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Models of Strategic Reasoning Lecture 3 Eric Pacuit University of Maryland, College Park ai.stanford.edu/~epacuit August 9, 2012 Eric Pacuit: Models of Strategic Reasoning 1/43 Lecture 1: Introduction, Motivation and Background Lecture 2:

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Models of Strategic Reasoning Lecture 3

Eric Pacuit University of Maryland, College Park ai.stanford.edu/~epacuit August 9, 2012

Eric Pacuit: Models of Strategic Reasoning 1/43

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Lecture 1: Introduction, Motivation and Background Lecture 2: The Dynamics of Rational Deliberation Lecture 3: Reasoning to a Solution: Common Modes of Reasoning in Games Lecture 4: Reasoning to a Model: Iterated Belief Change as Deliberation Lecture 5: Reasoning in Specific Games: Experimental Results

Eric Pacuit: Models of Strategic Reasoning 2/43

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General comments

◮ Extensive games, imprecise probabilities, other notions of stability,

weaken common knowledge assumptions,...

◮ Generalizing the basic model ◮ Why assume deliberators are in a “information feedback

situation”?

◮ Deliberation in decision theory.

Eric Pacuit: Models of Strategic Reasoning 3/43

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  • J. McKenzie Alexander. Local interactions and the dynamics of rational deliberation.

Philosophical Studies 147 (1), 2010.

Eric Pacuit: Models of Strategic Reasoning 4/43

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Consider a social network N, E (connected graph)

Eric Pacuit: Models of Strategic Reasoning 5/43

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Consider a social network N, E (connected graph) Convention: If there is a directed edge from A to B, then A always plays row and B always play column, and the interactions of Row and Column are symmetric in the available strategies.

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Consider a social network N, E (connected graph) Convention: If there is a directed edge from A to B, then A always plays row and B always play column, and the interactions of Row and Column are symmetric in the available strategies. Let νi = {i1, . . . ij} be i’s neighbors

Eric Pacuit: Models of Strategic Reasoning 5/43

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Consider a social network N, E (connected graph) Convention: If there is a directed edge from A to B, then A always plays row and B always play column, and the interactions of Row and Column are symmetric in the available strategies. Let νi = {i1, . . . ij} be i’s neighbors p′

a,b(t + 1) is represents the incremental refinement of player a’s state

  • f indecision given his knowledge about player b’s state of indecision

(at time t + 1).

Eric Pacuit: Models of Strategic Reasoning 5/43

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Consider a social network N, E (connected graph) Convention: If there is a directed edge from A to B, then A always plays row and B always play column, and the interactions of Row and Column are symmetric in the available strategies. Let νi = {i1, . . . ij} be i’s neighbors p′

a,b(t + 1) is represents the incremental refinement of player a’s state

  • f indecision given his knowledge about player b’s state of indecision

(at time t + 1). Pool this information to form your new probabilities: pi(t + 1) =

k

  • j=1

wi,ijp′

i,ij(t + 1)

Eric Pacuit: Models of Strategic Reasoning 5/43

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Billy Boxing Ballet Maggie Boxing (2,1) (0,0) Ballet (0,0) (1,2)

  • Fig. 7 The game of Battle of the Sexes.

80.7, 0.3< 80.7, 0.3< 80.7, 0.3< 80.4, 0.6< 80.4, 0.6< 80.4, 0.6<

(a) Initial conditions

81., 0< 81., 0< 81., 0< 80.4134, 0.5866< 80, 1.< 80, 1.<

(b) t = 1,000,000

  • Fig. 8 Battle of the Sexes played by

Nash deliberators (k = 25) on two cy- cles connected by a bridge edge (val- ues rounded to the nearest 10−4).

Eric Pacuit: Models of Strategic Reasoning 6/43

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The value of information

Why is it better to make a “more informed” decision? Suppose that you can either choose know, or perform a costless experiment and make the decision later. What should you do?

  • I. J. Good. On the principle of total evidence. British Journal for the Philosophy of

Science, 17, pgs. 319 - 321, 1967.

