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This template displays the payoff table and first Nash equilibrium for a zero-sum, two player strategic game.
The game is defined in a .gbt file in the Mixup namespace. These files are used by the game theory software Gambit.
In other words, this template takes a well-defined mixup and shows the optimal strategy for both players, i.e. how often they should use each option. See the example for how to interpret this.
Usage
{{Mixup|Kazumi-fly-infinite.gbt}}
Example
Kazumi FLY mixup in mirror after 1+2, ws3, and u/f+2 transitions on infinite stage.
| d | b | SWR | 4 | |
|---|---|---|---|---|
| FLY.1 | 29 | -31 | -68 | 34 |
| FLY.2 | -75 | -31 | 75 | 79 |
| FLY.4,2 | -80 | 36 | 36 | 39 |
| FLY.1+2 | 24 | 2 | -68 | -68 |
| FLY.3+4 | 64 | -68 | 13 | 65 |
Nash equilibrium with payoff -4.79
- FLY.4,2
- 0.39
- FLY.1+2
- 0.33
- FLY.3+4
- 0.28
- d
- 0.35
- b
- 0.44
- SWR
- 0.21
Payoff for dominated options
- FLY.1
- -17.58
- FLY.2
- -24.48
- 4
- -11.17
Explanation
- Table
- Every possible outcome for the options considered. Each cell is the damage done in this outcome from the blue player's perspective. Negative numbers mean damage in red's favor.
- If the attacker does FLY.1 and the defender does d (crouching guard), the outcome is 29 damage in the attacker's favor. (The attack actually does 25 damage, but the okizeme is approximated to be worth about 4 damage.)
- Nash equilibrium
- The optimal strategy given the above payoffs. If either player deviates from this strategy they can only worsen their payoff, so there's no way to force an opponent to change from it. However, it's possible for a player to get a better payoff by changing their strategy if their opponent is also not using this strategy, i.e. there might be a better way to exploit a weak opponent.
- ...with payoff -4.79
- When both players follow the optimal strategy, the average outcome is -4.79 damage from the blue player's perspective. It being negative means it's in the red player's favor.
- FLY.4,2: 0.39 ...
- The attacker uses FLY.4,2 ~39% of the time in the optimal strategy.
- Payoff for dominated options
- Some options aren't good enough to ever be used. For these, the average outcome of using it against the optimal strategy is shown.
- FLY.1: -17.58
- If the attacker used FLY.1, it'd result in an average outcome of -17.58, without the defender changing their strategy at all.
Limitations
A Nash equilibrium for a particular mixup is useful for getting a general idea for which options are better than others, but there are many limitations in the context of Tekken:
- Payoffs are defined in terms of damage, so any extra payoffs from frame advantage or okizeme need to be approximated.
- Life advantages aren't considered.
- An outcome where both players trade damage is considered an even outcome, but it actually favors the player that's ahead.
- A high risk option might be dominated because it has bad payoff on average, but if you're on 1 life, then taking 5 damage is the same as taking 80.
- Outcomes can vary a lot depending on execution.
- In particular, whiff punishment is a big problem. The punisher used and whether it's fast enough can lead to dramatically different outcomes. In general, determine the attack from what caused the whiff (e.g. Kazumi would use 1,1,2 with sidestep and 3,2 with backdash) and if the whiff is very big then approximate how likely it is the player would react and punish optimally. Be realistic—optimal whiff punishment shouldn't be expected.