Triple or Bust Paradox

Expectation values failing to predict long-term gains


Today I have a decision problem for you.

Alice offers Bob participation in a simple coin toss game. It’s called triple or bust. Alice start the game by writing an IOU to Bob for an amount of $ 1.00. Alice then makes at least six subsequent tosses with a fair coin. On each ‘heads’ Alice triples the IOU amount. On ‘tails’ she sets the IOU to zero. How much should Bob be prepared to pay Alice to participate in  this game, knowing that he can repeat this game as often as he likes?

Okay, let’s see: on each coin toss there is a 50:50 chance for tripling and for voiding the IOU. So on average, after a coin toss the IOU increases to 3/2 times the amount before the toss. That means that after the first coin toss, the expectation value for the IOU is $ 1.50. After two tosses the expectation is 1.50 times $ 1.50 or $ 2.25. This exponential growth continues, and after six tosses the IOU dollar amount has risen to 1.50^6 or 11.39. Any coin tosses after the sixth, will obviously continue the exponential growth of the expected IOU. In the long run, the game will yield returns closing in on the expectation value. So paying an amount less than $ 11.39 per game will make it advantageous to participate.

Bob has worked out the same logic and decides to offer Alice $ 10.00 per game.

Alice immediately accepts.

Bob pays Alice $ 10.00, Alice writes an IOU of $ 1.00, and starts tossing. Heads shows. Alice changes the IOU into $ 3.00. Again heads. The IOU is now $ 9.00. Then tails appears. “No need for any further coin tosses, okay?” Alice looks at Bob. Bob nods. Alice rips the IOU in pieces.

Bob decide to go for another game. Alice pockets another $ 10.00. Now tails shows in the first round. Once more an IOU gets shredded.

Bob is in it for the long haul, chasing a very profitable expectation value. He keeps playing.

After 37 games Bob has lost $ 370.00. Bob pays another $ 10.00. This time he is more lucky. After 5 heads in a row the IOU reads $ 243. Alice makes a sixth coin toss. Again a head. “Yes!! That’s 729 dollars!” Bob blurts out.

Alice writes down $ 729 on the IOU and prepares for a seventh coin toss.

“Wait a second” Bob intervenes. “Don’t throw another coin, just give me the 729 dollars.”

“I will give you another coin toss for free”, Alice replies. “As agreed upfront, I am entitled to give you additional coin tosses. I am sure you have incorporated this game feature into your decision to offer me $ 10. Isn’t it?”

Bob nods silently and stares at Alice’s hand containing the coin. She makes a seventh coin toss. Again heads. The IOU now reads $ 2187. An eight toss follows. A tail. Alice rips the IOU in pieces.

Bob shakes his head and quits the game.


What went wrong?  We have not made an error in our math, and neither has Bob. Something must be wrong in the logic.

It is correct that the expectation value for this game increases exponentially with the number of coin tosses. And for a fixed number of tosses per game, this expectation value does describe the returns that Bob will be make in the long run. It is also true that, having agreed at least six tosses, in the long run it is disadvantageous for Alice to add a seventh coin toss. And, again in the long run, it is even more disadvantageous for her to add an eight toss. Yet, giving Alice full liberty in adding any number of additional tosses gives her the power to make a killing in this game. Bob is guaranteed to lose every penny he puts into this game.

What is going on here?

The challenge is to understand the role of the expectation value for this game. Centuries of statistics research is based on applying expectation values as predictors for long-term gains. Putting your brain to sleep by ignoring the expectation value is not going to eliminate the paradox.

Zee’s Nutshell Trilogy

The books you wished you had when you were a student

Today I took delivery of my copy of Tony Zee’s third contribution to the Princeton University Press In a Nutshell series: “Group Theory in a Nutshell for Physicists”. With this book Tony has delivered a trilogy on fundamental physics. The earlier two books cover quantum field theory (“Quantum Field Theory in a Nutshell“) and general relativity (“Einstein Gravity in a Nutshell“).

Having the book in my possession for just a few hours and having thumbed through it, I am happy to report that, this book lives up to my high expectations and can stand shoulder to shoulder with Zee’s other two Nutshell books. I am not going to review the book here, but I do feel triggered to say a few words on Zee’s Nutshell trilogy in general. Hopefully, for those of you not familiar with Zee’s writing but eager to self-study fundamental physics, this will allow you to decide if these books are right for you.

For a start, if not clear already, I plead guilty to being a Zee fanboy. Not only is Zee’s Nutshell trilogy present on my bookshelves, I also own Zee’s popular science writings “An Old Man’s Toy“, and “Fearful Symmetry“. Stretching things a bit, these latter two books can be seen as the popular science versions of book 2 (Einstein Gravity) and book 3 (Group Theory) from the Nutshell trilogy. Stretching a bit more, one could classify Feynman’s “QED“, which contains an introduction by Zee, as the popular science version of Zee’s first Nutshell‘s book (Quantum Field Theory). You can find Zee’s introduction to “QED” at his homepage. This short text provides you with an appetizer for Zee’s writing style: informal, peppered with anecdotes, and rich in lighthearted remarks.

Throughout the Nutshell trilogy, Zee focuses on building physics intuition. If you are looking for strict mathematical rigor and formal derivations: forget about Zee’s Nutshell and look elsewhere. If you are looking for practical advice and eye-opening perspectives that help you build physics intuition on rather abstract subjects, the Nutshell trilogy is probably for you. Having said this: I don’t want to leave you with the impression that anywher in his Nutshell books Zee is sloppy in his math. Rather, I would judge these books as striking a healthy balance between handwaving and mathematical rigor.

The level addressed by Zee is that of an advanced undergraduate to graduate fundamental physics course. In self-study terms I’d say that Zee’s trilogy assumes a solid understanding of The Feynman Lectures in Physics, as well as a firm basis in linear algebra and analysis. From this, the three tomes bring you pretty close to research level. Quite a tall order, and it should not surprise you Zee’s trilogy counts a total of 2000 pages. I suspect that the majority audience for these books is not those whose self-study brings them in first-time contact with the areas studied. Rather it is an audience of readers who studied physics or a related discipline a while ago, but who feel the wish to revisit ‘some of that stuff’ to build a deeper and more up-to-date (and likely better) understanding. This group of readers is well served, as Zee set himself the standard to write the books “I wished I had when I was a student”.