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 A i . – A set of “other things that could happen in the world,” denoted A − i – This induces the set of states of the world A = A i × A − i – The individual’s preferences over states characterized by a function: U i : A → R • Terminology: – U i ( · ) 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 U i ( a ) > U i ( a ′ ) In case U i ( a ) = U i ( a ′ ) player i is “indifferent” 2
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: The Weather Rain Sun Umbrella 5 2 Alice’s Choice: No Umbrella 0 3 • Action sets (Alice is i , the weather is − i ): A i = { 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: U i ( U, R ) = 5 U i ( U, S ) = 2 U i ( ¬ U, R ) = 0 U i ( ¬ U, S ) = 3 • Question: What should Alice do? 3
If Alice knows what will happen The Weather Rain Sun Umbrella 5 2 Alice’s Choice: No Umbrella 0 3 • Simple Question: If Alice knows the weather, what should she do? • Definition: The best response function of player i , B i ( · ) , is B i ( a − i ) = { a i : U i ( a i , a − i ) ≥ U i ( a ′ i , a − i ) for all a ′ i ∈ A i } Note that the best response “function” is actually a set • To visualize, focus on a column: If raining, Alice’s payoff matrix is Rain Umbrella 5 Alice’s Choice: No Umbrella 0 Thus, B i ( R ) = U . If raining, bring umbrella. • If sunny, Alice’s payoff matrix is Sun Umbrella 2 Alice’s Choice: No Umbrella 3 Thus, B i ( 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 : The Weather Rain Sun Umbrella 5 2 Alice’s Choice: No Umbrella 3 0 • Alice’s pessimistic “payoff matrix” is Worst-case Umbrella 2 Alice’s Choice: No Umbrella 0 • 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 5 p. – If always sunny, expected payoff is 2 p + 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 ∆( A i ) to denote the set of all probability distributions over player i ’s action set. • We call s i = ( p 1 , p 2 , . . . , p |A i | ) ∈ ∆( A i ) a mixed strategy for player i . That is, – p k ≥ 0 for each k and – � |A i | k =1 p k = 1 . • Similarly, write ∆( A − i ) to denote the set of all probability distributions over other states of the world. • Given a joint mixed strategy s ∈ ∆( A i ) × ∆( A − i ) , player i ’s expected utility is � � U i ( s i , s − i ) = E s U i ( a i , a − i ) = p a i × p a − i × U i ( a i , a − i ) a i ∈A i a − i ∈A − i • A mixed strategy s i guarantees a payoff of v is for any s − i ∈ ∆( A − i ) : U i ( s i , s − i ) ≥ v • A player’s security value v is the highest payoff that the player can guarantee for any strategy s i ∈ ∆( A i ) • 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?” 8
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