Games and Decision Theory

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Games and Decision Theory
or how to win without actually cheating
Alison Simmons
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Overview
Games and Maths
Probability
Decision Theory
Wider Applications
Essay
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Games and Maths
Dynamical Programming
Probability Theory
Decision Theory
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Probability
The Monty Hall Problem
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Probability
The Monty Hall Problem
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2
3
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The Monty Hall Problem
The Solution
Now that youve seen the goat, you know that
whatever is in the other box is the opposite of
what you have already picked.
Two Strategies
Stick
Swap
Initial Probabilities Goat 2/3 Ferrari1/3 So
swapping Goat 1/3 Ferrari 2/3
Initial Probabilities Goat 2/3 Ferrari1/3
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Probability
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Probability
5 dice
15 dice. Expect 15/62.5 of each number Add them
to what you have (remember aces are wild). So
(your 3 5 more) 8 fives would be a reasonable
bid to make.
5 dice
5 dice
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Probability
Basic probability can really help us in winning
simple games and knowing when to gamble.
But imagine you were playing Perudo for money.
E.g. the winner gets paid 1 million by the
loser. Would you be so willing to take the gamble?
Thats where Decision Theory comes in.
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Decision Theory
Decision Theory is a wide-ranging subject.
The main idea of Decision Theory is making a
choice under uncertainty.
History C17th Expected Value Blaise Pascal
Pascals wager God either exists or doesnt
exist. You can choose to believe or not
believe. If God exists and you believe, the
payoff is infinite. gt Believe is the best
decision!
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Decision Theory
Decision Theory is a wide-ranging subject.
The main idea of Decision Theory is making a
choice under uncertainty.
History C18th Bernoulli Utility Theory
Doesnt use money, uses preferences in arbitrary
units (called utiles) Takes into account the risk
of each decision.
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Decision Theory
Decision Theory is a wide-ranging subject.
The main idea of Decision Theory is making a
choice under uncertainty.
History C20th Game Theory John Nash
A set of players, a set of strategies available,
a set of payoffs.
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Utility Theory
Why did it come about?
The St Petersburg Paradox
You toss a coin until the first time you get a
tail (after, say k tosses)
Reward 2k
1st toss win 2
158th toss win 3.653754093327257295509212081790
7x1047
How much are you willing to pay to enter this
game?
4?
EMV ½ x 2 ¼ x 4 ? x 8 8
So to maximise our win, we should be willing to
pay an infinite amount of money. Im certainly
not!
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Utility Theory
Utility Function
Its a function that takes gambles to rewards.
The rewards are real numbers, expressed in
utiles. The utility function follows our
preferences about the rewards.
0
30000
10000
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Utility Theory
A 10000 B ½ x 0 g ½ x 30000
EMV(A) 10000 EMV(B) ½ x 0
½ x 30000 15000
Utility Graph
U(0)0 U(10000)5 U(30000)7
U(A)5 U(B) ½ x 0 ½ x 7 3.5
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Game Theory
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Wider Application

• politics
• economics
• biology
• philosophy

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My Essay
Utility Functions
Game Theory
Use in schools
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Conclusion
Uses and limitations of probability theory
Gambles and Decision Theory
Game Theory and applications
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Questions?