Resource There is no way to outplay Sucker Punch

TLDR: If you're using Sucker Punch in a 1v1 last mon scenario, your best strategy is to calculate 1/(1+Sucker PP), generate a random number from 0 to 1, and use your non-sucker attack if the number is lower. For example, you should use your non-sucker attack 1/9 of the time if you have 8 Sucker Punch PP on that turn.

If you're playing against Sucker Punch in this scenario, nothing you do matters, so the best strategy to increase your odds is to get your opponent to waste as many Sucker Punches as possible before it comes to that point.

Note: Everything here assumes that the non-sucker punch player does not have moves like Encore, Disable, or Substitute, which significantly increase the odds of them winning.

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Why there is no way to outplay Sucker Punch
By bbg

burninator0 and I started arguing on Showdown this morning about the probability of winning a last-mon Sucker Punch mindgame and we ended up finding some very surprising results. Most notably, there is nothing the player facing Sucker Punch can do to have better odds of winning a Sucker Punch mindgame in a 1v1 scenario.

This is not just true statistically but also in practice. The only thing that affects the probability of either player winning is the strategy being used by the Sucker Punch player, and we've found the optimal strategy as well. It turns out that, if the Sucker Punch player is following the optimal strategy and has all 8 PP remaining, the probability of winning is 8/9 for the Sucker Punch player. There is nothing the non-Sucker Punch player can do to affect these odds.

We started off by thinking about the probabilities in the case where both players choose their moves randomly. There is a 50% chance that :Kingambit: uses Sucker Punch or Kowtow Cleave and a 50% chance that :Latios: uses Calm Mind or Aura Sphere. There are 4 possibilities:

- 25% of the time, :Kingambit: suckers and :Latios: uses Aura Sphere, so :Kingambit: wins.
- 25% of the time, :Kingambit: uses Kowtow Cleave and :Latios: uses CM, so :Kingambit: wins.
- 25% of the time, :Kingambit: uses Kowtow Cleave and :Latios: uses Aura Sphere, so :Latios: wins.



- 25% of the time, :Kingambit: suckers and :Latios: uses CM, so the mindgame continues and :Kingambit: loses 1 pp.



Every turn, there is a 50% chance that :Kingambit: wins and a 25% chance that :Latios: wins. There is also a 25% chance that the mindgame continues. So, before anything happens, :latios:'s chance of winning on turn T of the 1v1 situation is (.25)^T. Adding up these values from T=1 to 8 shows that :Latios:'s chance of winning approaches 1/3. Another way to think about this is that, on every turn that the game ends, Latios will win 1/3 of the time, and the only other scenario brings up that same scenario again, except for the small chance that Kingambit uses all 8 of its pp and Latios is guaranteed the win.

burninator then theorized that both players' optimal strategy is to attack (not sucker) 1/(1+remaining pp) of the time. Their intuition turned out to be almost exactly right. Burninator proved this calculating out the probability of :Kingambit: winning each turn with this strategy and considering the situation to be in a Nash Equilibrium, which is a concept I don't completely understand. All they had to do then was show that the next-closest strategies resulted in worse odds for the :Kingambit: player, which they did by instead having :Kingambit: use Sucker Punch 1/pp and 1/(2+remaining pp) of the time.

At the same time, I was coding a tiny simulation for this situation which allowed me to adjust the number of remaining PP and the likelihood of each player attacking on a given turn. I ran 100 million trials for a bunch of different likelihoods using this function:

Java:
public static boolean sucker(int pp) {
        // returns true if I win as the non-Sucker Punch player
        
        boolean they_suckered = false;
        boolean i_attacked = false;
        
        double opp_threshold = .5;
        double my_threshold = .5;
        
        while (pp > 0) {
            
            // how often the opponent attacks (does not sucker)
            opp_threshold = (double)1/(1+pp);
            
            // how often i attack
            // this doesn't seem to matter
            my_threshold = (double)1/(1+pp);
            
            
            double opp_choice = Math.random();
            double my_choice = Math.random();
            
            they_suckered = (opp_choice > opp_threshold);
            i_attacked = (my_choice < my_threshold);
            
            // i win if i attacked and they didn't sucker
            if (i_attacked && !they_suckered) return true;
            
            // mindgame keeps going if they suckered and i didn't attack, and they lose 1 pp
            pp--;
            if (!i_attacked && they_suckered) continue;
            
            // i lose in the two other scenarios
            return false;
            
        }
        
        // they're out of sucker pp so i win
        return true;
    }

I confirmed the same thing that burninator had observed--Kingambit should calculate 1/(1+remaining pp) every turn and choose to attack with something other than Sucker Punch that much of the time. For example, if I have 6 Sucker Punch PP left, I should use Kowtow Cleave 1/7 of the time.

The most interesting part of this is that I widely varied the likelihood that the non-sucker punch player attacks with a move that OHKOes Kingambit (i.e. :Latios: uses Aura Sphere). :Latios: wins just as often if it always attacks turn 1, if it does not attack until :Kingambit: has used up all of its Sucker Punch PP, if it attacks 50% of the time every turn, if it uses the same strategy as :Kingambit: based on the opponent's Sucker Punch PP, or if it uses the slight variants of that strategy that we showed to be worse for the :Kingambit: user. No matter what, Latios wins only 1/9 of the time.

