Game Theory Kumite

"All fixed set patterns are incapable of adaptability or pliability. The truth is outside of all fixed patterns. " - Bruce Lee1

Game theory has been defined in countless ways and been applied to many different things. Some of the definitions include, "A mathematical method of making decisions in which a competitive situation is analyzed to determine the optimal course of action for an interested party.2" and "a mathematical theory that deals with strategies for maximizing gains and minimizing losses within prescribed constraints, as the rules of a card game: widely applied in the solution of various decision-making problems, as those of military strategy and business policy. "3. It is this last definition that I will be focusing on because it has been applied to military strategy and the martial arts are, by nature, forms of military combat.
The Martial Arts are not like a traditional game that is usually analyzed by game theorists. Rather, it is actually a game that has certain elements of game theory installed within different styles so that depending upon which traditions you practice, you are actually taught to play a different game. In this paper, the focus will be to the different rules for game theory taught within different styles, and some interprations about what that means for the players of these games and further how we can extrapolate from these ideas something useful to Game Theory itself.
Before we can talk about what kind of game the players within the Martial Arts play, we must first look at different types of game strategies and learn a bit about why they are useful for the players involved. Because Game Theory is so extensive, we will look at a few common ones and compare them. According to the Stanford Encyclopedia of Philosophy,



"the Spanish conqueror Cortez, when landing in Mexico with a small force who had good reason to fear their capacity to repel attack from the far more numerous Aztecs, removed the risk that his troops might think their way into a retreat by burning the ships on which they had landed. With retreat having thus been rendered physically impossible, the Spanish soldiers had no better course of action but to stand and fight—and, furthermore, to fight with as much determination as they could muster. Better still, from Cortez's point of view, his action had a discouraging effect on the motivation of the Aztecs. He took care to burn his ships very visibly, so that the Aztecs would be sure to see what he had done. They then reasoned as follows: Any commander who could be so confident as to willfully destroy his own option to be prudent if the battle went badly for him must have good reasons for such extreme optimism. It cannot be wise to attack an opponent who has a good reason (whatever, exactly, it might be) for being sure that he can't lose. The Aztecs therefore retreated into the surrounding hills, and Cortez had his victory bloodlessly. "4

This quote demonstrates how people can alter the payoffs and motivations of a situation to create a game and control its outcome. Cortez wanted to make sure his army wouldn't retreat and desert him causing the battle to be lost, and at the same time he wanted the Aztecs to know that win or lose, they won't be leaving. It creates a two-fold payoff from a single action, and both payoffs work in Cortez favor. However, Cortez choice also causes the payoffs and motivations to change for every player involved in the game. When one is playing a game like this, it would appear that the most important choice is the first one made within the game (since this choice will effectively cause other choices to become available or not for the other players).
For instance, if we had someone from Cortez's army decide not to get off of the boat but rather to wait until the time was right, steal the boat and sail away with it. This would cause them to be in fear of Cortez who would surely kill them but would demotivate the rest of the army, might cause them to lose the fight and this one person could sail back, explain the story correctly, and be treated like a hero. Unfortunately, this would be at the expense of everyone else who served with them, people they know, people they grew up with, etc. However, if the player did decide to steal a boat, it would mean that Cortez wouldn't want to burn the other boats, might have ordered someone to go after the boat thief, may have given different orders, etc. Since Cortez chose to burn the boats before any of this could happen, Cortez controlled the game and forced the other players to make choices that were compatible with the outcome he wanted. In fact, when we take into account of the Aztecs leaving and not having any bloodshed, the situation turns into almost a Hicks optimal, as the best situation that could have happened did (although you could switch either party as the one who retreats).
One of the most basic elements in a game is that of the players themselves (or agents). Players are anyone that plays the game and/or who is influenced directly and/or indirectly of the game. If we are playing a game, and one of the outcomes is that you go cause something to happen, we can consider the person who is affected by what you caused as being an indirect player of the game. The indirect players are not as important as the direct ones because the direct ones are the ones are the ones who influence the game itself and cause certain outcomes to arise. Think of it this way, if we have a cold war arms race and two nations keep stockpiling weapons to compete with each other, these two nations are the direct players (or, rather, the ones who are in charge of making the decision to create more weapons are) and the indirect players are all of the other inhabitants of the nations and every other nation in the world (assuming that these weapons have the capacity for mass destruction). This is the difference between rational (direct) and irrational (indirect) players. The Stanford Encyclopedia of Philosophy says:

