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Epistemic Game Theory Lecture 1 ESSLLI12, Opole Eric Pacuit Olivier Roy TiLPS, Tilburg University MCMP, LMU Munich ai.stanford.edu/~epacuit http://olivier.amonbofis.net August 6, 2012 Eric Pacuit and Olivier Roy 1 The Guessing Game

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SLIDE 1

Epistemic Game Theory

Lecture 1

ESSLLI’12, Opole

Eric Pacuit Olivier Roy TiLPS, Tilburg University MCMP, LMU Munich ai.stanford.edu/~epacuit http://olivier.amonbofis.net August 6, 2012

Eric Pacuit and Olivier Roy 1

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SLIDE 2

The Guessing Game

Eric Pacuit and Olivier Roy 2

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SLIDE 3

Plan for the week

Eric Pacuit and Olivier Roy 3

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SLIDE 4

Plan for the week

  • 1. Monday Basic Concepts.
  • Basics of Game Theory.
  • The Epistemic View on Games.
  • Basics of Decision Theory

Eric Pacuit and Olivier Roy 3

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SLIDE 5

Plan for the week

  • 1. Monday Basic Concepts.
  • 2. Tuesday Epistemics.
  • Logical/qualitative models of beliefs, knowledge and

higher-order attitudes.

  • Probabilistic/quantitative models of beliefs, knowledge and

higher-order attitudes.

Eric Pacuit and Olivier Roy 3

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SLIDE 6

Plan for the week

  • 1. Monday Basic Concepts.
  • 2. Tuesday Epistemics.
  • 3. Wednesday Fundamentals of Epistemic Game Theory.
  • Common knowledge of Rationality and iterated strict

dominance in the matrix.

  • Common knowledge of Rationality and backward induction

(strict dominance in the tree).

Eric Pacuit and Olivier Roy 3

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SLIDE 7

Plan for the week

  • 1. Monday Basic Concepts.
  • 2. Tuesday Epistemics.
  • 3. Wednesday Fundamentals of Epistemic Game Theory.
  • 4. Thursday Puzzles and Paradoxes.
  • Weak dominance and admissibility in the matrix.
  • Russell-style paradoxes in models of higher-order beliefs. (The

Brandenburger-Kiesler paradox).

Eric Pacuit and Olivier Roy 3

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SLIDE 8

Plan for the week

  • 1. Monday Basic Concepts.
  • 2. Tuesday Epistemics.
  • 3. Wednesday Fundamentals of Epistemic Game Theory.
  • 4. Thursday Puzzles and Paradoxes.
  • 5. Friday Extensions and New Directions.
  • Nash Equilibrium and mixted strategies.
  • Forward Induction.
  • Are the models normative or descriptive?
  • Theory of play.

Eric Pacuit and Olivier Roy 3

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SLIDE 9

Practicalities

◮ Course Website:

  • ai.stanford.edu/~epacuit/esslli2012/epgmth.html

◮ There you’ll find handouts, reading material and additional

references.

◮ In case of problem:

  • Olivier Roy: Olivier.Roy@lmu.de
  • Eric Pacuit: E.J.Pacuit@uvt.nl

Eric Pacuit and Olivier Roy 4

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SLIDE 10

Basics of Game Theory

Eric Pacuit and Olivier Roy 5

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SLIDE 11

Basics of Game Theory

Key Concepts

◮ Games in Strategic (matrix) and Extensive (tree) form. ◮ Strategies (pure and mixed). ◮ Solution Concepts: Iterated Strict Dominance, Iterated Weak

Dominance, Nash Equilibrium,

Eric Pacuit and Olivier Roy 6

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SLIDE 12

Basics of Game Theory

The Matrix: games in strategic forms.

Eric Pacuit and Olivier Roy 7

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SLIDE 13

Basics of Game Theory

The Matrix: games in strategic forms.

Alexei Strangelove Players,

Eric Pacuit and Olivier Roy 7

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SLIDE 14

Basics of Game Theory

The Matrix: games in strategic forms.

Alexei Strangelove Disarm Arm Disarm Arm Players, Actions or Strategies, Strategy profiles,

Eric Pacuit and Olivier Roy 7

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SLIDE 15

Basics of Game Theory

The Matrix: games in strategic forms.

Alexei Strangelove Disarm Arm Disarm 3, 3 Arm 1, 1 Players, Actions or Strategies, Strategy profiles, Payoffs on profiles.

