![Page 1: CS344M Autonomous Multiagent Systemstodd/cs344m/slides/week8b-pp4.pdfAction 1 4,8 2,0 Player 1 Action 2 6,2 0,8 What if player 2 picks action 1 3/4 of the time? What if player 2 picks](https://reader036.vdocuments.net/reader036/viewer/2022063013/5fcdaa15ce10d3567f6ccd5b/html5/thumbnails/1.jpg)
CS344MAutonomous Multiagent Systems
Todd Hester
Department of Computer ScienceThe University of Texas at Austin
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Good Afternoon, Colleagues
Are there any questions?
Todd Hester
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Good Afternoon, Colleagues
Are there any questions?
Todd Hester
![Page 4: CS344M Autonomous Multiagent Systemstodd/cs344m/slides/week8b-pp4.pdfAction 1 4,8 2,0 Player 1 Action 2 6,2 0,8 What if player 2 picks action 1 3/4 of the time? What if player 2 picks](https://reader036.vdocuments.net/reader036/viewer/2022063013/5fcdaa15ce10d3567f6ccd5b/html5/thumbnails/4.jpg)
Logistics
• Progress reports due in 2 weeks
Todd Hester
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Logistics
• Progress reports due in 2 weeks
• Readings for next week
Todd Hester
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Game Theory Premises
• Simultaneous actions
• No communication
• Outcome depends on combination of actions
• Utility (payoff) encapsulates everything about preferencesover outcomes
Todd Hester
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Solution Concepts
• Dominant strategy
• Nash equilibrium
• Pareto optimality
• Maximum social welfare
• Maximin strategy
Todd Hester
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Prisoner’s Dilemma
Column
C(1) D(2)
C(1) 3,3 0,5
Row
D(2) 5,0 1,1
Todd Hester
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Chicken
Column
C(1) D(2)
C(1) 3,3 1,5
Row
D(2) 5,1 0,0
Todd Hester
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Bach/Stravinsky
• My wife and I agree to meet at a concert
Todd Hester
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Bach/Stravinsky
• My wife and I agree to meet at a concert
• Unfortunately, there are 2: Bach and Stravinsky
Todd Hester
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Bach/Stravinsky
• My wife and I agree to meet at a concert
• Unfortunately, there are 2: Bach and Stravinsky
• No time to get in touch with each other
Todd Hester
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Bach/Stravinsky
• My wife and I agree to meet at a concert
• Unfortunately, there are 2: Bach and Stravinsky
• No time to get in touch with each other
• I prefer Stravinsky, she prefers Bach
Todd Hester
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Bach/Stravinsky
• My wife and I agree to meet at a concert
• Unfortunately, there are 2: Bach and Stravinsky
• No time to get in touch with each other
• I prefer Stravinsky, she prefers Bach
• But most of all, we want to be together
Todd Hester
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Bach/Stravinsky
• My wife and I agree to meet at a concert
• Unfortunately, there are 2: Bach and Stravinsky
• No time to get in touch with each other
• I prefer Stravinsky, she prefers Bach
• But most of all, we want to be together
– If not, so distraught we don’t care what we’re listeningto
Todd Hester
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Bach/Stravinsky
• My wife and I agree to meet at a concert
• Unfortunately, there are 2: Bach and Stravinsky
• No time to get in touch with each other
• I prefer Stravinsky, she prefers Bach
• But most of all, we want to be together
– If not, so distraught we don’t care what we’re listeningto
• Propose a payoff matrix
Todd Hester
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Bach/Stravinsky
Wife
S B
S 2,1 0,0
Me
B 0,0 1,2
Todd Hester
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Nash Equilibrium
• Does every game have a pure strategy Nash equilibrium?
Todd Hester
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Matching Pennies• We each put a penny down covered
• If they match, I win, if they don’t, you win
Todd Hester
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Matching Pennies• We each put a penny down covered
• If they match, I win, if they don’t, you win
Player 2
H T
H 1,-1 -1,1
Player 1
T -1,1 1,-1
Todd Hester
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Matching Pennies• We each put a penny down covered
• If they match, I win, if they don’t, you win
Player 2
H T
H 1,-1 -1,1
Player 1
T -1,1 1,-1
Nash equilibrium?
Todd Hester
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Nash Equilibrium
• Every game has at least one Nash equilibrium
Todd Hester
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Nash Equilibrium
• Every game has at least one Nash equilibrium
– Nobel prize and academy award!
Todd Hester
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Nash Equilibrium
• Every game has at least one Nash equilibrium
– Nobel prize and academy award!
• Not known if complexity of finding one is NP-complete orin P
Todd Hester
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Some theory• Prove that if each player plays a dominant strategy, the
result is a Nash equilibrium
Todd Hester
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Some theory• Prove that if each player plays a dominant strategy, the
result is a Nash equilibrium
• Are all Nash equilibria the result of playing dominantstrategies?
Todd Hester
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Some theory• Prove that if each player plays a dominant strategy, the
result is a Nash equilibrium
• Are all Nash equilibria the result of playing dominantstrategies?
• Is the outcome of a Nash equilibrium necessarily Paretooptimal?
