A class in Game Theory for Management students

My host, Prof. Yukari Shirota, had arranged that whilst at Gakushuin University I would give an undergraduate class in some topic of mathematics to students in the Department of Management. For such a one-off, I had a fairly free hand as to subject matter; what seemed to be considered most valuable for the students was the (rare) opportunity to hear a native speaker of English (though actually I'm ethnically less than half English).

I chose Game Theory, the subject made famous by John von Neumann and Oskar Morgenstern in their landmark work, Theory of Games and Economic Behavior. I first came across Game Theory whilst at secondary school. In fact I became quite engrossed, venturing into Birmingham Central library to conduct research for an extended essay on a typewriter [yes, it was a long time ago]! I was fascinated by its confluence of mathematics, economics and psychology; the last of these particularly intrigues me. On this occasion my main reference was Games, Theory and Applications (1st edition) by Lyn Thomas, which I used during my own undergraduate studies at Southampton University. For anyone wanting to further their study, I would strongly recommend a textbook like this.

Would you like to explore this topic in this slightly extended post?

Good! So on a Wednesday morning I gave a class in the Multimedia Mathematics series to about 25 4th year undergraduates, kindly assisted by Prof. Shirota who gave explanations (sometimes in English, sometimes in Japanese) and encouragement. Normally there would be hands-on for students, but things were kept simple so that I only had to give a presentation with software demonstrations. I read somewhere some statistic indicating that the Japanese have more slides per presentation than any other nation, so I armed myself with 70+ slides. That's excessive, probably more suitable for 2-3 classes, but I had only one at my disposal and I wanted to share sufficient material to give a reasonable feel of the mathematical methods involved, including inductive reasoning and aspects of probability. I also included quite a few pictures and came up with simple examples to show how many situations in life can be treated as a game - such as growing tomatoes or commuting.

At the same time it was important for me not to rush delivery - speaking more slowly than usual, there was no way I would cover all this material in one go, so I just used a selection of slides, starting with the main concepts and proceeding via a few hops to a couple of famous non-zero-sum case studies, the Prisoner's [really prisoners'] Dilemma and The Battle of the Sexes.

I also had another prop with me - some reasonably authentic-looking notes in Pounds Sterling (Casdon PlayCash that I bought from the local Boswells store). I used it as an ice-breaker:

An introduction to Game Theory

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(Thanks to the American Mathematics Society for this idea, which I first saw on 'Who wants to be a mathematician?' roadshow).

Yes, a £1,000 giveaway! Except it was a bit credit-crunched: as only £937 cash was in the bank, I tried to explain that the British banks are struggling at the moment and a cheque could make up the remainder, £63.

The students were divided into 6 teams, A-F, one team per island, each with a representative. Each team was a 'player'. Teams had to choose a number N>=1; a team that picked the highest number would receive a share of £1000/N. This game had two rounds as follows:

• Round 1 [slide 3 above]: no communication
  We should have collected pieces of paper; as it was, numbers went something like: 20,50,80,100,50,52.
  Team D won £10.
• Round 2 [see slide 4]: communication allowed
  Teams chatted about this and quite quickly came to a decision, yielding the optimal result: 1,1,1,1,1,1, so each team gained £1000/6!

I was struck by the smoothness in reaching this outcome (and lack of betrayal among the teams) and think this may reflect a general culture in Japan of collective action and perhaps conformity, something that has helped the country to become such a productive and powerful economy. Quite different from an individualistic view, where it would be seen as problematic. It was later related to me that when someone says, "Ne...?" ("Isn't that so?"), there is often a feeling of obligation to say, "Ne!" ("Yes, that's so."). This would tend to support a culture of opinion leaders and followers. Ne?

In terms of software demonstration, I used mainly two tools, both released under open source licenses. The first was Gambit, which is a dedicated Game Theory suite that provides for the analysis of non-zero-sum games in both normal and extensive form. It has options to carry out computation, particularly of Nash Equilibrium, though one of its current limitations is that it restricts itself to games where players have to choose their moves independently. The other tool was Maxima, a Computer Algebra System, which I used for the graphical visualisation of payoff regions. Maxima by itself has only a command line interface, but it can invoke gnuplot to render graphical output and there is a choice of graphical interfaces: in my case I ran wxMaxima. All these are bundled together in the distribution.