Eric Pacuit and Olivier Roy 7

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SLIDE 16

Basics of Game Theory

The Matrix: games in strategic forms.

Alexei Strangelove Disarm Arm Disarm 3, 3 0, 4 Arm 4, 0 1, 1 Players, Actions or Strategies, Strategy profiles, Payoffs on profiles.

Eric Pacuit and Olivier Roy 7

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SLIDE 17

Basics of Game Theory

A three players game

Fidel - D Alexei Strglv D A D 3, 3, 3 1, 4, 5 A 4, 1, 1 2, 2, 2 Fidel - A Alexei Strglv D A D 3, 3, 2 1, 4, 4 A 4, 1, 0 2, 2, 2

Eric Pacuit and Olivier Roy 8

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SLIDE 18

Basics of Game Theory

The Tree: games in extensive forms.

S A A 3, 3 1, 4 4, 1 2, 2 D A D A D A Actions,

Eric Pacuit and Olivier Roy 9

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SLIDE 19

Basics of Game Theory

The Tree: games in extensive forms.

S A A 3, 3 1, 4 4, 1 2, 2 D A D A D A Actions, Players,

Eric Pacuit and Olivier Roy 9

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SLIDE 20

Basics of Game Theory

The Tree: games in extensive forms.

S A A 3, 3 1, 4 4, 1 2, 2 D A D A D A Actions, Players, Payoffs on leaves,

Eric Pacuit and Olivier Roy 9

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SLIDE 21

Basics of Game Theory

The Tree: games in extensive forms.

S A A 3, 3 1, 4 4, 1 2, 2 D A D A D A Actions, Players, Payoffs on leaves, Strategies

Eric Pacuit and Olivier Roy 9

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SLIDE 22

Basics of Game Theory

The Tree: games in extensive forms.

S A A 3, 3 1, 4 4, 1 2, 2 D A D A D A Actions, Players, Payoffs on leaves, Strategies

Eric Pacuit and Olivier Roy 9

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SLIDE 23

Basics of Game Theory

The Tree: games in extensive forms.

S A A 3, 3 1, 4 4, 1 2, 2 D A D A D A Actions, Players, Payoffs on leaves, Strategies

Eric Pacuit and Olivier Roy 9

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SLIDE 24

Basics of Game Theory

The Tree: games in extensive forms.

S A A 3, 3 1, 4 4, 1 2, 2 D A D A D A Actions, Players, Payoffs on leaves, Strategies

Eric Pacuit and Olivier Roy 9

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SLIDE 25

Basics of Game Theory

The Tree: games in extensive forms.

S A A 3, 3 1, 4 4, 1 2, 2 D A D A D A Actions, Players, Payoffs on leaves, Strategies

Eric Pacuit and Olivier Roy 9

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SLIDE 26

Basics of Game Theory

The Tree: games in extensive forms.

S A A 3, 3 1, 4 4, 1 2, 2 D A D A D A Actions, Players, Payoffs on leaves, Strategies

Eric Pacuit and Olivier Roy 9

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SLIDE 27

Basics of Game Theory

The Tree: games in extensive forms.

S A A 3, 3 1, 4 4, 1 2, 2 D A D A D A Actions, Players, Payoffs on leaves, Strategies

Eric Pacuit and Olivier Roy 9

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SLIDE 28

Basics of Game Theory

Extensive and strategic form games are related

A S D A D 3, 3 1, 4 A 4,1 2, 2 S A A 3,3 1,4 4,1 2,2 D A D A D A

Eric Pacuit and Olivier Roy 10

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SLIDE 29

Basics of Game Theory

Extensive and strategic form games are related

A S D A D 3, 3 1, 4 A 4,1 2, 2 S A A 3,3 1,4 4,1 2,2 D A D A D A

Eric Pacuit and Olivier Roy 10

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SLIDE 30

Basics of Game Theory

Some types of non-cooperative games of interest

◮ 2 players games. ◮ 2 players, zero-sum: if one player “wins” x then the other

“looses” −x.

◮ 2 players, win-loose games. ◮ Perfect/imperfect information.

Eric Pacuit and Olivier Roy 11

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SLIDE 31

Basics of Game Theory

Pure and mixed strategies.

Eric Pacuit and Olivier Roy 12

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SLIDE 32

Basics of Game Theory

Pure and mixed strategies.

