Game Theory: Lecture #8 Outline: Individual Optimization Security - - PDF document

Game Theory: Lecture #8

Outline:

• Individual Optimization
• Security Strategies

Optimization and Strategy

• Previous focus:

– Given individual preferences, compute group preference? – Given individual preferences, compute satisfactory matching? – Given (sub-)group costs, divide costs so all are happy?

• Not addressed: Individual strategy.

– Given social choice or matching rule, do individuals willingly share preferences? – How to model individual choice?

• Overarching question: How does individual choice impact collective behavior?
• First challenge: how to think about individual choice?

1

Individual Optimization

• Setup: a single decision-maker i

– A set of actions for the individual, denoted by Ai. – A set of “other things that could happen in the world,” denoted A−i – This induces the set of states of the world A = Ai × A−i – The individual’s preferences over states characterized by a function: Ui : A → R

• Terminology:

– Ui(·) referred to as “payoff” or “utility” or “reward” function – The individual i is referred to as an “agent,” “player,” “decision-maker,” or “user”

• Player i prefers state a to state a′ if and only if

Ui(a) > Ui(a′) In case Ui(a) = Ui(a′) player i is “indifferent”

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Alice and the Umbrella

• Example: Alice leaves home. Bring umbrella?

– Loves walking in the rain with umbrella – Hates walking in the rain without umbrella – If no rain, better to leave umbrella at home

• How to model this setup with a utility function?
• Matrix form is a convenient representation:

Alice’s Choice: The Weather Rain Sun Umbrella 5 2 No Umbrella 3

• Action sets (Alice is i, the weather is −i):

Ai = {U, ¬U} A−i = {R, S}

• Possible states of the world:

A = {(U, R), (U, S), (¬U, R), (¬U, S)}

• Utility function gives Alice a payoff for each possible state:

Ui(U, R) = 5 Ui(U, S) = 2 Ui(¬U, R) = 0 Ui(¬U, S) = 3

• Question: What should Alice do?

3

If Alice knows what will happen

Alice’s Choice: The Weather Rain Sun Umbrella 5 2 No Umbrella 3

• Simple Question: If Alice knows the weather, what should she do?
• Definition: The best response function of player i, Bi(·), is

Bi(a−i) = {ai : Ui(ai, a−i) ≥ Ui(a′

i, a−i) for all a′ i ∈ Ai}

Note that the best response “function” is actually a set

• To visualize, focus on a column: If raining, Alice’s payoff matrix is

Alice’s Choice: Rain Umbrella 5 No Umbrella Thus, Bi(R) = U. If raining, bring umbrella.

• If sunny, Alice’s payoff matrix is

Alice’s Choice: Sun Umbrella 2 No Umbrella 3 Thus, Bi(S) = ¬U. If sunny, leave umbrella at home.

• Note: to decide, Alice only compares numbers from a single column!

4

What if Alice Can’t Predict the Weather?

• Harder Question: If Alice can’t predict weather, what should she do?
• Why not use best response function here?
• New idea: Alice could “play it safe,” and try to limit her losses.
• Conceptually: assume the worst possible weather.
• Visually: optimize based on the smallest number in each payoff row:

Alice’s Choice: The Weather Rain Sun Umbrella 5 2 No Umbrella 3

• Alice’s pessimistic “payoff matrix” is

Alice’s Choice: Worst-case Umbrella 2 No Umbrella

• Terminology:

– Security value: v = 2 – Security strategy: Umbrella

• Interpretation: If Alice always brings her umbrella, the worst payoff she’ll ever get is 2.
• Question: can she guarantee any better?

5

What if Alice Can’t Predict the Weather?

• Question: can Alice guarantee a better payoff than 2?
• Thought experiment: what if Alice occasionally left umbrella at home?
• Setup:

– bring umbrella a fraction p of the days. – leave umbrella a fraction 1 − p of the days.

• What is the worst thing that could happen?

– If always rainy, expected (average) payoff is 5p. – If always sunny, expected payoff is 2p + 3(1 − p) = 3 − p. – Plot of these as a function of p: – (p = 1) corresponds to “umbrella” payoff row – (p = 0) corresponds to “no umbrella” payoff row

• Note: to maximize guaranteed expected payoff, p = 1/2.
• When p = 1/2, No matter what the weather does, expected payoff is at least 2.5.

6

Security Strategies

• No matter what the weather does, expected payoff is at least 2.5.
• How do we formalize this?
• Need a notion of probabilistic strategies.
• Write ∆(Ai) to denote the set of all probability distributions over player i’s action set.
• We call si = (p1, p2, . . . , p|Ai|) ∈ ∆(Ai) a mixed strategy for player i. That is,

– pk ≥ 0 for each k and – |Ai|

k=1 pk = 1.

• Similarly, write ∆(A−i) to denote the set of all probability distributions over other states
• f the world.
• Given a joint mixed strategy s ∈ ∆(Ai) × ∆(A−i), player i’s expected utility is

Ui(si, s−i) = EsUi(ai, a−i) =

• ai∈Ai
• a−i∈A−i

pai × pa−i × Ui(ai, a−i)

• A mixed strategy si guarantees a payoff of v is for any s−i ∈ ∆(A−i):

Ui(si, s−i) ≥ v

• A player’s security value v is the highest payoff that the player can guarantee for any

strategy si ∈ ∆(Ai)

• A player’s security strategy s∗

i is any strategy that guarantees the payoff v

7

Interpret

• Revisit Alice:
• Alice has a security value v = 2.5
• Alice has a security strategy s∗

i = (1/2, 1/2)

• Question: what is Alice assuming about the chance of rain to obtain this guarantee?
• Alice is assuming nothing! No matter what the weather does, she’ll always get at least

this amount.

• Next lecture: What if the weather were “out to get her?”

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