The Further Mathematics Support Programme Our aim is to increase the uptake of AS and A level Mathematics and Further Mathematics to ensure that more students reach their potential in mathematics. The FMSP works closely with school/ college maths departments to provide professional development opportunities for teachers and maths

promotion events for students. To find out more please visit www.furthermaths.org.uk A Level Maths & Further Maths Highly regarded and popular A levels Facilitating subjects for universities Many A level and university subjects require maths knowledge Opens doors to a variety of careers, many of which are well paid

Enjoyable Challenging Useful Careers that need maths qualifications

Engineer Scientist Teacher Computer programmer Games designer Internet security

Logistics Doctor Dentist Meteorologist

Finance Accountancy Business management Nursing Veterinary Sports science Physical Therapist

Pharmacist Medical statistician And many more Doves and Hawks Golden Balls A few years ago a TV show called golden balls was popular as tea time viewing. It ended with a split or steal game.

You can watch a clip here https://www.youtube.com/watch?v=cOH65fz-Dt8 What is the best strategy to win? What would you do? In your workbook I decide to (split/steal)

They decide Do I to win/lose? (split/steal) My winnings What do you notice about how you both play? Payoff matrix.

A splits B splits B steals A steals Payoff matrix A splits

A steals B splits A wins 2050 B wins 2050 A wins 4100 B loses B steals

B wins 4100 A loses A loses B loses Conclusions Both players are most likely to steal for exactly the same reasons. The players always choose (Steal, steal) as their strategy.

However, a strategy of (split, split) is better for both players. This is a conflict between individual rationality and group rationality: people often act only in their own self-interests, rather than for the group good. How this works in nature. Consider a population of birds... Suppose there are two fighting strategies, these are named hawks and doves Hawks always fight as hard as possible, quitting a fight

only when seriously injured. Doves merely posture, threatening in a dignified way, but never hurt anybody. If a hawk fights a dove then the doves runs away and so doesnt get hurt. If a hawk fights a hawk then an almighty brawl ensues, continuing until one of them is seriously injured. If a dove meets a dove then they just posture until one of them retreats (like a staring-out competition). No-one in the population can tell whether an individual is

a hawk or a dove. Furthermore, no memory of previous encounters with individuals is assumed. Let us make a game of the situation by allocating points. A win earns 50 points for the victor. A lose earns 0. A serious injury gains -100 points. For wasting time there is a penalty of -10 points. How many points do you get? If a hawk fights a dove, then the hawk wins and the dove loses. Therefore, the hawk gets 50 points, and the dove gets 0 points.

If a dove meets a dove, then one dove wins and one dove loses. However, that this process takes time, so each dove wastes time. So the winning dove gets 50 points, but loses 10 points for wasting time, giving 40 points in total. The loser gets 0 points and loses a further 10 points for wasting time, giving -10 points overall. If a hawk meets hawk, then one hawk wins and one hawk loses and is seriously injured. The winner gets 50 points and the loser gets 0 and also loses a further 100 points for their injuries. If Hawk meets a Hawk or a Dove meets a Dove, then the chances of winning are 50:50.

The payoff matrix. I am.. Dove Dove They are. Hawk Hawk

The payoff matrix Dove I am.. Hawk Dove 50:50 chance of winning. Hawk wins 50 points

Winner 50pts 10 pts for Dove 0 points wasting time. Total 40 points. Loser 0 pts -10 for wasting time. Total -10 points. Hawk Hawk wins 50 points Dove 0 points

They are. 50:50 chance of winning. Winner 50pts Loser 0 pts -100 for injuries. Total -100 points. To play this game: Use the table on p3 of the workbook.

Before you start to play you will be told whether you are a hawk or a dove. Walk around, when you meet someone, they will either be a hawk or a dove. If both of you are the same then you flip a coin to decide who wins the encounter. In your work book record who wins and who loses, and the points. Walk off and meet someone else and repeat. In your workbook. I am....

They are..... Do I win/lose? My points If we want to maximise the number of points we have, is it better to be a hawk or a dove?

The points an individual earns are convertible into gene survival points, everyone wants these; they all want to pass their genes on. So an individual who scores highly will pass on many of their genes. What is the average score for the Dove-Dove game? What would happen if we introduced a Hawk? What is the average score for a Hawk-Hawk game?

And we introduce a Dove? Conclusions? If the population consists of all doves then the introduction of a hawk causes the population of doves to decrease. On the other hand, if the population consists of only hawks, then the introduction of a dove causes the population of hawks to decrease. This means at some point in the middle, where there is

a mixture of hawks and doves, the number of hawks and doves remains the same. To put it another way, the average number of points earned by the doves and hawks is the same. The prisoner dilemma This work has all been based on a mathematical problem called the prisoners dilemma. The Prisoners dilemma 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. They hope to get both sentenced to a year in prison on a lesser charge. The Prisoners dilemma Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity to: Betray the other by testifying that the other committed the crime

Cooperate with the other by remaining silent The offer: 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) What should they do? A Level Maths & Further Maths

Highly regarded and popular A levels Facilitating subjects for universities Many A level and university subjects require maths knowledge Opens doors to a variety of careers, many of which are well paid Enjoyable Challenging Useful Careers that need maths qualifications

Engineer Scientist Teacher Computer programmer Games designer Internet security Logistics Doctor Dentist Meteorologist

Finance

Accountancy Business management Nursing Veterinary Sports science Physical Therapist Pharmacist Medical statistician And many more The Further Mathematics Support

Programme Our aim is to increase the uptake of AS and A level Mathematics and Further Mathematics to ensure that more students reach their potential in mathematics. The FMSP works closely with school/ college maths departments to provide professional development opportunities for teachers and maths promotion events for students.

To find out more please visit www.furthermaths.org.uk