The I followed the same procedure from there..
E[Cele] = .0811p + 1 - p = 1 - .9189p
E[Bliss] = .9189p

1-.989p = .9189p
p=54.41%

To be honest I'm not 100% confident I'm right about this, but I am pretty sure you need a different variable for the chance of fireblasting on the second turn than the chance of fireblasting on the first.

But also note its such a slight difference, that you wont really feel the impact of it, which goes to show that estimating these things may not be all that difficult.

Have a nice day.

 

[mtr]

I'm obviously not 100% confident either if I tried to model my calculations after yours regarding Heracross and Garchomp.

I think you need a different variable if the chance of the second is dependent on the first. I assumed independence, I believe. Perhaps I was wrong.

 

[MrIndigo]

This thread is getting really intense...

Let's try another simple endgame scenario: but this time, there's an element of recursion, which complicates the math.

Aggro player has Infernape on the field @ 25% health. Stall player has Celebi @ 70% and Blissey (Aromatherapy/Softboiled)@ 65% (so a crit makes no difference to ease calculation). Infernape has Life Orb, Celebi and Blisey have Leftovers. A Sandstorm is raging.

So Infernape has Fire Blast (the miss chance complicates things) and Close Combat, and will be dead in 2 attacks (Shoddy Battle doesn't count Life Orb recoil until after the turn). It basically has to match the correct attack on the first turn to win (unless it misses), and otherwise it loses. What is the best strategy for the aggro and stall players?

-If Infernape uses Fire Blast and the defender uses Celebi, the aggro player has an 85% chance of winning, and a 15% chance of being forced to repeat this cycle, as he is then @18.75% from sandstorm, which still allows him 2 attacks (each successful attack takes 16.25%, counting sandstorm).
-If Infernape uses Close Combat and the defender uses Celebi, the aggro player loses.
-If Infernape uses Fire Blast and the defender uses Blissey, the aggro player has a 15% chance of losing and an 85% chance of repeating the cycle (if Fire Blast misses, the cycle repeats, but only once more).
-If Infernape uses Close Combat and the defender uses Blissey, the aggro player wins.

So let's assume that Fire Blast misses either Celebi or Blissey(a 15% chance). The attacker still has 2 attacks, and must repeat the cycle. But this time, it's somewhat simpler:
-If Infernape uses Fire Blast and defender uses Celebi, aggro player has 85% chance to win, 15% chance to lose (if he misses this attack, the residual damage only allows him one more attack, meaning that he can't kill 2 people)
-If Infernape uses Close Combat and defender uses Celebi, aggro player loses.
-If Infernape uses Fire Blast and defender uses Blissey, aggro player loses (even with a miss, the sandstorm damage ensures he can only fire off another attack).
-If Infernape uses Close Combat and the defender uses Blissey, the aggro player wins.

So what are the best strategies for the aggro and stall players?

I'll write up an attempted solution later. I might need some help with this, so thanks. Remember that Fire Blast misses, which is the whole point of this. Assume that players do not have a conservative or liberal playstyle.

Use the same mathematical process layout I outlined above, assigning payoff values to each outcome as a score out of 100.

 

[twash]

If Infernape does miss Celebi with Fire Blast, Celebi is likely to be Grass Knotting or Thunder Waving anyway. If Celebi stays in and Fire Blast misses, it's pretty much an instant loss for Infernape..

Personally, I would argue that some prediction is skill, some are just educated guesses, or as close as one can be. Once you are into the battle (as in, more than a few moves), you know more about the opponent's playstyle, and their team. From this you can make skillful decisions and plan your route of play from the resulting move. Often, offensive teams work around resistances, and so wild out-of-place moves may not seem like such a bad idea in these cases. Against stall teams, one may decide to try and limit prediction for both sides as much as possible, meaning they can go with their gameplan with less risk of odd moves by the opponent.

The more information you have about your opponent's team, the easier it is to make a decision (at least, in my experience). Generally, the same is said about the opponent, but then it's harder to get information about how the opponent plays consistently over time. Early-game, it's more likely to be a guess than a solid prediction of what you expect to happen.

 

[mtr]

Crap, I didn't take that into account. That would have eliminated the element of recursion, which would have made the calculation a lot easier.

I should redraft an example among similar lines: An Infernape was on the same team as a Roserade who has used Sleep Powder. He is facing a Skarmory (asleep, for some bizarre reason) and a Blissey. There is a sandstorm raging, and everything else is the same as before.

