Pokemon Battling: A Lesson in Game Theory

[McGrrr]

This is actually my third attempt (in seven years) at a thread discussing the application of Game Theory to Pokemon battling. Despite good intentions, my first thread was compromised by a very basic understanding of key concepts, and I had trouble articulating my thoughts second time around. I hope that now I can finally offer something coherent, educational, and perhaps even interesting, to the average reader.

The Case for Jolly

During ancient times (the first and second generations), it was accepted that a perfect speed DV (a hexadecimal "deter value" for the uninitiated!) was absolutely necessary for competitive battling. Yet, this need for speed was forgotten during the early days of the third generation, as early innovators took awhile to understand the new IV/EV mechanics. However, everybody would once again fall in love with speed, this time manifesting as the dominance in usage of Jolly and Timid natures over Adamant and Modest.

Fast forward to the sixth generation, and we remain infatuated with speed, and rightly so. To illustrate, consider the following scenario:

1. We have Mega Charizard X, base 100 speed, 299 Adamant, 328 Jolly.
2. Our opponent (villain) has a Lucario (that can Mega Evolve this turn), base 90 speed, 279 Adamant, 306 Jolly.
3. Assume that half of all Mega Charizard X are Adamant, and the other half are Jolly. Assume also that half of all Lucario are Adamant, and the other half are Jolly. According to October statistics, Adamant Charizard X and Jolly Lucario are more common, but this assumption is fine for our purposes.
4. Neither player knows the opposing Pokemon's nature.
5. Assume that villain is competent. Therefore, he knows that switching out risks giving us a free Dragon Dance, potentially costing the battle. He uses a Jolly Lucario.

Exhibit A: The Adamant Pay-Off Matrix

Villain will certainly switch sometimes, and possibly the majority of the time, but he is actually making a significant mistake if that percentage is much higher than 50%, because of our assumption that 50% of Charizard X are Adamant (and therefore slower than Jolly Lucario).

Villain should play a mixed strategy where he sometimes Mega Evolves and uses Close Combat:

252+ Atk Adaptability Mega Lucario Close Combat vs. 0 HP / 4 Def Mega Charizard X: 284-336 (95.62 - 113.13%)

Not only is this a dominant strategy when Charizard is Adamant (meaning that the pay-off is superior regardless of our decision), but we sometimes uses Dragon Dance, and Mega Lucario can KO next turn with Bullet Punch. If Charizard uses Flare Blitz, it faints to recoil anyway.

If we are using Adamant Charizard, and we know that 50% of Lucario are Jolly, and that villain does not always switch, then we must sometimes switch as well, unless we can afford to lose Charizard.

Exhibit B: The Jolly Pay-Off Matrix

The game changes with a Jolly Mega Charizard X. Flare Blitz is now a dominant strategy, and we benefit regardless of villain's decision.

In short, being faster than the opposing Pokemon means that we use fewer mixed strategies, and therefore have fewer opportunities to make mistakes. This is because we are more often in a position where a dominating strategy exists.

The Illusion of Choice

Choice items are the embodiment of risk/reward in competitive battling, but the word "choice" is a misnomer. Sometimes, one (or more) attacks are never an option. The best illustration of this idea is Earthquake. Consider the following scenario:

1. We have a Choice Scarf Garchomp.
2. Our opponent has a Heatran. Villain also has a 100% healthy Dragonite. Stealth Rocks is not in play.

Unless we can deal with +1 Dragonite, Earthquake is never an option in this situation. A good opponent will know this.

Turn Zero

On Pokemon Showdown, the battle begins before the first turn. You can view your opponent's Pokemon, and can therefore tailor your lead to gain an advantage. For example, against a team of physical threats, I lead with Gyarados for an immediate Intimidate. I otherwise never lead with Gyarados.

You should also identify your most valuable Pokemon based on likely favourable match ups. For example, if I see a Gliscor, Gengar is automatically my MVP (unless I have even more compelling options), because I can switch in with impunity, and use Substitute as Gliscor switches out. From turn one onwards, it is a sequential game to engineer this precise match up.

Additional content to come when I'm more inspired and less lazy.

