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CSCI 3160 Design and Analysis of Algorithms Tutorial 9

CSCI 3160 Design and Analysis of Algorithms Tutorial 9. Yuanming Yu. A Game. A game consists of Players (at least 2) Strategies (at least 2 for each player) Payoffs. Strategy. Joint strategy s is a tuple s represents strategies made by all players

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CSCI 3160 Design and Analysis of Algorithms Tutorial 9

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  1. CSCI 3160 Design and Analysis of AlgorithmsTutorial 9 Yuanming Yu

  2. A Game • A game consists of • Players (at least 2) • Strategies (at least 2 for each player) • Payoffs

  3. Strategy • Joint strategy s is • a tuple s represents strategies made by all players • s = s1s2 is a joint strategy • si is the strategy made by player i • e.g. si can be confess, or silent • s-i represents strategies made by all players except i • s = sis-I • strategies made by all players except i combined with the strategy made by player i

  4. Payoffs • Given a joint strategy s. A payoff function ui(s) • indicates how many points player i gets if the strategy s is made • e.g. Let Blue be player 1 • u1(confess, silent) = 1

  5. A Game • A game can be describe by a matrix • Each row represents a strategy made by one player • Similarly for columns • So, each entry represents the strategies made by each player • Each player gets a payoff for each entry • e.g. • If Blue confesses and Red Silent • Blue gets -1 points • Red gets -5 points

  6. Nash Equilibrium • A joint strategy that achieves the goal • If anyone person deviates from the strategy • this person cannot be any better off • e.g. s = s1s2 = confess confess • is a (pure) Nash Equilibrium • u1(silent, confess) = -5 ≤ -4 =u1(confess, confess) • u2(confess, silent) = -5≤ -4 =u1(confess, confess)

  7. Different Equilibriums • Pure equilibriums • e.g. s = s1s2 • s1s2 are deterministic • Mixed equilibriums • Each player follows a probability distribution pi • e.g. s = s1s2 • s1 follows a probability distribution p1 • Confess with probability p, silent with prob. 1-p • s2follows another probability distribution p2 • Confess with probability q, silent with prob. 1-q

  8. Different Equilibriums • Correlated equilibrium • the joint strategy s itself follows a joint distribution • each joint strategy has a probability • p1,1 + p1,2 + p2,1+ p2,2= 1 • Following this distribution, If anyone person only knows his own strategy and deviates from the strategy In expectation this person cannot be any better off

  9. Different Equilibriums • Pure equilibriums • May not exist • Mixed equilibriums • Always exist • Cannot be computed efficiently (polynomial time) • Correlated equilibriums • Always exist • Can be solved in polynomial time • By linear programming

  10. Example: Traffic Game • Pure Nash Equilibrium • s = cross, stop or s = stop, cross • Mixed Nash Equilibrium • Blue crosses with probability 1/101 • Redcrosseswith probability 1/100 • Correlated Nash Equilibrium

  11. End • Questions?

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