[chessmind] Dennis Monokroussos: Last-round draws: an analysis

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Posted by Dennis Monokroussos:
Last-round draws: an analysis
http://chessmind.powerblogs.com/posts/1184284567.shtml


   In the comments section of [1]my post reporting on the World Open's
   results, "Inky" writes this:

     I don't know why anyone is amazed that they all drew - there are no
     rules against short draws in the World Open. Furthermore none of
     them wanted to chance a loss and get even less money than splitting
     1st through 9th.
     Not only did they all draw early, but they hung around like
     vultures to see if they had to split the prize money with one more.
     Shabalov and Perelshteyn fought a long hard game. They are two GMs
     worthy of our admiration. Of course they were both fighting for a $
     share in the top group.

   ([2]show)

   First, a preliminary point. If Inky's claim is that Shabalov and
   Perelshteyn are admirable because of their last round fight, it's hard
   to agree. A draw meant around $1000 - and that's before the entry fee
   and hotel (and airfare?) were deducted; in short, next to nothing. A
   win, however, meant nearly $8,000, so the decision for them to fight
   it out was a no-brainer.
   Now to the main event: were the eight leaders irrational for making a
   quick draw - should we be "amazed" by their actions? We shouldn't rush
   to answer - the question is insufficiently precise. There are
   different things we might, or might not, be amazed about.
   (1) Psychologically, their action isn't amazing. At the end of a
   tournament players are tied, and as Vince Lombardi (or George Patton)
   said, fatigue makes cowards of us all. Further, many people are
   risk-averse, and the thought of forsaking a nearly $8k bird in the
   hand for $1k in the bush, if one is unlucky, is an unhappy one.
   (2) Our focus, therefore, will be on the economic rationality of the
   decision, abstracted from questions of risk aversion and energy
   (though I will say more about those issues later). But we should make
   the question more precise still, because in working out the
   percentages, color matters. As Garry Kasparov writes in How Life
   Imitates Chess, the stats for GM games average 29% White wins, 18% for
   Black, and 53% draws. So the question I want to focus on is this:
   (Q) Should a last-round leader with White take a quick draw in a
   normal, still-living position?
   To solve this, we'll have to do some number crunching, and I'll make
   some simplifying assumptions. (Those who are willing to do a more
   thorough job are invited to write in with their more accurate
   conclusions.) Here goes.
   Assumption 1: There were ten prizes in the Open Section (I'm going to
   assume the numbers given on the website):
   $30,000, $15,000, $7,000, $3,000, $2,500, $2,000, $1,500, $1,000,
   $800, $700.
   The actual prize fund was reduced, but Iâm going to assume the
   reductions maintained the proportions and number of prizes originally
   offered.
   Assumption 2: I'll assume that if any of the eight leaders lost,
   they'd make nothing. This is in fact untrue â there are scenarios in
   which they could make over $1000 â but Iâll generously assume they
   wonât. (Itâs not that generous though, since Goichberg extracted the
   entry fee and (I think) the room costs from GMsâ prize money. So in
   many if not most cases they probably would have netted a goose egg.)
   Assumption 3: Another kindness on my part: Iâll assume that no one
   could catch the leaders if they all drew. (In fact, Shabalov did catch
   them and two others might have.) This too improves the financial
   incentive for a draw, though perhaps not enough to justify not trying
   for a win. Stay tuned.
   Assumption 4: One final generosity. While the odds of a decisive game
   are only 47% (29% + 18%), Iâll bump it up to 50% for ease of
   calculation â but only for our rivals. Since that increases the
   chances that oneâs rivals will reach the higher score group, this too
   offers some support to the âletâs all drawâ scenario.
   Ok, letâs go to the numbers (I do a little rounding up and down on the
   cents, so the results are approximate).
   There are three cases for our single player going for a win â letâs
   call him âHNâ, for no apparent reason â each with four sub-cases:
   (A) HN wins, and of the other three games either all are drawn, two
   are drawn, one is drawn or none are drawn.
   (B) HN draws, andâ¦the same four cases.
   (C) HN loses, andâ¦there are the same four cases, but we donât care,
   since weâll assume HN makes nothing â that was assumption 2.
   Case (A1): HN makes $30,000
   Case (A2): HN makes $22,500
   Case (A3): HN makes $17,333
   Case (A4): HN makes $13,750
   Case (B1): HN makes $7,750
   Case (B2): HN makes $5,167
   Case (B3): HN makes $3,625
   Case (B4): HN makes $2,750
   Case (C): HN makes $0
   There are still two steps left. We have to weight the cases by
   multiplying the totals by their likelihood. Thus in Case (A1), the
   odds of all three other games winding up drawn = 1/8 (1/2 x ½ x1/2),
   and his chance of winning is .29. So the expected value in this
   scenario = $30,000 x .125 x .29. Weâll leave the last multiplicand for
   the second step, though.
   Case (A1): $30,000 x 1/8 = $3,750
   Case (A2): $22,500 x 3/8 = $8,438
   Case (A3) $17,333 x 3/8 = $6,500
   Case (A4) $13,750 x 1/8 = $1,719
   Case (B1) $7,750 x 1/8 = $969
   Case (B2) $5,167 x 3/8 = $1,938
   Case (B3) $3,625 x 3/8 = $1,359
   Case (B4) $2,750 x 1/8 = $344
   Case (C) $0
   Now we sum up sub-cases and multiply by the appropriate percentages:
   Case (A) $20,407 x .29 = $5,918
   Case (B) $4,610 x .53 = $2,443
   Case (C) $0 x .18 = $0
   Add it all up, and HNâs expected value is $5,918 + $2,443 = $8,361.
   If youâre still awake, hereâs the bottom line: by going for a
   collective pre-arranged draw (which is against the rules, but common
   and capable of being arranged in a way that obeys the letter of the
   law while skirting its spirit) the player with White is, on average,
   tossing $611 out the window. (And thatâs with the generous assumptions
   given above, and also ignoring that the actual HN outrated his
   opponent by 75 points, which would also improve his expected value.)
   In sum, if one has a reasonably high tolerance for risk, he or she
   should play for a win with White in the last round. Or should he? The
   argument has one last twist.
   Some of you may be wondering, what about the player with Black? The
   calculation is easy â we keep the summed sub-case numbers and invert
   the multipliers: ($20,407 x .18) + ($4,610 x .53) + ($0 x .29) =
   $3,673 + $2,443 + $0 = $6,116. The answer this time is no. All things
   being equal, if the Black player goes for the win, heâs throwing away
   an average of $1,634 for his efforts. Note that on average, the
   drawing plan makes more for the players than going for a win, by an
   average of $512. If this is typical, then if one assumes the
   correctness and applicability of the game-theory strategy of
   [3]tit-for-tat, then maybe Inky is right after all â at least for a
   player whose rating is likely to be near the average in such a
   situation.
   ([4]hide)

References

   1. http://chessmind.powerblogs.com/posts/1183608614.shtml
   2. file://localhost/var/www/powerblogs/chessmind/posts/1184284567.html
   3. http://en.wikipedia.org/wiki/Tit_for_tat
   4. file://localhost/var/www/powerblogs/chessmind/posts/1184284567.html