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 : q i – Reported rankings for each player i : ˜ q i – Central system processes reported rankings (˜ q 1 , . . . , ˜ q n ) and constructs matching – Preference for each player i over constructed matching – Implies preferences over joint reports: (˜ q 1 , . . . , ˜ q n ) • Example: Single-item auction (Ebay) – Set of players: N = { 1 , ..., n } – True valuation of good for each player i : v i – Reported bid for each player i : b i – Central system processes bids ( b 1 , . . . , b n ) , awards good and charges prices – Preference for each player over awarded good and price charged – Implies preferences over joint bids: ( b 1 , . . . , b n ) • Features: – Players have choices – Preferences over joint choices 1
Strategic games • Setup: Strategic form games – Set of players, N = { 1 , ..., n } – A set of actions for each player i ∈ N , denoted by A i . – This induces the set of action profiles A = A 1 × A 2 × ... × A n – For each player, preferences over action profiles characterized by a function: U i : A → R • Terminology: – U i ( · ) 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 U i ( a ) > U i ( a ′ ) In case U i ( a ) = U i ( a ′ ) player i is “indifferent” • An action profile a ∈ A may be written in different ways: – The combination of all player actions: a = ( a 1 , ..., a n ) – The combination of the i th player’s action and everyone else’s: a = ( a i , a − i ) 2
Matrix form • Matrix form is a convenient representation for two player strategic games col L C R 1 , 9 2 , 8 T v, w 3 , 7 4 , 6 5 , 5 M row 6 , 4 7 , 3 B x, y • Player set: { row , col } • Action sets: A row = { T, M, B } A col = { L, C, R } • Action profiles: A = { ( T, L ) , ( T, C ) , ( T, R ) , ( M, L ) , ( M, C ) , ..., ( B, R ) } • Payoff functions: U row ( T, L ) = v & U col ( T, L ) = w U row ( T, C ) = 1 & U col ( T, C ) = 9 . . . U row ( B, R ) = 7 & U col ( B, R ) = 3 • Example: Prisoner’s dilemma: C D − 1 , − 1 − 4 , 0 C 0 , − 4 − 3 , − 3 D 3
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 0 , − 4 − 3 , − 3 D • 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 4
Example: Routing game High road S D Low road • Assume 2 players • Congestion: – High road: c H (1) , c H (2) – Low road: c L (1) , c L (2) – Note: Number in ( · ) highlight number of players using road • Cost Matrix (as opposed to Payoff Matrix): H L H c H (2) , c H (2) c H (1) , c L (2) c L (1) , c H (1) c L (2) , c L (2) L • Convention: Players focus on minimizing costs as opposed to maximizing negative utility. 5
Example: Security Strategies • What constitutes a reasonable prediction of strategic behavior in games? • Example: L R T 0 , 0 1 , 1 1 , 1 0 , 0 B • 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. 6
Dominant strategies • New focus: What does not constitute reasonable strategic behavior • Re-visit: Prison’s Dilemma C D − 1 , − 1 − 4 , 0 C 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 a i if U i ( a ′ i , a − i ) > U i ( a i , a − i ) for all a − i (alternatively, a i is strictly dominated ) • Definition: The action a ′ i weakly dominates a i if U i ( a ′ i , a − i ) ≥ U i ( a i , a − i ) for all a − i and U i ( a ′ i , a − i ) > U i ( a i , a − i ) for some a − i (alternatively, a i is weakly dominated ) • If action a ′ i is strictly dominated by a i , then a ′ i does not constitute reasonable strategic behavior 7
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: v 1 > v 2 > ... > v n – Players make bids: b 1 , b 2 , ..., b n – Highest bidder wins and pays second highest bid • Player i payoff: Let b = max { b − i } – If b i > b : v i − b – If b i < b : 0 Assume for convenience that ties never happen. • Claim: The bid b i = v i is a weakly dominant strategy • Cases: – b i > v i : ∗ b < v i < b i ∗ v i < b < b i ∗ v i < b i < b – b i < v i : ∗ b < b i < v i ∗ b i < b < v i ∗ b i < v i < b For each case, can show that changing the bid to v i performs just as well or bet- ter... regardless of other player bids (and valuations). 8
Example: First price sealed bid auction • Setup: – Players have internal valuations of item: v 1 > v 2 > ... > v n – Players make bids: b 1 , b 2 , ..., b n – Highest bidder wins and pays the highest bid • Player i payoff: Let b = max { b − i } – If b i > b : v i − b i – If b i < b : 0 Assume for convenience that ties never happen. • Question: Is the bid b i = v i a weakly dominant strategy? • Inspect: – Clearly having b i > v i is never advantageous – What about having b i < v i ? – Suppose b < b i < v i . Then U i ( b i , b − i ) = v i − b i > 0 U i ( v i , b − i ) = v i − v i = 0 – Therefore, b i = v i is not a dominant strategy • Could there be other dominant strategies? • Thought process: How does Ebay work? Notice any connection? 9
Iterated elimination of strictly dominated strategies • Successively eliminating dominated strategies can (sometimes) lead to a reasonable pre- diction of social behavior L C R 4 , 3 5 , 1 6 , 2 T M 2 , 1 8 , 4 3 , 6 3 , 0 9 , 6 2 , 8 B • Row player has no (strictly) dominated strategies • Column player can eliminate C • Reduced game: L R 4 , 3 6 , 2 T 2 , 1 3 , 6 M B 3 , 0 2 , 8 • Row player can now eliminate both M and B : L R 4 , 3 6 , 2 T • Column player can now eliminate R • ( T, L ) is the sole survivor 10
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