CS344M Autonomous Multiagent Systems Patrick MacAlpine Department - - PowerPoint PPT Presentation

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CS344M Autonomous Multiagent Systems Patrick MacAlpine Department - - PowerPoint PPT Presentation

Jan 05, 2024 265 likes •735 views

CS344M Autonomous Multiagent Systems Patrick MacAlpine Department of Computer Science The University of Texas at Austin Good Afternoon, Colleagues Are there any questions? Patrick MacAlpine Good Afternoon, Colleagues Are there any

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SLIDE 1

CS344M Autonomous Multiagent Systems

Patrick MacAlpine Department of Computer Science The University of Texas at Austin

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SLIDE 2

Good Afternoon, Colleagues

Are there any questions?

Patrick MacAlpine

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SLIDE 3

Good Afternoon, Colleagues

Are there any questions?

  • How can we apply game theory to RoboCup?
  • Examples of game theory that aren’t modeled as a

matrix?

  • What about irrational agents?
  • Pure vs mixed strategy?

Patrick MacAlpine

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SLIDE 4

Logistics

  • Progress reports due in 2 weeks

Patrick MacAlpine

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SLIDE 5

Logistics

  • Progress reports due in 2 weeks

Patrick MacAlpine

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SLIDE 6

Game Theory Premises

  • Simultaneous actions: (mutual exclusivity?)

Patrick MacAlpine

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SLIDE 7

Game Theory Premises

  • Simultaneous actions: (mutual exclusivity?)
  • No communication
  • Outcome depends on combination of actions
  • Utility (payoff) encapsulates everything about preferences
  • ver outcomes

Patrick MacAlpine

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SLIDE 8

Solution Concepts

  • Dominant strategy
  • Nash equilibrium
  • Pareto optimality
  • Maximum social welfare
  • Maximin strategy

Patrick MacAlpine

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SLIDE 9

Prisoner’s Dilemma

❈♦❧✉♠♥ ❈✭✶✮ ❉✭✷✮ ❈✭✶✮ ✸✱✸ ✵✱✺ ❘♦✇ ❉✭✷✮ ✺✱✵ ✶✱✶

Patrick MacAlpine

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SLIDE 10

Chicken

❈♦❧✉♠♥ ❈✭✶✮ ❉✭✷✮ ❈✭✶✮ ✸✱✸ ✶✱✺ ❘♦✇ ❉✭✷✮ ✺✱✶ ✵✱✵

Patrick MacAlpine

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SLIDE 11

Bach/Stravinsky

  • My wife and I agree to meet at a concert

Patrick MacAlpine

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SLIDE 12

Bach/Stravinsky

  • My wife and I agree to meet at a concert
  • Unfortunately, there are 2: Bach and Stravinsky

Patrick MacAlpine

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SLIDE 13

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

Patrick MacAlpine

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SLIDE 14

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

Patrick MacAlpine

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SLIDE 15

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

Patrick MacAlpine

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SLIDE 16

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 listening to

Patrick MacAlpine

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SLIDE 17

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 listening to

  • Propose a payoff matrix

Patrick MacAlpine

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SLIDE 18

Bach/Stravinsky

❲✐❢❡ ❙ ❇ ❙ ✷✱✶ ✵✱✵ ▼❡ ❇ ✵✱✵ ✶✱✷

Patrick MacAlpine

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SLIDE 19

Nash Equilibrium

  • Does every game have a pure strategy Nash equilibrium?

Patrick MacAlpine

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SLIDE 20

Matching Pennies

  • We each put a penny down covered
  • If they match, I win, if they don’t, you win

Patrick MacAlpine

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Matching Pennies

  • We each put a penny down covered
  • If they match, I win, if they don’t, you win

P❧❛②❡r ✷ ❍ ❚ ❍ ✶✱✲✶ ✲✶✱✶ P❧❛②❡r ✶ ❚ ✲✶✱✶ ✶✱✲✶

Patrick MacAlpine

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SLIDE 22

Matching Pennies

  • We each put a penny down covered
  • If they match, I win, if they don’t, you win

P❧❛②❡r ✷ ❍ ❚ ❍ ✶✱✲✶ ✲✶✱✶ P❧❛②❡r ✶ ❚ ✲✶✱✶ ✶✱✲✶

Nash equilibrium?

Patrick MacAlpine

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SLIDE 23

Nash Equilibrium

  • Every game has at least one Nash equilibrium

Patrick MacAlpine

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SLIDE 24

Nash Equilibrium

  • Every game has at least one Nash equilibrium

– Nobel prize and academy award!

Patrick MacAlpine

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SLIDE 25

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 or

in P

Patrick MacAlpine

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SLIDE 26

Some theory

  • Prove that if each player plays a dominant strategy, the

result is a Nash equilibrium

Patrick MacAlpine

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SLIDE 27

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 dominant

strategies?

Patrick MacAlpine

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SLIDE 28

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 dominant

strategies?

  • Is the outcome of a Nash equilibrium necessarily Pareto
  • ptimal?

Patrick MacAlpine

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SLIDE 29

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 dominant

strategies?

  • Is the outcome of a Nash equilibrium necessarily Pareto
  • ptimal?
  • Is a Pareto optimal outcome necessarily the result of Nash

equilibrium strategies?

Patrick MacAlpine

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SLIDE 30

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 dominant

strategies?

  • Is the outcome of a Nash equilibrium necessarily Pareto
  • ptimal?
  • Is a Pareto optimal outcome necessarily the result of Nash

equilibrium strategies?

