Algorithmic Game Theory at TU München

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Money Pump

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A preference relation is called rational if it is?

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Transitivity means ... (in one word)

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Continuity (definition)

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Continuity implies that, ...

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Independence (definition)

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Independence implies that ...

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What do we need to get a  von Neumann expected utility theorem?

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Risk Aversion

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Allais Paradox

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why are there either 1 ore infinitely many best responses? 

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Pareto-Optimality

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Exemplary flashcards for Algorithmic Game Theory at the TU München on StudySmarter:

Algorithmic Game Theory

Money Pump

a < b < c < a
if you pay one euro because you prefer the next version someone can trade you the the Better versions  for the lower ones +1 euro and can distract infinite euros from you. 

Algorithmic Game Theory

A preference relation is called rational if it is?

complete (i.e., ∀x,y ∈ A: (x ≿ y) ∨ (y ≿ x))

and

transitive (i.e., ∀x,y,z ∈ A: (x ≿ y) ∧ (y ≿ z) ⇒ (x ≿ z)).

Algorithmic Game Theory

Transitivity means ... (in one word)

Indistinguishability

Algorithmic Game Theory

Continuity (definition)

For all L1 ≻ L2 ≻ L3, there is some p ∈ (0,1) such that L2 ~ [p : L1, (1-p) : L3].

Algorithmic Game Theory

Continuity implies that, ...

Continuity implies that, if the probability of a plane crash (L3) is sufficiently small, you still prefer flying to Hawaii.

Algorithmic Game Theory

Independence (definition)

For all L1, L2, L3 and all p ∈ (0,1), L1 ≿ L2 ⇔ [p : L1, (1-p) : L3] ≿ [p : L2, (1-p) : L3]

Algorithmic Game Theory

Independence implies that ...

Independence implies that an equal probability of a plane crash (L3) for each trip (no matter how large) will not affect your preference.

Algorithmic Game Theory

What do we need to get a  von Neumann expected utility theorem?

if take all 4 axioms(transitivity, completeness, continuity, independence), then we get von Neumann expected utility theorem

Algorithmic Game Theory

Risk Aversion

Utility can be very different from monetary value.

People buy insurances because their utility is concave in value. They are risk-averse.

People buy lottery tickets because their utility is convex in value. They are risk-seeking.

Algorithmic Game Theory

Allais Paradox

Which lottery would you prefer?

  • L1 = [1: €1million]
  • L2 = [0.98: €3million, 0.02: nothing]
  • L3 = [0.050: €1million, 0.950: nothing] 
  • L4 = [0.049: €3million, 0.951: nothing]
  • Numerous experiments have shown that most people prefer L1 to L2 and L4 to L3.
  • These preferences cannot be explained by expected utility!
  • The independence axiom implies that L1 ≻ L2 ⇔ L3 ≻ L4.

Algorithmic Game Theory

why are there either 1 ore infinitely many best responses? 

As son as there are 2 best responses, you can linearly combine those responses with different weights on each response (as long as Sum(weights) = 1 ) to form new responses.

Algorithmic Game Theory

Pareto-Optimality

An outcome is Pareto-optimal if it is not Pareto-dominated.

An outcome is (weakly) Pareto-dominated if there exists another outcome in which all players obtain at least as much utility and one player is strictly better off.

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