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gtintro[1]

gtintro[1] - Brief Comments on Game Theory Ron...

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Unformatted text preview: 1/28/10 Brief Comments on Game Theory Ron Parr CSP 170 What is Game Theory •  Very general mathema?cal framework to study situa?ons where mul?ple agents interact, including: –  Popular no?ons of games –  Everything up to and including mul?step, mul?agent, simultaneous move, par?al informa?on games –  Can even including nego?a?ng, posturing and uncertainty about the players and game itself •  von Neumann and Morgenstern (1944) was a major launching point for modern game theory •  Nash: Existence of equilibria in general sum games 1 1/28/10 Covered Today •  2 player, zero sum simultaneous move games •  Example: Rock, Paper, Scissor •  Linear programming solu?on Linear Programs (max formula?on) •  Note: min formula?on also possible •  LP tricks –  Min: cTx –  Subject to: Ax≥b –  Mul?ply by  ­1 to reverse inequali?es –  Can easily introduce equality constraints, or arbitrary domain constraints 2 1/28/10 Rock, Paper, Scissors Zero Sum Formula?on •  In zero sum games, one player’s loss is other’s gain •  Payoﬀ matrix: •  Minimax solu?on maximizes worst cast outcome Rock, Paper, Scissors Equa?ons •  R,P,S = probability that we play rock, paper, or scissors respec?vely (R+P+S = 1) •  U is our expected u?lity •  Bounding our u?lity: –  Opponent rock case: U ≤ P – S –  Opponent paper case: U ≤ S – R –  Opponent scissors case: U ≤ R – P •  Want to maximize U subject to constraints •  Solu?on: (1/3, 1/3, 1/3) 3 1/28/10 Rock, Paper, Scissors LP Formula?on •  Our variables are: x=[U,R,P,S]T •  We want: –  Maximize U –  U ≤ P – S –  U ≤ S – R –  U ≤ R – P –  R+P+S = 1 •  How do we make this ﬁt: ? Rock, Paper, Scissors Solu?on •  If we feed this LP to an LP solver we get: •  Solu?on for the other player is: –  The same… –  By symmetry –  R=P=S=1/3 –  U=0 •  This is the minimax solu?on •  This is also an equilibrium –  No player has an incen?ve to deviate –  (Deﬁned more precisely later in the course) 4 1/28/10 Minimax Solu?ons in General •  Minimax solu?ons for 2 ­player zero ­sum games can always be found by solving a linear program •  The minimax solu?ons will also be equilibria •  For general sum games: –  Minimax does not apply –  Equilibria may not be unique –  Need to search for equilibria using more computa?onally intensive methods 5 ...
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