by Rune

# -3

Gil Kalai, a well known mathematician, has this to say on the topic of chess and luck:

http://gilkalai.wordpress.com/2009/07/05/chess-can-be-a-game-of-luck/

I didn't follow his argument at all, but it seems like something other LW posters may understand, so I decided to post it here. Do comment on his arguments if you agree or disagree with him.

New Comment

Consensus so far (to which I add my voice) is that none of us can even detect an argument there, let alone agree or disagree with it. Solved!

[-][anonymous]10

I think I do detect an argument.

Most people would say that the difference between a game of luck and a game of skill is the degree to which luck and skill contribute to the outcome, proportionally. If more than half of it is luck, it's a game of luck.

Gil Kalai seems to be saying that it's really a matter of risk exposure. If the chance of losing multiplied by the dollars lost is high even for skilled players, it's gambling.

ETA: I guess I was seeing faces in the clouds.

Considering the number of things he didn't mention that he's since endorsed as exactly what he meant, I've joined your consensus.

well, brian, what you wrote is not exactly what I was saying

the problem with your statement Most people would say that the difference between a game of luck and a game of skill is the degree to which luck and skill contribute to the outcome: is that I am not aware of any definite way to quantify the degree to which luck and skill contribute to the outcomes

People often assume that the most skillful the player need to be the higher the contribution of skill to the outcome is but this does not seem to be true

"We can suppose that the higher the stakes are, the higher the ingredient of luck of the combined game."

That's called "assuming what you are trying to prove"!

I think the argument is that if the stakes are high enough people's betting patterns create a zero expectation on the bit itself. This seems wrong on the face of it. It assumes that the bettors on the chess match are perfectly evaluating their skills at making perfect bets with expectation of zero, that there is no skill in determining the bet. Thus with an expectation of zero, the winner of the bet is determined by luck.

This becomes more absurd in the poker game. The difference in skill of betting for action is a large part of the game. Most poker books try to teach it. Most people can't do it.

"Most people can't do it"

This is precisely my point and probably the basis for the judge's rationale in the old case. The situation of those" most people" who cannot do it but still take parts in betting on poker is similar to those playing the roulete. if this accounts for a large percent of participants than it is justified to regard the activity as primarily - gambling (or game of luck) I think there are additional ingredients that will push the situation towards a game of luck when the stakes are high.

I think I see... for positive-expectation games (e.g. scientific research) it is possible that the majority of people involved play with justified reason to believe they will make money/good/utility/whatever without implying that the minority without that justified reason to believe will lose tons of money. For zero or negative sum games this is not true. The majority of people cannot have justified reason to believe their individual game is positive expectation (it's not justified 'cause information that the game is zero/negative sum, and information about how good at the game other people are, is widely available), and are therefore relying on "luck" to select them to win rather than others. Or if the majority know they will win, that implies the minority are losing a lot.

I couldn't discern why the magnitude of the stakes matters at all. By the article's reasoning, each player has some probability of winning => chess is a "game of luck".

It seems that Kalai's argument is that high stakes can transform a game of luck through its effects on the "incentives and motives of players", though he never explains what those effects are, or describes what he thinks the "incentives and motives of players" are in any situation. In short, I don't think he presented an actual argument to be followed.

Precisely, my argument is that high stake bettings can transform a game of skill inti a game of luck I tried to decribe some mechanisms why it can occur in a further comment on my blog.

Folks, I got Gil's argument at last. Woohoo!

A chess tournament produces bits of information: who won which game. If most of those bits can be causally linked to skill differentials between players, rather than chance, Gil calls it a game of skill. Otherwise, luck. If stakes are high and zero-sum, players start trying to find weaker opponents. This means each game will be between players of roughly equal ability, so skill differentials cease to contribute bits to the bottom line.

Gil, is that right? If so, I apologize for misunderstanding at first. This sounds interesting.

Precisely, this is what I meant. Sorry for not being clearer.

It's not a matter of being clearer; you didn't say anything like the paragraph cousin_it wrote above. You have to show some intermediate steps here, because it looked to me that you were claiming that the same two opponents would have different probabilities of outcomes depending on the stakes of the game, without any explanation of why this should be so.

If you're indeed talking about the selection bias for players as stakes change (and more or less an "efficient market" hypothesis near the top), you needed to say something in that direction.

