Game theory

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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?

“Game Theory attempts to mathematically capture behaviour in games where an individual’s success in making choices depends on the choices of others.”

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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

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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?

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Cooperative vs. non-cooperative
Zero sum
Perfect vs. non-perfect information
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John von Neumann published the first real paper on Game Theory, in 1928.
He focused on 2-person zero-sum games.

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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.

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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.

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?

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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?

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In Game Theory we make the following assumptions:
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That the players are rational
That players do not trust each other
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The behaviour predicted by Game Theory for the Ultimatum Game is rarely seen.
Does this mean that humans are irrational?

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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.

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Nash studied the theory of non-cooperative games and the idea of equilibria.
In his thesis, Nash attacked the longaccepted 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…

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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.

Bill
1. Blue
2. White
3. Pink
Phil
1. Pink
2. Blue
3. White
Gill
1. White
2. Blue
3. Pink
Why is it in Gill’s best interest to vote for pink ?

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A payoff matrix records the winnings of each player in every possible outcome of the game.
Player 2 odd even
3 6 odd
3 0
Player 1
0 even
1
6 2 equilibri um

run
Player 1 stay
0 run
0
Player 2
-1 stay
5
-1 -4
5 -4

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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.

run
0
Player 2 run stay
0 5
-1
Player 1
-1 -4 stay
5 -4
Nash equilibrium
Nash equilibrium

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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.

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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.

Player 2
Stay quiet Defect
-2 0
Stay quiet
-2 -6
Player 1
-6 -5
Defect
0 -5 equilibrium

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

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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?
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Do people act morally because it is in their selfinterest to do so?

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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.
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Stealing
Combating climate change
Price fixing in supermarkets
Animals sharing a resource
Athletes taking drugs

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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.

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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.

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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.