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But the REAL problem is that:
1) Estimates are never exactly right; they are always just estimates, and
2) WAR uses an analytical system to process those estimates which has the potential to enormously magnify whatever inaccuracies are included.
In a WAR estimate, there are dozens and dozens of internal estimates—estimates of runs created, estimates of runs saved by fielding, estimates of the run value of a single, a double, a triple or a double play, estimates of the park effect, etc.
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If you want to compare players on a single overall value like WAR, it seems like the best you could do is just take the averages for the season across all situations, like how many runs a single adds on average. But maybe there is room for more situational values, because I think a player's value on one team or in one place in the batting order or whatever might be different if they were on a different team or place in the batting order.
Maybe a single is more or less valuable for different teams, on average. For example, a weak offensive team might get less value from a single because there's likely less runners on base and less chance of the player who got the single to be batted in. Or a walk is more valuable on a team that hits more HRs. Or a more defensively capable player is less valuable to a team with better pitching that is already less likely than average to give up hits, compared to a team whose pitchers can't get a K. Or a player's batting line might be more valuable batting 1st instead of 4th. Even the field shape could benefit some players more than others, and could significantly affect their production.
This kind of information could be useful to teams in finding players better suited to particular spot on their team than how they are currently be used in another team. So WAR is a nice general number for overall comparison, but there's a lot of potential for more specific situational comparison calculations.