I find the topics of communication and cooperation to be of philosophical interest. A standard definition of a cooperative game is couched in terms of business contracts (in the UK we can think of the Co-op supermarket) and so in such games players are said to enter binding agreements. It is used accordingly as a basic binary categorization and its importance is evident in e.g. providing assurance for the mathematical calculations. However, it means communication becomes secondary and I'd argue that [human] communication is more fundamental - it's what made the huge difference in the giveaway of slides 3 and 4 and to my lay-person's thinking, co-operation was established through a collectively agreed strategy before it became binding. No ties are needed to work together! Communication doesn't imply cooperation, but it usually precedes it.

So, I should issue the caveat that my slides exhibit a natural personal bias to this voluntary sense of cooperation, illustrated, for example in the Battle of the Sexes, in which a young married couple have free time at the weekend for an outing. The only issue is that the husband prefers a sporting venue, whilst his wife prefers a concert (so the story goes), but the bottom line is that they'd both prefer to be together than go there separate ways - see slides 65-71.

In Gambit (using this source file), we can compute the Nash equilibrium points. If we assume x is the probability with which the husband choose the first venue and y is similarly the probability that the wife chooses the first venue, then the expected returns e1 and e2 are given respectively by:

• e1(x,y)=5xy -4x -4y +4 - (eq1)
• e2(x,y)=5xy -4x -4y +4 -(eq2)
• where 0 <=x <=1, 0<= y <=1.

For minimax we set both of these equations to equal the value of the game. Gambit can do the calculation for us. The following screenshot shows the matrix used and underneath three equilibrium points.

The first of the equilibrium points are the respective the minimax strategies. But a value of 4/5 seems rather poor and would suggest - if the payoff matrix is a true reflection - that both 'battlers' will reason that settling on any venue would be better. Indeed, underneath are two other equilibrium points that return expected returns of 1 and 4 and vice versa.

However, the computation of individual points doesn't give a full picture. Just a few lines of Maxima instructions enables us to compute the region covered by all mixed strategies. It generates a 3D parametric plot, and we can initially set the z-axis to be constant, so with a bit of dexterity, you can rotate it to show the following:

The x and y-axes denote the respective expectations for husband and wife. I'm fascinated by the shape: the attentuation to the corners (1,4) and (4,1) - this particular graph reminds me of someone sitting in a hammock! Note that the point (4/5,4/5), which is the expected value of the game under minimax, lies a long way from those corners and it's also nested deeply in the region. It's certainly not on the boundary since in equations 1 and 2 above, if we set x=y=0.5, we get e1(x,y)=e2(x,y)=1.25.

It's a graph that assumes no cooperation, which is not a very optimistic view of a newly wed couple. We'd expect them to work something out in the form of a cooperative strategy, pure or mixed, so that whenever they have an outing they will go to one of these attractions together. If that's the case, then we can simplify the equations so that the (0,0) outcomes are factored out. The resulting graph is a line, the convex closure of the original region:

(Incidentally, I wonder if there is some metric indicating how far one is from cooperation in choosing minimax, perhaps defined in terms of the angle created by the expected returns with pairs of 'pure cooperative' vertices - the smaller the angle, the greater the missed opportunity for cooperation...?)

In the case of the Prisoner's Dilemma there is no minimax strategy. Geometrically, if you plot that graph you get a triangle, i.e. the set of points in Euclidean space is already convex.

Student Response and Feedback

In the event the class listened attentively and concentrated well. The opening game helped to stimulate interest, which they seemed to sustain for the duration. I was informed that they could understand most of what I said, which was a relief since preparing this class felt a bit like navigating in the dark. Certainly a few of the students gave responses that indicated they understood particular concepts. Although I didn't receive questions at the end (same kind of traditional response as Thai and other oriental students), facial expressions were not blank or bemused. This may have been helped in no small measure by Prof. Shirota, who produced (in one evening/night!) a translation into Japanese of some (possibly all?) of the slides. This would also encourage students in further reading and assist them in an assignment - a write-up about the Prisoner's Dilemma.