Alexei Strangelove Head Tail Head 1, -1

  • 1, 1

Tail

  • 1, 1

1, -1

Eric Pacuit and Olivier Roy 12

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SLIDE 33

Basics of Game Theory

Pure and mixed strategies.

Alexei Strangelove Head Tail Head 1, -1

  • 1, 1

Tail

  • 1, 1

1, -1

◮ Strangelove has two pure strategies: Head and Tail.

Eric Pacuit and Olivier Roy 12

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SLIDE 34

Basics of Game Theory

Pure and mixed strategies.

Alexei Strangelove Head Tail Head 1, -1

  • 1, 1

Tail

  • 1, 1

1, -1

◮ Strangelove has two pure strategies: Head and Tail. ◮ A mixed strategy is a probability distribution over the set of

pure strategies. For instance:

  • (1/2 Head, 1/2 Tail)
  • (1/3 Head, 2/3 Tail)
  • ...

Eric Pacuit and Olivier Roy 12

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SLIDE 35

Basics of Game Theory

Pure and mixed strategies.

Alexei Strangelove Head Tail Head 1, -1

  • 1, 1

Tail

  • 1, 1

1, -1

◮ Strangelove has two pure strategies: Head and Tail. ◮ A mixed strategy is a probability distribution over the set of

pure strategies. For instance:

  • (1/2 Head, 1/2 Tail)
  • (1/3 Head, 2/3 Tail)
  • ...

◮ Additional subtleties in extensive games. (mixing at a node vs

mixing whole strategies).

Eric Pacuit and Olivier Roy 12

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SLIDE 36

Basics of Game Theory

Interpretation of mixed strategies

Eric Pacuit and Olivier Roy 13

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SLIDE 37

Basics of Game Theory

Interpretation of mixed strategies

  • 1. Real randomizations:
  • Side of goal in penalty kicks.
  • Serving side in tennis.
  • Luggage check at the airport.

Eric Pacuit and Olivier Roy 13

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SLIDE 38

Basics of Game Theory

Interpretation of mixed strategies

  • 1. Real randomizations:
  • Side of goal in penalty kicks.
  • Serving side in tennis.
  • Luggage check at the airport.
  • 2. Epistemic interpretation:
  • Mixed strategies as beliefs of the other player(s) about what

you do.

Eric Pacuit and Olivier Roy 13

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SLIDE 39

Basics of Game Theory

Solution Concepts

Eric Pacuit and Olivier Roy 14

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SLIDE 40

Basics of Game Theory

Solution Concepts

◮ Set of profiles or outcome of the game that are intuitively

viewed as “rational”.

Eric Pacuit and Olivier Roy 14

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SLIDE 41

Basics of Game Theory

Solution Concepts

◮ Set of profiles or outcome of the game that are intuitively

viewed as “rational”.

◮ Three well-known solution concepts in the matrix:

  • Nash Equilibrium.
  • Iterated elimitation of:

◮ Strictly dominated strategies. ◮ Weakly dominated strategies. Eric Pacuit and Olivier Roy 14

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SLIDE 42

Basics of Game Theory

Solution Concepts

◮ Set of profiles or outcome of the game that are intuitively

viewed as “rational”.

◮ Three well-known solution concepts in the matrix:

  • Nash Equilibrium.
  • Iterated elimitation of:

◮ Strictly dominated strategies. ◮ Weakly dominated strategies.

◮ In the tree we will focus on one:

  • Backward induction.

Eric Pacuit and Olivier Roy 14

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SLIDE 43

Basics of Game Theory

Nash Equilibrium

A B a 1, 1 0, 0 b 0, 0 1, 1

◮ The profile aA is a Nash equilibrium of that game.

Eric Pacuit and Olivier Roy 15

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SLIDE 44

Basics of Game Theory

Nash Equilibrium

A B a 1, 1 0, 0 b 0, 0 1, 1

◮ The profile aA is a Nash equilibrium of that game.

Definition

A strategy profile σ is a Nash equilibrium iff for all i and all s′

i = σi:

ui(σ) ≥ ui(si, σ−i)

Eric Pacuit and Olivier Roy 15

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SLIDE 45

Basics of Game Theory

Some Facts about Nash Equilibrium

◮ Nash equilibria in Pure Strategies do not always exist. ◮ Every game in strategic form has a Nash equilibrium in mixed

strategies.

