You live in a world with 1000 other people, only one of whom can beat you at chess. You can beat the other 999. This gives you great prestige, because this is a world where chess skill is exalted above all else.
Now your chess skills atrophy, and all of a sudden you find that 100 people can beat you at chess; you can beat the other 900. You’ve lost some prestige.
I want to quantify the fraction of your prestige that’s gone missing. Of course the answer could be anything at all depending on how you choose to quantify “prestige”, but I’m looking for a definition that most people will agree captures their intuitions (or at least doesn’t grate too harshly against their intuitions).
Attempt One: If N people can beat you, then your prestige is measured by 1/N. Therefore your prestige has fallen from 1/1=1 to 1/100 = .01. You’ve lost 99% of your prestige.
Attempt Two: Your prestige is measured by the number of people you can beat. Therefore your prestige has fallen from 999 to 900. You’ve lost just under 10% of your prestige.
Which of these seems more “right” to you? And do you have an “Attempt Three” that seems even better?
In a few days, I’ll tell you why I asked.
Okay, first off attempt 1 seems obviously wrong to me—it doesn’t take into account the total number of people who you could potentially lose to. If you lived in a world with 101 people and you went from ‘one person can beat you’ to ‘100 people can beat you’ you’ve gone from the absolute top of the pile to the bottom, whereas in this 1000-person world you’re still 90th percentile. I’m tempted to say ‘discard any method that doesn’t take into account the total size of the population’.
Attempt 2 also feels wrong, by the very qualitative reasoning that when calculating prestige I don’t *care* all that much about the ability of the below-average members of the population. The difference between the person who’s second-worst at chess and the person who’s worst at chess doesn’t feel qualitatively significant. On the other hand, the difference between best and second-best seems fairly important, prestige-wise (gold medal athletes, etc.)
So for an attempt 3, we could take the assumption that the difference between first and second best is greater than the difference from second to third best and so on, and assume that prestige is some sort of exponentially decaying function. If we say that the best chess player in the world has a prestige of 1,000, you originally had a prestige of 500 and now have a prestige of… 7.9*10^-28. I’m not even going to bother to calculate the percentage there, but it could potentially be accurate? I could namecheck a couple of the best chess players of all time (Bobby Fischer, etc), but certainly not any chess player in the 90th percentile unless they were like someone I knew personally. On the other hand, chess in this world is far more important than chess in real life, so I don’t think ‘being 90th percentile in chess’ is necessarily as unremarkable as I calculated. So maybe use a different function? Really
Last attempt: standard deviation. I’m thinking about the way we quantify IQ here. Assume that your chess population has a normal distribution, set ‘average’ prestige to 100, and quantify prestige by standard deviations from the mean. Some very quick and loose calculations based on the weschler iq scale (average is 100, +15 points per standard deviation) and the table of standard deviations available on wikipedia because I’m lazy: you started with a chess iq of 149.35 (3.29 standard deviations/99.9th percentile) and ended up with a chess iq of 124.66 (1.644 standard deviations/90th percentile). That’s a 16.5% loss in chess iq/prestige.
Obviously that prestige number will vary a lot depending on where you set your average value and how big you make your standard deviations, plus it assumes that chess skill in this world exists in a normal distribution.
I don’t know whether I have a good model for population behavior in any of these, but it’s an interesting problem!
“this is a world where chess skill is exalted above all else”
I for one welcome our robot overlords. They get all the prestige; humans get none. So there’s no loss.
OK, now a serious answer. I think Attempt 1 is close. In sports, prestige is measured in money (salary + endorsements). It’s very heavily concentrated at the top. Nike won’t hire a 50th ranked athlete as a spokesperson, not at any price.
In Attempt 1 the top player has infinite prestige, more than all others combined. That seems unrealistic. I would change your formula to 1/(N+1), so that the top player has prestige 1, the second player 1/2, the third best 1/3, and so on. In this formulation, as in sports, you really don’t want to fall out of the top 10. In your example, falling from rank 2 to rank 101 you lose a little over 98% of your prestige.
