![Nash equilibrium [Nash 1950]
• A profile (= strategy for each player) so that no
player wants to deviate
0, 0 -1, 1
1, -1 -5, -5
D
S
D S
• This game has another Nash equilibrium in
mixed strategies…](https://image.slidesharecdn.com/cps570gametheory-240416172139-fa1a3081/85/AI-subject-cps-570-_-game-_-theory-ppt-15-320.jpg)
AI subject - cps 570 _ game _ theory.ppt
- 1. CPS 570: Artificial Intelligence Game Theory Instructor: Vincent Conitzer
- 2. Penalty kick example probability .7 probability .3 probability .6 probability .4 probability 1 Is this a “rational” outcome? If not, what is? action action
- 3. Rock-paper-scissors 0, 0 -1, 1 1, -1 1, -1 0, 0 -1, 1 -1, 1 1, -1 0, 0 Row player aka. player 1 chooses a row Column player aka. player 2 (simultaneously) chooses a column A row or column is called an action or (pure) strategy Row player’s utility is always listed first, column player’s second Zero-sum game: the utilities in each entry sum to 0 (or a constant) Three-player game would be a 3D table with 3 utilities per entry, etc.
- 4. A poker-like game 1 gets King 1 gets Jack raise raise check check call fold call fold call fold call fold “nature” player 1 player 1 player 2 player 2 2 1 1 1 -2 -1 1 1 0, 0 0, 0 1, -1 1, -1 .5, -.5 1.5, -1.5 0, 0 1, -1 -.5, .5 -.5, .5 1, -1 1, -1 0, 0 1, -1 0, 0 1, -1 cc cf fc ff rr cr cc rc
- 5. “Chicken” 0, 0 -1, 1 1, -1 -5, -5 D S D S S D D S • Two players drive cars towards each other • If one player goes straight, that player wins • If both go straight, they both die not zero-sum
- 6. “2/3 of the average” game • Everyone writes down a number between 0 and 100 • Person closest to 2/3 of the average wins • Example: – A says 50 – B says 10 – C says 90 – Average(50, 10, 90) = 50 – 2/3 of average = 33.33 – A is closest (|50-33.33| = 16.67), so A wins
- 7. Rock-paper-scissors – Seinfeld variant 0, 0 1, -1 1, -1 -1, 1 0, 0 -1, 1 -1, 1 1, -1 0, 0 MICKEY: All right, rock beats paper! (Mickey smacks Kramer's hand for losing) KRAMER: I thought paper covered rock. MICKEY: Nah, rock flies right through paper. KRAMER: What beats rock? MICKEY: (looks at hand) Nothing beats rock.
- 8. Dominance • Player i’s strategy si strictly dominates si’ if – for any s-i, ui(si , s-i) > ui(si’, s-i) • si weakly dominates si’ if – for any s-i, ui(si , s-i) ≥ ui(si’, s-i); and – for some s-i, ui(si , s-i) > ui(si’, s-i) 0, 0 1, -1 1, -1 -1, 1 0, 0 -1, 1 -1, 1 1, -1 0, 0 strict dominance weak dominance -i = “the player(s) other than i”
- 9. Prisoner’s Dilemma -2, -2 0, -3 -3, 0 -1, -1 confess • Pair of criminals has been caught • District attorney has evidence to convict them of a minor crime (1 year in jail); knows that they committed a major crime together (3 years in jail) but cannot prove it • Offers them a deal: – If both confess to the major crime, they each get a 1 year reduction – If only one confesses, that one gets 3 years reduction don’t confess don’t confess confess
- 10. “Should I buy an SUV?” -10, -10 -7, -11 -11, -7 -8, -8 cost: 5 cost: 3 cost: 5 cost: 5 cost: 5 cost: 5 cost: 8 cost: 2 purchasing + gas cost accident cost
- 11. Back to the poker-like game 1 gets King 1 gets Jack raise raise check check call fold call fold call fold call fold “nature” player 1 player 1 player 2 player 2 2 1 1 1 -2 -1 1 1 0, 0 0, 0 1, -1 1, -1 .5, -.5 1.5, -1.5 0, 0 1, -1 -.5, .5 -.5, .5 1, -1 1, -1 0, 0 1, -1 0, 0 1, -1 cc cf fc ff rr cr cc rc
