"Students play games while learning the connection between these games and Game Theory in computer science or Rock-Paper-Scissors and Poker what s
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1 "Students play games while learning the connection between these games and Game Theory in computer science or Rock-Paper-Scissors and Poker what s the connection to computer science?
2 Game Theory Noam Brown Carnegie Mellon TechNights
3 Prisoner s Dilemma Need 2 volunteers Don t Betray Betray Don t Betray 2, 2 0, 3 Betray 3, 0 1, 1
4 Nash Equilibrium Nash Equilibrium: A set of strategies where no player has incentive to deviate Don t Betray If opponent betrays, 1 > 0 If opponent does not betray, 3 > 2 Betray Don t Betray 2, 2 0, 3 Betray 3, 0 1, 1
5 What are Games? Requirements for a game: At least 2 players Actions available for each player Payoffs to each player for those actions Examples?
6 What is Game Theory? Assumptions in Game Theory: 1) All players want to maximize their utility (candy) 2) All players are rational 3) It is common knowledge that all players are rational
7 Tragedy of the Commons Groups of 5-6 Game 1: To pollute or not pollute? Everyone receives 3 fish Anyone may pollute. If you pollute, you receive +2 fish and everyone else receives -1. Game 2: Same as Game 1, except players first vote on whether polluting is allowed. If all players vote against polluting, it is not allowed
8 Tragedy of the Commons
9 2/3 Average Everyone plays! Everyone writes down a number in [0, 100]. Let X be the average of all guesses Whoever s guess is closest to 2 X wins. 3
10 2/3 Average Analysis Range of guesses is [0,100], so range of winning guesses is [0, ]
11 2/3 Average Analysis Range of guesses is [0,100], so range of winning guesses is [0,66 2 ] 3 Therefore, range of rational guesses is [0,66 2 ], 3 so range of winning guesses is [0,44 4 ] 9
12 2/3 Average Analysis Range of guesses is [0,100], so range of winning guesses is [0,66 2 ] 3 Therefore, range of rational guesses is [0,66 2 ], 3 so range of winning guesses is [0,44 4 ] 9 Therefore The game-theoretic solution is 0
13 Pirate s Treasure Groups of 5-6 Pirates are ordered A, B, C, D, and E. A starts as captain. The captain must divide 20 pieces of treasure among the pirates. Then, all pirates vote on the division (the captain breaks ties) If the vote succeeds, the treasure is divided. Otherwise, the captain is thrown overboard and the next pirate in the order becomes the captain.
14 Pirate s Treasure A B C D E Captain A: Captain B: Captain C: Captain D: Captain E:
15 Pirate s Treasure Backward induction! A B C D E Captain A: Captain B: Captain C: Captain D: Captain E: X X X X 20
16 Pirate s Treasure Backward induction! A B C D E Captain A: Captain B: Captain C: Captain D: X X X 20 0 Captain E: X X X X 20
17 Pirate s Treasure Backward induction! A B C D E Captain A: Captain B: Captain C: X X Captain D: X X X 20 0 Captain E: X X X X 20
18 Pirate s Treasure Backward induction! A B C D E Captain A: Captain B: X Captain C: X X Captain D: X X X 20 0 Captain E: X X X X 20
19 Pirate s Treasure Backward induction! A B C D E Captain A: Captain B: X Captain C: X X Captain D: X X X 20 0 Captain E: X X X X 20
20 Rock, Paper, Scissors Does Rock, Paper, Scissors have a Nash equilibrium? 0,0-1,1 1,-1 1,-1 0,0-1,1-1,1 1,-1 0,0
21 Poker Artificial Intelligence I do research on Poker AI Defending world champion among poker AIs Played against the world s best human poker players in 2015
22 Half-Street Kuhn Poker 3 cards: J, Q, and K. Highest card wins Two players are dealt one of the cards Both players ante 1 chip P1 may bet or check If P1 checks, the higher card wins 1 chip If P1 bets, P2 may call or fold - If P2 folds, P1 wins 1 chip - If P2 calls, the highest card wins 2 chips ±1 P1 P2 1, 1 ±2 21
23 Half-Street Kuhn Poker Strategy P1 should always bet with K P2 should always call with K P2 should never call with J Therefore P1 should never bet with Q! ±1 P1 P2 1, 1 ±2 22
24 Half-Street Kuhn Poker Strategy P1 should always bet with K P2 should always call with K P2 should never call with J Therefore P1 should never bet with Q! Call with Q ±1 P1 P2 1, 1 ±2 Fold with Q Bet with J Check with J 23
25 Half-Street Kuhn Poker Strategy P1 should always bet with K P2 should always call with K P2 should never call with J Therefore P1 should never bet with Q! Optimal strategy: Randomize! P1: Bets with J 33% of the time P2: Call with Q 33% of the time ±1 P1 P2 1, 1 ±2 24
26 Other Topics: Braess Paradox Can building a road increase traffic? Suppose there are 4000 drivers We construct a superfast highway from A to B
27 Other Topics: Elections Are there election systems where everyone wants to vote truthfully?
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