It uses most, if not all, logical operations.
Basically how it works is when the button is pressed inside the arena or the 5 minute timer (item on wood pressure plate) expires, it activates a series of T-flip flops, converting the number of times the it receives a signal to a 4 digit binary code. The binary code is then separated out into 16 individual paths and shortened to 2 ticks. It is then multiplied to the paths before it, over a set delay, and each signal is passed on to a main path, causing it to pulse the number of times the button/times has been activated. Increasing 1 pulse per wave, the current is then sent to a few note blocks and dispensers. In summary, the wave/round number equals the total number of times the dispensers are activated.
The funny thing is that a couple weeks ago, I made a contraption that has exactly the same final product, but it only uses a line of normal pistons and repeaters, and produces little to none multiplayer lag.