Kingambit's optimal strategy does not change no matter how much PP it has. However, its winrate does decrease according to the number of PP it starts with. If Kingambit has only 3 Sucker Punch PP, it will win 3/4 of the time from the beginning of the scenario. More generally, Kingambit wins PP/(1+PP) of the time, and Latios wins 1/(1+PP) of the time.

burninator and I also ran this scenario in a 1v1 Custom Game 31 times before we got bored, with them using this strategy on :Kingambit: and me choosing between my two options according to "intuition." They won 27/31 games. I'm personally more convinced by the numbers from the 100 million trials.

Thanks for reading and thanks again to burninator0.
 
This seems to vindicate all my experiences of finding that people aren't prepared for me to use Sucker Punch 7-8 times in a row in ending 1-vs.-1s (the one opponent I had who dodged 7 Sucker Punches in a row, I switched to a regular attack the 8th time (counting on me having lured them into a false sense of security) and I won). Note that the strategy in the OP is fairly likely to output 7 Sucker Punches in a row (or at least some degenerate number like 5 or more).
 
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Just to add to this, the big takeaway for me about this is that the sucker punch player shouldn't even bother trying to play any mind games.
Pick an amount of PP you're willing to waste and just carry this out. The strategy is basically equivalent to just saying "I am going to not sucker punch on turn (X)" (picked at random 1-9) and never changing that plan.

It's possible for the non-sucker punch player to exploit a weaker strategy; in other words, winning the mind game.

This strategy cannot be exploited.

Sure, you can try and get in the other player's head, but why? 8/9 is about 89% which is pretty solid odds. Your chances of losing are about the same as missing overheat or leaf storm. I'll gladly take those odds rather than pretend I can reliably win the mind games.

Also it's worth nothing that substitute completely flips this dynamic. Now the substitute player is the one favored with a chance to win of (SUBS left) / (SUBS left + 1). In fact, it's slightly higher than that since that assumes they automatically lose when they can't sub anymore where in reality they get to now play the sucker punch mind game (with a small chance of winning).
 
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Your solution agrees with my own analysis from a couple years back. The sucker punch "50/50" (not quite evenly weighted as you discovered) is one of the simplest examples of a mixed Nash Equilibrium in pokemon.
Simultaneous move games (regardless of whether or not they are otherwise perfect information) have solution concepts in the form of Nash Equilibrium strategies. In alternating move games, these NE are usually pure which means there is a 'best move' which can play deterministically. In simultaneous move games you may have to sample your moves according to a probability distribution in order to be unexploitable.
I've written an article that goes a bit more into this and also explains how to calculate the sucker punch optimal strategy. Good work with your independent discovery and testing!
 
it's worth nothing that substitute completely flips this dynamic. Now the substitute player is the one favored with a chance to win of (SUBS left) / (SUBS left + 1). In fact, it's slightly higher than that since that assumes they automatically lose when they can't sub anymore where in reality they get to now play the sucker punch mind game (with a small chance of winning).
Some other moves like Encore, Disable, and Torment also turn the situation into a pure 50/50 if your opponent knows you have them. Substitute is the best one of these moves for sure (besides like Spore and such obviously). Protect also helps.

Pick an amount of PP you're willing to waste and just carry this out. The strategy is basically equivalent to just saying "I am going to not sucker punch on turn (X)" (picked at random 1-9) and never changing that plan.
Exactly--if you want to hit the highest possible success rate over time you'll need to calculate 1/(1+PP) every turn and decide according to that chance, or have some process to automate that. But it's less time consuming just to pick a number at the beginning and choose the middle-late numbers more often and would still be significantly better than mind games.

I've written an article that goes a bit more into this and also explains how to calculate the sucker punch optimal strategy. Good work with your independent discovery and testing!
Awesome, would love to see a link to this! I was thinking maybe someone had done it before but I googled around and couldn't find anything. This 8/9 number is not common knowledge so I was hoping to spread that around.
 

Nashrock

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check with backward induction
at turn 8, we have the matrix
Kingambit\LatiosAttackDon't Attack
Sucker Punch1,-1-1,1
Don't Sucker Punch-1,11,-1
this is clearly a coinflip so both players play 50/50
at turn 7, we now know the only thing changed is that sucker and don't attack will make the game go to turn 8 with expected payoff 0,0
Kingambit\LatiosAttackDon't Attack
Sucker Punch1,-10,0
Don't Sucker Punch-1,11,-1
Latios wants to pick p such that -p = p - (1-p) so they're indifferent from their opponent's choice. This gives p=1/3, so Latios clicks attack 1/3 of the time and don't attack for 2/3.
Kingambit follows the same logic. Pick p such that -p+(1-p) = -(1-p) gives p = 2/3, so they click sucker punch for 2/3 of the time.
Same logic follows back to turn 1 and the result should be the same with yours.
Kingambit should click sucker punch every turn with probability 1- (1/ (pp left +1))
Latios should click attack every turn with probability 1/ (pp left +1)
 
