Mathematical Ideas that Shaped the World

Game theory

Plan for this class

What is the idea of Game Theory? What kind of situations is it useful in? Find out how nice the person next to

you is! How can Game Theory explain human

altruism? How (and why!) does eBay work?

What is Game Theory?

“Game Theory attempts to mathematically capture

behaviour in games where an individual’s success in making

choices depends on the choices of others.”

Examples of where to use game theory A shop owner deciding the price of

biscuits Competing with your mates for a

girl/boyfriend Khrushchev and Kennedy in the Cuban

Missile Crisis Combating climate change Deciding whether to help someone Auctions Voting systems Animal behaviour and evolution

Nim

There are n objects 2 players On your turn, you may take either 1, 2

or 3 of the objects The person who takes the last object

wins the game.

Can you find a winning strategy?

Types of game

Cooperative vs. non-cooperative Zero sum Perfect vs. non-perfect information

John von Neumann published the first real paper on Game Theory, in 1928.

He focused on 2-person zero-sum games.

John von Neumann (1903 – 1957) Born in Budapest to a

Jewish family. By the age of 6 could

speak classical Greek and recite telephone directories.

Got PhD at the age of 22 and worked in Berlin before emigrating to the US in 1930.

John von Neumann (1903 – 1957) Worked out the key

steps in the nuclear physics needed to create the hydrogen bomb.

Was on the committee which decided where to drop the first atomic bomb.

Died of cancer under military security.

I know that you know that I know...

The mischievous leprechaun

You and a friend get to the end of a rainbow and find a leprechaun with a pot of gold.

It has 1 billion pounds.

The leprechaun offers you £1. If you refuse, he offers £10 to your friend. If they refuse, he offers £100 to you, etc. When should you take the money?

The Ultimatum Game

Two players. Player 1 has £100. They can choose

how to split this between themselves and Player 2.

If Player 2 accepts the split, they get the money.

Otherwise, they both get nothing. How much should Player 1 offer to

Player 2?

Assumptions

In Game Theory we make the following assumptions: That the players are rational That players do not trust each other

The behaviour predicted by Game Theory for the Ultimatum Game is rarely seen.

Does this mean that humans are irrational?

John Nash (1928 – now)

Born in West Virginia. The letter of

recommendation for his PhD was a single line: “This man is a genius.”

Completed his thesis at Princeton – it consisted of only 28 pages.

Nash’s thesis

Nash studied the theory of non-cooperative games and the idea of equilibria.

In his thesis, Nash attacked the long-accepted theories of Adam Smith, whose maxim was “every man for himself”.

This is highlighted in his biographical film A Beautiful Mind, in which he explains why going for your first choice is not always a good idea…

Strategic voting

Bill, Phil and Gill are thinking of painting the living room. They are going to vote on whether to keep the walls white or paint them blue.

Phil proposes instead to paint them pink. They decide to vote first between blue

and pink. They then vote between the winning

colour and white.

Strategic voting

Bill Phil Gill

1.Blue

2.White

3.Pink

1.Pink

2.Blue

3.White

1.White

2.Blue

3.Pink

Why is it in Gill’s best interest to vote for pink?

Payoff matrices A payoff matrix records the winnings

of each player in every possible outcome of the game.

odd

even

odd even

3

3

6

6

0

0

2

1

Player 1

Player 2

equilibrium

Chicken

run

stay

run stay

0

0

5

5

-1

-1

-4

-4Player

1

Player 2

Nash equilibria

In Chicken, there is no dominant strategy.

But there are two places where a Nash equilibrium occurs.

A Nash equilibrium is where, if you think your opponent will keep the same strategy, you will do best if you keep using the strategy that you currently have.

Nash equilibria

run

stay

run stay

0

0

5

5

-1

-1

-4

-4Player

1

Nash equilibriu

m

Nash equilibriu

m

Player 2

Golden Balls

This was a game show where the finale involved two people and a jackpot of money.

Each person could choose whether to share or steal the money.

If they both steal, they get nothing If they both share, they get 50% each. If one steals and one shares, the

stealer gets everything.

Prisoners Dilemma

Two people are on trial for a crime. Each of them has a chance to grass on the

other one. If both stay quiet, there is little evidence to

convict and they only get a 2 year jail term. If they both grass, they get 5 years in jail. If one grasses and the other doesn’t, he

gets released while the other does 6 years.

Prisoners payoff

Defect

Stay quiet Defect

-2

-2

0

0

-6

-6

-5

-5

Player 1

Player 2

equilibrium

Stay quiet

Prisoners analysis

The game-theoretic analysis of the Prisoners Dilemma seems counterintuitive – clearly both players would be better off if they cooperated with each other.

In experiments, 40% of subjects cooperated.

What if the game is played a number of times and the scores added?

The iterated Prisoners Dilemma Did you have a different strategy in the

extended version of the game? What kinds of strategies do you think

are likely to be effective? Can this explain anything about human

behaviour? Do people act morally because it is in their

self-interest to do so?

Applications

The Prisoners Dilemma applies in any situation where it is in the collective interest to work together, but where there is always an incentive to defect. Stealing Combating climate change Price fixing in supermarkets Animals sharing a resource Athletes taking drugs

John Nash - illness

Nash was admitted to hospital in 1959 for paranoid schizophrenia.

Upon release he sought asylum in France but was deported back to the US.

Spent another 9 years on medication and visiting hospitals.

John Nash - recognition

After 1970 he refused to take any more medication, and began doing mathematics again.

In 1994 he finally received a Nobel Prize for his work on Game Theory.

In total 8 game theorists have won Nobel prizes in economics.

Lessons to take home

Game Theory can be applied to almost any situation: in politics, sociology, economics, biology, philosophy, law.

Standard game theory does not always accurately predict human behaviour...

But on the other hand it can go some way towards explaining deep philosophical concepts such as morality.