"An economically rational player is one who can (i) assess outcomes, in the sense of rank-ordering them with respect to their contributions to her welfare; (ii) calculate paths to outcomes, in the sense of seeing which sequences of actions would lead to which outcomes; and (iii) select actions from sets of alternatives (which we'll describe as ‘choosing’ actions) that yield her most-preferred outcomes, given the actions of the other players. "5

While there are many examples of direct players being irrational players, for the scope of this paper when we discuss direct players we can use the terms interchangeably because within every martial art has as a goal to turn all direct players of games into rational players. So to discuss briefly the three premises put forth for determining rational players, we have the first one which is the ability to assess outcomes. In other words, when someone is a rational player they have the ability to discern between an outcome that they would want to occur and an outcome that would be detrimental to them. Suppose we had someone who was in the middle of a championship basketball game. The team they are on is ahead by 40 points and no matter what they do the outcome is the same. There are only 30 seconds left in the game and our rational player has the ball. They can either choose to shoot, to throw it at an audience member, to pass it to another team mate to make a shot, to drop the ball, to reverse, etc. There are many different outcomes that occur, but all of them have the same result for the team: They win. A rational player would look at these different outcomes and be able to say, "X looks like a much better outcome than Y because within Y are conditions that negatively affect me and within X are conditions that lead to both my teams winning and other preferable outcomes" (suppose they take the shot and make it?). Therefore, a rational player would choose one of the outcomes where they take the shoot, or pass it to a teammate, rather than throw it to the crowd because they can see the difference between outcomes (eventhough several of them have positive payouts) and can see that this is the outcome that affects their personal welfare the most.
The second important aspect of a rational player is the fact that they can calculate and associate certain outcomes with the choices and moves that lead them to this outcome. This is demonstrated with Cortez who was a rational player who decided that in order to get the outcome of making sure the soldiers didn't retreat, he needed to make the move of burning the ships. He also needed to make the first move to make sure and limit the counter movements. So, when we get a rational player, we mean someone who is able to not just add 1 + 1 to get 2, we need someone that when we say get us as quickly as you can from 9 to 81, they don't add, they simply square it. While being able to calculate outcomes for yourself is good, a true rational player also needs to be able to do this to the other players in the game (rational or not).
The third important aspect is that a rational player needs to be able to quickly determine alternative courses of action that will help them cultivate the outcome that they want. They need to be prepared, not in a static sense where they wait for other players to make a move, but in the sense that should the player make a move that they are not equipped to handle, they will be able to ascertain the best counter movement so that they can set themselves up for the best outcomes. Now that we have an idea of the most integral part of the game, the players, we can begin talking about martial arts.
First, there are literally thousands (possibly millions) of different martial arts, but for the most part if an art is not eclectic, then we can classify them into one of  3 basic categories (even if the art is eclectic, usually they base themselves on one of the 3 categories and then mix attacks from some of the others to make it more well-rounded). The 3 basic categories are: Striking, Grappling, and Weapons. A few examples of striking arts are: Karate, Tae Kwan Doe, Kung Fu. A few examples of the grappling arts are: Hapkido, Akido, Judo, JuJutsu, Ninjutsu. And finally, a few examples of  the Weapon arts are Eskrima, Kendo, and Fencing. This list is not exhaustive, and it doesn't mean that one art completely excludes movements from another category, or isn't as good as one from another category, etc. The point is only to show that these arts, usually, base their techniques with certain qualities that can be used to differentiate between systems and lineages and these qualities usually fall into these 3 categoeis.
Aside from the players within the game,  one of the most important aspects is the game itself. The game is almost always the same for all of these arts, in Karate it is referred to as Kumite or sparring. This game can take many forms, but for the purposes of this paper we are going to focus in on sparring that is not martial combat (so both players are not trying to kill their opponents, simply get them to submit and/or score a point). Presumably, we can derive principles of game theory that apply to martial combat from analyzing sport sparring, but in an effort to keep this paper focused on a more general analyses, we will stick to sport sparring.
Within the three categories are different aspects of the same game. Each art teaches you to approach things differently, to score in different ways, and to prefer different outcomes. In other words, each art has as its goal to make you a rational player within the game, but each one has its own unique applied training methods to get you there. We will start out analyzing the Striking arts.
