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BGonline.org Forums
Blundering early vs. blundering late: UBK's example revisited
Posted By: Timothy Chow
Date: Thursday, 8 September 2011, at 1:31 a.m.
I've advocated MWC difference as a way of running a skill-based tournament. Recently, UBK posed a test case that I'd like to examine here in some detail.
I will assume for simplicity, as UBK did, that gammons and backgammons don't exist, and neither does the cube. People play 11-point matches, and each game is worth precisely one point.
UBK asked, suppose one player blunders a win into a loss early in the match. The other player blunders a win into a loss late in the match (at a score of 1a1a). In a skill-based tournament, who should be awarded more points?
The MWC-difference method that I advocate would award more points to the player who blundered early. UBK's implicit argument, I believe, is that since each player blundered away exactly one point, they played equally skilfully, so intuitively they should be awarded equal points.
I'd like to examine UBK's example more closely by elaborating on it a little. Instead of examining the results of just one match, let's imagine two players playing a series of matches. Nervous Nellie always loses the first game of the match, but otherwise plays perfectly. Choking Charlie always loses the Crawford game, regardless of whether he or his opponent is the leader at Crawford, but otherwise plays perfectly. ("Crawford" doesn't mean much when there's no cube in play anyway, but this is the most succinct way I could think of to describe Charlie's behavior.) What will happen when Nervous Nellie and Choking Charlie play a long series of matches?
It's easy to see that both Nellie and Charlie will blunder away exactly one game per match. Less obvious, perhaps, is that Nellie and Charlie will win equally many matches in the long run, but this is true.
Now Nellie blunders early and Charlie blunders late (or at least later than Nellie). One might think, therefore, that the MWC-difference scoring method would punish Charlie and award Nellie more points in the long run. If so, then this would indeed highlight a problem with MWC-difference scoring.
However, it turns out that this is not the case. Perhaps surprisingly, the MWC-difference scoring method will award both players the same number of points in the long run. The point is that even though Charlie always blunders later than Nellie does, this doesn't mean that Charlie always blunders away more match equity than Nellie does. For example, if Charlie is hopelessly behind when the Crawford game rolls around, then his blunder costs him relatively little ME since he had so little ME to begin with.
More generally, no matter what kinds of players you dream up, if in the long run the players win equally many matches, then the MWC-difference method will also award the players equally many points in the long run (assuming of course that the bot doing the scoring is perfect). So the scoring method does not suffer from this particular potential weakness.
There remains the question, is the MWC-difference method scoring Charlie and Nellie's individual matches "correctly"? This is a matter of opinion, but I think so. Charlie's blunders sometimes cost him a lot of ME and sometimes they don't. It strikes as reasonable to penalize him more when he blunders away more ME. Maybe UBK thinks Charlie and Nellie should be awarded equal points in every single match; this is a defensible position, and it underlies the preference that many people have for measuring errors using an EMG-based score like ER or PR. However, while I like using PR for self-training purposes, I think that for match scoring purposes, it makes sense that Choking Charlie be penalized more when he chokes when it really counts.
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