Fighting, Games, and Game Theory
“Dominant strategy”, “mixed strategy”, “payoff”, et cetera are terms that are thrown around often in discussion about basic game theory, but very rarely are the examples extrapolated to a discipline that also uses these terms: video games. You’ll hear terms like “Most Efficient Tactic”, “Mix-Ups”, and “Reward” in lieu of the game theory variants, however.
To be fair, game theory is often applied to games like chess and sports. In other words, competitions where people are in head-to-head opposition, whose choices are better or worse based on other’s choices. Video games are just the modern adaptation of this to a digital space.
In the course so far, we’ve mostly looked at game theory between two people in direct opposition through the lens of payoff matrices and equilibrium. The most applicable genre of video games currently, then, would be fighting games. Street Fighter, Tekken, King of Fighters, these games were born in the era of arcade games, where it wasn’t feasible to fit more than 2 players on an expensive electronic cabinet.
Even better, these games involve a set of fairly limited strategies at any time; at the core of most fighting games are three options: attacking, blocking, and moving. The appeal in fighting games is in the speed at which these choices must be performed at; you have to do the risk-reward calculation intuitively in a matter of milliseconds, before your opponent acts. As Core-A Gaming, a fighting game analysis channel on Youtube, puts it, it’s “competitive speed math.” (Source: Hard Reads: Why Fighting Games are Worth It).
The Theory in the Game
Let’s consider a common situation in fighting games, where neither person is in range to hit each other, say, at the start of the match. And let’s also assume that they are close enough, however, that both players cannot react to the other player; they will execute their strategies at the same time. This is called “ the neutral”, because neither player is in range to threaten the other; it is a “neutral situation”.
Let’s see if we can apply game theory to the 3 most fundamental fighting game strategies used to play the neutral. In Japan, they are called “atewaza”, “okiwaza”, and “sashigaeshi-waza”, but we will call them “Move-in attack”, “Pre-emptive attack”, and “Punishing attack”.
A Move-in attack refers to moving into the enemy range and launching an attack.
A Pre-emptive attack refers to predicting a Move-in attack, and launching an attack that will hit them as they move-in, but before they are able to attack.
A Punishing attack refers to predicting a Pre-emptive attack, and hitting the opponent’s out-stretched limb as they are recovering from their missed attack.
And to come full circle, a Move-in attack will beat a Punishing attack. An opponent waiting for a Pre-emptive attack will be caught off guard by a Move-in attack.
Finally, in the case that both players choose the same strategy, they will mutually miss each other, with the exception of Move-in attacks. Different fighting games handle this differently, but for the purposes of simplification, we will consider this a 0 payout for both players.
So now we have 2 players, and 3 strategies each, with certain payoffs. We can model this with a payoff matrix. We will describe “getting hit” with a -1, and “landing a hit” as 1. We’ll have Player A’s strategies in the rows, and Player A’s payoffs as the first number in each cell.
| Move-in | Pre-emptive | Punishing | |
| Move-in | 0, 0 | -1, 1 | 1, -1 |
| Pre-emptive | 1, -1 | 0, 0 | -1, 1 |
| Punishing | -1, 1 | 1, -1 | 0, 0 |
Even in a simple fighting game like this, there is no pure strategy Nash equilibrium or any dominant strategies. The mixed equilibrium also has every player playing each strategy equally.
However, no fighting game is this simple. Let’s take a very simple addition to the game: different characters with different movesets. Let’s say Player A is playing Abel, who has an extremely good Move-in move that does 3 times as much damage. And let’s say that Player B is playing Baal, who has a Punishing move and a Pre-emptive move that both deal 2 times as much damage.
| Move-in | Pre-emptive | Punishing | |
| Move-in | 0, 0 | -2, 2 | 3, -3 |
| Pre-emptive | 1, -1 | 0, 0 | -2, 2 |
| Punishing | -1, 1 | 1, -1 | 0, 0 |
Now the mixed strategy Nash equilibrium completely shifts. Baal doesn’t want to use his Move-in move, because it is relatively weak compared to other ones. However, if he uses his Punishing move, and gets hit by Abel’s Move-in move, that will be extremely bad for him. Thus, he might fall back on his Pre-emptive move instead, but if Abel knows this, he will just hit him with a Punishing move.
Fighting games are so interesting because there are a variety of other complex factors that will affect this payoff matrix, which, don’t forget, has to be calculated in the heat of battle. Which character is faster? Which character has more health? Does one character have more available strategies than the other? The numbers are crunchable, but not without difficulty.
If you’re interested in reading more about game theory in fighting games, I’ve linked an article down below, “A poke at fighting games with game theory.” It does an in-depth breakdown of a single move, of a single character, in a single fighting game, but still ends up being quite a long analysis.
A poke at fighting games with game theory.
Hard Reads: Why Fighting Games are Worth It.