For the sake of simplicity, let us consider a one on one interaction between Urshifu and Dragapult. The available actions for Urshifu are Sucker Punch, or use some other attack which will KO Urshifu. On the other hand, the available actions for Dragapult are Dragon Dance (a non-damaging move) or use some other damaging move that will KO Urshifu. Assume that Sucker Punch’s max PP of 8 is the limiting factor, i.e. the sub-game will continue so long as Sucker Punch has remaining PP or one of the participants faints. Define the payoff of each set of actions as the probability that the player wins, and let V n denote the total payoff for Dragapult when there are n Sucker Punch PP remaining. This gives us the following payoff matrix for G n , the scenario that arises when there are n Sucker Punch PP remaining. Urshifu Sucker Punch (S) Other Attack (O) Dragapult Dragon Dance (D) ( V n − 1 , 1 − V n − 1 ) (0 , 1) Attack (A) (0 , 1) (1 , 0) Table 1: Payoff matrix for G n (n¿1), where the tuple ( x, y ) denotes the payoffs for Dragapult and Urshifu respectively Let d n and a n denote the probability that Dragapult uses Dragon Dance and Attacks, respectively, as a part of their optimal strategy. In a Nash Equilibrium, both Sucker Punch and using the Other Attack should result in equal payoff for Urshifu. Using this fact, we have the following relation: payoff S = payoff O d n (1 − V n − 1 ) + a n (1) = d n (1) + a n (0) d n V n − 1 = a n Note that we also have d n + a n = 1, thus we have d n = 1 1 + V n − 1 Similarly, let s n and o n denote the probabilities that Urshifu should use Sucker Punch and the Other Attack, respectively as a part of their optimal strategy. Equating Dragapult’s payoffs for each action, we have payoff D = payoff A s n V n − 1 + o n (0) = s n (0) + o n (1) s n V n − 1 = o n Since s n + o n = 1, we then have s n = 1 1 + V n − 1 = d n 1 Also note that in a Nash Equilibrium, we have V n = payoff D = payoff A = d n since both actions should yield equal payoff for Dragapult. All that remains is to consider the base case G 1 For G 1 , we have the following payoff matrix: Urshifu Sucker Punch (S) Other Attack (O) Dragapult Dragon Dance (D) (1 , 0) (0 , 1) Attack (A) (0 , 1) (1 , 0) Table 2: Payoff matrix for G n , where the tuple ( x, y ) denotes the payoffs for Dragapult and Urshifu respectively Equating the payoffs for Dragapult and Urshifu, we have s 1 = o 1 = 1 2 d 1 = a 1 = 1 2 V 1 = 1 2 With these values, we can then solve for all of the remaining values and figure out each player’s optimal strategy. The final results are listed below n d n a n s n o n V n 8 8/9 1/9 8/9 1/9 1/9 7 7/8 1/8 7/8 1/8 1/8 6 6/7 1/7 6/7 1/7 1/7 5 5/6 1/6 5/6 1/6 1/6 4 4/5 1/5 4/5 1/5 1/5 3 3/4 1/4 3/4 1/4 1/4 2 2/3 1/3 2/3 1/3 2/3 1 1/2 1/2 1/2 1/2 1/2 Table 3: Optimal strategy for each player given n remaining Sucker Punch PP, as well as the total payoff V n for Dragapult 2