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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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 Spring '11
 Parr
 Artificial Intelligence

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