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Programming to investigate Quarto

I was invited to contribute to a special issue of The Mathematics Enthusiast on ‘Risk – Mathematical or Otherwise‘, guest edited by Egan J Chernoff. I wrote about the Maths Arcade and programming strategies for a game we play there called Quarto. Really, I was sketching an outline of an idea to encourage student project work.

My title is ‘Developing Strategic and Mathematical Thinking via Game Play: Programming to Investigate a Risky Strategy for Quarto‘ and the abstract is below.

How to Win at Pointless

For the benefit of overseas readers, or British readers in full-time employment, I should briefly explain the concept of daytime TV quiz phenomenon Pointless. The pinnacle of British public service broadcasting, it’s shown at 5.15pm every weekday on BBC One and is hosted by Alexander Armstrong of comedy double-act Armstrong & Miller, and Richard Osman of comedy double-act Armstrong & Osman. We shall investigate how we can use maths to analyse the show, improve our chances of winning it, and ultimately perhaps improve the show itself.

The aim of the game is in each round to give the most obscure correct answer to a given question. Each question ($Q$) has a large set of valid answers $A_Q$, questions perhaps asking contestants to name “Films starring Bruce Willis” or “Countries without an O in their name”. All the questions have been asked to 100 members of the public prior to the quiz (call this set $P$), and they each have 100 seconds to name as many examples as they can (giving rise to a set $A_p\subseteq A_Q$ for each $p\in P$. The contestant gets a point for every one of the 100 people who named their answer $a$:

\[ \mbox{score}(a) = \begin{cases}
| \{p\in P : a\in A_p \} | & \mbox{if}\ a\in A_Q \\
100 & \mbox{if}\ a\not\in A_Q.
\end{cases} \]

So an obvious answer like Die Hard or France will score a lot of points, and an obscure answer like Striking Distance or Central African Republic will score fewer points. Points are bad (hence the title) so it’s better to dredge up an obscure answer than stick with something safe. However an incorrect answer like Avatar or Mexico scores the maximum 100 points. At the end of the round the contestant with the most points is eliminated.

Applied game theory: Students boycott exam and all get an A

A piece on the New York Times Economix blog casts a story from Inside Higher Ed as a piece of applied game theory. Professor Peter Fröhlich of Johns Hopkins University has a grading system in which

each class’s highest grade on the final counts as an A, with all other scores adjusted accordingly. So if a midterm is worth 40 points, and the highest actual score is 36 points, “that person gets 100 percent and everybody else gets a percentage relative to it,” said Fröhlich.

Can you spot the problem? The Economix posts points out that this allows “at least two Bayesian Nash equilibria”:

Equilibrium #1 is that no one takes the test, and equilibrium #2 is that everyone takes the test. Both equilibria depend on what all the students believe their peers will do.

In equilibrium #1, everyone scores the same mark – zero – and the calibrated marking scheme maps this onto 100%. The students, realising this, arranged a complete boycott and were all awarded grade A.

Alas, this was not a game theory class! Prof Fröhlich has since changed his grading scheme.

Further information:

Dangerous Curves (Inside Higher Ed) has some detail of how the boycott was arranged (had just one taken the test, all would have been forced to follow suit).

Gaming the System (Economix) discusses the related game theory and economics concepts.

via @Tony_Mann on Twitter.

Unusual prime number competition results

Author Robin Sloan offered a simple competition:

give me a prime number of your choosing. I’ll send books to the five people who choose the lowest unique prime numbers. So, if you pick 2 but seven other people pick 2, no book for you. If you pick 3 and no one else picks 3, you get a book.

Which number would you pick?

The primes people chose, including the five winners, and more background information are given in a blog post The Penumbra primes.

New strategy for winning the iterated prisoner’s dilemma

A new strategy for the iterated prisoner’s dilemma allows, over the very long run, one player to unilaterally claim an unfair share of the rewards.