Whilst at Gakushuin, I also met Prof. Jun WAKO, who is a specialist in Game Theory. I hope he would approve of my presentation, but at least he may now have a few more enquiries from interested students...

Research Visit to Gakushuin University

Located towards the northwest of central Tokyo, Gakushuin is a private academic corporation that comprises schools and colleges as well as a university that currently has 9000 students. It is famous in Japan because of its historical connections with the Imperial Court. When I arrived a couple of weeks ago, I found the Mejiro campus spacious and verdent, surprisingly so given its centrality; I was informed that among the universities inside the JR Yamanote loop, it's the second largest after the University of Tokyo.

The present campus, like much of Tokyo, has seen many changes, but among the modern blocks there still remain a few of the older buildings, a little over 100 years old:



The modern multi-storey block on the right (East building no. 1?) obviously offers more capacity, but in the distance are some well-maintained old classrooms, cherished by staff and students even today. To the left is the former library building, now housing the Museum of History, graced by a venerable old tree in front of a small pool and foundation:


However, I wasn't here to be a tourist (although during my week's stay I did have a chance to wander), but rather to explore aspects of e-learning. The opportunity had arisen following earlier exchanges of ideas in the UK: in 2005, Oxford University Computing Services (OUCS) received a visit from Professor Yukari Shirota of the Department of Management, Faculty Economics, Gakushuin University. I arranged for her to give a presentation on some interactive software she had developed that guided students through the study of some topics in mathematics. The system's architecture was based on solution plans to word problems and delivered using an intelligent agent (animated by the Microsoft wizard).

Prof. Shirota is a computer scientist of long-standing - for instance, she co-authored an introduction to UNIX in 1984. During the past decade, Prof. Shirota has been developing e-learning systems to aid in the teaching of mathematics to her Management students. Inspired by George Pólya, her research has been focused on problem solving, invoking techniques in A.I. and especially visualisation, to help make sense of how the formulae and equations are used in word problems in Economics. I was particularly struck by her idea that A.I. might be able replicate the rhythm of instruction from teacher to pupil, quite similar perhaps to the rhythm of communication between a mother and baby. I hadn't come across anything like this thinking in the UK.

More recently, Prof. Shirota's research has concerned the provision of integrated tools that enable academic staff to create a range of online materials to direct students step by step in tackling certain types of questions, particularly in the field of bond mathematics. These systems are typically Web applications with scripts that invoke computer algebra systems such as Maple and Maxima to deliver step by step instructions. Some recent examples of this work are evident in overview of activities for 2011.

With regard to the financial mathematics, Prof. Shirota and her colleagues have used a conceptual approach based on entity-relationship diagrams to relate variables to formulae and equations. So the tasks of understanding may be characterised by being able to visualise and understand this map and its relations. How may that process be best aided online? In its entirety, the complete diagram is too extensive and detailed to show all at once, so any online implementation will need navigation - to focus on particular formulae and relations, but also to 'zoom out' and see the model as a whole. So does this suggest navigating it like, say, Google Earth or some other way based more closely on the relationships?

It's in addressing these considerations that I have been offering some input with my background in mathematics, somewhat distant now, and more recently e-learning and Web development, gained mainly whilst in the Learning Technologies Group at OUCS. Determining effective solutions is a multi-faceted task: its design, especially in terms of user interface, should be soundly rooted in principles of cognition. As I am not trained in educational psychology, I have to deal more with the nuts and bolts of the learning context itself, primarily in terms of the current and emerging technologies. In this regard, I'd say that whilst the emphases in pedagogy vary from country to country, the predominance of personally own computing devices has presented new variables to factor into the education at all levels. It's particularly this phenomenon, which I tend to call mobile and ubiquitous computing, that I was encouraged to explore ahead of my visit in the form of a survey paper on e-learning systems for mathematics, with particular reference to business and economics.

