1. By Matt Bush

2. Mixed Martial Arts (MMA) is a relatively new sport, combining elements of many different traditional martial arts The sport has evolved from its early days, when fighters used only one style that they trained in. For example, some typical fights in early MMA would be karate vs. boxing, sumo wrestling vs. taekwando, or Brazilian jiu-jitsu vs. wrestling In contemporary MMA, most fighters incorporate many different martial arts and fighting styles into their training

3. With the evolution of MMA, fighters are now skilled in a variety of fight styles. In any fight, a fighter has a choice to use one of three fight styles as their primary fight styles Grappling – Combines elements of Brazilian jiu-jitsu, judo, sambo, and other martial arts. The goal is to use joint locks and chokes to make the opponent tap out (submit). Usually grapplers want to take the fight to the ground Striking - Uses boxing and Muay Thai kickboxing. Strikers try to knockout their opponent or force the referee to stop the fight (TKO). Strikers want to stay standing Ground and Pound – Relatively new. Uses wrestling and brawling. The goal of ground and pound is to take the opponent to the ground and then use punches to end the fight, usually by TKO. The fighters choice of what style to use depends both on their own strengths and the strengths of the opponent, and also the risk of injury

4. The three fight styles correspond to three strategies that each fighter has, resulting in a 3x3 game matrix The payoffs are based on three primary factors: the offensive strength of the fighter in the style, the defensive strength of his opponent in the style, and the injury risk from the combination of the styles

5. ScoreOffensive DescriptionDefensive Description 1 Never trained or fought in the style Never seen the style, unaware of the style 2No formal training in the styleRarely sees the style in training 3 Some formal training, but no regular training No regular training against the style 4Occasionally trains in the styleOccasional training against the style 5 Trains with the style regularly, never actually used in a fight Trains against the style regularly, beat often by it in fights or never seen in a fight 6Used the style in a fight beforeFought against the style in a fight 7 Skilled in the style, trains with it extensively and used in a few fights Skilled in the style, may have been beaten once or twice by it 8 Very skilled in the style, used the style extensively in fights Very skilled against the style, never been beaten by it 9 Extremely skilled in the style, top 10 in the world Extremely skilled against the style, top 10 in the world 10Unquestioned best in the world (top 3 best)

6. Injury rate if fighter uses style Injury rate if opponent use style Grappling.724%2% Striking6.97%4.54% Ground and Pound 3.4%3.5% Data on injuries taken from studies of injuries in other martial arts

7. Payoff = (Fighters offensive strength in the style the fighter chooses – Opponents defensive strength in the style fighter chooses) – [(Risk of injury (%) from style chosen + Risk of injury (%) from opponents style)/10] Payoff = Offensive Advantage – Overall Injury Risk/10

8. To see what predictions the model makes and if it holds up in the real world, the model is applied to an actual MMA tournament. This tournament is the Pride Fighting Championships 2004 Heavyweight Grand Prix It was a 16 fighter tournament, but only 4 of the fights are examined in the paper The 4 fights are the 4 fights of the tournament champion, Fedor Emelianenko

9. The first fight was the Russian Fedor Emelianenko vs. the American Mark Coleman Fedor, a very balanced and skilled fighter, is ranked an 8 or 9 in each strength category Coleman invented the ground and pound technique, and his rated very highly in that style, but lower in the other two, especially submissions Because of these strengths, the game looks like this: Coleman Fedor GrapplingStrikingGround and Pound Grappling0.73, -3.27.47, -2.9-.58,.46 Striking-.9, -3.52.-1.15, -3.15-1.047,.21 Ground and Pound -.54, -3.42-.8, -3.05-.69,.31

10. Both have dominant strategies, Fedor should grapple no matter what, and Coleman should ground and pound The Nash equilibrium is (Grapple, Ground and Pound) This is what actually happened in the fight. Coleman took Fedor down and tried ground and pound and Fedor used an armbar to submit Coleman and end the fight

11. Fedors next opponent was a training partner of Coleman, and had similar strengths, although he was a better striker. Randleman Fedor GrapplingStriking Ground and Pound Grappling.73, -4.27.47, -1.9.58, -.54 Striking.1, -4.53-.15, -2.15-.05, -.8 Ground and Pound -.46, -4.42.2, -2.05-.69, -.69

12. The game makes the same predictions of the last fight. The NE is (grapple, ground and pound). This is what happened, apparently Randleman didnt learn anything from Coleman, and he was submitted with a kimura armlock

13. Naoya Ogawa was a world champion in Judo. Because of this, his grappling skills were very good. Ogawa Fedor GrapplingStriking Ground and Pound Grappling-.27, -.27-.53, -3.9-.42, -2.54 Striking2.1, -.521.85, -4.15-1.95, -2.8 Ground and Pound 1.46, -.421.2, -4.05-1.3, -2.69

14. Unlike the previous fights, Fedor does not have a dominant strategy. The style he had been using, grappling, is now dominated. Ogawa has a dominant strategy of grappling. The NE is (strike, grapple). Once again, the model predicts reality. Fedor came out striking in the fight and hit Ogawa early. Although Fedor ends the fight with a submission, strikes were what allowed him the opportunity and were his primary fight style.

15. The championship fight was between Fedor and the Brazilian Antonio Rodrigo Nogueira, a Brazilian jiu-jitsu expert Nogueira is a 10 both offensively and defensively in grappling. He is also very good defensively in the other two styles, and has some offensive skill in striking and ground and pound Since the championship is the last round of the tournament, injury becomes irrelevant.

16. Fedor GrapplingStrikingGround and Pound Grappling-1, 1-1, -1-1, -2 Striking0, 10, -10, -2 Ground and Pound 0, 10, -10, -2 Nogueira Fedors grappling is dominated, and Nogueiras ground and pound is dominated There are two Nash Equilibria, (Strike, Grapple) and (Ground and Pound, Grapple). In the first fight between the two, in 2003, Fedor used ground and pound to beat Nogueira, who tried to grapple. This time, Fedor kept the fight standing, countering Nogueiras takedowns and damaging Nogueira with strikes. Fedor would win the fight by unanimous decision.

17. Potential counterarguments: The payoffs should be based on fighters perceptions of their strengths and opponent strengths. This would bring issues of overconfidence into the model. But since trainers and coaches help make the decision on what style to use, this is not a very strong argument The payoffs values are arbitrary. Why use a 10 point scale, and why divide the injury rate by 10 The injury variable is not precise, it relies on studies of other martial arts. But because of the lack of MMA studies, this is a necessary weakness Conclusion: The model predicts reality in every fight it is applied to. It appears that fighters choose what style to use based on relative strength advantage and injury risk (in a tournament). Knowing this, fighters can predict what their opponent will do, bringing a whole new aspect into MMA strategy.