Nash-Equilibria Relating to Competitive Strategy Games
One of the more personally compelling applications of Nash-Equilibria relates a personal hobby of mine, competitive card games such as Magic: The Gathering. These competitions are strategy games where a careful use of resources, timing, synergy, and a bit of luck can lead to competition wins of hundreds of thousands of dollars . However, in many ways the most important decision to be made happens before you step foot into the tournament, what deck of cards you want to use. In Magic, and other games of the genre, much of the appeal and strategy of the game is that unlike games such as Chess, players do not use the same pieces, instead, they choose sixty cards out of a pool of tens of thousands, to use. While there are a seemingly infinite amount of combinations of cards one could choose, oftentimes cards can complement each other and form strategies, leading to a number of leading archetypes to be popular before a tournament. In a simple metagame, there might be three popular strategies, A, B, C, which have different relative strengths against each other. Say A wins against B 70% of the time, B wins against C 65% of the time, and C wins against A only 60% of the time. If you were to assume a balanced metagame where an equal amount of players choose each strategy, it would be pretty clear that A is best, but if A is more dominant, we could expect less people to play B and more to play C, which would counteract A. This is obviously solvable through a mixed strategy Nash equilibria, but things get much, much more interesting from here.
The most important tournaments, and those yielding the greatest cash prizes, take place over 3 days. Beginning on a Friday, with a very large field of players, narrowing to around 1/5 the field on day 2, Saturday, and having the top 8 players play on Sunday for the greatest prizes. Those sporting anything but a great record on Day 1 will not make it to Day 2, and only those with elite win-loss ratios will make it to the Sunday Showdown. This is really interesting from a modeling perspective, because it becomes not about the sheer best strategy to play for the metagame, but the best strategy to play such that you do well enough on Day 1 to make it into Day 2, but really are favored on Day 2 as it is the more important day, with a much higher bar to entry to day 3. Assume our matchups from earlier, and that 50% of the metagame brings deck A, 10% deck B, and 40% deck C. What we are likely to see is that deck C will dominate on day 1, crushing a field with more favored matchups, Deck A will fair middlingly, and B will do really poorly on Day 1. But, given that C will do amazingly on day 1 and very likely make up more of the field on day 2, we might expect that it will be players of deck B, those who survived the dismal day 1, to dominate on a C swarmed day 2. And perhaps in the top 8, which could likely be swarmed with a plurality of B, it would be A that wins the event. This is fascinating to me, and would be an interesting situation to model via a more advanced mathematical approach or a monte-carlo simulation