## What’s my Kasparov number?

A colleague writes:

Personally my Kasparov number is two: I beat ** in a regular tournament game, and ** beat Kasparov!

That’s pretty impressive, especially given that I didn’t know this guy played chess at all! Anyway, this got me thinking, what’s my Kasparov number? OK, that’s easy. I beat Magnus Carlsen the other day when he was passing through town on vacation, Carlsen beat Anand, . . .

OK, just kidding. What is my Kasparov number, though? Note that the definition, unlike that of the Erdos or Bacon numbers, is asymmetric: it has to be that I had a victory over person 1, and person 1 had a victory over person 2, etc., and ultimately person N-1 had a victory over Kasparov. The games don’t have to be in time order, they just all have to be victories. And we’ll further require that the games all be played after childhood and before senility (i.e., it doesn’t count if I happened to play someone who happens to be a cousin of some grandmaster whom he beat when they were both 7 years old). I’d also require the these be official tournament games, but I’ve never played in a tournament so we’ll relax that rule.

So what is N? How can I do this? I beat my dad, who on occasion beat his dad, who was really good. Grandpa Moses probably beat some serious tournament players from time to time (he played at coffeehouses etc., not tournaments, but I think some competitive players would show up), and one of the people he beat must have had a win against one of the top U.S. players in that era (1920s-1930s I guess is when Grandpa was strongest). To get from one of those guys to Kasparov . . . how many steps would it take? I don’t know, but that one must not be too hard to answer. Just take the oldest guy who Kasparov ever lost to, then the oldest American player that guy lost to, . . . then maybe 2 more steps after that? So then my Kasparov number would be 9 (dad, grandpa, the (hypothetical) tournament player my grandpa beat, the top guy he beat, then the next 2 guys in the chain, then the American player who beat the old guy who beat Kasparov). So that’s my guess.

What else could it be? Phil is better than me but I’ve beaten him on occasion. I think Phil played in a tournament once or twice, and he might have won a game against someone who played in tournaments more frequently, . . .

In any case, I have a feeling that my Kasparov number is much much lower than my Muhammed Ali number. The last time I got into a fight was in 8th grade, but I wouldn’t call it a win, it was more of a draw. If I were to count it though, then I’m pretty sure this guy got into a lot of fights after that, but it would take a long long chain of slugging to get to Michael Spinks or whatever. If my Ali number (or even my Chuck Wepner number) is less than infinity, it must be in the hundreds.

P.S. Phil reports that the best opponent he ever beat in a tournament had a rating of something like 1562 1800, so maybe we could get to Kasparov that way. Someone who was 1800 when he played Phil might ultimately have reached 1900 and beat someone who at some point reached 2100, etc etc etc. I don’t really have a good sense of whether the Phil path or the Grandpa path would get me faster to that elusive Kasparov victory. But I’m pretty sure these are the only options I have.

1. Chip Lynch says:

Curse you! Now I have to spend all day trying to figure this out…

I LOST to Kasparov once, directly… sorry that doesn’t help, otherwise I’d offer to play you in a game. ;-)

—Chip

2. Beliavsky says:

“Carlsen” not “Carlson”.

3. keith says:

I beat my dad who beat Nigel Short who took about one game off Kasparov in their non-epic world title clash.

4. I would guess my Kasparov number is almost certainly less than or equal to 3, achieved probably by the 1990s. My reasoning involves young rather than old players: as a kid in the 1990s, I beat at least one or two other peers who later became much stronger than me (including becoming US champion). I don’t know if any of these personally beat Kasparov, but among the wider range of peers including all the top US juniors of the day (many of whom ended up in my college math and physics classes!), surely they must have beaten someone who beat Kasparov. I’m pretty confident, for example, that I beat someone who beat Patrick Wolff, who beat Kasparov in a simul.

• I meant “as a kid in the 1980s”: I basically “retired” from competitive chess when I reached high school, playing my last serious game in 1985 (until my return to the game as an adult twenty years later).

• Andrew says:

But see above: I wrote, “we’ll further require that the games all be played after childhood . . .”

You need to define childhood more clearly, and even then it’s a dubious restriction. I suspect very many players reach their all time peak at around age 10 or 12, playing competitively in primary school before losing interest.

• Andrew says:

I don’t think it’s a dubious restriction! If person A at age 20 beats person B at age 7, I wouldn’t want to count that as a win in the chain.

But what if person B is 12 rather than 7, and at the peak of their chess ability?

• I only think about games I played at age 10 or higher, e.g., when I beat a future US champion when we were both 14 and he was already a master.

5. Z says:

There were multiple national chess champions at my high school (two had been champion when they were younger, one during high school), but I never played any of them when they were drunk so my Kasparov number is probably infinite/undefined.

6. Jay says:

If you’ve ever been a tournament player in the US, the US Chess Federation has all the data on your opponents and your opponents’ opponents going back several decades (http://www.uschess.org/component/option,com_wrapper/Itemid,181/). ChessGames.com has more data on elite play.

I rarely play in tournaments anymore and am not particularly high-rated (1853), but my Kasparov number seems to be 6. I once beat a 2100 who beat a 2300 who beat a strong American GM who beat Loek Van Wely who beat Nigel Short who beat Kasparov.

