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Finding the Optimal Fighting Style: A Game Theoretic Model of Mixed Martial Arts Tournament Fighting. By Matt Bush. Introduction. Mixed Martial Arts (MMA) is a relatively new sport, combining elements of many different traditional martial arts

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finding the optimal fighting style a game theoretic model of mixed martial arts tournament fighting

Finding the Optimal Fighting Style: A Game Theoretic Model of Mixed Martial Arts Tournament Fighting

By Matt Bush

  • 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
fight styles
Fight Styles
  • 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 fighter’s choice of what style to use depends both on their own strengths and the strengths of the opponent, and also the risk of injury
the model
The Model
  • 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

Data on injuries taken from studies of injuries in other martial arts

payoff function
Payoff Function
  • Payoff = (Fighter’s offensive strength in the style the fighter chooses – Opponent’s 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
applying the model
Applying the Model
  • 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, FedorEmelianenko
fedor emelianenko vs mark coleman
FedorEmelianenko vs. Mark Coleman
  • The first fight was the Russian FedorEmelianenko 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 emelianenko vs mark coleman1
FedorEmelianenko vs. Mark Coleman
  • 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
fedor emelianenko vs kevin randleman
FedorEmelianenko vs. Kevin Randleman
  • Fedor’s next opponent was a training partner of Coleman, and had similar strengths, although he was a better striker.
                  • Randleman
fedor emelianenko vs kevin randleman1
FedorEmelianenko vs. Kevin Randleman
  • The game makes the same predictions of the last fight. The NE is (grapple, ground and pound).
  • This is what happened, apparently Randleman didn’t learn anything from Coleman, and he was submitted with a kimuraarmlock
fedor emelianenko vs naoya ogawa
FedorEmelianenko vs. Naoya Ogawa
  • Naoya Ogawa was a world champion in Judo. Because of this, his grappling skills were very good.
                  • Ogawa
fedor emelianenko vs naoya ogawa1
FedorEmelianenko vs. Naoya Ogawa
  • 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.
fedor emelianenko vs antonio rodrigo nogueria
FedorEmelianenko vs. Antonio Rodrigo Nogueria
  • 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.
fedor emelianenko vs antonio rodrigo nogueria1
FedorEmelianenko vs. Antonio Rodrigo Nogueria


Fedor’s grappling is dominated, and Nogueira’s 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 Nogueira’s takedowns and damaging Nogueira with strikes. Fedor would win the fight by unanimous decision.

  • 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.