Use the same mathematical process layout I outlined above, assigning payoff values to each outcome as a score out of 100.

I got it, I think, although I made a mistake in assuming independent probabilities of Fire Blast. According to Hipmonlee, it's only a .35% difference to the correct answer, though.

 

[The SPrinkLer]

I do a little bit of both. As of late, I've been trying to not predict. As for your answer, I believe it's a bit of both. Based on how opponents play, which can be reasoned over the course of the battle, you can make smart moves. Kudos to mtr12, he summed it up perfectly. As such, I don't believe anything else needs to be said on my part. He took the words out of my mouth while considering what to post in this thread.

 

[Pim]

I am a big fan of game theory, but naturally, it's unpractical to use in normal battles. However, I do think that skilled players can estimate their chances as such that they will actually play a nash equilibrium, over the long run. (As you can view the mixed NE as randomization, but also as different choices when repeating.) Therefore, I do think that prediction is a skill.

 

[mtr]

I don't know. It's only practical to calculate a Nash Equilibrium in a simple endgame scenario like Infernape v. SkarmBliss or Hipmonlee's Heracross and Garchomp example. Theoretically speaking, a Nash Equilibrium would exist for any situation, as Hip said, but in anything but a simple endgame, it's very complex. Not to mention that the optimal probabilities very rarely reach over 55%, meaning that its still practically a coin toss.

 

[The SPrinkLer]

What's a Nash Equilibrium?

 

[mattj]

I hope this doesn't come across wrong, but it seems you guys are taking this way too far (no offence). When you guys are playing PBR, or are in a tournament, do you really whip out a TI-90 and do all this math, or do you just go with what you've memorized about Commonly Used Movests and Combos and your gut feeling about the skill level of your opponent? I personally trash the calculator for my memory and my personal evaluation of my opponent. Sometimes it works, sometimes it doesn't.

For an example of it working, the other day I played a very good friend of mine in the final round of a PBR tourney for 1st Place. It was kind of weird, because we're trading/EVtraining/battling partners already, so we share lots of pokemon and ideas/strategy and we battle each other a lot already, so I was'nt sure how the game was going to go. We actually started the match with one of the same pokemon, though with different items. However, because I know him well, and because of his other lead pokemon (doubles) I predicted a protect and totally pwned one of his leads. I then knew he had only a 50% chance of getting protect off on that pokemon, so I focused on it and KO'd it. I later got 2 lucky guesses on who he would protect with and won 6/1.

It all came down to lucky guesses (he could have attacked instead of protecting, but the circumstances favored protecting) + my evaluation of him.

Hope this doesn't derail your math party. :/

 

[MrIndigo]

What's a Nash Equilibrium?

A Nash equilibrium (named for the late John Nash, the mathematician in "A Beautiful Mind") is a situation that arises in zero-sum games (i.e. games where one player's payoff is equivalent to another player's loss).

The Nash equilibrium is the situation where neither player can change their strategy to yield a better return.

The classic example is the Prisoner's Dilemma. Let's say you, an innocent man, and Jeff, another innocent man you have never met, are captured by the Gestapo. You are put in separate rooms and told:

"We have captured your accomplice Jeff. We are willing to be lenient if you confess to us, however.

If you confess, and Jeff does not, you will get one year in prison, and he gets 10 years in prison.
If you confess and Jeff confesses, you both get 6 years in prison.
If you don't confess and Jeff does, you get 10 years and he gets 1.
If you don't confess and neither does Jeff, you both get 3 years in prison."

The best option for both of you together is to stay silent, and get 3 years each. But wait! Lets say that Jeff stays silent. You now have the choice between staying silent and getting three years, or confessing and getting 1 year. So obviously, it is better for you to confess.

Let's say Jeff confesses. You now have the choice between staying silent and getting 10 years, or confessing and getting 6 years. Again, you are better off if you confess. This makes confession a "dominant strategy"; in any situation, confession always makes you better off than not confessing.

Of course, Jeff comes to this same conclusion too. The end result is that both of you confess, and both get 6 years in prison. The confess-confess situation is a stable Nash equilibrium. Neither player can improve his position by changing strategy (even though a better outcome, namely the silent-silent situation, is actually a better outcome!).

The silent-silent outcome is sometimes called an unstable Nash equilibrium. Assuming both parties are rational actors, they cannot improve their position by changing strategy (because their opponent does the same and the end up with 3 more years each to their sentence). The reason it is unstable is because while changing strategy together yields a worse result, there is no incentive not to change individual strategy.

