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Introduction to Network Mathematics (3) - Simple Games and applications Yuedong Xu 16/05/2012

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Outline Overview Prison’s Dilemma Curnot Duopoly Selfish Routing Summary

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Overview What is “game theory”? – A scientific way to depict the rational behaviors in interactive situations – Examples: playing poker, chess; setting price; announcing wars; and numerous commercial strategies Why is “game theory” important? – Facilitates strategic thinking!

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Overview Olympic Badminton Match 2012 – Four pair of players expelled because they “throw” the matches – Why are players trying to lose the match in the round-robin stage?

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Overview Chinese VS Korean – If Chinese team wins, it may encounter another Chinese team earlier in the elimination tournament. (not optimal for China) Best strategy for Chinese team: LOSE – If Korean team wins luckily, it may meet with another Chinese team that is usually stronger than itself in the elimination tournament. Best strategy for Korean team: LOSE

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Overview Korean VS Indonesian – Conditioned on the result: China Lose – If Korean team wins, meet with another Korean team early in the elimination tournament. (not optimal for Korea) Best strategy for Korean team: LOSE – If Indonesian wins, meet with a strong Chinese team in the elimination tournament. Best strategy for Indonesian team: LOSE

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Overview What is “outcome”? – Ugly matches that both players and watchers are unhappy – By studying this case, we know how to design a good “rule” so as to avoid “throwing” matches

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Outline Overview Prison’s Dilemma Curnot Duopoly Selfish Routing Summary

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Prison’s Dilemma Two suspects are caught and put in different rooms (no communication). They are offered the following deal: – If both of you confess, you will both get 5 years in prison (-5 payoff) – If one of you confesses whereas the other does not confess, you will get 0 (0 payoff) and 10 (-10 payoff) years in prison respectively. – If neither of you confess, you both will get 2 years in prison (-2 payoff)

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Prison’s Dilemma Prisoner 2 Prisoner 1 Confess Don’t Confess Confess -5, -50, -10 Don’t Confess -10, 0-2, -2

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Prison’s Dilemma Prisoner 2 Prisoner 1 Confess Don’t Confess Confess -5, -50, -10 Don’t Confess -10, 0-2, -2

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Prison’s Dilemma Game – Players (e.g. prisoner 1&2) – Strategy (e.g. confess or defect) – Payoff (e.g. years spent in the prison) Nash Equilibrium (NE) – In equilibrium, neither player can unilaterally change his/her strategy to improve his/her payoff, given the strategies of other players.

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Prison’s Dilemma Some common concerns – Existence/uniqueness of NE – Convergence to NE – Playing games sequentially or repeatedly More advanced games – Playing game with partial information – Evolutionary behavior – Algorithmic aspects – and more ……

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Prison’s Dilemma – Two NEs Prisoner 2 Prisoner 1 Confess Don’t Confess Confess -5, -5-3, -10 Don’t Confess -10, -3-2, -2

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Prison’s Dilemma – No NE Rock-Paper-Scissors game: If there exists a NE, then it is simple to play!

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Outline Overview Prison’s Dilemma Curnot Duopoly Selfish Routing Summary

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Curnot Duopoly Basic setting: Two firms: A & B are profit seekers Strategy: quantity that they produce Market price p: p = 100 - (q A + q B ) Question: optimal quantity for A&B

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Curnot Duopoly A’s profit: Strategy: quantity that they produce Market price p: p = 100 - (q A + q B ) Question: optimal quantity for A&B

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Curnot Duopoly A’s profit: π A (q A,q B ) = q A p = q A (100-q A -q B ) B’s profit: π B (q A,q B ) = q B p = q B (100-q A -q B ) How to find the NE?

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Curnot Duopoly A’s best strategy: dπ A (q A,q B ) —————— = 100 - 2q A – q B = 0 dq A B’s best strategy: dπ B (q A,q B ) —————— = 100 - 2q B – q A = 0 dq B Combined together: q A * = q B * = 100/3

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Curnot Duopoly Take-home messages: – If the strategy is continuous, e.g. production quantity or price, you can find the best response for each player, and then find the fixed point(s) for these best response equations.

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Outline Overview Prison’s Dilemma Curnot Duopoly Selfish Routing Summary

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Selfish Routing Braess’s Paradox st x1 x 1 0 st x1 x 1 Traffic of 1 unit/sec needs to be routed from s to t Want to minimize average delay Braess 1968, in study of road traffic

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Selfish Routing Before and after st x1 1 x 1 0 0 1 0 1 st x1.5 x 1 Think of green flow – it has no incentive to deviate Adding a 0 cost link made average delay worse!!!

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Selfish Routing Braess’s paradox illustrates non- optimality of selfish routing Think of the flow consisting of tiny “packets” Each chooses the lowest latency route This would reach an equilibrium (pointed out by Wardrop) – Wardrop equilibrium = Nash equilibrium

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Summary Present the concept of game and Nash Equilibrium Present a discrete and a continuous examples Illustrate the selfish routing

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