Related: Pokemon Breeding: A Lesson in Economics

 

[Myzozoa]

I would be curious to know what you think about entry hazards and 'solving' pokemon. If you have Stealth Rocks on the field, the game can now only go on for a finite number of turns (as before you and your opponent could double switch an infinite number of times so in theory the game never has to end) , this doesn't mean that entry hazards solve pokemon battles, as even with hazards there is not usually a 100% certain win based on a sequence of moves, but it does have a huge change on the game in going from a potentially infinite series of double switches (and I could attempt to contrive a situation where each player has 3 pokemon and it is always 'most correct' for both players to double switch) to a certainty that the game will end in x turns. Basically I think entry hazards are the most powerful moves in pokemon because they lead to situations where there is a 'correct and advantageous ' move on every turn for the player with hazards, ultimately leading to his victory. Hazards aren't a guaranteed win for the player with entry hazards, but they are the closest thing to a move that 'solves' the game.

 

[Calm_Mind_Latias]

I really do not know how to quantify and formalize this in game theory language... but it seem somewhat intuitive that at critical speed tiers, such as base 108 in gen 5 OU, most people would try to leverage that speed to at least tie other 108 speed. I would suppose that for OU Pokemon that base speed is inversely correlated with "bulk", and that for a base 108 Pokemon which is unlikely to take many hits and speed, the best form of "defense" (striking first before they have an opportunity to strike) against similar offensive Pokemon. Is there a Nash equilibrium for Jolly for various speed tiers? But it would seem difficult if not impossible to prove a given "Nash equilibria" exists given the diversity of possible strategies in the metagame. If there is no Nash equilibria for a given situation in battle, the best strategy is usually a mixed strategy unless one can predict the opponent using psychological considerations.

Since offense was more prevalent in gen 5, it would seem one would have a greater expected payoff with Jolly by getting the jump on other high speed Pokemon as opposed to the additional 2HKOs an adamant nature would yield against stall teams.

 

[vonFiedler]

Maybe it took three attempts to make this thread because you aren't saying anything really new. Even though game theory predates games as we generally see them today (tabletop rpgs, tcgs, video games etc.), there's a reason it's called game theory and some elements of game theory can be found in any kind of game. If anything the lesson you're offering is just how much people take that stuff for granted, but then part of game theory is that everyone is already an unwitting player.

 

[Myzozoa]

I agree that there isn't anything 'new' information here, but though many people teach these principles to new players, it is rare to see the information presented through the frame work of game theory the way mcGrrr has done in this thread. The value of doing this type of work is the clarity of the presentation, the demonstration of this knowledge in a very graspable way or at least a differing way than Smogon normally articulates the information which may be easier for some new players or even current ones to understand.

 

[Shurtugal]

I have always felt that winning most pokemon battles depends on how well you know your opponent. The opponent that knows the other one best can often times set themselves in a winning scenario much more efficiently. It's what defines who is "in control."

For me, it's purely instinct. There are other factors like how many risks they do or don't take that I subconsciously store in the back of my mind or how well my opponent has read me, but for the most part, its what compels me which move or risk to take.

What I'm saying is that for me, the most dominant choice is the one that I know will put me in the lead against my opponent despite the risk factor.

Sometimes I manipulate my move choices and take greater risks to put pressure on my opponents to catch up with me.

I was wondering about your opinion on that. I also agree with Myo that hazards are the single best move in the game as its the move with the greatest rewards.

I also look at team symmetry to make assumptions on my opponents sets so I have a greater chance of assuming the right course of action, etc., which really factors in on what you are saying! (Like assuming Excadrill carries Rapin Spin and is therefore max jolly most likely when paired with one or more SR weak pokemon with no other way if removing rocks but with Exca's Spin).

 

[Mr.E]

It doesn't take a genius to notice Speed is more important than other stats. Speed is absolute; you either outspeed the opposition or you don't. (Or you tie and...) An extra turn is an oh-so-valuable extra attack, +100% damage dealt or getting your utility move off first. Heck, it's also worth -100% damage taken if you kill the opposing mon with that extra turn (-50%, -33%... for multi-turn KOs) , priceless if it lets you do something like pull off a faster Quiver Dance to tank the hit and go on to sweep! A non-Speed nature is, well, a 10% change in damage dealt or received, as advertised. Even the difference between 0 and 252 EVs is only 20-30% unless you're talking Blissey's Defense.

Regardless, I don't think the "need for speed" was ever "forgotten."

 

[Darknut07]

In the scenario with the Heatran against Garchomp, should we leave tran in then? I know you said unless you can deal with +1 Dragonite, but most of the time, we have at least 1 option to deal with +1 Dragonite (at least a check like scarfchomp lol). I know this would change radically depending on whether your opponent has a Dragonite hard counter or not, but lets say the only thing your opponent had was the Garchomp check (so he will switch out on the DD turn and threaten with his next pokemon, and upon its fainting switch Chomp back). What would you do?

