gametheory[1]

# gametheory[1] - Intro to Game Theory CPS 170 Ron...

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Unformatted text preview: 4/1/10 Intro to Game Theory CPS 170 Ron Parr (with many slides courtesy of Vince Conitzer) What is game theory? •  Game theory studies seFngs where mulGple parGes (agents) each have –  diﬀerent preferences (uGlity funcGons), –  diﬀerent acGons that they can take •  Each agent’s uGlity (potenGally) depends on all agents’ acGons –  What is opGmal for one agent depends on what other agents do •  Very circular! •  Game theory studies how agents can raGonally form beliefs over what other agents will do, and (hence) how agents should act –  Useful for acGng as well as (potenGally) predicGng behavior of others •  Game theory does not directly aim to be a descripGve theory 1 4/1/10 Real World Game Theory Examples War AucGons Animal behavior Networking protocols, peer to peer networking behavior •  Road traﬃc •  •  •  •  •  Mechanism design: Suppose we want people to do X? How do we engineer the situaGon so that they will act that way? Penalty kick example probability .7 probability .3 ac#on probability 1 ac#on probability .6 probability .4 Is this a “ra*onal” outcome? If not, what is? 2 4/1/10 Rock ­paper ­scissors Column player AKA player 2 (simultaneously) chooses a column 0, 0 -1, 1 1, -1 Row player AKA player 1 chooses a row 1, -1 0, 0 -1, 1 -1, 1 1, -1 0, 0 A row or column is called an acGon or (pure) strategy Row player’s uGlity is always listed ﬁrst, column player’s second Zero ­sum game: the uGliGes in each entry sum to 0 (or a constant) Three ­player game would be a 3D table with 3 uGliGes per entry, etc. “Chicken” •  Two players drive cars towards each other •  If one player goes straight, that player wins •  If both go straight, they both die S D D S D D S S 0, 0 -1, 1 1, -1 -5, -5 not zero ­sum 3 4/1/10 Rock ­paper ­scissors – Seinfeld variant 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. 0, 0 1, -1 1, -1 -1, 1 0, 0 -1, 1 -1, 1 1, -1 0, 0 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) strict dominance weak dominance  ­i = “the player(s) other than i” 0, 0 1, -1 1, -1 -1, 1 0, 0 -1, 1 -1, 1 1, -1 0, 0 4 4/1/10 Prisoner’s Dilemma •  Pair of criminals has been caught •  District anorney has evidence to convict them of a minor crime (1 year in jail); knows that they commined 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 reducGon –  If only one confesses, that one gets 3 years reducGon confess confess don’t confess don’t confess -2, -2 0, -3 -3, 0 -1, -1 “Should I buy an SUV?” accident cost purchasing + gas cost cost: 5 cost: 3 cost: 5 cost: 5 cost: 8 cost: 2 cost: 5 cost: 5 -10, -10 -7, -11 -11, -7 -8, -8 5 4/1/10 “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 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 6 4/1/10 “2/3 of the average” game revisited 100 dominated (2/3)*100 (2/3)*(2/3)*100 dominated a8er removal of (originally) dominated strategies … 0 Mixed strategies •  Mixed strategy for player i = probability distribuGon over player i’s (pure) strategies •  E.g. 1/3 , 1/3 , 1/3 •  Example of dominance by a mixed strategy: 1/2 3, 0 0, 0 1/2 0, 0 3, 0 1, 0 1, 0 7 4/1/10 Nash equilibrium [Nash 50] •  A vector of strategies (one for each player) is called a strategy proﬁle •  A strategy proﬁle (σ1, σ2 , …, σn) is a Nash equilibrium if each σi is a best response to σ ­i –  That is, for any i, for any σi’, ui(σi, σ ­i) ≥ ui(σi’, σ ­i) •  Note that this does not say anything about mulGple agents changing their strategies at the same Gme •  In any (ﬁnite) game, at least one Nash equilibrium (possibly using mixed strategies) exists [Nash 50] •  (Note  ­ singular: equilibrium, plural: equilibria) Nash equilibria of “chicken” S D D S D S D S 0, 0 -1, 1 1, -1 -5, -5 •  (D, S) and (S, D) are Nash equilibria –  They are pure ­strategy Nash equilibria: nobody randomizes –  They are also strict Nash equilibria: changing your strategy will make you strictly worse oﬀ •  No other pure ­strategy Nash equilibria 8 4/1/10 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 acGon •  If the other player does this, every acGon will give you expected uGlity 0 –  Might as well randomize Nash equilibria of “chicken”… D D S S 0, 0 -1, 1 1, -1 -5, -5 •  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 uGlity for playing D =  ­pcS •  Player 1’s uGlity for playing S = pcD  ­ 5pcS = 1  ­ 6pcS •  So we need  ­pcS = 1  ­ 6pcS which means pcS = 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 uGlity  ­1/5 for each player 9 4/1/10 ComputaGonal Issues •  Zero ­sum games can be solved eﬃciently as linear programs (see slides from earlier in the semester) •  General sum games may require exponenGal Gme (in # of acGons) to ﬁnd a single equilibrium (non known eﬃcient algorithm and good reasons to suspect that none exists) •  Some bener news: Despite bad worst ­case complexity, many games can be solved quickly Game Theory Issues •  How descripGve is game theory? –  Some evidence that people play equilibria –  Some evidence that people act irraGonally –  If it is computaGonally intractable to solve for equilibria of large games, it would seem unlikely that people are doing this •  How reasonable is game theory? –  Are payoﬀs known? –  Are situaGons really simultaneous move with no informaGon about how the other player will act? –  Are situaGons really single ­shot 10 4/1/10 Extensions •  ParGal informaGon ( just as MDPs are extended to POMDPs) •  Uncertainty about the game parameters, e.g., payoﬀs (Bayesian games) •  MulGstep games with distribuGons over next states (game theory + MDPs = stochasGc games) •  MulGstep + parGal informaGon (ParGally observable stochasGc games) •  Game theory is so general, that it can encompass essenGally all aspects of strategic, mulGagent behavior, e.g., negoGaGng, threats, bluﬀs, coaliGons, bribes, etc. 11 ...
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