Game Theory: Lecture #10 Outline: Strategic form games Dominated - - PDF document

Game Theory: Lecture #10

Outline:

• Strategic form games
• Dominated strategies
• Examples

Strategic agents

• What is a reasonable description of strategic behavior?
• Previous focus:

– Single-agent scenarios: Unknown environment – Zero-sum games: Adversarial environment – Unifying theme: Worst-case guarantees

• What about strategic behavior outside of zero-sum games?
• Example: Matching with strategic agents

– Set of players: N = {1, ..., n} – True rankings for each player i: qi – Reported rankings for each player i: ˜ qi – Central system processes reported rankings (˜ q1, . . . , ˜ qn) and constructs matching – Preference for each player i over constructed matching – Implies preferences over joint reports: (˜ q1, . . . , ˜ qn)

• Example: Single-item auction (Ebay)

– Set of players: N = {1, ..., n} – True valuation of good for each player i: vi – Reported bid for each player i: bi – Central system processes bids (b1, . . . , bn), awards good and charges prices – Preference for each player over awarded good and price charged – Implies preferences over joint bids: (b1, . . . , bn)

• Features:

– Players have choices – Preferences over joint choices

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Strategic games

• Setup: Strategic form games

– Set of players, N = {1, ..., n} – A set of actions for each player i ∈ N, denoted by Ai. – This induces the set of action profiles A = A1 × A2 × ... × An – For each player, preferences over action profiles characterized by a function: Ui : A → R

• Terminology:

– Ui(·) referred to as “payoff” or “utility” or “reward” function – A player is referred to as an “agent” or “actor” or “decision-maker” or “user”

• Player i prefers action profile a to action profile a′ if and only if

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

• An action profile a ∈ A may be written in different ways:

– The combination of all player actions: a = (a1, ..., an) – The combination of the ith player’s action and everyone else’s: a = (ai, a−i)

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Matrix form

• Matrix form is a convenient representation for two player strategic games

row col L C R T v, w 1, 9 2, 8 M 3, 7 4, 6 5, 5 B 6, 4 x, y 7, 3

• Player set: {row, col}
• Action sets:

Arow = {T, M, B} Acol = {L, C, R}

• Action profiles:

A = {(T, L), (T, C), (T, R), (M, L), (M, C), ..., (B, R)}

• Payoff functions:

Urow(T, L) = v & Ucol(T, L) = w Urow(T, C) = 1 & Ucol(T, C) = 9 . . . Urow(B, R) = 7 & Ucol(B, R) = 3

• Example: Prisoner’s dilemma:

C D C −1, −1 −4, 0 D 0, −4 −3, −3

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More examples

• Matrix games can capture elaborate complex setups where “actions” represent sophisti-

cated strategies.

• Example: Prisoner’s dilemma with multiple stages and sophisticated strategies

C D C −1, −1 −4, 0 D 0, −4 −3, −3

• New setup:

– Play for N stages – Strategies: Comprehensive plan of action ∗ GT “grim trigger”: Play C until the opponent plays D, then play D ever afterwards ∗ TfT “tit for tat”: At stage k, repeat the opponents move at stage k − 1 ∗ Cy “cycle”: Play sequence {C, D, C, ...} – Overall payoff is the sum of stage payoffs

• We can recast new setup in the standard framework

– The “action” set for each player is {GT, TfT, Cy} – The payoff functions require filling in the new matrix game: GT TfT Cy GT ?,? ?,? ?,? TfT ?,? ?,? ?,? Cy ?,? ?,? ?,?

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Example: Routing game

S D High road Low road

• Assume 2 players
• Congestion:

– High road: cH(1), cH(2) – Low road: cL(1), cL(2) – Note: Number in (·) highlight number of players using road

• Cost Matrix (as opposed to Payoff Matrix):

H L H cH(2), cH(2) cH(1), cL(2) L cL(1), cH(1) cL(2), cL(2)

• Convention: Players focus on minimizing costs as opposed to maximizing negative utility.

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Example: Security Strategies

• What constitutes a reasonable prediction of strategic behavior in games?
• Example:

L R T 0, 0 1, 1 B 1, 1 0, 0

• Reasonable description: (B, L) or (T, R)
• What about security strategies?

– row: T or B – col: L or R – Prediction: Anything?

• Fact: Security strategies do not necessarily constitute reasonable strategic behavior out-

side of zero-sum games.

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Dominant strategies

• New focus: What does not constitute reasonable strategic behavior
• Re-visit: Prison’s Dilemma

C D C −1, −1 −4, 0 D 0, −4 −3, −3 What does not constitute reasonable behavior?

• Observation: D is better than C regardless of behavior of other player
• Definition: The action a′

i strictly dominates ai if

Ui(a′

i, a−i) > Ui(ai, a−i) for all a−i

(alternatively, ai is strictly dominated)

• Definition: The action a′

i weakly dominates ai if

Ui(a′

i, a−i) ≥ Ui(ai, a−i) for all a−i

and Ui(a′

i, a−i) > Ui(ai, a−i) for some a−i

(alternatively, ai is weakly dominated)

• If action a′

i is strictly dominated by ai, then a′ i does not constitute reasonable strategic

behavior

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Example: Second price sealed bid auction

• Fact: Dominant strategies are not common, but very powerful in predicting behavior
• Example: Second price sealed bid auction
• Setup:

– Players have internal valuations of item: v1 > v2 > ... > vn – Players make bids: b1, b2, ..., bn – Highest bidder wins and pays second highest bid

• Player i payoff: Let b = max {b−i}

– If bi > b: vi − b – If bi < b: 0 Assume for convenience that ties never happen.

• Claim: The bid bi = vi is a weakly dominant strategy
• Cases:

– bi > vi: ∗ b < vi < bi ∗ vi < b < bi ∗ vi < bi < b – bi < vi: ∗ b < bi < vi ∗ bi < b < vi ∗ bi < vi < b For each case, can show that changing the bid to vi performs just as well or bet- ter...regardless of other player bids (and valuations).

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Example: First price sealed bid auction

• Setup:

– Players have internal valuations of item: v1 > v2 > ... > vn – Players make bids: b1, b2, ..., bn – Highest bidder wins and pays the highest bid

• Player i payoff: Let b = max {b−i}

– If bi > b: vi − bi – If bi < b: 0 Assume for convenience that ties never happen.

• Question: Is the bid bi = vi a weakly dominant strategy?
• Inspect:

– Clearly having bi > vi is never advantageous – What about having bi < vi? – Suppose b < bi < vi. Then Ui(bi, b−i) = vi − bi > 0 Ui(vi, b−i) = vi − vi = 0 – Therefore, bi = vi is not a dominant strategy

• Could there be other dominant strategies?
• Thought process: How does Ebay work? Notice any connection?

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Iterated elimination of strictly dominated strategies

• Successively eliminating dominated strategies can (sometimes) lead to a reasonable pre-

diction of social behavior L C R T 4, 3 5, 1 6, 2 M 2, 1 8, 4 3, 6 B 3, 0 9, 6 2, 8

• Row player has no (strictly) dominated strategies
• Column player can eliminate C
• Reduced game:

L R T 4, 3 6, 2 M 2, 1 3, 6 B 3, 0 2, 8

• Row player can now eliminate both M and B:

L R T 4, 3 6, 2

• Column player can now eliminate R
• (T, L) is the sole survivor

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