 

[MrIndigo]

Actually, for purity's sake, the mathematical formalism I gave at the top of this page is valid only for various instances of the EXACT SAME situation.

In terms of sequential moves, you would have to modify the probabilities involved using Bayesian reasoning as you obtained new data (like Fire Blast missing), or alternatively just restart the problem with new probabilities for the second turn, changing the payoff values after the first turn's results.

 

[WarriorPrince]

Prediction can and will only be guessing, and both players will more than likely try to predict, so there is never a sure choice. Predicting is attempting to choose the most probable of all the possibilities. Sometimes things work out, sometimes you end up in a world of hurt.

 

[capefeather]

*hasn't read other people's posts yet*

Prediction is definitely a skill. It involves lots of knowledge to gain an accurate appraisal of the risks and rewards and, as people have already said (OK, so I skimmed through a couple of posts) a Nash equilibrium. It also involves an ability to understand your opponent, trying to figure out any behaviour patterns in the short time in which it matters. A Pokémon battle is comparable to repeated weighted rock-paper-scissors games (or Prisoner's Dilemma "games") and my second point is what makes those potentially interesting. Lastly, it involves your willingness to take a risk every once in a while, because if you never take a risk, you may be forced into losing Nash equilibria or some-such.

 

[mtr]

I hope this doesn't come across wrong, but it seems you guys are taking this way too far (no offence). When you guys are playing PBR, or are in a tournament, do you really whip out a TI-90 and do all this math, or do you just go with what you've memorized about Commonly Used Movests and Combos and your gut feeling about the skill level of your opponent? I personally trash the calculator for my memory and my personal evaluation of my opponent. Sometimes it works, sometimes it doesn't.

For an example of it working, the other day I played a very good friend of mine in the final round of a PBR tourney for 1st Place. It was kind of weird, because we're trading/EVtraining/battling partners already, so we share lots of pokemon and ideas/strategy and we battle each other a lot already, so I was'nt sure how the game was going to go. We actually started the match with one of the same pokemon, though with different items. However, because I know him well, and because of his other lead pokemon (doubles) I predicted a protect and totally pwned one of his leads. I then knew he had only a 50% chance of getting protect off on that pokemon, so I focused on it and KO'd it. I later got 2 lucky guesses on who he would protect with and won 6/1.

It all came down to lucky guesses (he could have attacked instead of protecting, but the circumstances favored protecting) + my evaluation of him.

Hope this doesn't derail your math party. :/

Well, all this math is fun and all, but purely theoretical. I mean, with my answer, I only got that the Infernape user should use Fire Blast 54.05% if we assume independent probabilities for the Fire Blast, 54.41% if we do not make such an assumption (according to Hipmonlee). That's hardly better than coin tossing, and operating on the assumption that both players play "optimally" is not fair to do. So basically, here's what I try to do.

Well first of all, I try not to involve too many predictive elements on my team. If I'm playing offense (I usually play stall), I try to stay with 1 Choice item, maybe 2 at max (I play Singles OU and Singles Ubers). Some Ubers players are confident in themselves enough to use 4 Choice Items. I am not.

I try not to make predictions that exhibit rather abundant quantities of temerity early in the match. This means that I will attempt to avoid risky double switching if at all possible unless I'm losing and I need to make a bid for counterplay. I will also attempt to strike for neutral damage against various threats rather than making the obvious move or a ballsy prediction. I mean, If I have a ScarfGon on the field and I need to revenge kill a Mixmence who just blasted one of my dudes, I won't use Outrage. Otherwise, I might have just lost it to a Scarftran. I won't use Earthquake unless I need to, or otherwise I'm giving a setup chance to a Gyarados/whatever, or worse, letting his Salamence blast me. I might use Stone Edge because it has an 80% chance of killing the Salamence anyways after SR and LO recoil (and accuracy), and even if I guess wrong, I still have a Flygon to spare, and I've blocked his Gyarados from gaining momentum (Stone Edge 2HKOes even with Intimidate and SR, a crit OHKOes, and I outspeed Gyara, so it's the other dude's risk).

And if I'm in dire straits, I will try to guess aggressively. On stall, this might mean making a blind double switch to try to get in Forretress if Spikes are raping me up the ass. On offense, this might mean using Dragon Pulse with Heatran predicting the Latias if I'm down 5-3.

