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Debunking Gigabet's Dilemna

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  1. jay_shark jay_shark is offline

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    Sep 17th, 2007 (9/17/2007 11:09am)

    I just finished reading Jennifears article which I thought was a fantastic read but I have a few differences . One part that caught my attention was about taking the worst of it because of your increased equity in attaining a large stack . I believe many players grossly over-estimate their added increase in "real" ICM equity as the big stack or chip leader . While it is definitely true that you have more fold equity and yadda yadda as the chip leader , most of this added equity happens when the large stack is confronting the small stack . A medium stack player will not and should not be too intimidated by the large stack because it is a perfect opportunity for the medium stack to become the chip leader . I will however say this : If you're a great player and you have the chip lead , then there is some added "real" ICM equity which is undeniable . This does not make it correct to take the worst of it because of your added equity .

    If we let f(x) denote your ICM equity and x represents the % of chips in play , then it's been proven (In fact , I've done this myself) that f(x) is an increasing function but it increases at a decreasing rate . That is , f(x) is a convex function and f''(x) is a negative number which means that as we increase chips , our added equity is increasing but at a decreasing rate . So in turn ,if we consider ourselves to be better than the average player , then by doubling up we have increased our equity but by a lesser amount than an inferior player would have with respect to our initial starting chip position .

    ie , a great player buys in for $10 but his chip stack is really worth $15. When he doubles up , his equity may be worth $28(excluding the added equity as the chip leader) which is ~1.8666*15 . An average player's stack may be worth $10 and when he doubles up , his equity may be $19(excluding the added equity) which is 1.9*$10 . So the inferior player has actually gained more utility with respect to his initial starting stack than the great player would have .

    So I think there are two opposing forces here and I believe they closely cancel each other out . For one , the great player has increased his equity even more than ICM would suggest because of his big stack . On the other hand , he loses some equity because of the simple fact that f(x) is a convex function which means the inferior player gains more utility by doubling up than a great player would .

    I would like to coin this phrase as jay_sharks equilibrium lol .
  2. jay_shark jay_shark is offline

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    Sep 17th, 2007 (9/17/2007 3:07pm) in reply to jay_shark

    Bump
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  3. Suthereader Suthereader is offline

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    Sep 17th, 2007 (9/17/2007 4:59pm) in reply to jay_shark

    nice post jay
  4. Doctor Razz Doctor Razz is offline

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    Sep 17th, 2007 (9/17/2007 5:27pm) in reply to jay_shark

    Interesting post Jay, but it's not clear how you feel you're debunking the Gigabet dilemma. Your analysis deals with continuous functions, but Gigabet's theory is that ICM equity is not continuous rather it's discreet or quantized. This means that sometimes winning or losing a certain amount of chips smaller than the important quanta at the time (or "block") does not change your ICM equity significantly. So if you can risk a relatively meaningless amount of chips to add an extra block to your arsenal, you should do so with only a modest chance of success. The converse to this (which usually gets overlooked) is that risking chips that would break up a "block" to win chips that don't gain you an additional block is a bad idea, even if you have the best of it in a cEV sense.
  5. hanker307 hanker307 is offline

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    Sep 17th, 2007 (9/17/2007 5:36pm) in reply to Doctor Razz

    i.e. equity in terms of ICM operates similar to a step function
  6. EyeKnows EyeKnows is online now

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    Sep 17th, 2007 (9/17/2007 5:38pm) in reply to jay_shark

    "So the inferior player has actually gained more utility with respect to his initial starting stack than the great player would have."

    Inferior gains more equity but the utility (assuming a great player uses the big-stack to more of an advantage than the inferior player would) is still much greater for the great player I think.
  7. jay_shark jay_shark is offline

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    Sep 17th, 2007 (9/17/2007 5:44pm) in reply to EyeKnows

    This is the whole point of this thread . Attaining a large stack is not so much advantageous as one would think . You gain some additional equity for obvious reasons but some of this additional equity is offset by the convexity of the ICM function ;(f(x)) , where x represents the percentage of chips in play .
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  8. Suthereader Suthereader is offline

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    Sep 17th, 2007 (9/17/2007 6:07pm) in reply to jay_shark

    Is there any difference in the application of this theory in an MTT versus an SNG?
  9. TMLMS13 TMLMS13 is offline

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    Sep 17th, 2007 (9/17/2007 6:23pm) in reply to jay_shark

    http://forumserver.twoplustwo.com/sh...art=1&vc=1

    just fyi
  10. jay_shark jay_shark is offline

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    Sep 17th, 2007 (9/17/2007 6:33pm) in reply to Suthereader

    In either situation(sng or mtt) I believe it's never correct to take the worst of it for all your chips in the early going of a tourney . If all players have equal abilities then you should make the call with a 54% edge or so in a sttsng . If you're a great player then I believe it's incorrect to call when you're a 49% favorite to win . The additional leverage of attaining a large stack is simply not a good enough reason to make the call .

    A unique situation happens when you double up . If we compare two players in a $10 sng , and player 1's stack is really worth $15 and player2's is $10 , then if each player doubles up , it's player 2 who increases his utility the most(excluding his big stack equity) . However ,player 1's utility becomes closer and closer to player 2's utility simply because of the hidden large stack equity . I firmly believe that these two situations closely off-set each other and the situation becomes comparable to if each player has the same skill set .
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