Consider the following game: you have a box that contains one white ball and one black ball. You choose a ball at random and then return it to the box. If you chose a white ball then a white ball is added to the box, and if you chose a black ball then a black ball is added to the box. Played over time, what is the probability that more than 80 percents of the chosen balls are white?

This question is related to the fact that professional players can emerge in pure games of luck that was mentioned in the previous post.

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: 

E) Two players play chess and bet 5 dollars.

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. 

 Andrei Raigorodskii

(This post follows an email by Aicke Hinrichs.)

In a previous post we discussed the following problem:

Problem: Let be a measurable subset of the -dimensional sphere . Suppose that does not contain two orthogonal vectors. How large can the -dimensional volume of be?

Setting the volume of the sphere to be 1, the Frankl-Wilson theorem gives a lower bound (for large ) of  ,
2) The double cap conjecture would give a lower bound (for large ) of .

A result of A. M. Raigorodskii from 1999 gives a better bound of . (This has led to an improvement concerning the dimensions where a counterexample for Borsuk’s conjecture exists; we will come back to that.) Raigorodskii’s method supports the hope that by considering clever configurations of points instead of just -vectors and applying the polynomial method (the method of proof we described for the Frankl-Wilson theorem) we may get closer to and perhaps even prove the double-cap conjecture.

What Raigorodskii did was to prove a Frankl-Wilson type result for vectors with coordinates with a prescribed number of zeros. Here is the paper.

Now, how can we bit  beat the record???

This is not as clear cut a question as the earlier ones, and if you do not know an answer then it will be difficult to figure one out just based on intuition. (But perhaps possible).

If you are intrigued by the question and would like to explore what an answer could be, I would be interested to know how you tried to find an answer.  Asked a colleague? Looked at a book (which?)? Looked online (where?)? 

Here is the question:

A differentiable complex function automatically has derivatives of every order. (In contrast to differentiable real functions that need not have even second derivatives at any point.) 

Can you describe this “miracle” as part of a more general phenomenon?

Praise for: ” ‘Gina Says,’ Adventures in the Blogsphere String War

(Below the dividing line:  Greg Kuperberg, Scott Aaronson, Clifford Johnson, Peter Woit, Motty Perry, Caterina Calsamiglia, Yuval Peres, Eva Illouz, and (right from the comment section)  Luca Trevisan, Thomas Love, John Sidles, Jacques Distler, Marni D Sheppeard, (and from other journals/blogs) Hamish Johnston, Lance Fortnow: .

Download the first part of the book (pdf file) (See also this post)

  Shmuel Weinberger (August 07): Very much enjoyed the story of Gina’s involvement in the blog world– i read it through on my flight back to America. It was a very interesting if occasionally difficult read. Probably the part that resonated most was the advice (i think it came from your father) that every subject is fascinating after you study it deeply.

Avi Wigderson: (August 07)  I expected no less from the author of the immortal translation of the classic book “Where is Pluto?”

Oded Schramm (Dec 2007): Though it is somewhat uneven, there were some definite enjoyable highlights. (Feb. 08 ) What about a sequel? I’m really curious what’s happening with Gina these days?

Itai Benjamini It was such a joy to listen to your unique voice (music) yet again. A typo: Section 20 “children’s teaching disabilities” should be “children learning disabilities”. (Feb 08 ):

Elchanan Mossel (Feb 08): I read it when I was sick and couldn’t do other things, and it cheered me up

Olle Haggstrom (March 08): I found it a real page-turner, and read the entire thing for four straight hours last night! Very interesting stuff, on several levels. And very original, of course. May I ask what is your relation to Gina? You seem to have remarkable insight into her mind…

Ken Binmore (June 08) Dear Gil, I like your book a lot. If you get it published, it could do with some pictures. Best wishes, Ken

Greg Kuperberg: (Oct 08)  …So the assertion that spacetime is 10-dimensional comes with the important asterisk that it might only look 10-dimensional when it is near a classical limit in the sense of standard semiclassical perturbation theory. In the realistic limit it looks effectively 4-dimensional, and in more exotic limits it may look 11-dimensional (say). If you do not take a limit at all, then maybe spacetime is some non-geometric algebraic object that doesn’t have a dimension.

  Scott Aaronson (Nov 08): It is one of the strangest documents I have ever read…  

At the beginning, the greatest charm of your manuscript is its promise to bring an “outsider perspective” to the string wars: unlike the partisans on both sides (Motl, Distler, Smolin, Woit…), Gina presents herself as a curious outsider who’s just trying to pose questions and understand what’s going on.  By the end, though, she’s become (at least in my eyes) basically another partisan, for the pro-string side…    

 there’s lots of food of thought in this unusual manuscript — [despite the above concerns,] I certainly found myself reading till the end.

Clifford Johnson (Nov 2008): I find the Gina character rather well done…. I actually really like your idea… it is fresh and amusing. Well done.

Peter Woit (January 09) : I read through it quickly, amused to relive again some battles of the string wars…. In your fantasy of the future, you mention my book being translated into Czech. Funny, a publishing company there did buy the rights a year or so ago, and I think they will be bringing it out. Sometimes reality and fantasy are indistinguishable in this story… (A post from “Not Even Wrong” with more)

Motty Perry (Jan 09): I started reading your book on my way to the UK (will be back in 3 days) and was forced to stop when the laptop’s battery died. So far (50 pages), it is fascinating!!!; I find it educating and I wait to my next flight to continue reading.