- 12. Iterated dominance • Iterated dominance: remove (strictly/weakly) dominated strategy, repeat • Iterated strict dominance on Seinfeld’s RPS: 0, 0 1, -1 1, -1 -1, 1 0, 0 -1, 1 -1, 1 1, -1 0, 0 0, 0 1, -1 -1, 1 0, 0
- 13. “2/3 of the average” game revisited 0 100 (2/3)*100 (2/3)*(2/3)*100 … dominated dominated after removal of (originally) dominated strategies
- 14. Mixed strategies • Mixed strategy for player i = probability distribution over player i’s (pure) strategies • E.g. 1/3 , 1/3 , 1/3 • Example of dominance by a mixed strategy: 3, 0 0, 0 0, 0 3, 0 1, 0 1, 0 1/2 1/2
- 15. Nash equilibrium [Nash 1950] • A profile (= strategy for each player) so that no player wants to deviate 0, 0 -1, 1 1, -1 -5, -5 D S D S • This game has another Nash equilibrium in mixed strategies…
- 16. Rock-paper-scissors 0, 0 -1, 1 1, -1 1, -1 0, 0 -1, 1 -1, 1 1, -1 0, 0 • Any pure-strategy Nash equilibria? • But it has a mixed-strategy Nash equilibrium: Both players put probability 1/3 on each action • If the other player does this, every action will give you expected utility 0 – Might as well randomize
- 17. Nash equilibria of “chicken”… 0, 0 -1, 1 1, -1 -5, -5 D S D S • Is there a Nash equilibrium that uses mixed strategies? Say, where player 1 uses a mixed strategy? • If a mixed strategy is a best response, then all of the pure strategies that it randomizes over must also be best responses • So we need to make player 1 indifferent between D and S • Player 1’s utility for playing D = -pc S • Player 1’s utility for playing S = pc D - 5pc S = 1 - 6pc S • So we need -pc S = 1 - 6pc S which means pc S = 1/5 • Then, player 2 needs to be indifferent as well • Mixed-strategy Nash equilibrium: ((4/5 D, 1/5 S), (4/5 D, 1/5 S)) – People may die! Expected utility -1/5 for each player
- 18. The presentation game Pay attention (A) Do not pay attention (NA) Put effort into presentation (E) Do not put effort into presentation (NE) 2, 2 -1, 0 -7, -8 0, 0 • Pure-strategy Nash equilibria: (E, A), (NE, NA) • Mixed-strategy Nash equilibrium: ((4/5 E, 1/5 NE), (1/10 A, 9/10 NA)) – Utility -7/10 for presenter, 0 for audience
- 19. Back to the poker-like game, again 1 gets King 1 gets Jack raise raise check check call fold call fold call fold call fold “nature” player 1 player 1 player 2 player 2 2 1 1 1 -2 -1 1 1 0, 0 0, 0 1, -1 1, -1 .5, -.5 1.5, -1.5 0, 0 1, -1 -.5, .5 -.5, .5 1, -1 1, -1 0, 0 1, -1 0, 0 1, -1 cc cf fc ff rr cr cc rc 2/3 1/3 1/3 2/3 • To make player 1 indifferent between rr and rc, we need: utility for rr = 0*P(cc)+1*(1-P(cc)) = .5*P(cc)+0*(1-P(cc)) = utility for rc That is, P(cc) = 2/3 • To make player 2 indifferent between cc and fc, we need: utility for cc = 0*P(rr)+(-.5)*(1-P(rr)) = -1*P(rr)+0*(1-P(rr)) = utility for fc That is, P(rr) = 1/3
- 20. Real-world security applications Airport security Where should checkpoints, canine units, etc. be deployed? US Coast Guard Which patrol routes should be followed? Federal Air Marshals Which flights get a FAM? Milind Tambe’s TEAMCORE group (USC) Wildlife Protection Where to patrol to catch poachers or find their snares?