Nakamura and McShane's big mistake
I love following this Millionaire Chess tournament. It's really quite a spectacle: untitled players can pick up tens of thousands of dollars, one top player gets to go home with $100,000, and there's even the bizarre "Win a million" lottery to keep us interested. And with so many 'novelties', it's not surprising that there's always the potential for controversy.
The biggest of these happened yesterday in the final round of qualification for Millionaire Monday. The main organiser GM Maurice Ashley was visibly irate when discussing the nine-move draw between the top seed Hikaru Nakamura and English GM Luke McShane. (You can see all the interviews of the draw controvery here.) He called short draws "a stain on our game". Poor Hikaru and Luke suffered a fair bit of backlash in the chat channels and on Twitter for their performance, although they handled their interviews extremely well (particularly Luke, one of the real gentlemen of chess).
Hikaru also spoke well, aside from two notable exceptions. In the interview you'll hear these two sentences: "the risk wasn't worth the reward, frankly", followed later by "I don't think I did anything wrong". At this point, the economist in me was highly dubious, although I have no doubt that both Hikaru and Luke actually believed this to be true.
I don't know Hikaru personally, but in recent years I've been impressed by his interviews, and particularly how gracious and appreciative he is about being able to play chess for a living. And in games where the result is clearly not prearranged and either player would have to make a concession to avoid the repetition, I have no moral problems with a draw - so in this, the players are right. But in my opinion, one of the greatest innovations of Millionaire Chess is that its unique prize structure should naturally prevent boring draws. This is because the risk really is worth the reward in most cases.
So at this point, I did what any math/chess geek would do: I wrote down the problem :) And without going into too many details, it turns out that the short draw was almost certainly the wrong decision for the players to make for themselves. Even under some very tolerant assumptions, the expected payoff from playing on, for either player, was greater than the expected payoff from accepting the repetition.
In my analysis, I had to make a bunch of assumptions, although I think they're all pretty reasonable. I took into account that by playing on, the players would most likely have a very long game that would sap their energy somewhat (Luke had had a couple of really tiring previous games, while Hikaru said he had been feeling unwell). This would decrease their performance in the tie-breaks (if they occured) and the rest of the event. I also assumed that whoever chose to avoid the repetition would have to make a concession that would decrease their chances in the game from what they were at the outset. I assumed that, all else being equal, Hikaru's chances in tie-breaks and the final-four were above that of an average competitor, while Luke's were average (...after a short draw, while a bit lower if he played on). Finally, as it turned out, almost the maximum number of players on 4.5 points who could get to a tie-break with 5.5 points did so, while really made Hikaru's and Luke's decision look silly - but they couldn't have known that when they took the draw. So I relaxed this assumption a bit so that a normal number (five out of eight potentials) reached the 'tie-break score' of 5.5.
The analysis is a lot more complicated than this, but you can already get a rough idea of things by checking out the prize list. It's incredibly top-heavy, and so under almost any realistic assumptions, a player in their shoes would want to maximise their chances of making the final four, above all else. If Luke played on, his chances of beating Hikaru were slim - but they were still much higher than making it through a tie-break with seven other players, including Hikaru. And for HIkaru himself, despite being one of the best rapid players out there, the odds still suggested the same decision.
(For those interested: my final numbers suggested that Luke's expected payoff was roughly $4,000 higher from playing on, while for Nakamura, avoiding the repetition was worth about $8,000 in expectation.)
Of course, 'in expectation' is such an economist thing to say; probabilities are one thing, but only one outcome can actually occur in real life. For Nakamura, he made it through the tie-breaks (though not without some very bumpy moments!), and so it looks like things have paid off. But that's not the right way to think about things. It's like winning your first ever spin of roulette: just because you got paid doesn't mean you made the right decision. I would definitely advise Hikaru in future to do these sorts of calculations (or better yet, get someone else to!) before crucial money clashes.
(Luke, on the other hand, is not a professional chess player and probably doesn't care that much about the money. While he didn't make it through the tie-breaks, he's still had a good tournament and has good chances of picking up a big consolation prize in the rest of the open. But still, from a purely academic perspective, the decision-making was dubious!)
Of course, this was mainly just an academic exercise for a bit of fun (although professional players may want to take note - I'm open for consultation ). But there is one policy implication, and here I'm specifically talking to Maurice and organisers like him. The lesson is: Don't be discouraged! The Millionaire Chess team have done exactly the right thing in their structure to promote fighting chess. It's hardly their fault if the players haven't yet worked out how to act in their own best interests. But this will happen through experience (and maybe through posts like this...), so there's no need to panic.
For the time being, I'm going to sit back, relax and watch the final fight - in which, typically, I expect Hikaru to defy the odds, win the tournament and thereby blow a big, fat raspberry at my analysis