Caterina Calsamiglia (Jan 09) Every once in a while there are interesting thoughts about what the aim of science should be, that are nice…! Physicists and mathematicians are quite special…!

Yuval Peres (Jan 09) I spent several hours reading through it, trying to separate fact from fiction. The statement “Why should I be surprised if I can simply disbelieve.” on page 17 is not due to Gina’s great uncle “Lena” but rather to my grandmother Malka Heller. (GK: Correct!)

Eva Ilous (Apr 09) I kept on reading (there were 2 dense pages that stretched me) and I now find it very entertaining.

New from the remarks sections:

 Luca Trevisan (June 09)  It was a wonderful idea, and very well executed; I look forward to the second part, but the Comic Sans hurts my eyes.

Thomas Love: Great fun!

   John Sidles: Congratulations Gil … this book is a very interesting read!  

   Jacques Distler: I recall rather enjoying the Gina character at the time. She seemed a curious mix of occasionally penetrating insights, and a sometimes rather befuddling layman’s (lack of) understanding of the scientific issues under discussion. … the book (at least, the excerpt) has a quaintly antiquarian feel. Comic Sans notwithstanding.

Marni D Sheppeard (Kea) Loved the book, thanks! Best of luck with the publishing. From Gina’s sister.

From Physicsworld

Hamish Johnston: I couldn’t help feeling sorry for Gina as she tried to ingratiate herself back into the conversation.

From Computational Complexity

  Lance Fortnow: It certainly was a fun read but I finished not sure of the purpose of the book… Kalai’s book does have the big advantage of being a free download.

On Just Notes:

“Gina says” on String Wars  reminds me of the Bourbaki phenomenon in math. Though, so far, the endings are quite different, the initial pourpouses are (were) quite similar: to clarify the topic on one hand and to set foundational grounds on the other hand. I wonder if having internet and blogs those days, Bourbaki would have ended up being “Bourbaki says”  as well.  (!!!)

I wrote a book. It is a sort of a popular science book and it is also about blogging and debating.

You can download the first part of the book : It is a 94 page pdf file.

 “Gina Says,”

Adventures in the

Blogsphere String War

selected and edited by Gil Kalai

Praise for “Gina Says” 

Preface 

Debates portrayed in books, are the worst sort of readings,                        Jonathan Swift.

In the summer of 2006 two books attacking string theory, a prominent theory in physics, appeared. One by Peter Woit called “Not even wrong” and the other by Lee Smolin called “The trouble with Physics.” A fierce public debate, much of it on weblogs, ensued.

Gina is very curious about science blogs.  Can they be useful for learning about, or discussing science? What happens in these blogs and who participates in them? Gina is eager to learn the issues and to form her own opinion about the string theory controversy. She is equipped with some academic background, even in mathematics, and has some familiarity with academic life. Her knowledge of physics is derived mainly from popular accounts. Gina likes to debate and to argue and to be carried by her associations. She is fascinated by questions about rationality and philosophy, and was exposed to various other scientific controversies in the past.

This book uses the blog string theory debate to tell about blogs, science, and mathematics. Meandering over various topics from children’s dyscalculia to Chomskian linguistics, the reader may get some sense of the chaotic and often confused scientific experience.  The book tries to show the immense difficulty involved in getting the factual matters right and interpreting fragmented and partial information.

Jeff Kahn

Jeff and I worked on the problem for several years. Once he visited me with his family for two weeks. Before the visit I emailed him and asked: What should we work on in your visit?

Jeff asnwered: We should settle  Borsuk’s problem!

I asked: What should we do in the second week?!

and Jeff asnwered: We should write the paper!

And so it was.

Karol Borsuk conjectured in 1933 that every bounded set in can be covered by sets of smaller diameter.  Jeff Kahn and I found a counterexample in 1993. It is based on the Frankl-Wilson theorem.

Let be the set of vectors of length . Suppose that and is a prime, as the conditions of Frankl-Wilson theorem require. Let . All vectors in are unit vectors.

Consider the set . is a subset of .

Remark: If , regard as the by matrix with entries . 

It is easy to verify that:

Claim: .

It follows that all vectors in are unit vectors, and that the inner product between every two of them is nonnegative. The diameter of is therefore . (Here we use the fact that the square of the distance between two unit vectors and is 2 minus twice their inner product.)

Suppose that has a smaller diameter. Write for some subset of . This means that (and hence also ) does not contain two orthogonal vectors and therefore by the Frankl-Wilson theorem

 .

It follows that the number of sets of smaller diameter needed to cover is at least . This clearly refutes Borsuk’s conjecture for large enough . Sababa.

Let me explain in a few more words how looks like when is large. the dominant binomial coefficient in the the sum defining is . (Since every binomial coefficient in the sum is smaller than half the next binomial coefficient.) is smaller than . When is large this binomial coefficient equal to , where is the entropy function.  To verify it and even to get better estimates you can use Stirling’s formula.

We will further discuss Borsuk’s conjecture and related problems in a later post.  

Are you aware of any? (current ones? old ones?)