 

[Shurtugal]

In the scenario with the Heatran against Garchomp, should we leave tran in then? I know you said unless you can deal with +1 Dragonite, but most of the time, we have at least 1 option to deal with +1 Dragonite (at least a check like scarfchomp lol). I know this would change radically depending on whether your opponent has a Dragonite hard counter or not, but lets say the only thing your opponent had was the Garchomp check (so he will switch out on the DD turn and threaten with his next pokemon, and upon its fainting switch Chomp back). What would you do?

If its Chomp vs. Heatran and DNite, you have to hope Outrage can crit more than once since its the only way you can possibly have a chance of winning. EQ won't affect DNite so your win chance is zero if you EQ

 

[Darknut07]

If its Chomp vs. Heatran and DNite, you have to hope Outrage can crit more than once since its the only way you can possibly have a chance of winning. EQ won't affect DNite so your win chance is zero if you EQ

Sorry that wasn't quite what I meant. Garchomp is the only pokemon left on that team that can check Dragonite at +1. But you do have other pokemon and so are not locked into one move.

 

[Explorer]

Being a math geek, this is an intriguing conversation for me. It would be nice to redo the calculations with the actual Adamant/Jolly ratios. Plus can't Lucario use Stone Edge for more power than Close Combat?

 

[blitzlefan]

Being a math geek, this is an intriguing conversation for me. It would be nice to redo the calculations with the actual Adamant/Jolly ratios. Plus can't Lucario use Stone Edge for more power than Close Combat?

Close Combat in this scenario is stronger - Mega Lucario's STAB Adaptability Close Combat hits 240 BP while a super effective Stone Edge hits Mega Charizard X for 200 BP, while is unreliable.

 

[alexwolf]

Js, but Charizard should always stay in against Lucario regardless of natures, as long as Charizard hasn't MEvolved. You just do whatever you want to do without MEvolving (attack or set up) so that you can have your Fighting resistance against Lucario to stomach one Close Comabt and either OHKO or set up with Dragon Dance.

 

[56k]

The real lesson here isn't about the deterministic options, but the shitty "opponent got a free hit" and "double switch" options that guaranteed happen in any real match. Suppose I'm the Lucario and he's the unknown Adamant or Jolly Charizard X or Y.

I need to build a team where "charizard gets a free hit on something" is actually bad for him, because I either have a true counter (Gyarados maybe?) or I have several checks lined up like mixed Landorus-I or Garchomp or Terrakion waiting (which can boost and should rightly scare him and get into his head). Or, I can pull off the 100% safe extremespeed for some damage, and finish it off with a fast Volt Switch if it's Y or a fast EQ or Stone Edge if it's X. However, I must finish it off with something that outspeeds AND puts questions to the opponent because I haven't revealed my item. Now it's 5-5 and even though I had a shit unlucky lead, black has parity because there are still questions for the opponent now (not 100% choices for him to make).

The worst thing I could do is switch to a slow bulky counter that has only exactly one option; attack, because if I don't attack then I get 6-0'd. I've already lost momentum and now we have here ANOTHER 50/50 coinflip and I can't gain any ground unless I attack him. But even if I kill Char, he has the next threat that gets a free hit on my guy. I think countering is very much less relevant nowadays and having multiple ace-up-your-sleeve checks against the metagame is the only real chance (which, on a related note, is why I'm not having as much fun playing and teambuilding with Gen 6 as I was in the more precision/stall/counter/you-activated-my-trap-card of Gen 3 and 4)

Sacrificing is so often the only answer to the game because it keeps your momentum and control of the match in order. Because you forgot something important in your game squares. What happens if you switch and he dragon danced.

 

[doughboy]

Overall power is sorta overrated. The key here is chip damage.

The game has really evolved over time to incorporate more and more chip damage, with lesser ways to combat it. Gen 2 introduced Spikes, but there (iirc) you could only have one layer that did 12%. Gen 3 introduced Sandstorm and improved Spikes that could stack to do 25% if you got all three layers. Gen 4 was icing on the cake with Stealth Rock and U-turn, the two best chip damage moves in the game (and what I and many consider to be the best moves in the game).

