Tuition and game theory
Tuition and game theory
The other day I was helping a student with their extended essay. This part of the IB Diploma asks students to write a 4,000-word essay on a topic that interests them. Selfishly I love them; the diverse interests of the students mean that each time I supervise an essay I learn an enormous amount. I have supervised essays ranging from relativistic time shift to Jamaican Dancehall music (though, admittedly, I was much more able to contribute helpfully to the first than the second). This time I was talking about a piece of Game Theory and the example I found myself discussing was tuition.
I should say at the outset that it is absolutely true that many of our students go for tuition. When we talk to our colleagues in other schools we can see that this is the situation across Singapore (and indeed beyond) rather than being a UWCSEA-only issue. There have been several reports in the press recently about the size of the tuition 'industry'. It is also worth noting that tuition does work. One of the earliest and still strongest pieces of education research is on the effect of 'time on task'. If you spend more time actively thinking about a particular piece of work you get better at it. Of course one should also note that more time on X is less time on Y and hence tuition in science, for example, comes at the cost of time for study on history or psychology or indeed just kicking a ball around.
From my conversations with parents and students there are a number of reasons that people seek tuition: for students it varies from remediation and support to getting ahead of the class and ensuring the 'top spot in the class'. From parents it varies from (again) remediation and support to avoiding family time being taken up with schoolwork to being a good parent by 'giving the best possible start in life'. It is perhaps this range of reasons that makes this a complex subject. One of the common complaints I hear is that people feel that they have to send their children to tuition ‘because everybody else does’.
And so to my Game Theory example. My profound apologies to mathematicians and political analysts everywhere for my appallingly simplistic treatment of the idea:
The prisoner's dilemma is a standard example of a game that is used to show why two rational individuals might not cooperate, even if it appears that it is in their best interests to do so. It is usually presented in the following way: two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge but they hope to get both sentenced to a year in prison on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner has the opportunity either to betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent.
The possible scenarios are:
- If A and B each betray the other, each of them serves 2 years in prison
- If A betrays B but B remains silent, A will be set free and B will serve 3 years in prison (and vice versa)
- If A and B both remain silent, both of them will only serve 1 year in prison (on the lesser charge)
Sometimes these possibilities are shown as a matrix:
A 1 year in prison
B 1 year in prison
B 3 years in prison
A 3 years in prison
A 2 years in prison
B 2 years in prison
Because betraying a partner offers a greater reward than cooperating with them, all purely rational self-interested prisoners would, according to the theory, betray the other. The interesting part of this result is that rationally pursuing individual reward logically leads both of the prisoners to betray, which results in a situation in which both suffer.
This model (or an extended version of the game where two classically rational players betray each other repeatedly) was a popular representation of the Soviet-US arms race. Each side could choose to ‘arm’ or to not with the logical conclusion being military escalation. However, it occurred to me as I chatted to the student that the game could equally well be applied in the following way:
The choice at any time for students and parents is to either get tutoring or to not and rely only on class teaching and personal study. Mirroring the prisoners’ dilemma logic would suggest that in any particular ‘move’ a student is personally better off getting tutoring regardless of what the other students decide to do. If he or she gets tutoring and others do not then we get the situation where the student is ahead of the other members of the class. If on the other hand the other members of the class get tutoring then at the very least the student is on par with everybody else and not disadvantaged. Like the prisoners’ dilemma (and the arms race) the outcome if everybody thinks in this way is a costly (and some would say damaging) tuition race. That and some very well off tutors.
Of course this is applying a simple 2 x 2 game to a far more complex situation.
In the prisoners’ dilemma both sides are assumed to ideally prefer unilateral advantage whereas in real life there will be a variety of desired states. Perhaps a particular person, or indeed many people, would prefer not to pay for tuition. Their desired state might be equity and ‘multilateral de-escalation’. Game Theory’s answer to the question of what prevents them is the perception that everybody else prefers unilateral escalation. I do it ‘because everybody else does.’
We cannot (should not) say to parents that they are not allowed to provide tuition for their children. That is of course a personal choice and there will be situations where tuition is a very sensible choice. However in a paper on the application of the Dilemma to the tension between the Soviet Union and the US, Scott Plous noted that ‘when enough redundant, expensive or destabilising weapons are accumulated the desirability of mutual arms reductions finally exceeds that of unilateral armament and the game is transformed’. Perhaps, extending my vastly overly simplified application of the game, when enough people agree that the costs of tuition outweigh the gains, that game too will be transformed.