Yes, indeed i am talking about selection bias for players as stakes change. When the stakes are higher if players are rational then the selection bias will lead to them to have similar skills, and this means the game turning more into a game of luck, unless.. some players without adequete skills are playing just by gambling effect, and this also pushes the game into a game of luck ---of a different nature

Well, as someone else has noted here, this still doesn't correspond to our usual meanings of skill and luck. I mean, I'm equally likely to beat or lose to you in a game of Candyland, and Federer and Nadal are (very roughly) equally likely to win at tennis, but the two situations aren't equivalent, since you could replace me with your 4-year-old niece at Candyland and have the same probabilities, but not so much with the tennis example. This is why most of us call Candyland a game of luck and tennis a game of skill (with a small amount of luck involved); the degree of skill is determined by how much the probabilities of outcomes depend on certain measurable and robust characteristics of the players.

I don't know, but I suspect that you have some strong ethical boundaries regarding the "skill/luck" line, so that you've identified it with a "good/bad" line; tennis is "good" because few people lose their livelihood trying to succeed at it, so it must be "skill", while poker is "bad" because a large number of people lose their savings playing it, so it must be "luck". But you need to be aware that there's a rather uncontroversial way that most of us use the terms "skill" and "luck", and that your usage violates it.

Dear Orthonormal, You are partially correct. it may be true that the way I use game of skill/luck it is not the ususal one, but it is a reasonable way, more solid, in my opinion, from the point of view of game theory/economics, and more importantly, it is relevant to the usage in the relevant laws which was meant to define what gambling is, and to common wisdon regarding gambling

The conventional way does not give you a way to measure the ingredients of skill and luck, and it treats the bare game without taking into account the entire scenario, the game, the betting, the winners, the enterance fees, etc

When you try to determine what is a gambling activity and what is not, looking just at the bare game is insufficient the common wisdom regarding these 50 years old poker clubs in J-m is that what was going on there was a gambling activity Aumann proposed a loophole in the legal definition of gambling based on luck and skill ingredients However, if you examine the situation carefully you realize that the usual meaning about what is game of luck/skill is insufficient, and is not compatible with the usual meaning of what is a gambling activity. A more careful analysis based on the entire scenario is more appropriate

In any case, the difference is not between tennis and poker, and it is not based on ethical boundaries (I do not have strong opinions about whether gambling should be legal). The difference is between high stakes negative- expected-rewards games and high stakes games where players expect positive rewards

In any case, the difference is not between tennis and poker, and it is not based on ethical boundaries (I do not have strong opinions about whether gambling should be legal). The difference is between high stakes negative- expected-rewards games and high stakes games where players expect positive rewards

That's absolutely fine. Just don't call it skill versus luck. Come up with new words to express the concepts you think are important; these names are taken.

The conventional way does not give you a way to measure the ingredients of skill and luck

Yes it does; one piece of evidence that tennis involves less luck than golf is that in match-play tournaments of about the same size, you're much more likely for a golfer outside the top 10 to win than for a tennis player outside the top 10 to win.

Or that baseball involves more luck than basketball, because a streak of (say) 27 wins in 30 games is much more common for a top team in basketball than baseball.

(Yes, there are differences in how level various playing fields are, but these phenomena seem to be robust across competition in high school, college and professional levels (AFAICT), so it seems like strong evidence to me. If it were important to me, I could start finding more and more objective metrics of skill versus luck in various games.)

Very good. The conventional ways you propose to measure luck and skill ingredients are precisely the same as what I would use.So my notions are consistent with the usual ones. When you say that baseball involves more luck that basketball your notion depend on the entire scenario and reward systems and not just on the rules of the games precisely as I suggest the new ingredient in my analysis is that these measures may not be robust if we change the reward system.

Perhaps a better way to explain myself is by looking at what I'd consider a sign of how much skill is involved in a game: roughly speaking, how many different strata of players are there for a widely played game, where each stratum can beat the stratum below (let's say) 90% of the time? There's only one stratum for Candyland, and at most three (humanly speaking) for tic-tac-toe, but numerous strata for a game like tennis.

So we can think of luck-vs-skill also in the sense that tennis is more stratified than golf, because a few PGA players have a 10% or better chance of beating Tiger on a given weekend, and many PGA players have a 10% or better chance of beating one of those players, and a top amateur or college golfer has a 10% chance of beating one of those players, and so on down; while there seem to be more layers than that in tennis.

