Additional Strategies and Equilibrium in Fighting Games
Several months ago, math-focused blogger Ryan R (www.suburbanlion.com) discussed a payoff matrix for play-styles in the fighting game Marvel Vs Capcom 3, as well as a method of evolving the mixed equilibrium to match developing playstyles. The article focused on the development of strategies over time, using complex system modeling to simulate the evolution of a metagame over time. For simulation purposes, three ‘strategies’ were defined, most easily summarized as Close, Far, and Defensive. Each is neutral towards itself, disadvantaged towards another, and advantaged towards the third (for example, Close:Close is 0:0, Close:Far is 10:0, and Close:Defensive is 0:10). The demonstration using evolving generations of mixed-equilibrium strategies shows that no universally dominant strategy emerges: over time, preferred strategies will change in cycles.
When analyzing conclusions, he points out something important about the simplifications: while the model assumes specific strategies, the game is far more complex. Something in particular that had not yet fully developed is the use of a team which aimed to play a new strategy entirely. This particular type of team does not use the standard three styles and instead has a fourth, best summarized as Meter. This option has the given advantages and disadvantages: Close:Meter is 10:0, Far:Meter is 0:10, Defensive:Meter is 0:10, and Meter:Meter is 0:0. While this appears to provide more options, it in fact centralizes the metagame. Defensive and Far both have an estimated negative payoff if played equally against an equally random opponent, while Meter and Close both have an estimated positive payoff. In practice, this is roughly equivalent to what happened across the board – the existence of this fourth strategy pushed more people to use Close, while Far, disadvantaged against both Close and Meter, was almost eliminated.
Though this is still a simplification, it is interesting to note the negative effects another option directly has on the game. Ryan’s observation that players can simultaneously play multiple styles (noted in his article’s Further Investigations) served as a check to this option. Meter also made it more difficult to play Defensive, since the strategy required one to reduce other payoffs in order to pick it. Even with such balance in place, the current metagame months later is dominated by Close and Meter-type teams.
As the game’s developers are specifically weakening this strategy, it is interesting to note that increasing the diversity of options actually hurt the diversity of the mixed strategy. More options, as it turns out, can lead to a more limited equilibrium.
(Initial analysis can be found at http://www.suburbanlion.com/?p=200 .)