The striking arts have quite a few differences between them when you look at the techniques that are taught and are thought of as important, but they all focus upon the strike: That is, at the heart of what makes a certain art a striking art is that they focus quite a bit of their time on learning how to strike their opponent (punch, kick) and less time on things like grabbing or throwing. Within the striking arts are applied theories about how one must attack.
Within the numerous ways to attack are usually different strategies. One of the most famous ones is what is known to game theorists as tit for tat. In the martial arts, when you are learning how to spar one of the first things you learn is to use control. To control how hard you hit someone and to control how you react to them hitting you. When we spar in the striking arts, one of the best ways to score a point is hitting them when they are hitting you. If someone punches you in the stomach, at the exact moment of impact, they are usually open to be hit in their stomach or their face, so one begins by using the tit for tat strategy. If someone hits me, then I hit them back. If someone kicks me, then I kick them back. Usually in the same place and with the same amount of power. We do this both because it is effective, but it also forces the other player to begin considering other strategies and to see the weakness involved in this method. This also involves the information aspect of game theory and the idea of rational players that was mentioned previously.
When people learn to spar, it is usually with the tit for tat method. You learn phrases like 'attack the attack', or 'meet the attack', which basically means to do to them what they do to you. Slowly this causes the players to gain a rhythm within the match, and they begin to recognize when someone is about to attack and when they aren't. We use this as a training method to prepare the fighter to learn more advanced fighting strategies. Tit for tat is useful because it teaches players to understand the rules of the game, and to understand when they are most likely to score a point. It teaches them to realize the steps necessary to reach certain outcomes. It also helps instill in them the idea that a better outcome for them would be one in which they don't lose. Tit for tat is only one of the introductory lessons because, once you get to the next level of training, you can telegraph the other players movements and move beyond the strategy.
Suppose we have two players who are both using tit for tat. If one of them comes in for a punch, the other one will punch in the spot that is now open from the first player. Now suppose that the first player was only testing the game to see if the other was using tit for tat. Now, when we repeat the game during the next movement, it would be smarter for the first player to go first and then quickly strike again as the second player will be trying to strike the original spot that was open and won't be expecting another strike so quickly. This is the next level of kumite game theory.
If you remember back to our definition of what a rational player was, one of the steps was being able to force your opponent to make certain moves that give you a strategic edge over them. The way that Martial Arts are set up (any), they are designed to turn what would ordinarily be irrational players (instinctual fighters) into rational players (trained fighters). So by applying tit for tat strategy in order to illicit a response that leads to the original player having the chance at a better outcome, is one example of game theory in martial arts. Another example comes from the grappling arts.
One of the most recognized of the grappling arts is that of Judo. Judo means the gentle way and consists mostly of techniques where you grab and/or throw. When Judo players begin to spar, they operate under a rule of "when they push, you pull; when they pull, you push". This is an example of tat for tit, where you do the opposite of what the other player is doing. If the first player throws out a strike, rather than strike them back and leaving yourself open, you evade the strike and take away the opportunity for the point that the other player was trying to create. If the other player begins to back away and doesn't attack you, then you should attack. But what if someone who is trained in games that have both of these maxims? What would a rational player do if they began playing with one who shows knowledge of both ways? And furthermore, how does one tell which strategy the other player is using?
The simplest way would be to start out striking, as the grapplers usually have a strategy of pacifism until attacked. This way, upon seeing what action the second player takes, you could see what strategy they are using and adjust your strategy accordingly. If you were the second player, it would seem that the best course of action would be to fake that you are playing tit for tat, and then switch to the grappling tat for tit in order to completely throw off your opponent. It would also seem in your best interests (no matter which player you are) to eventually switch back and forth between the strategies on a random basis.
Because of the way that most martial artists are trained (through repetition), they would easily be able to see which patterns the other player is using. Since the players of the game will most likely be another martial artist (as someone who isn't wouldn't be able to play long enough to make a game out of it), the best strategy would ultimately be one that combines a random approach in a non-patterned and non-linear way. Certain movements would have to be used in a way that would indicate to the other party the opposite of your intentions and yet at other times you would have to play completely transparently (think poker bluffs).