APRIL 2013 VOL 1 NO 4

AB INITIO Because all Mathematics must have a beginning …

FROM WHENCE IT COMES … This reminds me of a saying my mother often used when I was a child: "Take it from whence it comes." Meaning, of course, that EVERYTHING has a context that's as important as the OBJECT / CONCEPT themselves.

KEY PHRASES

GAME THEORY

DECISION MAKING

DETERMINISTIC BEHAVIOUR

MATHEMATICAL MODELING

CONFLICT AND COOPERATION

ZERO–SUM GAMES

In this edition of the newsletter, we will look at how understanding GAME THEORY can help guide and advise the DECISION MAKING process by analyzing the

strategies of other players in order to maximize the positive outcome of your game.

Can you win at “ROCK – PAPER – SCISSORS” each and every time you play? In order to make an effective decision (choice) on your game strategy, one has to be able to look at several things concurrently, that will collectively affect the outcome or the impact of the outcome on any future move.

This is made even more complicated if the intended strategy requires the cooperation of other people, because then, the psychological perspective of others and their perceived impact will need to be aligned with yours in order for the maximum benefit to be obtained. This requires some element of cooperation towards a collective “benefit”.

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How might HARRY POTTER defeat VOLDERMORT with the help of some simple Mathematics and mathematical logical thinking?

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Had Hogwarts offered classes in Mathematics, Harry Potter could have used game theory to defeat Lord Voldemort.

The key to defeating Voldermort could be found in a simple phenomenon – that because no one dared to even speak the name of Lord Voldemort, this made the tracking of people who did use his name easy. In “Harry Potter and the Deathly Hallows”, Voldemort and his followers realized that they could track Harry, Ron and Hermione by simply monitoring the use of his name (and only members of this little band were “brave” enough to verbalise that name).

Had Harry and his team thought about this more clearly, they would have realized that if they could convince everyone in the magical world to overcome their (irrational) fear of Voldermort and bravely speak his name, then Voldemort would not have been able to trace every individual, let alone be able to find Harry and his team.

This is a co-ordination problem – the best result is if everyone can coordinate and agree to say Lord Voldemort’s name, then there is safety in numbers.

The outcome will only be dire if Harry failed to co-ordinate with the rest of the population and he ended being the only one who says the name. In this situation, the safest option would be to not to say his name.

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If Harry could have coordinated enough people to speak the name of ‘He-who-must-not-be-named’ then he would have been safe from being discovered. But the risk of failing to co-ordinate is so great that no one wanted to take the risk.

THE ISSUE OF TRUST –

This is identical to the hunting problem – as a cave man, you have 2 options, you can either hunt alone and work on catching rabbits, or team up with another caveman and hunt bison. You don’t need help to catch rabbits and you cannot catch bison alone.

So, the night before the hunt, you meet with the other hunter to discuss strategy. At the end of the meeting you return home before the next morning’s hunt.

Because you cannot contact the other caveman before you set off, you must decide whether you trust the other caveman enough to stick with the agreed upon strategy of hunting bison together.

This is the decision that you’ll have to make – go to the designated area and prepare the hunt bison (but you may go hungry if the other hunter doesn’t appear) or head towards the rabbits (and break the pack, but at least you’ll have food for your family).

The rabbit is a low risk option, but the rewards are a lot higher if you take the higher risk option.

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So, what exactly is GAME THEORY and how can this help us make more effective decisions based on the risk analysis and efficacy of the expected reward?

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BACKGROUND

Game Theory can trace its history back to the early 18th century with letters describing a MINIMAX STRATEGY for a card game between two British nobles.

However, despite several ventures into the topic at various times, it wasn't distinguished as a discreet field until mathematician John von Neumann published a paper in his 1928 paper "Zur Theorie der Gesellschaftsspiele" in Mathematische Annalen.

Neumann later expanded these ideas in a book published in 1944 and co-authored with Oskar Morgenstern titled “Theory of Games and Economic Behavior”.

Shortly thereafter, in 1950, Game Theory gained its most well known discovery with the rise of John Nash and his contribution of the so-called NASH EQUILIBRIUM, which eventually won him the Nobel Prize for Economics in 1994.