7. ben hyde says:

Presumably many players have multiple Kasparov numbers, but choose to mention only the best one they are aware of. Which allows us to ponder what is Kasparov’s worse Kasparov number?

8. Corey says:

AG describes the “starting-from-me” search and its likely results starting from him. The recurrence of Nigel Short in the comments suggests that the way the “starting-from-Kasparov” search works would be: list the players who ever beat Kasparov, weakest first (perhaps measured by some sort of time- or number-of-games-weighted average ELO score); repeat the process on the resulting list. Keep all results on one sorted list instead of creating a tree of lists.) Stop when you find yourself.

• If I had the graph at hand, I’d just use Dijkstra’s algorithm.

Corey’s algorithm isn’t guaranteed to get the right result, because the strength heuristic doesn’t guarantee you have explored all the possible shorter paths. The priority queue needs to be sorted to always explore a shortest path first (i.e., perform a breadth-first search). The trick of Dijkstra’s algorithm is to eliminate redundancy while still exploring the nodes breadth first.

• Corey says:

I was thinking I should write, “This gives an upper bound for your Kasparov number, so you can make an efficient start-with-me search” but then I thought I might look foolish if the algorithm was exact. Hmm…

9. Phil says:

Andrew says “P.S. Phil reports that the best opponent he ever beat in a tournament had a rating of something like 1562, so that doesn’t seem to help. And in my whole life I don’t think I’ve ever beaten anyone better than Phil” but this isn’t true at all! I beat some guys in the 1700s or maybe low 1800s when I played a few tournaments in college.

But I played a few of tournament games after 1991, and among THOSE games the strongest guy I beat was 1562. I can’t look up the earlier records because the records before ’91 aren’t available on the USCF website.

10. C Ryan King says:

So does anyone have a finite Bacon-Erdos-Kasparov number?

• I spent an afternoon a while back figuring my Erdos-Bacon-Sabbath number, and I suspect that if I’d ever played chess I would have just lost the rest of the evening after reading this post. So for the sake of their valuable time, I hope nobody does.

11. Jake says:

With the chess graph, I’d wonder what the circumference is, i.e. what’s the longest A defeats B defeats C etcetcet defeats ??? defeats A chain you can make.

12. Steve Sailer says:

At Rice U. in 1977, we used to compete to see who could come up with the most parsimonious argument for why Rice’s 1-10 football team was, when you stop and think about it, the number #1 in the country. Rice beat Idaho, which beat Montana, which beat Oregon St., which beat Arizona, which beat Oregon, which beat Washington, which beat Arkansas, which beat Mississippi, which beat Notre Dame, which beat Earl Campbell’s previously undefeated Texas to win the National Championship. (Or something like that.)

So, Rice deserved to be #1.

Are you going to argue with logic?

An equally fun (and equally valid) undertaking was to come up with the least parsimonious pathway for Rice to claim to be #1, with extra points for involving Division II teams.

13. zbicyclist says:

Pretty sure my Kasparov number would be higher than my Elo number.

• James Annan says:

Actually, the Elo system itself is a pretty big clue. You have a decent chance of beating someone about 200 points above your grade, so most people who play at all regularly against a decent range of opponents will have done this a few times. A basic grade of 1000 (which is pretty low) would suggest a K number of about ~10. Of course this relies on there actually being a chain in existence, but chess is probably quite well mixed (again, for people who play reasonably regularly).

14. Mark says:

This was fun to look up and now I know my number is no greater than 5, even with the strict rules. I’m about 1700 and going to the USCF website I can see every rating game I’ve played and so found the highest rated player I beat, the highest rated player he beat, etc, until I found a GM I recognized and looked up his Kasparov games. I beat:

MATTHEW R TRALDI (2258, Candidate Master) who beat
ANDREI A ZAREMBA (2418, Life Master) who beat
ALEXANDER IVANOV (2611, Life Senior Master) who lost to Kasparov but did beat
ALEXANDER SHABALOV (2613, Life Senior Master) who beat
GATA KAMSKY (2813, Life Senior Master) who beat Kasparov (http://www.chessgames.com/perl/chessgame?gid=1066681)
GARY KASPAROV (2851*, Retired World Champion) who beat the world (http://www.chess.com/forum/view/game-showcase/gary-kasparov-vs-the-world2)

*I think USCF ratings are higher than FIDE and I switched rating systems here.

15. jyyh says:

If speed chess is taken into account, my Kasparov number would probably be around 8, cannot prove it as I cannot trace the club speed chess games. Kasparov beaten by Tal beaten by Iivo Nei beaten by T.Ristoja beaten by Luukkonen beaten by Holmsten who in his youth likely lost to someone in the club I played speed chess for 1 year. As this is now about on the right age group, add couple of links and the result is 8-9. Thanks for the idea. (My Club rating was about 1350, so it fits to James Annan approximation :-))

16. Let’s see — I’d start with a win over my father, who had a win over Reshevsky. Oh wait, that was at ping pong. They were at U of Chicago and made a deal: my father would teach Reshevsky to play ping pong, and Sammy would teach my father to play chess. My father would cite this as evidence of being the much better teacher because Reshevsky eventually could play a decent game of ping pong.

17. […] more of slow grinds rather than Kasparov style flourishes. Speaking of Kasparov, following Andrew Gelman, one defines the Kasparov number as the length of the shortest (ordered) chain of people (starting […]