The runs to win conversion is the cornerstone of understanding how to measure value with modern metrics. It's also not quite as clear as it might appear.
Prerequisites from derivation: Environment, data, expected win/loss.
How Do We Get There?
Ten. Ten runs per win. I suppose that's all you really need to know, but it's always good knowing how we come up with things otherwise they seem totally arbitrary. First of all, ten runs for every win isn't strictly true. It's close enough to suit us, for the most part, but knowing some of the little details will come in handy later. Let's start what we know about the run/win relationship already: Pythagorean win expectancy.
With Pythagorean expectancy as part of our toolbox, we can derive some interesting results. First of all, it's obvious that the league as a whole will have a .500 winning percentage, will score a league average number of runs a game, and concede a league average number of runs per game. What happens when you bump up runs per game and keep runs conceded static? The result is an increase in winning percentage (and therefore wins by the same amount, if we're just looking at one game). If a certain change in runs is equivalent to a change in wins, we have our runs per win. This value changes slightly as we go up and down the win percentage scale, but it sits around ten for the overwhelming majority of the time, so that's what we'll use from now on. Except when...
A Special Case
... a player has a major impact on the local run environment. When a game features a #5 pitcher, there will probably be more total runs score in said game. The opposite is true if an ace is on the mound for one team. Since pitchers can have such a large impact on the environment, and our method of deriving the runs per win relationship depends on the environment, the numbers will ideally be re-run for pitchers. The effect serves to amplify the distance a pitcher is from the mean (in wins). Note also that a change in league run environment will alter (perhaps significantly) the runs to wins conversion.
If we are using a statistic based on runs, why do we need to convert to wins? Apart from the special case for pitchers outlined above, it seems a bit strange to put so much weight on a translation that essentially involves dividing by ten. The reason this is done is to keep our focus on what teams actually value: winning games. There's no logical reason for runs to have a value outside of their power to win or lose games, so in order to actually assign a value to a player, the win conversion is necessary, either implicitly or explicitly. Keeping the unique pitchers on the same scale as the hitters and average pitchers is another consideration, although I believe that it's less fundamentally important.
To Sum Up
- Using ten runs per win suits us perfectly for the most part (thanks to the typical run environment we operate in).
- Extremely good or bad pitchers have an effect on the runs per win conversion, lowering the amount of runs per win for good pitchers and raising them for bad ones.
- We convert to wins because there's no logical reason that runs might reflect player value, while wins clearly benefit teams.