Can chess be a game of luck?
Let us consider the following two scenarios:
A) We have a chess tournament where each of forty chess players pay 50 dollars entrance fee and the winner takes the prize which is 80% of the the total entrance fees.
B) We have a chess tournament where each of forty chess players pay 20,000 dollars entrance fee and the winner takes the prize which is 80% of the the total entrance fees.
Before dealing with these two rather realistic scenarios let us consider the following more hypothetical situations.
C) Suppose that chess players have a quality measure that allows us to determine the probability that any one player will beat the other. Two players play and bet. The strong player bets 10 dollars and the waek player bets according to the probability he will win. (So the expected gain of both player is zero.)
D) Suppose again that chess players have a quality measure that allows us to determine the probability that any one players will beat the other. Two players play and bet. The strong player bets 100,000 dollars and the weak player bets according to the probability he will wins. (Again, the expected gain of both players is zero.)
When we analyze scenarios C and D the first question to ask is “What is the game?” In my opinion we need to consider the entire setting, so the “game” consists of both the chess itself and the betting around it. In cases C and D the betting aspects of the game are completely separated from the chess itself. We can suppose that the higher the stakes are, the higher the ingredient of luck of the combined game. It is reasonable to assume that version C) is mainly a game of skill and version D) is mainly a game of luck.
Now what about the following scenarios:
Here the main ingredient is skill; the bet only adds a little spice to the game.
F) Two players play chess and bet 100,000 dollars.
Well, to the extent that such a game takes place at all, I would expect that the luck factor will be dominant. (Note that scenario F is not equivalent to the scenario where two players play, the winner gets 300,000 dollars and the loser gets 100,000 dollars.)
Let us go back to the original scenarios A) and B). Here too, I would consider the ingredients of luck and skill to be strongly dependant on the stakes. The setting of scenario A) can be quite compatible with a game of skill where the prizes give some extra incentives to participants (and rewards for the organizers), while in scenario B) it stands to reason that the luck/gambling factor will be dominant.
One critique against my opinion is: What about tennis tournaments where professional tennis players are playing on large amounts of prize money? Are professional tennis tournaments games of luck? There is one major difference between this example and examples A and B above. In tennis tournaments there are very large prizes but the expected gain for a player is positive, all (or at least most) players can make a living by participating. This changes entirely the incentives. This is also the case for various high level professional chess tournaments.
For mathematicians there are a few things that sound strange in this analysis. The luck ingredient is not invariant under multiplying the stakes by a constant, and it is not invariant under giving (or taking) a fixed sum of money to the participants before the game starts. However, these aspects are crucial when we try to analyze the incentives and motives of players and, in my opinion, it is a mistake to ignore them.
So my answer is: yes, chess can be a game of luck.
Now, what about poker?
This post was triggered by a discussion with a young brilliant probabilist (and an amateur poker player) in Sweden who gave an expert opinion regarding poker in Swedish appeals court. It turned out that there were similar cases in Israel as well. I am not sufficiently informed about current poker disputes to offer any opinion about current cases, but let me tell you about a case from fifty years ago.
About fifty years ago, the police raided a poker club in Jerusalem, arrested the operators, and proceeded to prosecute them for running an illegal gambling operation. Israel law defines gambling as participation in what is primarily a game of luck; i.e., where the element of skill is absent or inconsequential. At the request of the defense, Bob Aumann, young game theorist at the time, gave an expert opinion that poker is primarily a game of skill. Aumann gave a detailed explanation, and cited, among others, John von Neumann, Oskar Morgenstern, and John Nash. The basic point is that even when your hand is very good, if you bid accordingly you won’t win much more than the ante, because the other players will catch on and fold; and the other side of the coin is that if you have a poor hand, you can still make a lot of money by skillful bluffing. To Aumann’s surprise, the judge’s verdict was that poker clubs are illegal; he explained by writing: “In these clubs people lose their monthly wages, leaving their families with nothing to live on.” A few weeks later, Aumann met the judge at a party (Jerusalem was then a small town, and perhaps still is). He asked the judge how he could make his ruling after hearing Aumann’s detailed explanation that poker is a game of skill. The judge listened to what Aumann said and replied: “But you see, these clubs cause people to lose everything, leaving their families destitute.”
For years Aumann regarded this case as an example of Israeli judicial activism, where judges ignore the law, ruling by their own personal conception of what is right and what is wrong.
I respectfully disagree.
The issue is what “the game” is. Is it just the pure abstract game, be it chess or poker, or is it the entire setting. The judge was correct to consider the entire setting. 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).
The law is vague and open to different interpretations and, in any case, I am not a legal expert, but from the point of view of game theory (and economics) I think the judge was correct!
We always need to have a large rather than a narrow interpretation of what “the game” is.
Some more discussion can be found in lesswrong.
Update: Here is a related post over freaconomics. Stephen Dubner bring a simple argument why poker is primarily a game of skill: It is possible to intentionally lose a poker game. One comentator sais: “one could also intentionally lose at blackjack (continue drawing cards) but it is still primarily a game of chance.” Another commentator asserts: “Poker, as played in brick & mortar casinos and online is not a zero-sum game because of the rake (the miniscule percentage that the house takes from every pot). Therefore, for the average poker player, poker is a losing proposition. (In fact, around 8% of online players are long-term winners. We know this because we can datamine the billions of hands that have been played online and compute such statistics with a high level of confidence).” And here is a post from “new scientist“.
Update: Here (from the comments section) is my proposal for a definition of the term ” A game which is primarily a game of luck”.
A game of luck is a betting game where the short terms outcomes depend on luck and the participants bet in spite of being able to rationally conclude that their expected returns are negative.
A betting game is primarily a game of luck if this definition applies to most bets in the game.