I like attempt 2, because it’s like looking at your percentile rank. If I can beat 90% of the people, that seems pretty good. (Forget about the absolute number of people I can beat.)
1. I’m going to suggest 1/(N+1) instead of 1/N. Otherwise the best player in the world gets an undefined/infinite amount of prestige.
2.This (in spirit attempt 1) seems to do a pretty good job of capturing prestige to me.
3. Empirical idea, look up the worlds top 100 chess players by Elo. Find how many Instagram followers and regress both curves. Repeat this for other competitive activities and see how results vary.
“You live in a world with 1000 other people, only one of whom can beat you at chess. You can beat the other 999.”
I’m confused by your use of the word “can”. At the beginning of this thought experiment, is it true that you always beat the 999 and the one other person always beats you?
If it is literally that you “can” beat someone else but not always, this will affect how we will discuss the merits of the attempts.
Dirk (#6): For “can”, you should read “always does”.
Daniel R. Grayson (#4) (and this is also relevant to others): Certainly the absolute number is irrelevant here; I fixed the population size for illustration, but everything is meant to scale with that size.
So let the population size be P, and let N be the number of people who can beat you. Then (up to a multiplicative constant) Attempt One defines prestige as 1/(N/P) and Attempt Two defines it as (1-N)/P. And the multiplicative constant doesn’t matter, because I’m not actually interested in the raw measure of prestige; I’m interested only in the fraction of prestige that gets lost when N changes.
Jonathan Kariv and AMTBUff (#5 and #3): 1/N+1 won’t work because then the measure won’t scale with population size. But I’m not worried about the infinity problem, because I’ll eventually be applying this to a model where we have some uncertainty about how many people can beat you, so that N is replaced by the expected value of N, and this will always be positive.
Summarizing the past few comments, here is a more careful formulation of the problem.
In a world with population P, the expected number of people who can beat you at chess (conditional on a bunch of publicly observable facts about your skill) is N. That expected number is always greater than zero.
Attempt One: Prestige is 1/(the expected fraction of the population who can beat you) = 1/(N/P) = P/N. Then when N changes to M, your fraction of prestige lost is 1-N/M.
Attempt Two: Prestige is (the expected fraction of people you can beat) =(P-N)/P. Then when N changes to M, your fraction of prestige lost is (N-M)/(N-P).
Note that both cases scale with population size (i.e. if we multiply N, M, and P by the same constant, then the measure of prestige lost is unchanged).
Any reasonable attempt should have this scaling property.
Jonathan Kariv (#5): I love your empirical idea #3, though I’m not sure I’ll find the time/energy to pursue it. I’ll be thrilled if someone else does, though.
This is a great question. The term `prestige’ can mean very different things. Joe Henrich uses it to mean something like `honor’ or `status’, with everyone having a relative level. But a more everyday definition has a handful of prestigious people and everyone else is below the radar and has basically zero prestige.
I think the world you describe here is more like Henrich’s: if chess matters above all, then Chess Schlub 1 and Chess Schlub 2 care intensely about even relatively small differences in their relative prestige. That is, the world you’re describing is like a chess tourney, not a society in which chess is one of many pursuits.
With that in mind, prestige differences shouldn’t evaporate outside the top couple percent. And people at the median should feel themselves intensely, not marginally, more prestigious than people at the very bottom. This is more like Henrich’s concept, and more like Attempt Two above.
[Attempt One above seems rather arbitrary to me. Why not let your prestige be -(N^pi), where N is the number of people who can beat you? You can stick N into any decreasing function f(N), but I don’t see a reason to prefer one such f(N) over another. Attempt Two has a much clearer intuition.]
In a more realistic application, an Attempt Three would be like alan’s “Last Attempt” based on standard deviations, since the middle of most distributions is compressed. But you’ve explicitly given us a world where the 433rd-best player always beats the 434th-best, no matter how many times they play (in the short run). This is a uniform distribution, not an [observable] bell curve. In a uniform distribution, there’s going to be no difference between relative and absolute standing, so Attempt Two is sufficient.