Now, high-stakes games can cause a selection bias wherein all lower strata will withdraw from a competition, but this doesn't change the components of skill and luck for that game; if you forced the withdrawing players to play for those high stakes, you'd see similar results as you'd see for low stakes (neglecting psychological effects of the fear of a bigger loss, but your analysis doesn't seem to incorporate these). The usual concepts of skill and luck are defined by what would happen if X played against Y; your analysis is about something else, which determines whether X plays in the first place. So please call it something else.

So my notions are consistent with the usual ones. When you say that baseball involves more luck that basketball your notion depend on the entire scenario and reward systems and not just on the rules of the games.

No. Orthonormal said nothing of the sort. He just listed ways to check whether a game is skill or luck based, and didn't claim that the critical factor is the reward system.

I suggest you spend less effort trying to look like you were right all along and just admit, "Okay, I used the terms differently from how they're normally used." Is it really that hard? It's the least you could do given such a misleading title.

I think you mean, "Sorry for not being clearer despite using over 15 times as many words."

[-][anonymous]00

There is no reason for bets to be of zero expected money. Just bet around zero expected utility of money. The whole argument might be a characterization of this bizarre betting rule, but not of the games themselves.

But what do you mean by "luck"? I am wondering if there is a translation problem here, and a Hebrew word that does not quite map onto the English one. In English it simply means random factors outside the control of the participants. (It is often accompanied by a superstition that it isn't random but a fickle force that can be attracted or repelled by suitable behaviours.) In that case roulette is entirely luck, poker is partly luck and partly skill (but less luck and more skill the better you are at it and the longer you play), and chess has very little luck. All regardless of the stakes.

The case of the poker club in Jerusalem was presumably conducted in Hebrew. Did the judge simply decide that the activity in question was a social evil that had to be found illegal whatever the letter of the law, or did the law allow itself to be read as supporting the ruling?

Well, in hebrew the meaning is the same. Regarding the poker club in Jerusalem indeed Aumann felt that the judge did not follow the letter of the law but rather his own perception of right and wrong. I offered an explanation why the judge ruling is consistent with the law

I think the difference between our opinions is that you regard the part of poker that is luck and the part that is skill as intinsic property of poker, and perhaps also of how long is the game. In my opinion, just as longer games make the skill element higher there are other ingredients (such as high stakes; winner takes all, and more) that push the skill element down

[-][anonymous]00

Gil, how exactly do you define game of luck vs game of skill? For example, you write:

For example, if players are playing even-bets chess games then a major effort of the players will be to choose opponents with lower or equal skill level. This will push the betting to be primarily between players of equal skills with roughly the same probability to win.

So does your definition imply that if players are evenly matched, the game is about luck? Doesn't look very intuitive to me.

Perhaps a better way to put the argument in the post is as follows: High-stake chess is either a game of luck (which is in tension with no-gambling laws) or a fraud (which is also in conflict with the law). The same holds (even more so) for high-stake poker.

Another example: If people bet on the outcome of a fair coin this is a 'game of chance' (and this in conflict with the no-gambling laws). If people bet (evenly) on the outcome of a biased coin this is fraud (and also in conflict with law). High stake chess and high-stake poker are either a  'game of chance' or fraud.

I'm glad you brought this to discuss here. I read that article earlier this morning before I saw this post and was similarly confused.

People are risk averse, so a zero-sum bet (in money) would have negative utility for the weaker player, unless the player causes the probability of winning to increase given a fixed bet (this is "application of skill", that is performance with exceptionally good concentration/preparation/etc. that makes the a priori estimation of player's skill inaccurate in the particular game). The amount on which this probability must be increased in order to make the expected utility of the bet positive, depends on the absolute values of bets. The bigger the bets, the more the weaker player must increase the probability of winning in order to cross over into positive expected utility.

A game can then be called "a game of skill" if effort can sufficiently increase player's probability of winning, and "a game of luck" if an impossible increase in this probability is necessary. Normal lotteries can't be turned to positive expected utility, so they are always games of luck. Poker can't be turned to positive expected utility by weak players, so for them it's game of luck, while for stronger players it's game of skill. Chess for smaller bets is game of skill, and for bigger bets game of luck for the weaker player.

