A Brief Introduction to Game Theory · (Tarleton State University) Brief Intro to Game Theory...
Transcript of A Brief Introduction to Game Theory · (Tarleton State University) Brief Intro to Game Theory...
A Brief Introduction to Game Theory
Jesse Crawford
Department of MathematicsTarleton State University
November 20, 2014
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The Penny Game
Two players.Start with 4 pennies in center of table.Each player can take 1 penny or 2 pennies on his/her turn.Player to take the last penny wins.First player = blueSecond player = red
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Backwards Induction for Penny Game
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Backwards Induction for Penny Game
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Backwards Induction for Penny Game
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Backwards Induction for Penny Game
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Backwards Induction for Penny Game
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Backwards Induction for Penny Game
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Backwards Induction for Penny Game
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Backwards Induction for Penny Game
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Backwards Induction for Penny Game
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Game Tree for Tic-Tac-Toe
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Tic-Tac-Toe and Checkers
With optimal play, tic-tac-toe is a draw.
Schaeffer et al. (2007) showed that checkers is also a draw.
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Chess
First turn white moves = 20
First turn black moves = 20
(20)(20) = 400
400 variations on first twomoves!
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Chess
Allis (1994) estimated number of chess variations:
# of variations > 10123 > Number of atoms in visible universe!
Chess programs do use the game tree.I Only plot to finite depth.I Use an evaluation function to evaluate positions.
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Scratch-off Lottery Ticket
Ticket costs $1
Prize $0 $2 $10Probability 0.8 0.15 0.05
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Scratch-off Lottery Ticket
Ticket costs $1
x $0 $2 $10p(x) 0.8 0.15 0.05
Expected Value
E(X ) =∑
xp(x) = 0(0.8) + 2(0.15) + 10(0.05) = 0.8
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Texas Hold’em
Pot = 23Bet = 2Probability of hitting our straight = p = 4
46 = 0.087.
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Expected Value for Hold’em Problem
If we fold
x $0p(x) 1 E(X ) = 1(0) = 0
If we call
x $23 −$2p(x) p (1 − p) E(X ) = 23p − 2(1 − p)
We should call if
23p − 2(1 − p) ≥ 0
p ≥ 223 + 2
= 0.08
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Rock-Paper-Scissors
Two playersEach one chooses Rock, Paper, or Scissors simultaneously.
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Payoff Matrix for RPS
First player = blueSecond player = red
Rock Paper ScissorsRock (0,0) (-1,1) (1,-1)Paper (1,-1) (0,0) (-1,1)
Scissors (-1,1) (1,-1) (0,0)
RPS is a zero sum game.
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Randomized Strategy for RPS
Randomized strategy.I pR = probability of choosing RockI pP = probability of choosing PaperI pS = probability of choosing Scissors
Example:I pR = 0.7I pP = 0.2I pS = 0.1
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Randomized Strategy for RPS
Example:I pR = 0.7I pP = 0.2I pS = 0.1
If opponent chooses Paper, his expected utility is
0.7(1) + 0.2(0) + 0.1(−1) = 0.6
This is the maximum utility our opponent can achieve.We want to minimize his maximum utility.
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Minimax Strategy for RPS
Minimax strategy:I pR = 1
3I pP = 1
3I pS = 1
3
Now if opponent chooses Paper, his expected utility is
13(1) +
13(0) +
13(−1) = 0
No matter what he does, his expected utility will be 0.
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Equilibrium Strategies
In zero sum games with finite strategy spaces, minimax strategiesalways exist for both players.Theory can be generalized to multiplayer games, cf. Nash (1949).
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King-Two-Jack Game
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King-Two-Jack Game
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Expected Value for Player 1
p = probability that Player 1 bluffs with 2♦.q = probability that Player 2 calls when Player 1 betsPlayer 1’s expected value is
EV1 = −1 + 12 [3q + 2(1 − q)] + 1
2p[−1q + 2(1 − q)]
EV1 = −1 + 12 [q + 2] + 1
2p[2 − 3q]
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Optimal Calling Frequency
EV1 = −1 + 12 [q + 2] + 1
2p[2 − 3q]
Claim: Player 2 should choose q = 23 .
If q < 23 , Player 1 can choose p = 1, and
EV1 = 1 − q > 13 .
If q > 23 , Player 1 can choose p = 0, and
EV1 = 12q > 1
3 .
If q = 23 , then
EV1 = 13 .
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A Bit of Algebra
EV1 = −1 + 12 [q + 2] + 1
2p[2 − 3q]
EV1 = −1 + 12 [q(1 − 3p) + 2 + 2p]
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Optimal Bluffing Frequency
EV1 = −1 + 12 [q(1 − 3p) + 2 + 2p]
Claim: Player 1 should choose p = 13 .
If p < 13 , Player 2 can choose q = 0, and
EV1 = p < 13 .
If p > 13 , Player 2 can choose q = 1, and
EV1 = 12 − 1
2p < 13 .
If p = 13 , then
EV1 = 13 .
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King-Two-Jack Solution
Player 1 should always bet with K♦.Player 1 should bluff 1
3 of the time with 2♦.
Player 2 should call 23 of the time when Player 1 bets.
Player 1 will win about 33 cents per hand on average.
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A Non-zero-sum Game: Prisoner’s Dilemma
Two criminalsInterrogated in separate rooms
Stay Silent ConfessStay Silent (-1,-1) (-10,0)
Confess (0,-10) (-5,-5)
General principle: individuals acting in their own self interest canlead to a negative outcome for the group.Related problems:
I Pollution/“Tragedy of the Commons”I Cartels/MonopoliesI Taxation and public goods
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Areas of Application for Game Theory
Economics/Political ScienceI Bargaining problems
BiologyI Competition between organismsI Sex ratiosI Genetics
Philosophy
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References
Allis, V. (1994). “Programming a Computer for Playing Chess”.Philosophical Magazine 41 (314).Chen, B., and Ankenman, J. (2006). The Mathematics of Poker.Conjelco.Luce, R.D., and Raiffa, H. (1989). Games and Decisions:Introduction and Critical Survey. Dover.Nash, J.F. (1949). Equilibrium Points in N-Person Games.Proceedings of the National Academy of Sciences 36 48-49.Schaeffer, et al. (2007). Checkers is Solved. Science 141518-1522.
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Thank You!
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