  • Is the maximum social welfare outcome necessarily Pareto
  • ptimal?

Patrick MacAlpine

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SLIDE 31

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 dominant

strategies?

  • Is the outcome of a Nash equilibrium necessarily Pareto
  • ptimal?
  • Is a Pareto optimal outcome necessarily the result of Nash

equilibrium strategies?

  • Is the maximum social welfare outcome necessarily Pareto
  • ptimal?
  • If both players play maximin, is it necessarily a Nash

equilibrium?

Patrick MacAlpine

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SLIDE 32

Mixed strategy equilibrium

P❧❛②❡r ✷ ❆❝t✐♦♥ ✶ ❆❝t✐♦♥ ✷ ❆❝t✐♦♥ ✶ ✹✱✽ ✷✱✵ P❧❛②❡r ✶ ❆❝t✐♦♥ ✷ ✻✱✷ ✵✱✽

Patrick MacAlpine

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SLIDE 33

Mixed strategy equilibrium

P❧❛②❡r ✷ ❆❝t✐♦♥ ✶ ❆❝t✐♦♥ ✷ ❆❝t✐♦♥ ✶ ✹✱✽ ✷✱✵ P❧❛②❡r ✶ ❆❝t✐♦♥ ✷ ✻✱✷ ✵✱✽

  • What if player 2 picks action 1 3/4 of the time?

Patrick MacAlpine

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SLIDE 34

Mixed strategy equilibrium

P❧❛②❡r ✷ ❆❝t✐♦♥ ✶ ❆❝t✐♦♥ ✷ ❆❝t✐♦♥ ✶ ✹✱✽ ✷✱✵ P❧❛②❡r ✶ ❆❝t✐♦♥ ✷ ✻✱✷ ✵✱✽

  • What if player 2 picks action 1 3/4 of the time?1 = 3.5, 2 =

4.5

Patrick MacAlpine

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SLIDE 35

Mixed strategy equilibrium

P❧❛②❡r ✷ ❆❝t✐♦♥ ✶ ❆❝t✐♦♥ ✷ ❆❝t✐♦♥ ✶ ✹✱✽ ✷✱✵ P❧❛②❡r ✶ ❆❝t✐♦♥ ✷ ✻✱✷ ✵✱✽

  • What if player 2 picks action 1 3/4 of the time?1 = 3.5, 2 =

4.5

  • What if player 2 picks action 1 1/4 of the time?

Patrick MacAlpine

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SLIDE 36

Mixed strategy equilibrium

P❧❛②❡r ✷ ❆❝t✐♦♥ ✶ ❆❝t✐♦♥ ✷ ❆❝t✐♦♥ ✶ ✹✱✽ ✷✱✵ P❧❛②❡r ✶ ❆❝t✐♦♥ ✷ ✻✱✷ ✵✱✽

  • What if player 2 picks action 1 3/4 of the time?1 = 3.5, 2 =

4.5

  • What if player 2 picks action 1 1/4 of the time?1 = 2.5, 2 =

1.5

Patrick MacAlpine

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SLIDE 37

Mixed strategy equilibrium

P❧❛②❡r ✷ ❆❝t✐♦♥ ✶ ❆❝t✐♦♥ ✷ ❆❝t✐♦♥ ✶ ✹✱✽ ✷✱✵ P❧❛②❡r ✶ ❆❝t✐♦♥ ✷ ✻✱✷ ✵✱✽

  • What if player 2 picks action 1 3/4 of the time?1 = 3.5, 2 =

4.5

  • What if player 2 picks action 1 1/4 of the time?1 = 2.5, 2 =

1.5

Patrick MacAlpine

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SLIDE 38
  • Player 1 must be indifferent between actions 1 and 2

Patrick MacAlpine

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SLIDE 39
  • Player 1 must be indifferent between actions 1 and 24q+2-

2q = 6q

Patrick MacAlpine

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SLIDE 40
  • Player 1 must be indifferent between actions 1 and 24q+2-

2q = 6q

  • Player 2 must be indifferent between actions 1 and 2

Patrick MacAlpine

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SLIDE 41
  • Player 1 must be indifferent between actions 1 and 24q+2-

2q = 6q

  • Player 2 must be indifferent between actions 1 and 28p+2-

2p = 8-8p

Patrick MacAlpine

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SLIDE 42
  • Player 1 must be indifferent between actions 1 and 24q+2-

2q = 6q

  • Player 2 must be indifferent between actions 1 and 28p+2-

2p = 8-8p

Patrick MacAlpine

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SLIDE 43

Rock/Paper/Scissors

  • Nash equilibrium?

Patrick MacAlpine

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SLIDE 44

Rock/Paper/Scissors

  • Nash equilibrium?
  • Why is anything else not an equilibrium?

Patrick MacAlpine

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SLIDE 45

Correlated Equilibria

Sometimes mixing isn’t enough: Bach/Stravinsky

❲✐❢❡ ❙ ❇ ❙ ✷✱✶ ✵✱✵ ▼❡ ❇ ✵✱✵ ✶✱✷

Patrick MacAlpine

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SLIDE 46

Correlated Equilibria

Sometimes mixing isn’t enough: Bach/Stravinsky

❲✐❢❡ ❙ ❇ ❙ ✷✱✶ ✵✱✵ ▼❡ ❇ ✵✱✵ ✶✱✷

Want only S,S or B,B - 50% each

Patrick MacAlpine