Teaching kids to gamble

I do a workshop of couple of hours each summer with kids who have finished their Standard Grades (GCSEs in England, i.e they are 16) and are moving on to their Highers (A-level) labelled "Deal or no Deal".  My aim is to encourage students to think a bit differently about maths, in particular to appreciate the relevance tof maths to areas other than the physical / natural sciences.

The workshop is popular, I had thought it was because the TV show was popular but it turns out only a couple of kids watch the show regularly.  The students had been told

The world is uncertain and mathematics is exact, so how can maths help in making decisions about the future?

The workshop will look at a number of simple games that involve making decision when you don't know what will happen, and investigate how maths can help you make the winning choice.  The workshop will show how maths is as important in understanding economics, biology and psychology as in understanding physics or engineering.

I started the workshop by talking about the role of gambling in mythology/religion, referring to Hellenic, Hindu, Chinese and Biblical examples.  I then suggested that gambling (or casting lots) is something common to all cultures, like language, mathematics, music and art are, but physics and agriculture are not.  To explore this point further I split the group into groups of five and ask them to gamble for smarties.  One of the 5 is given more smarties than the rest and the idea is that by playing fair games this imbalance is corrected.  There is a simple java simulation (don't play it too long as one player usually wipes out the others) and I go on to talk about how anthropologists think that gambling prevents the establishment of hierarchies in neolithic societies (eg Mitchell or Altman).

The group was of 25 and so I got them to order themselves by date of birth - there were two pairs sharing the same birthday.  The purpose of this was to enable the group to split into pairs who did not know each other (well).  I then got the pairs to participate in the Ultimatum Game and then talk about how "fairness" is a learnt human concept (Murnighan and Saxon, Henrich et al., Jensen et al.).

I then talk about how Cardano undertook the first mathematical analysis of gambling in response to trying to address the issue of the ethics of gambling.  I give one of the key quotes and then explain how the idea of mathematical expectation comes out of this.  When the group were happy that this was all OK, I present the Petersburg game and ask for offers to play it.  The best offer is 2 I then calculate the game's expectation.

Some of the students identify the risk as the key issue, and after a brief discussion I offer some more games based on dice and ask which one each participant prefers.  Un-remarkably almost half prefer B, the lowest variance game, and I explain that variance is a measure of uncertainty and is associated with risk.

I then ask whether taking a risky decision is always a bad idea.  I present the case of a bird in winter who needs to find food to survive the night and outline this as a game.  The students work out there is a risky and safe strategy (based on variance) and I ask them to try and work out if there is a good strategy, playing safe or taking risks.  After the students play about and we discuss their suggestions I run through a java simulation. The simulation (after a few runs) enables me to demonstrate that the re are two regions in the time-berry state space were taking risks is better than playing safe.

I then discuss the Deal or No Deal game, and, with the results of 22 games, I discuss how a mathematician might analyse the game.  I finish off with Cardano's 350 year old advice, maths is of little practical value but does help in understanding, and the only way to be ethical in gambling is through science.