  • The proof of this make use of Kakutani’s Fixed point thm.

◮ Some games have multiple Nash equilibria.

Eric Pacuit and Olivier Roy 16

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SLIDE 46

Basics of Game Theory

von Neumann’s minimax theorem

For every two-player zero-sum game with finite strategy sets S1 and S2, there is a number v, called the value of the game such that: v = max

p∈∆(S1)

min

q∈∆(S2) u1(s1, s2)

= min

q∈∆(S2) max p∈∆(S1) u1(s1, s2)

Furthermore, a mixed strategy profile (s1, s2) is a Nash equilibrium if and only if s1 ∈ argmaxp∈∆(S1) min

q∈∆(S2) u1(p, q)

s2 ∈ argmaxq∈∆(S2) min

p∈∆(S1) u1(p, q)

Finally, for all mixed Nash equilibria (p, q), u1(p, q) = v

Eric Pacuit and Olivier Roy 17

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SLIDE 47

Basics of Game Theory

Strictly Dominated Strategies

Eric Pacuit and Olivier Roy 18

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SLIDE 48

Basics of Game Theory

Strictly Dominated Strategies

A S D A D 3, 3 1, 4 A 4,1 2, 2

Eric Pacuit and Olivier Roy 18

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SLIDE 49

Basics of Game Theory

Strictly Dominated Strategies

A B

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SLIDE 50

Basics of Game Theory

Strictly Dominated Strategies

A B

Eric Pacuit and Olivier Roy 19

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SLIDE 51

Basics of Game Theory

Strictly Dominated Strategies

A B > > > > >

Eric Pacuit and Olivier Roy 19

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SLIDE 52

Basics of Game Theory

Strictly Dominated Strategies

A B > > > > > In general, the idea applies to both mixed and pure strategies.

Eric Pacuit and Olivier Roy 19

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SLIDE 53

Basics of Game Theory

Iterated Elimination of Strictly Dominated Strategies

Bob Ann

U L R U

1,2 0,1

U D

0,1 1,0

U

Eric Pacuit and Olivier Roy 20

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SLIDE 54

Basics of Game Theory

Iterated Elimination of Strictly Dominated Strategies

Bob Ann

U L R U

1,2 0,1

U D

0,1 1,0

U

Eric Pacuit and Olivier Roy 20

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SLIDE 55

Basics of Game Theory

Iterated Elimination of Strictly Dominated Strategies

Bob Ann

U L R U

1,2 0,1

U D

0,1 1,0

U

Eric Pacuit and Olivier Roy 20

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SLIDE 56

Basics of Game Theory

Iterated Elimination of Strictly Dominated Strategies

Bob Ann

U L R U

1,2 0,1

U D

0,1 1,0

U

Eric Pacuit and Olivier Roy 20

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SLIDE 57

Basics of Game Theory

Facts about IESDS

◮ The algorithm always terminates on finite games. Intuition:

this is a decreasing (in fact, monotonic) function on sub-games. It thus has a fixed-point by the Knaster-Tarski thm.

◮ The algorithm is order independent: One can eliminate SDS

  • ne player at the time, in difference order, or all
  • simultaneously. The fixed-point of the elimination procedure

will always be the same.

◮ All Nash equilibria survive IESDS. But not all profile that

survive IESDS are Nash equilibria.

Eric Pacuit and Olivier Roy 21

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SLIDE 58

Basics of Game Theory

Weak Dominance

A B

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SLIDE 59

Basics of Game Theory

Weak Dominance

A B

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SLIDE 60

Basics of Game Theory

Weak Dominance

A B > = > = =

Eric Pacuit and Olivier Roy 22

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SLIDE 61

Basics of Game Theory

Weak Dominance

A B > = > = =

◮ All strictly dominated strategies are weakly dominated.