To give some structure in this rather broad landscape, I initially made reference to some recent features of e-learning in the UK. There's a strong focus on learner-centred education; within that I'm particularly interested in processes of deep reflection, stemming from the RAMBLE project I led in mobile blogging and learning environments. I'm not sure about the merits of focusing so much on individual predispositions, but I felt that the work on open educational resources and podcasting as a delivery mechanism was a useful vehicle to illustrate how learning has emerged from being concentrated in a classroom or workshop and flowed out into less formal environments. It was with this image in mind that I developed a thread to show how computer assisted learning for mathematics has similarly emerged from the laboratory into the open and is now squarely aiming at handheld tablet and multi-touch devices.

The paper, whose full title is 'Ubiquitous e-Learning: Designing Web Systems for Economics and Business Mathematics', has been published in Gakushuin Economic papers and is now available online - in HTML and PDF formats.

Arrival in Japan

This month I had the opportunity to spend a week in Tokyo (1-7 July), my first time in Japan. I came on a research visit kindly arranged by Professor Yukari Shirota at Gakushuin University. I'll describe the research aspect in a later post, but here I'll just share some initial impressions on my arrival.

Japan has a very distinct cultural identity; it's one of the few countries that retains - at least in many people's perception - a uniqueness that has persisted in spite of its immersion in modern industrialisation and particularly global markets and consumer products. It's famed for its etiquette and politeness and it was as though the whole trip was couched in such ethos from the moment I dropped off my bags at Heathrow, where I had a friendly conversation with the staff of Virgin Atlantic.

We know Japan as 'the land of the rising sun', which is a translation of Nippon. It is fitting in many ways; the heat and humidity in the summer months is quite palpable, certainly sub-tropical, feeling not much different from Thailand. (I feel sympapthy for 'cool biz' workers who have to trade in their jackets and ties for reduced air conditioning, with the government advising units to be set to a minimum temperature of 28 degrees. Even in a land used to construction and reconstruction, there's been a lot of discomforting changes, faced with admirable forbearance.) But it's particularly as the emergence of the hi-tech society, that the sun it such a resonant symbol. It wasn't long before I was struck by its manifestation in rail transport.

On arrival at Narita Airport, there are many options to proceed to the centre of Tokyo. With the aid of a Lonely Planet guide, I had perused various routes to my destination of Mejiro and settled on catching the fastest train service available, the Keisei Skyliner, which can whisk you into heart of the capital in under 40 minutes, followed by a trip on the circular JR Yamanote line. So after collecting my baggage (probably the shortest wait I've had), I bought a ticket for the Skyliner, complete with a seat reservation, another one for the local service, and made my way down to the platform. The train duly arrived:

As this is the terminus, the train is cleaned before boarding, but there is also a wait for something else: the repositioning of the seats. Just like synchronised swimmers, every passenger seat is rotated in unison, through 180 degrees to face the direction of travel. Once on the train, pre-recorded announcements are given in Japanese (in a singsong voice) and more regular US English. The driver(?) makes only occasional announcements to inform passengers of the location of toilets and where to find refreshments - not the buffet car, but vending machines!

My train was surprisingly not on time. There had been an incident on the line causing congestion, but I was in no hurry, and there was no visible response from the other passengers. I disembarked at Nippori station, (mis)fed my Skyliner ticket into a turnstile, plucked out the other ticket and after enunciating "Me-ji-ro" to a couple of station staff I found the right platform. Shortly before midday I emerged from Mejiro station, into broad daylight and my first steps on Tokyo soil outside the transport system!

Initially a little disorientated, I established my bearings once I spotted the Northwest entrance to Gakushuin University.

This made me feel I really had reached my destination. With the aid of a map and directions from the porter at the gate, I subsequently made my way to the Faculty of Economics, met Prof. Shirota, and was later shown to the guest accommodation.

Time for a bit of rest, before the preliminary discussions later that afternoon...