In the arena, the more you win (i.e., kill), the higher you rise in rank. The game sets it up so that opponents of equal rank compete, so when you start out as a "pit dog," you can only fight other pit dogs. After you've killed three dogs, you're promoted to brawler and must kill three fellow brawlers to rise another rank. Sounds simple, right? Well, each of your opponents has three kills of his own; if he didn't, he wouldn't be a brawler. Between you four, there have been 15 deaths in all. Perhaps you see where this is going.
The next rank is bloodletter, and each of your three opponents traveled his own 15-death path to reach that rank. We're now up to 63 total deaths. This goes on and on, with the death toll rising higher and higher, until you reach the championship rank, when all of your opponents are there due to a river of blood left in their wake -- by this point, you and your opponents have contributed to an astounding 16,383 deaths. The grand champion, whom you challenge and kill next, is responsible for at least that many deaths himself, since he fought to become grand champion the same way you did.
Bethesda Game Studios
And we're not even counting his early murder years as a bastard Imperial orc vampire lord.
These numbers, however, assume that each of your opponents has only the minimum number of kills necessary for the title. In practice, many will have more; a first-time pit dog could fight a pit dog who already has two kills, for instance. So, if we assume your opponents always have the same kill count as you, using the humble formula of 2^n - 1 to determine the total number of deaths after each match, we find that your 22-match rise to grand champion results in ...
... 4,194,303 deaths. For a sport. That's just you, by the way -- what of the many people who challenged the grand champion and failed? Their paths resulted in almost as much death, and they didn't even get a cool title to show for it. It's no wonder there's less than a thousand people in the world. Genocide is their international pastime.