Todd Hester
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Some theory• Prove that if each player plays a dominant strategy, the
result is a Nash equilibrium
• Are all Nash equilibria the result of playing dominantstrategies?
• Is the outcome of a Nash equilibrium necessarily Paretooptimal?
• Is a Pareto optimal outcome necessarily the result of Nashequilibrium strategies?
Todd Hester
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Some theory• Prove that if each player plays a dominant strategy, the
result is a Nash equilibrium
• Are all Nash equilibria the result of playing dominantstrategies?
• Is the outcome of a Nash equilibrium necessarily Paretooptimal?
• Is a Pareto optimal outcome necessarily the result of Nashequilibrium strategies?
• Is the maximum social welfare outcome necessarily Paretooptimal?
Todd Hester
![Page 30: CS344M Autonomous Multiagent Systemstodd/cs344m/slides/week8b-pp4.pdfAction 1 4,8 2,0 Player 1 Action 2 6,2 0,8 What if player 2 picks action 1 3/4 of the time? What if player 2 picks](https://reader036.vdocuments.net/reader036/viewer/2022063013/5fcdaa15ce10d3567f6ccd5b/html5/thumbnails/30.jpg)
Some theory• Prove that if each player plays a dominant strategy, the
result is a Nash equilibrium
• Are all Nash equilibria the result of playing dominantstrategies?
• Is the outcome of a Nash equilibrium necessarily Paretooptimal?
• Is a Pareto optimal outcome necessarily the result of Nashequilibrium strategies?
• Is the maximum social welfare outcome necessarily Paretooptimal?
• If both players play maximin, is it necessarily a Nashequilibrium?
Todd Hester
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ActivityPlayer 2
Rock Paper Scissors
Rock 0,0 -1,1 1,-1
Player 1
Paper 1,-1 0,0 -1,1
Scissors -1,1 1,-1 0,0
Todd Hester
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ActivityPlayer 2
Rock Paper Scissors
Rock 0,0 -1,1 1,-1
Player 1
Paper 1,-1 0,0 -1,1
Scissors -1,1 1,-1 0,0
Todd Hester
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Mixed strategy equilibriumPlayer 2
Action 1 Action 2
Action 1 4,8 2,0
Player 1
Action 2 6,2 0,8
Todd Hester
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Mixed strategy equilibriumPlayer 2
Action 1 Action 2
Action 1 4,8 2,0
Player 1
Action 2 6,2 0,8
• What if player 2 picks action 1 3/4 of the time?
Todd Hester
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Mixed strategy equilibriumPlayer 2
Action 1 Action 2
Action 1 4,8 2,0
Player 1
Action 2 6,2 0,8
• What if player 2 picks action 1 3/4 of the time?• What if player 2 picks action 1 1/4 of the time?
Todd Hester
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Mixed strategy equilibriumPlayer 2
Action 1 Action 2
Action 1 4,8 2,0
Player 1
Action 2 6,2 0,8
• What if player 2 picks action 1 3/4 of the time?• What if player 2 picks action 1 1/4 of the time?• Player 1 must be indifferent between actions 1 and 2
Todd Hester
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Mixed strategy equilibriumPlayer 2
Action 1 Action 2
Action 1 4,8 2,0
Player 1
Action 2 6,2 0,8
• What if player 2 picks action 1 3/4 of the time?• What if player 2 picks action 1 1/4 of the time?• Player 1 must be indifferent between actions 1 and 2• Player 2 must be indifferent between actions 1 and 2
Todd Hester
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Mixed strategy equilibriumPlayer 2
Action 1 Action 2
Action 1 4,8 2,0
Player 1
Action 2 6,2 0,8
• What if player 2 picks action 1 3/4 of the time?• What if player 2 picks action 1 1/4 of the time?• Player 1 must be indifferent between actions 1 and 2• Player 2 must be indifferent between actions 1 and 2
Do actual numbers matter?
Todd Hester
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Rock/Paper/Scissors
• Nash equilibrium?
Todd Hester
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Rock/Paper/Scissors
• Nash equilibrium?
• Why is anything else not an equilibrium?
Todd Hester
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Rock/Paper/Scissors
• Nash equilibrium?
• Why is anything else not an equilibrium?
• Rock Paper Scissors tournament
Todd Hester
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Rock/Paper/Scissors
• Nash equilibrium?
• Why is anything else not an equilibrium?
• Rock Paper Scissors tournament
• Poker
Todd Hester
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Discussion• What is an example game within robot soccer?
Todd Hester
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Discussion• What is an example game within robot soccer?
Goalie
Block
Right Left
Left 1,-1 -1,1
Kicker
Right -1,1 1,-1
Todd Hester
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Discussion• What is an example game within robot soccer?
Goalie
Block
Right Left
Left 1,-1 -1,1
Kicker
Right -1,1 1,-1
• Can we use game theory to devise better strategies?
Todd Hester
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Correlated Equilibria
Sometimes mixing isn’t enough: Bach/Stravinsky
Wife
S B
S 2,1 0,0
Me
B 0,0 1,2
Todd Hester
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Correlated Equilibria
Sometimes mixing isn’t enough: Bach/Stravinsky
Wife
S B
S 2,1 0,0
Me
B 0,0 1,2
Want only S,S or B,B - 50% each
Todd Hester