### Game Theory

```Mathematical Ideas that
Shaped the World
Game theory
Plan for this class
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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?
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
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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
Nim
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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?
Types of game
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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)
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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.
John von Neumann (1903 – 1957)
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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.
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
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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?
Assumptions
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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
The behaviour predicted by Game Theory for
the Ultimatum Game is rarely seen.
Does this mean that humans are irrational?
John Nash (1928 – now)
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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.
Nash’s thesis
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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…
Strategic voting
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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.
Strategic voting
Bill
Phil
Gill
1. Blue
1. Pink
1. White
2. White
2. Blue
2. Blue
3. Pink
3. White
3. Pink
Why is it in Gill’s best interest to vote for pink?
Payoff matrices
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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
odd
Player 1
3
6
0
0
even
6
1
2
equilibri
um
Chicken
Player 2
stay
run
0
run
0
Player 1
stay
5
-1
-1
5
-4
-4
Nash equilibria
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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.
Nash equilibria
Player 2
run
stay
0
run
0
Player 1
5
-1
-1
stay
5
Nash
equilibrium
-4
-4
Nash
equilibrium
Golden Balls
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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.
Prisoners Dilemma
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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.
Prisoners payoff
Player 2
Stay
quiet
Stay quiet
Defect
-2
0
-2
-6
Player 1
-6
Defect
0
-5
-5
equilibrium
Prisoners analysis
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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.
What if the game is played a number of times
The iterated Prisoners Dilemma
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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?
Applications
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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
John Nash - illness
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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
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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.
Lessons to take home
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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.
```