In the 1970's, due to the work of John Maynard Smith, Game theory was applied to the field of biology and the study of genetics and evolution.

In 2005, Thomas Schelling joined John Nash as a Nobel Laureate with the advent of what has since to become known as EVOLUTIONARY GAME THEORY.

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INTRODUCTION & DEFINITION

According to the Stanford Encyclopedia of Philosophy, GAME THEORY is "the study of the ways in which strategic interactions among rational players produce outcomes with respect to the preferences (or utilities) of those players, none of which might have been intended by any of them."

It involves the study of how the final outcome of a competitive situation is dictated by interactions among people involved based on the goal and preferences of those players, and on the strategy that each player employs. It was developed on the premise that for whatever circumstance, or for whatever game, there is a strategy that will allow one to win.

DECISION MAKING PROCESS

Decision theory can be viewed as a theory of one-person games, or a game of a single player against nature.

The focus is on preferences and the formation of beliefs. The most widely used form of decision theory argues that preferences among risky alternatives can be described by the maximization of the expected value of a numerical utility function, where utility may depend on a number of things, but in situations of interest to economists often depends on money income.

Probability theory is heavily used in order to represent the uncertainty of outcomes, decision theory is often used in the form of decision analysis, which shows how best to acquire information before making a decision."

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A (very generous!) casino offers you a game where the pot starts at $1 and on each turn a (fair) coin is tossed. If it comes up heads then the pot is doubled, if it comes up tails then you win whatever is in the pot.

How much would you pay to play this game?

Half the time a tail comes up on the first coin toss and you win $1. Half the time you get a head, the pot doubles to $2, and you get to toss the coin again.

On the second toss, half the time you get a tail and win the $2 and half the time you get a head again and the pot doubles to $4 and you get to toss again.

Overall you get the following pattern. Half the time you win $1, a quarter of the time you win $2, one eighth of the time you win $4, etc. The amount you win gets bigger and bigger, but the chance of winning that amount gets smaller and smaller.

This means that you should expect to win:

(1/2 x $1) + (1/4 x $2) + (1/8 x $4) + (1/16 x $8) …

There are an infinite number of terms in this equation,

50c + 50c + 50c …

which is the same as an infinite number of 50 cents, which is infinity.

So using an expected value argument you would pay an infinite amount to play the game because you will win an infinite amount.

However you almost certainly wouldn’t pay a massive amount to play the game. There have been a number of arguments put forward to explain this:

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Explanation (1) – Daniel Bernoulli in 1738 was the first to attempt to resolve the paradox. He suggested that because each extra bit of money means less to you, then you won’t value the later money you could win as much. Basically, if you have nothing then $1,000 is a lot of money, if you have $100 million then an extra $1,000 is irrelevant.

Explanation (2) – Another argument is that people don’t imagine that such a long string of heads is possible. Most people instinctively imagine that after a long run of heads a tail is more likely, even though it isn’t. That’s why casinos show all the recent numbers that have come up on a roulette table, because people are looking for patterns to base their next bet on.

Explanation (3) – One more argument is that the casino only has a limited amount of money and cannot payout an infinite amount. If the casino is willing to risk $1,000,000 on the bet then the expected payout drops dramatically. It takes a run of 20 heads to take the payout to over $1,000,000, this means that the expected payout drops to a bit over $10. The high expected value normally comes from having some incredibly rare but extraordinarily high payouts, without these the value drops substantially.

Which explanation do you prefer for the paradox, or do you have your own? You can submit your explanation as part of the weekly class problem solving series. Refer to the end of this edition of the newsletter for the email address to submit your response to this analysis.

EXAMPLE –THE ST PETERSBURG PARADOX

ANOTHER EXAMPLE

THE PROBLEM – The beautiful city of Bath is renowned for its Roman Baths and Georgian architecture but its council has fallen into a PRISONERS’ DILEMMA.

The city has three main recycling centres to encourage people to recycle as much waste as possible. At the moment anyone can use the centres whether they live in Bath or not. Equally anyone in Bath can also use a centre in another city if they want to.