The most interesting part of this is that I widely varied the likelihood that the non-sucker punch player attacks with a move that OHKOes Kingambit (i.e. :Latios: uses Aura Sphere). :Latios: wins just as often if it always attacks turn 1, if it does not attack until :Kingambit: has used up all of its Sucker Punch PP, if it attacks 50% of the time every turn, if it uses the same strategy as :Kingambit: based on the opponent's Sucker Punch PP, or if it uses the slight variants of that strategy that we showed to be worse for the :Kingambit: user. No matter what, Latios wins only 1/9 of the time.
One thing I'd like to add is that it's in the nature of Nash Equilibrium strategies to have these properties. When you play your Nash Equilibrium strategy as either player in a game like this, there isn't anything the other player can do to make you do worse than the value of the game. Consider rock-paper-scissors: playing each option 1/3 of the time is a Nash equilibrium strategy for either player. If I'm playing this strategy, it doesn't matter what my opponent is doing, if they throw rock 100% of the time, or any arbitrary distribution, whatever they throw they'll win a third of the time and lose a third of the time because of my 1/3 1/3 1/3 strategy.

The logic also cuts the other way. If Latios attacks with probability 1/(Sucker PP + 1), Kingambit won't be able to act in such a way that Latios will win less than 1/9 of the time. So the moral of the story isn't that the sucker punch player is unilaterally in control (that's closer to the lesson you learn from calculating the value of the game and realizing that when everyone plays optimally the sucker punch player is 8/9 to win: but they aren't in control, just ahead). The takeaway is that Nash equilibria are strong, and playing these strategies is a way to capture value regardless of your opponent's decisions and be less exploitable.
 
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Just want to highlight the exact meaning of this. Playing by this strategy lets you become unexploitable (meaning your odds of winning stay constant no matter what the opponent does). This doesn't mean however that the strategy is optimal. Being unexploitable doesn't mean you can't yourself exploit knowledge/beliefs about your opponent.
For example when you're very low on the ladder one would (usually) always use Sucker Punch on the first turn to account for the possibility that the opponent doesn't know how Sucker Punch works. In the case that the opponent actually doesn't know this, this strategy wins 100% of the times while the randomization will win less than 100%.

Also would like to add that to truly play the Nash equilibrium both players need to play with a randomization strategy. In a Nash equilibrium both players are unexploitable. One can show equivalently that the non Sucker Player becomes unexploitable by doing the same randomization as the Sucker Player with probability 1/(1+Sucker PP) of clicking the attacking move. When both players play by this strategy, neither of them can improve by playing a different strategy.

In general there is no optimal strategy. Let's define the strategy of Player A as the probability (sequence) of clicking Sucker Punch as the Sucker player and the strategy of Player B as the probability (sequence) of clicking the status move. If Player B plays by his Nash strategy, all strategies of A) become equally good (giving him the same odds of winning). There is not an optimal one. If Player B clicks the status move with probability 1, Player As optimal strategy is to click Sucker with probability 0. If Player B clicks the status move with probability 0, Player As optimal strategy is to click Sucker with probability 1. So the optimal strategy of Player A generally depends on the action of Player B and therefore there is no single optimal strategy for Player A.

Maybe a bit nitpicky but I wanted to highlight the difference between the Nash strategy (which is just a global equilibrium concept) and an optimal strategy. The Sucker player can still guarantee himself odds of winning 8/9 times in case of full Sucker PP. From experience I'd say it's hard to exploit "mindgames" to win more than that so this randomization strategy is still useful in practice.
 

CaffeineBoost

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Exactly--if you want to hit the highest possible success rate over time you'll need to calculate 1/(1+PP) every turn and decide according to that chance, or have some process to automate that. But it's less time consuming just to pick a number at the beginning and choose the middle-late numbers more often and would still be significantly better than mind games.
The distribution is uniform, you shouldn't place any bias towards middle/late numbers.

Suppose we start with 8 PP, then on turn 1 we are clicking Not Sucker 1/(1+8) or 1/9 times and Sucker 8/9 times.
If we clicked Non Sucker, the minigame is (for the purpose of this vacuum) over.
There's an 8/9 chance you clicked Sucker though, so we're back in the tank. 7 PP left, we're clicking Non Sucker 1/8 times and Sucker 7/8 times.
This means that if we started with 8 PP, then for the rest of the game we are doing the sequence Sucker, Non Sucker (8/9)*(1/8) times. This is 1/9, the same as an immediate Non Sucker.
If you start with 8 PP, the chance you reach
This iteratively keeps on going to yield the result that, at the start of the minigame when you have 8 PP and you plan to use this method, it is equivalent to randomly generating an integer between 0 and 8 and clicking Sucker Punch that number of times. The same (intuitively) applies for when you start with fewer than 8 PP since the probability distribution doesn't change. While the chances you click Non Sucker increase as you lose PP with this method, the chance that you will be faced with that choice in the first place with this method decrease in an inversely proportional way.

tl;dr, this strategy is equivalent to randomly generating a number between 0 and N when you have N PP and clicking Sucker that many times.



That said, this is a good thread for demonstrating how some 50/50's just... aren't.
 

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