Eric Pacuit and Olivier Roy 22

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SLIDE 62

Basics of Game Theory

Iterated Elimination of Weakly Dominated Strategies

Bob Ann

U L R U

1,2 0,1

U D

0,1 1,1

U

Eric Pacuit and Olivier Roy 23

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SLIDE 63

Basics of Game Theory

Iterated Elimination of Weakly Dominated Strategies

Bob Ann

U L R U

1,2 0,1

U D

0,1 1,1

U

Eric Pacuit and Olivier Roy 23

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SLIDE 64

Basics of Game Theory

Iterated Elimination of Weakly Dominated Strategies

Bob Ann

U L R U

1,2 1,1

U D

0,1 1,1

U

Eric Pacuit and Olivier Roy 23

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SLIDE 65

Basics of Game Theory

Iterated Elimination of Weakly Dominated Strategies

Bob Ann

U L R U

1,2 0,1

U D

0,1 1,1

U

Eric Pacuit and Olivier Roy 23

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SLIDE 66

Basics of Game Theory

Facts about IEWDS

◮ The algorithm always terminates on finite games. ◮ The algorithm is order dependent!: Eliminating simultaneously

all WDS at each round need not to lead to the same result as eliminating only some of them.

◮ Not all Nash equilibria survive IESDS.

Eric Pacuit and Olivier Roy 24

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SLIDE 67

The Epistemic View on Games

Hey, no, equilibrium is not the way to look at games. Now, Nash equilibrium is king in game theory. Absolutely

  • king. We say: No, Nash equilibrium is an interesting

concept, and its an important concept, but its not the most basic concept. The most basic concept should be: to maximise your utility given your information. Its in a game just like in any other situation. Maximise your utility given your information! Robert Aumann, 5 Questions on Epistemic Logic, 2010

Eric Pacuit and Olivier Roy 25

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SLIDE 68

The Epistemic View on Games

Hey, no, equilibrium is not the way to look at games. Now, Nash equilibrium is king in game theory. Absolutely

  • king. We say: No, Nash equilibrium is an interesting

concept, and its an important concept, but its not the most basic concept. The most basic concept should be: to maximise your utility given your information. Its in a game just like in any other situation. Maximise your utility given your information! Robert Aumann, 5 Questions on Epistemic Logic, 2010 Two views on games:

Eric Pacuit and Olivier Roy 25

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SLIDE 69

The Epistemic View on Games

Hey, no, equilibrium is not the way to look at games. Now, Nash equilibrium is king in game theory. Absolutely

  • king. We say: No, Nash equilibrium is an interesting

concept, and its an important concept, but its not the most basic concept. The most basic concept should be: to maximise your utility given your information. Its in a game just like in any other situation. Maximise your utility given your information! Robert Aumann, 5 Questions on Epistemic Logic, 2010 Two views on games:

◮ Based on solution Concepts.

Eric Pacuit and Olivier Roy 25

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SLIDE 70

The Epistemic View on Games

Hey, no, equilibrium is not the way to look at games. Now, Nash equilibrium is king in game theory. Absolutely

  • king. We say: No, Nash equilibrium is an interesting

concept, and its an important concept, but its not the most basic concept. The most basic concept should be: to maximise your utility given your information. Its in a game just like in any other situation. Maximise your utility given your information! Robert Aumann, 5 Questions on Epistemic Logic, 2010 Two views on games:

◮ Based on solution Concepts. ◮ Classical, decision-theoretic.

Eric Pacuit and Olivier Roy 25

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SLIDE 71

The Epistemic View on Games

Component of a Game

A game in strategic form: Ann/ Bob L R T 1, 1 1, 0 B 0, 0 0, 1 A coordination game: Ann/ Bob L R T 1, 1 0, 0 B 0, 0 1, 1 G = Ag, {(Si, πi)i∈Ag}

◮ Ag is a finite set of

agents.

◮ Si is a finite set of

strategies, one for each agent i ∈ Ag.

◮ ui : Πi∈AgSi −

→ R is a payoff function defined on the set of outcomes of the game. Solutions/recommendations: Nash Equilibrium, Elimination of strictly dominated strategies, of weakly dominated strategies...

Eric Pacuit and Olivier Roy 26

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SLIDE 72

The Epistemic View on Games

A Decision Problem: Leonard’s Omelette

Egg Good Egg Rotten Break with other eggs 4 Separate bowl 2 1

Eric Pacuit and Olivier Roy 27

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SLIDE 73

The Epistemic View on Games

A Decision Problem: Leonard’s Omelette

Egg Good Egg Rotten Break with other eggs 4 Separate bowl 2 1

◮ Agent, actions, states, payoffs, beliefs.