Assuming that the Bath city council spend £100,000 on the recycling service (it’s always good to use a nice round number!), if 10% of people come from other areas and they are banned from using the sites then the cost will fall to £90,000, but the council will now have to pay the cost of monitoring who is using the sites. Let’s say monitoring cost is £5,000 then the cost for the council becomes £95,000. The council has saved £5,000 by stopping people from other areas from using the service. The neighbouring council now has more people to deal with because no one from their area can go to Bath anymore. Their cost goes up to £110,000.

This is fine for Bath until the neighbouring councils do the same thing. When that happens then the 10% of people who were going from Bath to another area can only use the Bath centres. This now adds £10,000 back onto the Bath cost bringing it up to £105,000. This is the original cost of the service plus the cost of monitoring who is using it.

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The GRID looks like this –

ACCEPT OTHERS

RESIDENT ONLY

ACCEPT OTHERS

100k / 100k

110k / 95k

RESIDENT ONLY

95k / 110k

105k / 105k

We can see that this is a classic prisoners dilemma – If the other council is accepting others then Bath is better off if it restricts its service to residents only (a cost of £95,000 rather than £100,000). The alternative is that the other council limits its service and then Bath’s best response is also to limit its service (a cost of £105,000 rather than £110,000).

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Either way the best response is to limit the service even though this leads to a worse result for both than if they both kept an open service.

That’s explains why they have decided to restrict the service to residents, but why has this only just started to happen when the service has been running for years? The answer is because the councils have changed the game they are playing. Before the ‘Great Recession’ the councils based their service around maximizing the amount of waste that is recycled. When this is their priority then they have no reason to restrict who uses the service, all that matters is that the waste is recycled Now the council’s are playing the game using money as the measure of success. This has changed the game into a prisoners’ dilemma and the councils are now trapped.

This is a real life example where changing what is important in the game has changed the outcome.

AB INITIO – Vol 1 No 4 APRIL 2013

Closing Remarks … According to Wikipedia, game theory is the "attempt to mathematically capture behavior in strategic situations, in which an individual's success in making choices depends on the choices of others."

This leads to many examples in a wide range of fields of study and there are examples of the direct application of game theory in the areas of simple children games (for instance “rock, paper, scissors”), competitive martial arts and sports, Economics and Human Altruistic patterns, biology and evolution.

There is also instances of game theory in video games, for instance “The Javelin Glitch” in Call of Duty: Modern Warfare 2 – This “glitch” comprises of a bizarre sequence of button presses that enable the player to "glitch" the game so that when he/she dies in multiplayer, the player would self destruct and murder everyone within 30 feet.

John von Neumann (1903 – 1957) was born in Hungary and obtained a PhD in Mathematics at 22 years old. He was a polymath and made significant contributions to the areas of Mathematics, Economics (game theory) and Computer Science .

As this newsletter is to highlight the applicability of Mathematics in your daily lives, YOU are welcomed to write articles, thoughts and reflections for the future editions of the newsletter.

Email ideas and submissions of the CLASS PROBLEM SOLVING COMPETITION to –

jason_ingham@sst.edu.sg

CLASS PROBLEM SOLVING COMPETITION PROBLEM SET –

THE GENES OF GILGAMESH

If three of my grandparents were French and one Russian then I

would be said to be one quarter Russian and three quarters French

(RFFF). If half of my great-grandparents were English and half

other nationalities then I would be said to be half English (EX)

The king of Sumeria around 2600 BC was a great warrior called

Gilgamesh . The tale of Gilgamesh the hero stated that Gilgamesh

was "Two Thirds God and One Third Man".

Assuming normal reproductive behaviour between a set of ancestors

of type pure G and a set of anscestors of type pure M, could you

create an offspring of type two-thirds G and one-third M?

How many generations would it take to create a genetic stock to

within 1% of (GGM)?

John Maynard Smith (1920 – 2004) was an evolutionary biologist and geneticist from the University of Cambridge (Engineering) and University College London (Genetics) and he was awarded the Crafoord Prize (1999) for his work in using Game Theory in the study of evolution and the evolution of sex and signaling theories.