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Ideal Values and Practical Values (part 1). A discussion of the values of chess pieces.
Robert Shimmin wrote on 2002-11-19 UTC
<i> Remember, the principle is that Pawn and move may be 2:1 odds if the stronger player is rated 1800 USCF and the weaker is 1600; but if the stronger player is 2600, he can only give P+move to a 2200 (numbers are made-up examples for rhetorical effect). </i> <p> If chess is a theoretical draw, then this principle won't always apply, or rather it applies in a weaker form to small advantages than to big ones. <p> Let's assume chess is a theoretical draw. Two equally matched and very strong chess players (stronger than any grandmaster, but not perfect -- they usually lose when they play against God, but draw often enough to make things interesting) play at odds. The odds are small enough that the game is still a theoretical draw, but large enough that any larger advantage would be a theoretical win. Half the time, the side giving odds will make that infinitessimal slip-up that allows the other side to win, but the other half of the time, the side given odds will make that infinitessimal slip-up that gives the other side enough breathing space to ensure a draw. So the value of these odds to these inhumanly strong players is 3:1 money, or 190 ratings points. <p> When God plays Himself at these same odds, the game is always a draw, since He plays perfectly and the game is a theoretical draw. When weaker chess players play at these odds, the side giving odds may occasionally win, and the odds are worth somewhat less than 190 points. <p> The mathematics that inspired this thought experiment yields the following results: <p> (1) If chess is a theoretical draw, then no odds small enough to keep the game a theoretical draw are worth more than 190 points at any level of play. <p> (2) All such small odds have some level of play at which they are worth a maximum. At weaker levels of play, the side given odds is too weak to fully exploit them, and at stronger levels of play, the side giving odds is strong enough to overcome its disadvantage. <p> (3) As the odds increase, this critical level of play above which stronger players actually notice the odds <i>less</i> also increases. <p> (4) For even the smallest odds, however, this critical level of play is stronger (not too much stronger, but stronger nonetheless) than any human being plays.

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