But all that doesn’t measure the “percentage” question, which is what you’re actually driving at! Here, I think Attempt Two has nice properties. Given a 10% drop in Type Two Prestige
1) If you meet someone at random, the chances that you’re more prestigious than them has decreased 10%
2) Prestige could be relabeled ‘relative chess ability’ [Where Type One Prestige would be ‘relative chess inability’, which is kind of weird.]
But it also has a potentially bad property, which is that a move from 999th place to 998th place is a 100% increase in prestige.
You could define prestige as exp(N) so that the percent increase in prestige is equal for an increase from 999th to 998th as it is for an increase from 2 to 1. But why bother – you’re just finding a complicated way to get back to levels.
But what if you wanted a form of prestige that gave large & symmetric prestige rewards & penalties at the very top or bottom of a distribution? N is the number of people who can beat you. Prestige P could be:
P = exp[(1/(N+1)) + (1/(N-1000))]
So the top player has a prestige of 2.72, the bottom player a prestige of 0.37. Dropping from 1st to 2nd place or from 999th to 1000th place yields a prestige loss of 39%. Dropping form 2nd to 3rd or from 998th to 999th docks you 15%. Dropping from 500th to 501st only lowers your prestige 0.0008%.
It is difficult to put oneself onto a world where chess (or anything else) matters above all to everyone, so that does colour my intuition.
My first instinct was that attempt 1 was better than attempt 2, since prestige in our world is usually concentrated on the very top. Sliding down to 90% is going to lose you a lot more than 10%. 99% seems a bit high, but if forced to choose only between these two, then 1 seems better.
But then I think that in this hypothetical world, chess skill is exalted by everyone above all else. That is going to change things. In our world no skill is universally exalted, so prestige is diluted. Maybe it makes no difference.
What if we reverse it and look at the lowest? One person is the worst at chess and has essentially zero prestige. the next person up can beat one person, so has presumably a lot more prestige.
Another idea is to reduce the size of the population and see how that fits with intuition. No time at the moment to explore this.
Intuitively, I think of prestige as something quite important for people at the very top of a scale, and much less so for most others. Being the top tennis player has a lot of prestige. Being the second-best tennis player has prestige because you’re constantly being associated with the top player. Being the 10th best player? It has a lot of prestige among tennis enthusiasts, but otherwise? I’m reminded of Prospect Theory, wherein the amount people are willing to pay to reduce the risk of a loss from 10% to 0% was greater than what they’d pay to reduce the risk of the same loss from 50% to 40%. In short, it’s not a matter of fixed ratios.
But WHY would being the 10th best tennis player matter so much less? Because people’s attentions are focused on other things, so that they can bear in mind few tennis players–and it makes sense that they’d focus on the top one or two.
In contrast, Landsburg specifies that we’re talking about a society in which “chess skill is exalted above all else.” Perhaps this means that there is no trade-off frontier, because there is only one desired good. In such an environment, I’d guess that people would model prestige in a manner similar to Attempt 2. People might be HIGHLY sensitive to their relative status, with no sense of equivalences.
Perhaps you should ask Irving, the 142nd Fastest Gun in the West. (“A hundred forty-one could draw faster than he / But Irving was looking for one forty-three….”)
#14 Irving sounds very much like Ernie, who drove the fastest milk cart in the west, by (or, as it turns out, ripped off by) Benny Hill. Well, the tune and tone are similar but the lyrics all Benny’s. After saying she wanted to bathe in milk Ernie says
“Do you want it pasteurised ‘cos pasteurised is best.” He says “Ernie I’ll be happy if it comes up to my chest!”
In the song Ernie is battling for prestige with two ton Ted from Teddington, who can offer treacle tarts and tasty wholemeal bread, and the size of his hot meat pies nearly turned her head.
Ernie was primarily concerned with having more prestige than Ted. Swapping one place made a big difference to him.
Your comment relates to mine – how do the people at the bottom value their chess prestige? If you are crap at chess you don’t derive any prestige at all from it, but maybe you are a good boxer and get your prestige from that.
