The Chess Mind

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 I’m 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 – there are scenarios in which they could make over $1000 – but I’ll generously assume they won’t. (It’s not that generous though, since Goichberg extracted the entry fee and (I think) the room costs from GMs’ prize money. So in many if not most cases they probably would have netted a goose egg.)

Assumption 3: Another kindness on my part: I’ll 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%), I’ll bump it up to 50% for ease of calculation – but only for our rivals. Since that increases the chances that one’s rivals will reach the higher score group, this too offers some support to the “let’s all draw” scenario.

Ok, let’s 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 – let’s call him “HN”, for no apparent reason – 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, and…the same four cases.
(C) HN loses, and…there are the same four cases, but we don’t care, since we’ll assume HN makes nothing – 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. We’ll 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 HN’s expected value is $5,918 + $2,443 = $8,361.

If you’re still awake, here’s 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 that’s 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 – 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, he’s 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 tit-for-tat, then maybe Inky is right after all – at least for a player whose rating is likely to be near the average in such a situation.