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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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