In our world, chess prestige effectively levels off to zero after a small fraction of people. Individuals get some prestige by winning in their local chess club, where they might be 100,000th in the world, but that prestige is very limited to the other members of their own club. I might beat Grandpa, and get a small amount of prestige in my family, but nobody else cares. So unless you are in the top 1000 or so in the world, effectively there is no prestige from chess. Going from 70th percentile to 60th percentile makes effectively no difference, which is nearly 1 billion people more who can beat you. Going from 10th 20th probably makes a lot of difference and that is only 10 more people that can beat you.
But if we use this chess thing as a stand-in for the total prestige you have in society, my intuition tells me it will be distorted towards the top and the bottom. Going from No.1 to No.2 seems like it may be a big thing. Going from the bottom to next-to-bottom also seems like it may be a big thing. Going from near the middle to slightly higher or slightly lower does not seem so important. Where the extremes are in sight, they seem to take on a greater importance. This may be because I am at least partially conflating prestige with something else, maybe power.
In the chess valuing world, one would imagine that prestige carries with it other direct benefits, such as power and influence. It is difficult to reduce these multi-dimensional things to a single dimension.
Perhaps, like Ted and Ernie, it makes sense to think of it as a mating game.
I think option one more closely matches my intuitions about how prestige works in the real world. Instead of chess, consider professional boxing. Plausibly, the current world heavyweight champion is the most prestigious boxer in the world, and the number one contender is plausibly the second most prestigious. (Disclaimer – I actually have no idea who either of these people actually are, but if you want to know my credentials for judging boxing, I’ll have you know that I’ve watched *all* of the Rocky movies.) Presumably, the top ranked half dozen or so boxers in line for a shot at the title are also quite prestigious. But what about the guy who’s ranked as the 100th best heavyweight boxer in the world right now? I’d be willing to be a fair amount of money that you could go to the most die hard boxing fan, and give him ten guesses at the name of this boxer, and he’d fail to name him. Being ranked in the top handful makes you a household name (for fans of whatever the activity is). Being ranked 100th makes you basically anonymous – your odds of being recognized in public are essentially zero. Lance Armstrong was a household name for winning the Tour de France 7 times. (Granted, that whole cheating scandal knocked him off his podium, but I’m simply talking about what gave him prestige in the first place.) Would anyone remember the name of a cyclist who came in 100th place seven times? I doubt it.
For the hell of it, I thought I’d take as stab at naming this boxer in ten guesses. I guessed–
Jesus Hernández
Alejandro Hernández
Miguel Ángel Hernández
Eduardo Hernández
José Hernández
Juan Hernández
Fernando Hernández
Diego Hernández
David Hernández
Jorge Hernández
I checked and … nope. Rodney Hernandez clocks in at 47. There’s also an Eduardo Hernandez, Genaro Hernandez, Nico Hernandez. And there’s a Ángel Hernández—not bad!—but he’s a MIDDLEweight boxer.
You win this time, DavidDC. But just you wait….
Oops–make that JesusDC. Or AlejandroDC.
No? I’ve still got seven more guesses….
It turns out to be Apti Davtaev. Not a single Smith or Jones in the top 100. There are two Fury’s and a Rock.
Since we now have discussion of both chess and boxing in the same post it is compulsory to mention chess boxing. Perhaps the relative lack of prestige enjoyed by chess-boxers is illuminating here. You need to have an ELO rating over 1600 and fought 50 fights to compete. Yet the combination of both skills does not give as much prestige as being top in either one. Maybe it is just a matter of time before chess boxing becomes the world’s No.1 sport.
I love this post. (Note: to avoid “cheating” I have refrained from reading Stephen’s follow-up post.) Is this not more of an aesthetic question, since either approach is “valid” from a purely mathematical point of view. That said, couldn’t we side step this aesthetic choice between competing ranking procedures with the Elo system?
What is he difference in prestige between the No.1 Elo ranked player and the number 2?