Eric Pacuit and Olivier Roy 27

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SLIDE 74

The Epistemic View on Games

A Decision Problem: Leonard’s Omelette

Egg Good Egg Rotten Break with other eggs 4 Separate bowl 2 1

◮ Agent, actions, states, payoffs, beliefs. ◮ Ex.: Leonard’s beliefs: pL(EG) = 1/2, pL(ER) = 1/2.

Eric Pacuit and Olivier Roy 27

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SLIDE 75

The Epistemic View on Games

A Decision Problem: Leonard’s Omelette

Egg Good Egg Rotten Break with other eggs 4 Separate bowl 2 1

◮ Agent, actions, states, payoffs, beliefs. ◮ Ex.: Leonard’s beliefs: pL(EG) = 1/2, pL(ER) = 1/2. ◮ Solution/recommendations: choice rules. Maximization of

Expected Utility, Dominance, Minmax...

Eric Pacuit and Olivier Roy 27

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SLIDE 76

The Epistemic View on Games

The Epistemic or Bayesian View on Games

◮ Traditional game theory:

Actions, outcomes, preferences, solution concepts.

◮ Decision theory:

Actions, outcomes, preferences beliefs, choice rules.

Eric Pacuit and Olivier Roy 28

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SLIDE 77

The Epistemic View on Games

The Epistemic or Bayesian View on Games

◮ Traditional game theory:

Actions, outcomes, preferences, solution concepts.

◮ Decision theory:

Actions, outcomes, preferences beliefs, choice rules.

◮ Epistemic game theory:

Actions, outcomes, preferences, beliefs, choice rules.

Eric Pacuit and Olivier Roy 28

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SLIDE 78

The Epistemic View on Games

The Epistemic or Bayesian View on Games

◮ Traditional game theory:

Actions, outcomes, preferences, solution concepts.

◮ Decision theory:

Actions, outcomes, preferences beliefs, choice rules.

◮ Epistemic game theory:

:= (interactive) decision problem and choice rule + higher-order information.

Eric Pacuit and Olivier Roy 28

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SLIDE 79

Basics of Decision Theory

Eric Pacuit and Olivier Roy 29

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Basics of Decision Theory

A Decision Problem: Leonard’s Omelette

ui P ¬P A 4 B 2 1 pi P ¬P A 1/8 3/8 B 1/8 3/8

◮ Actions, states, payoffs, beliefs.

Eric Pacuit and Olivier Roy 30

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SLIDE 81

Basics of Decision Theory

A Decision Problem: Leonard’s Omelette

ui P ¬P A 4 B 2 1 pi P ¬P A 1/8 3/8 B 1/8 3/8

◮ Actions, states, payoffs, beliefs.

Eric Pacuit and Olivier Roy 30

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SLIDE 82

Basics of Decision Theory

A Decision Problem: Leonard’s Omelette

ui P ¬P A 4 B 2 1 pi P ¬P A 1/8 3/8 B 1/8 3/8

◮ Actions, states, payoffs, beliefs.

Eric Pacuit and Olivier Roy 30

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SLIDE 83

Basics of Decision Theory

A Decision Problem: Leonard’s Omelette

ui P ¬P A 4 B 2 1 pi P ¬P A 1/8 3/8 B 1/8 3/8

◮ Actions, states, payoffs, beliefs.

Eric Pacuit and Olivier Roy 30

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SLIDE 84

Basics of Decision Theory

A Decision Problem: Leonard’s Omelette

ui P ¬P A 4 B 2 1 pi P ¬P A 1/8 3/8 B 1/8 3/8

◮ Actions, states, payoffs, beliefs.

Eric Pacuit and Olivier Roy 30

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SLIDE 85

Basics of Decision Theory

A Decision Problem: Leonard’s Omelette

ui P ¬P A 4 B 2 1 pi P ¬P A 1/8 3/8 B 1/8 3/8

◮ Actions, states, payoffs, beliefs. ◮ Solution/recommendations: choice rules.

  • Which choice rule is normatively or descriptively appropriate

depends on what kind of information are at the agent’s disposal, and what kind of attitude she has.