Ask yourself this: how meaningful is that 10% damage in the game? In gen 5 that meant crap. You could easily create a team and an environment where you could get chip damage on the opponent that would make up for the 10% extra power that you would be able to achieve with a +Atk or +SpA nature. Volt-Turn teams where the poster child of this phenomenon. First, sandstorm strips away Leftovers recovery from the opponent, the only real thing in Pokemon that counteracts chip damage. From there, you only need one SR switch in, one U-turn on a resisted target, or one Volt Switch on a resisted target to make up for the power loss on using a positive speed nature'ed pokemon.

My idea is this: you can create the opportunities for chip damage easily, but can not make up for the opportunities lost by using a neutral speed nature easily. All you need is that one SR switchin or Spikes switch, that one U-turn / Volt Switch attack, that one shot of priority (gen 6 will have this a lot), or those two rounds of permasand on the opponent. If you have the opportunity to create chip damage, do it since most of the time it will be the best move overall in the match (with some exceptions). My last team in gen 5 had all positive speed natures lol

I'll describe one exception to creating chip damage: your second layer of spikes. Going for a second layer of spikes is one of the worst moves in the game unless you can be assured you can set up the third. This is because the second layer of of spikes has the smallest damage footprint of all entry hazards.

SR: +12% damage
Spikes (layer 1): +12% damage
Spikes (layer 2): +4% damage
Spikes (layer 3): +8% damage
Toxic Spikes (layer 1): 12% end turn
Toxic Spikes (layer 2): 6% end turn 1 and increasing 6% every turn

So the second layer of Spikes does less damage overall than the first turn of Toxic damage. That's pretty small. So the turn spent setting up a second layer not very impactful on a match. To put it into perspective, let's say an opponent has a team that is completely vulnerable to Spikes (all grounded) and Deoxys-D's Night Shade (no normal types). Your Night Shade will do the same amount of damage to the team's "total health" as the opponent switching 25 many extra times with the second layer of Spikes down.

Edit: Stallion bulbapedia (go to gen 3) and smogon say otherwise. I thought that for a long time too.

1 /8 = .125
1/6 = .166
.166 - .125 = .0416 = 4.16%

Edit 2: realized I confused hp with percent w.q point still stands the second layer of spikes sucks

 

[Stallion]

Overall power is sorta overrated. The key here is chip damage.

The game has really evolved over time to incorporate more and more chip damage, with lesser ways to combat it. Gen 2 introduced Spikes, but there (iirc) you could only have one layer that did 12%. Gen 3 introduced Sandstorm and improved Spikes that could stack to do 25% if you got all three layers. Gen 4 was icing on the cake with Stealth Rock and U-turn, the two best chip damage moves in the game (and what I and many consider to be the best moves in the game).

Ask yourself this: how meaningful is that 10% damage in the game? In gen 5 that meant crap. You could easily create a team and an environment where you could get chip damage on the opponent that would make up for the 10% extra power that you would be able to achieve with a +Atk or +SpA nature. Volt-Turn teams where the poster child of this phenomenon. First, sandstorm strips away Leftovers recovery from the opponent, the only real thing in Pokemon that counteracts chip damage. From there, you only need one SR switch in, one U-turn on a resisted target, or one Volt Switch on a resisted target to make up for the power loss on using a positive speed nature'ed pokemon.

My idea is this: you can create the opportunities for chip damage easily, but can not make up for the opportunities lost by using a neutral speed nature easily. All you need is that one SR switchin or Spikes switch, that one U-turn / Volt Switch attack, that one shot of priority (gen 6 will have this a lot), or those two rounds of permasand on the opponent. If you have the opportunity to create chip damage, do it since most of the time it will be the best move overall in the match (with some exceptions). My last team in gen 5 had all positive speed natures lol

I'll describe one exception to creating chip damage: your second layer of spikes. Going for a second layer of spikes is one of the worst moves in the game unless you can be assured you can set up the third. This is because the second layer of of spikes has the smallest damage footprint of all entry hazards.

SR: +12% damage
Spikes (layer 1): +12% damage
Spikes (layer 2): +4% damage
Spikes (layer 3): +8% damage
Toxic Spikes (layer 1): 12% end turn
Toxic Spikes (layer 2): 6% end turn 1 and increasing 6% every turn

So the second layer of Spikes does less damage overall than the first turn of Toxic damage. That's pretty small. So the turn spent setting up a second layer not very impactful on a match. To put it into perspective, let's say an opponent has a team that is completely vulnerable to Spikes (all grounded) and Deoxys-D's Night Shade (no normal types). Your Night Shade will do the same amount of damage to the team's "total health" as the opponent switching 25 extra times with the second layer of Spikes down.

Good post, although the second layer of spikes actually adds 6.25% dmg, same as the third