The picture at the top of the article is of a chess board that has been set up incorrectly (the white queen and king are swapped). The rest of the article is just concentrated confusion; he switches unexpectedly from talking about chess to talking about betting on games in general, then rambles about luck for a bit without ever nailing down a definition of it.

[-][anonymous]00

I've always though that the difference was the frequency that skilled players win.

Incidentally, Richard Garfield, designer of Magic, once wrote on the topic of luck in games. He said that there is, in fact, luck in chess, because we cannot predict the outcome with certainty. He went on to explain that if he sat down to play chess against Kasparov or some other world-class chess champion, he'd expect to lose, but there's still a possibility (however small) that he could happen to stumble upon a superior line of play, perhaps without even realizing it, and end up winning.

Perhaps it would be useful to make a distinction between the game of chess, which is mostly skill, and the game of betting on chess, which is mostly luck with fairly well-known probabilities. It looks to me as though the original article is conflating the two very badly, and that this is the cause of much confusion.

Richard Garfield, designer of Magic, once wrote on the topic of luck in games. He said that there is, in fact, luck in chess, because we cannot predict the outcome with certainty. ... there's still a possibility (however small) that he could happen to stumble upon a superior line of play, perhaps without even realizing it, and end up winning.

Incidentally, when I saw the title of this top-level post, I thought the argument was going to be something like what you've described here: when you make a move, you're steering the game in a direction that has an element of randomness because you can't really review all possibilities. And so you end up surprised at how good or bad it was for you.

Alas, it turns out that Gkalai was simply using a non-standard meaning for his words. Bait-and-switch.

Correct, of course; but we can make some pretty strong quantitative distinctions. I'm more likely to win the lottery than to beat Kasparov (assuming he's healthy and playing at his usual level, etc). But I bet I could beat any human being or computer program at Candyland half the time.

Similarly, if you put me heads-up against a poker pro, I might stand as much as a 10% chance of knocking them out (by getting lucky on the river on an all-in); but that level of luck averages out to a great degree over a long tournament, so that again my chances of making the final table at the WSOP are order-of-lottery bad.

Correct, of course; but we can make some pretty strong quantitative distinctions. I'm more likely to win the lottery than to beat Kasparov (assuming he's healthy and playing at his usual level, etc). But I bet I could beat any human being or computer program at Candyland half the time.

Indeed. (Assuming nobody's cheating, of course.) Garfield's statement does not necessarily reflect my own opinions on things.

Similarly, if you put me heads-up against a poker pro, I might stand as much as a 10% chance of knocking them out (by getting lucky on the river on an all-in)

In this case, you can probably improve your chances by making the game more about luck -- just go all-in every hand.

I've heard that, if you go all-in on every hand in a heads-up poker match, the optimal counter-strategy still leaves you with a 1/3 chance of winning. (I don't know if this is correct or not.)

Sounds about right to me. Going all-in every hand (pre-flop, and blind, of course, so I can't be read) would definitely improve my odds if I were in a heads-up game against a pro. But at a table with more than (say) 3 others, unless they can read me as perfectly as Omega, I should probably start looking at my cards and following a simple memorized poker algorithm.

It depends -- in the limit where blinds are zero, you only call with aces and win 80% of the time. For more realistic values you may well be right.

(I had a truly marvelous bit about luck in chess in an unposted draft. Now I'll probably throw that bit away.)

Reminds me of making the system dumber when faced with a superior adversary.

If you're winning, simplify. If you're losing, complicate. Works in philosopher's football too -- if you expect to be the one to bolt first, you want a kink in the path.

Fails to define what it means to be a "Game of Luck" - AFAICS. In practice, human chess is a game of skill with a small element of luck.

The judge did not base his argument on an abstract mathematical modeling of the entire setting but rather on the consequences. If such clubs indeed cause people to lose everything leaving their families destitute we must conclude that the overall setting makes this activity mainly a gambling activity and that this game is mainly a game of luck (in the same way playing the roulette is).

By this "argument", financial investing is mainly a gambling activity and mainly a game of luck, since it's just as obviously the case that such markets "indeed cause people to lose everything leaving their families destitute."

This is an interesting comment. I think people are aware that financial investment involves some elements of luck, but carries the benefit that it advances the economy a major difference with our case is that financial investment is not (at least not obiously) a negative expectation activity