Eric Pacuit and Olivier Roy 30

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SLIDE 86

Basics of Decision Theory

Decision Under Risk

When the agent has probabilistic beliefs, or that her beliefs can be represented probabilistically. ui P ¬P A 4 B 2 1 pi P ¬P A 1/8 3/8 B 1/8 3/8 Expected Utility: Given an agent’s beliefs and desires, the expected utility of an action leading to a set of outcomes Out is:

  • ∈Out

[ subjective prob. of o] × [utility of o]

Eric Pacuit and Olivier Roy 31

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SLIDE 87

Basics of Decision Theory

Why don’t we just give our best guess of wet or dry? Often people want to make a decision, such as whether to put out their washing to dry, and would like us to give a simple yes or no. However, this is often a simplification

  • f the complexities of the forecast and may not be

accurate.

Eric Pacuit and Olivier Roy 32

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SLIDE 88

Basics of Decision Theory

Why don’t we just give our best guess of wet or dry? Often people want to make a decision, such as whether to put out their washing to dry, and would like us to give a simple yes or no. However, this is often a simplification

  • f the complexities of the forecast and may not be
  • accurate. By giving PoP we give a more honest opinion
  • f the risk and allow you to make a decision depending
  • n how much it matters to you.

Eric Pacuit and Olivier Roy 32

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SLIDE 89

Basics of Decision Theory

Why don’t we just give our best guess of wet or dry? Often people want to make a decision, such as whether to put out their washing to dry, and would like us to give a simple yes or no. However, this is often a simplification

  • f the complexities of the forecast and may not be
  • accurate. By giving PoP we give a more honest opinion
  • f the risk and allow you to make a decision depending
  • n how much it matters to you. For example, if you are

just hanging out your sheets that you need next week you might take the risk at 40% probability of precipitation, whereas if you are drying your best shirt that you need for an important dinner this evening then you might not hang it out at more than 10% probability.

Eric Pacuit and Olivier Roy 32

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SLIDE 90

Basics of Decision Theory

Why don’t we just give our best guess of wet or dry? Often people want to make a decision, such as whether to put out their washing to dry, and would like us to give a simple yes or no. However, this is often a simplification

  • f the complexities of the forecast and may not be
  • accurate. By giving PoP we give a more honest opinion
  • f the risk and allow you to make a decision depending
  • n how much it matters to you. For example, if you are

just hanging out your sheets that you need next week you might take the risk at 40% probability of precipitation, whereas if you are drying your best shirt that you need for an important dinner this evening then you might not hang it out at more than 10% probability. PoP allows you to make the decisions that matter to you. http: // www. metoffice. gov. uk/ news/ in-depth/ science-behind-probability-of-precipitation

Eric Pacuit and Olivier Roy 32

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SLIDE 91

Basics of Decision Theory

Maximization of Expected Utility

Let DP = S, O, u, p be a decision problem. S is a finite set of states and O a set of outcomes. An action a : S − → O is a function from states to outcomes, ui a real-valued utility function

  • n O, and pi a probability measure over S. The expected utility
  • f a ∈ A with respect to pi is defined as follows:

EUp(a) := Σs∈Sp(s)u(a(s)) An action a ∈ A maximizes expected utility with respect to pi provided for all a′ ∈ A, EUp(a) ≥ EUp(a′). In such a case, we also say a is a best response to p in game DP.

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Basics of Decision Theory

Decision under Ignorance

What to do when the agent cannot assign probabilities states? Or when we can’t represent his beliefs probabilistically? Many alternatives proposed:

◮ Dominance Reasoning ◮ Admissibility ◮ Minimax ◮ ...

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SLIDE 93

Basics of Decision Theory

Dominance Reasoning

A B > > > > >

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SLIDE 94

Basics of Decision Theory

Some facts about strict dominance

◮ Strict dominance is downward monotonic: If ai is strictly

dominated with respect to X ⊆ S and X ′ ⊆ X, then ai is strictly dominated with respect to X ′.

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SLIDE 95

Basics of Decision Theory

Some facts about strict dominance

◮ Strict dominance is downward monotonic: If ai is strictly

dominated with respect to X ⊆ S and X ′ ⊆ X, then ai is strictly dominated with respect to X ′.

  • Intuition: the condition of being strictly dominated can be

written down in a first-order formula of the form ∀xϕ(x), where ϕ(x) is quantifier-free. Such formulas are downward monotonic: If M, s | = ∀xϕ(x) and M′ ⊆ M then M′, s | = ∀xϕ(x)

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SLIDE 96

Basics of Decision Theory

Some facts about strict dominance

◮ Relation with MEU:

Suppose that G = N, {Si}i∈N, {ui}i∈N is a strategic game. A strategy si ∈ Si is strictly dominated (possibly by a mixed strategy) with respect to X ⊆ S−i iff there is no probability measure p ∈ ∆(X) such that si is a best response with respect to p.

