## Game theory: mathematics as metaphor

Last semester I offered my students $1,000,000 dollars. They turned me down. This was lucky, despite the money and glamour of academic mathematics, I do not have a million dollars. The game was simple. The class of 100 each had to write a number. The highest number won. Of course there was a catch, the prize was $1,000,000 divided by the winning number. The best outcome for the students as a whole would come if everyone wrote 1, $10,000 is not a bad return for a lecture. Of course if everyone is writing 1, the person who writes 2 wins and makes far more for themselves. What happened?

I did tell the students that they should all cooperate and write 1, explaining how this was the best outcome. Some very trusting students actually wrote 2. This was actually rather sweet, although they were out to win more for themselves they felt that everyone else would be looking out for the group. There were also more cynical souls, realising that they were not the only one they simply wrote the largest number that they could. As a result I did not have to pay out a single cent. I was slightly sad not to receive the answer “highest number written plus 1”, that others who ran the game have done. This gets even more interesting when two people do it!

Readers watching closely will recognise that this is a group version of the famous game prisoners dilemma. Another version was used in the final round of the UK TV series Goldenballs ^{1}. Watch these clips and try to guess what the people will do:

Having played the million dollar game, and watched the clips I asked the students what they would do. About 2/3 did say split, unfortunately for them only 1/3 of the students had written 1 for the previous game! This is not surprising, in the $1,000,000 game something was on the table. The high number writers still wanted the chance of winning the game, even if no money was involved. In a simple pole you get no benefit from admitting (even to yourself) that you would do over your neighbour.

Both these examples are compelling as they illustrate game theory in action. In the million dollar game the theory is actually being used to model the behaviour of a large group. A statistical study of the data from series of Goldenballs, reveals some subtleties. Even though they are playing the game just once over half of players actually did split. Tellingly, however, the average money in situations where both players split was lower than that on the table for stealers. Interestingly a higher proportion of people who used the word “promise” did split.

These examples can be studied using the mathematics of game theory, but they also reveal the problems, the exact pay off differs for each individual. It is not simply that it is hard to establish exact values, the values actually differ dramatically from one person to another. While it may be true, for example, that everyone has their price, the exact value of that price can dramatically change the game that is being played. Other factors (also varying from one individual to another) can also come into play. In a more far out example, in Goldenballs players might see themselves as playing primarily against the Television company. In this case part of the pay off would be seeing the company give out the money. This will definitely happen if they split, but might not if they steal. For these people the game changes from Prisoner’s dilemma to Chicken.

Does this mean that game theory is not worth studying, or even misleading? It certainly means that we have to treat it with caution. One of the founders of game theory ~~Dr Strangelove~~ John Von Neumann actually argued that it proved the necessity of the nuclear first strike during the cold war. Luckily for everyone his counsel was not followed!

We associate mathematics with the unreasonably effective models we find in Physics, Chemistry and even Biology. In fact “mathematical” has almost become synonymous with precise. These models are certainly impressive, even beautiful, but game theory is not one of them. Game theory becomes powerful not as model, but as metaphor. It can help us understand what behaviours come out of situations with different payoffs. The lesson from prisoner’s dilemma is that people rationally following individual benefit in the society can lead to the group as a whole suffering. Historical events can also be analysed in terms of certain games. Although, unlike the models of physics the mathematics of game theory cannot be used to predict the future it can be used to understand the past and the present.

For more on the history and development of game theory and its potential social applications I can recommend these books:

Prisoner’s Dilemma: John Von Neumann, Game Theory and the Puzzle of the Bomb by William Poundstone

Natural Justice by Ken Binmore

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Technically this is not quite Prisoners dilemma, as, assuming your opponent is stealing there is no difference between splitting (you receive nothing) and stealing (you receive nothing).

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A related reference that I listened to the day before reading your post. I’m not sure when I’ll read more about game theory, but you’ve definitely gotten me interested.

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I would’ve written 0. Why go for a million when you can have it all? XD

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That is very interesting!! I think playing a game like that in class would be very fun and interesting to see what people will do in those situations.