That's all I can think of right now, at any rate. Basically, I recognize that prediction, at its heart, is guesswork, and play accordingly. A chess player (ok, maybe not all chess players, but Tigran Petrosian at the least) rarely makes risky gambits to try to get an advantage early on, he does it when making a bid for counterplay. I try to play Pokemon the same way.

*hasn't read other people's posts yet*

Prediction is definitely a skill. It involves lots of knowledge to gain an accurate appraisal of the risks and rewards and, as people have already said (OK, so I skimmed through a couple of posts) a Nash equilibrium. It also involves an ability to understand your opponent, trying to figure out any behaviour patterns in the short time in which it matters. A Pokémon battle is comparable to repeated weighted rock-paper-scissors games (or Prisoner's Dilemma "games") and my second point is what makes those potentially interesting. Lastly, it involves your willingness to take a risk every once in a while, because if you never take a risk, you may be forced into losing Nash equilibria or some-such.

Well, you didn't miss much. Mostly the debate moved to Game Theory and stuff, with us discussing courses of action in endgame scenarios like Mixape v. SkarmBliss with incapacitated Skarm (for the convenience of calculation, lol).

 

[DCJ]

Although I have merely skimmed the previous posts within this thread, I can confidently sum up my opinion of prediction.

Prediction is the "skill" used by competitors to apply knowledge previously obtained through past experience and what has happened thus far within a battle to give yourself as positive of an outcome you can receive while outweighing the risk and reward of being correct or incorrect. Simply put, it is how well you can mentally assess your situation, outwitting your opponent. Within doing so alongside an opponent you create a web of choices that will either have a favorable or unfavorable outcome that wraps around both players. As decisions become harder and harder to pick, players will start (with a lack for a better word) guessing. They come to a conclusion of picking an action without having much logic behind having X done when Y seems almost equally viable. When intelligence and instinct intertwine you are given the concept that I simply call the tool of prediction: Playing off every piece of information (and possible deception) to leave yourself ahead of the opponent while handpicking each action you take as you contemplate a blend of impulse and intellect.

It is absolutely NOT guessing between 1 and 2. "Skill" is not exactly the word I'd use to describe prediction, but it can fit the bill for many, and I have no problem with it. Prediction is a very large factor in competitive Pokemon, and definitely a concept any battler would benefit from understanding.

 

[Lucalibur]

i prefer to do this at a easy style,let me explain,i will only try to predict what my enemy is going to do when i get enough info of they team,in the beggining i have no problem is going all out like if i was berserk(but normaly SR if first priority for me) so i will fight,and has the battle go on,i slow down and begin to think if the info i got is enough to begin the REAL mind games,lets say the enemy got a gliscor on the team,but they got tyranitar in,i got my lucario on the field,i am sure he would switch to gliscor if he never saw any pokemon in my team that carried ice beam,if i know he dont know enough about my team,i will put that pokemon with ice beam hoping he would switch to his gliscor,lets say he switched,he knows the pokemon i got in field now got ice beam so he switch or die,if he switch the pokemon he placed take SR damage,ice beam damage and give me a free turn to think and depending on how much we both know about the other team,i will make my move,if he dont switch,he give a free switch to his pokemon,but lose his gliscor,what would stop my lucario from coming back after some time and begun to sweep after SD

 

[boredomisbliss]

its glorified guesswork no doubt, ive u-turned heatrans and gyarados, but ive also had scarftran locked on FT stay in on his scarftran while he FBd and earth powered before i even knew he had a heatran.

it is my opinion that if two people sit down and think long enough, the choice that seemed so stupid to start now seems like a good option, and eventually it becomes 50-50= complete luck

 

[Eraddd]

It's a medley of both.

On one hand, you have people who can properly analyze their team, and think what they'll bring in, due to their playstyle (conservative, liberal, etc), their team (who counters who, and who's a bigger threat than who) and more.

On the other, it just usually comes down to playing mind games with your opponents, trying to mind fuck them. You have a specs jolteon locked on thunderbolt, against the gyarados and you know that he has a dugtrio. What do you do? Do you take a chance and Tbolt it, or do you take the safe route, and go out. The opponent, recognizing also that you would probably switch out due to the huge counter, might just dragon dance, completing a sweep. But on the same note, Jolteon can also choose to T bolt, recognizing that gyarados might take the chance to DD on the switch, thus going round in circles and circles. This is where it's glorified guesswork.