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SLIDE 97

Basics of Decision Theory

Some facts about admissibility

◮ Admissibility is NOT downward monotonic: If ai is not

admissible with respect to X ⊆ S and X ′ ⊆ X, it can be that ai is admissible with respect to X ′.

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SLIDE 98

Basics of Decision Theory

Some facts about admissibility

◮ Admissibility is NOT downward monotonic: If ai is not

admissible with respect to X ⊆ S and X ′ ⊆ X, it can be that ai is admissible with respect to X ′.

  • Intuition: the condition of being inadmissible can be written

down in a first-order formula of the form ∀xϕ(x) ∧ ∃xψ(x), where ϕ(x) and ψ(x) are quantifier-free. The existential quantifier breaks the downward monotonicity.

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SLIDE 99

Basics of Decision Theory

Some facts about admissibility

◮ Relation with MEU:

Suppose that G = N, {Si}i∈N, {ui}i∈N is a strategic game. A strategy si ∈ Si is weakly dominated (possibly by a mixed strategy) with respect to X ⊆ S−i iff there is no full support probability measure p ∈ ∆>0(X) such that si is a best response with respect to p.

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SLIDE 100

Road Map again

  • 1. Today Basic Concepts.
  • Basics of Game Theory.
  • The Epistemic View on Games.
  • Basics of Decision Theory

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SLIDE 101

Road Map again

  • 1. Today Basic Concepts.
  • Basics of Game Theory.
  • The Epistemic View on Games.
  • Basics of Decision Theory
  • 2. Tomorrow Epistemics.
  • Logical/qualitative models of beliefs, knowledge and

higher-order attitudes.

  • Probabilistic/quantitative models of beliefs, knowledge and

higher-order attitudes.

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Formal Definitions

Strategic Games

Definition

A game in strategic form G is a tuple A, Si, ui such that :

◮ A is a finite set of agents. ◮ Si is a finite set of actions or strategies for i. A strategy

profile σ ∈ Πi∈ASi is a vector of strategies, one for each agent in I. The strategy si which i plays in the profile σ is noted σi.

◮ ui : Πi∈ASi −

→ R is an utility function that assigns to every strategy profile σ ∈ Πi∈ASi the utility valuation of that profile for agent i.

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SLIDE 103

Formal Definitions

Extensive form games

Definition

A game in extensive form T is a tuple I, T, τ, {ui}i∈I such that:

◮ T is finite set of finite sequences of actions, called histories,

such that:

  • The empty sequence ∅, the root of the tree, is in T.
  • T is prefix-closed: if (a1, . . . , an, an+1) ∈ T then

(a1, . . . , an) ∈ T.

◮ A history h is terminal in T whenever it is the sub-sequence of

no other history h′ ∈ T. Z denotes the set of terminal histories in T.

◮ τ : (T − Z) −

→ I is a turn function which assigns to every non-terminal history h the player whose turn it is to play at h.

◮ ui : Z −

→ R is a payoff function for player i which assigns i’s payoff at each terminal history.

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Formal Definitions

Strategies

Definition

◮ A strategy si for agent i is a function that gives, for every

history h such that i = τ(h), an action a ∈ A(h). Si is the set

  • f strategies for agent i.

◮ A strategy profile σ ∈ Πi∈ISi is a combination of strategies,

  • ne for each agent, and σ(h) is a shorthand for the action a

such that a = σi(h) for the agent i whose turn it is at h.

◮ A history h′ is reachable or not excluded by the profile σ from

h if h′ = (h, σ(h), σ(h, σ(h)), ...) for some finite number of application of σ.

◮ We denote uh i (σ) the value of utili at the unique terminal

history reachable from h by the profile σ.

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Formal Definitions

Nash Equilibrium - General Definition

Definition

A profile of mixed strategy σ is a Nash equilibrium iff for all i and all mixed strategy σ′

i = σi:

EUi(σi, σ−i) ≥ EUi(σ′

i, σ−i)

Where EUi, the expected utility of the strategy σi against σ−i is calculated as follows (σ = (σi, σ−i)): EUi(σ) = Σs∈ΠjSj

  • (Πj∈Agσj(sj))ui(s)
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