“Never decide today what you might postpone until tomorrow in order to learn something new”

Eric Pacuit: Models of Strategic Reasoning 7/43

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Choose between n acts A1, . . . , An or perform a cost-free experiment E with possible results {ek}, then decide. EU(A) =

  • i

p(Ki)U(A & Ki)

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Choose between n acts A1, . . . , An or perform a cost-free experiment E with possible results {ek}, then decide. EU(A) =

  • i

p(Ki)U(A & Ki) Then, U(Choose now) = max

j

  • i

p(Ki)U(Aj & Ki) = max

j

  • k
  • i

p(Ki)p(ek | Ki)U(Aj & Ki)

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The value of an informed decision conditional on e: max

j

  • i

p(Ki | e)U(Aj & Ki)

Eric Pacuit: Models of Strategic Reasoning 9/43

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The value of an informed decision conditional on e: max

j

  • i

p(Ki | e)U(Aj & Ki) U(Learn, Choose ) =

  • k p(ek) maxj
  • i p(Ki | ek)U(Aj & Ki)

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The value of an informed decision conditional on e: max

j

  • i

p(Ki | e)U(Aj & Ki) U(Learn, Choose ) =

  • k p(ek) maxj
  • i p(Ki | ek)U(Aj & Ki)

=

  • k p(ek) maxj
  • i( p(ek | Ki)p(Ki)

p(ek)

)U(Aj & Ki)

Eric Pacuit: Models of Strategic Reasoning 9/43

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The value of an informed decision conditional on e: max

j

  • i

p(Ki | e)U(Aj & Ki) U(Learn, Choose ) =

  • k p(ek) maxj
  • i p(Ki | ek)U(Aj & Ki)

=

  • k p(ek) maxj
  • i( p(ek | Ki)p(Ki)

p(ek)

)U(Aj & Ki) =

  • k maxj
  • i p(ek | Ki)p(Ki)U(Aj & Ki)

Eric Pacuit: Models of Strategic Reasoning 9/43

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The value of an informed decision conditional on e: max

j

  • i

p(Ki | e)U(Aj & Ki) U(Learn, Choose ) =

  • k p(ek) maxj
  • i p(Ki | ek)U(Aj & Ki)

=

  • k p(ek) maxj
  • i( p(ek | Ki)p(Ki)

p(ek)

)U(Aj & Ki) =

  • k maxj
  • i p(ek | Ki)p(Ki)U(Aj & Ki)

Compare maxj

  • k
  • i p(Ki)p(ek | Ki)U(Aj & Ki) and
  • k maxj
  • i p(ek | Ki)p(Ki)U(Aj & Ki)

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The value of an informed decision conditional on e: max

j

  • i

p(Ki | e)U(Aj & Ki) U(Learn, Choose ) =

  • k p(ek) maxj
  • i p(Ki | ek)U(Aj & Ki)

=

  • k p(ek) maxj
  • i( p(ek | Ki)p(Ki)

p(ek)

)U(Aj & Ki) =

  • k maxj
  • i p(ek | Ki)p(Ki)U(Aj & Ki)

Compare maxj

  • k
  • i p(Ki)p(ek | Ki)U(Aj & Ki) and
  • k maxj
  • i p(ek | Ki)p(Ki)U(Aj & Ki)
  • k maxj g(k, j) is greater than or equal to maxj
  • k g(k, j), so the

second is greater than or equal to the first.

Eric Pacuit: Models of Strategic Reasoning 9/43

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The Cost of Thinking

“A person required to risk money on a remote digit of π would have to compute that digit in order to comply fully with the theory, though this would really be wasteful if the cost of computation were more than the prize involved.

Eric Pacuit: Models of Strategic Reasoning 10/43

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The Cost of Thinking

“A person required to risk money on a remote digit of π would have to compute that digit in order to comply fully with the theory, though this would really be wasteful if the cost of computation were more than the prize involved. For the postulates of the theory imply that you should behave in accordance with the logical implications of all that you know.

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The Cost of Thinking

“A person required to risk money on a remote digit of π would have to compute that digit in order to comply fully with the theory, though this would really be wasteful if the cost of computation were more than the prize involved. For the postulates of the theory imply that you should behave in accordance with the logical implications of all that you know. Is it possible to improve the theory in this respect, making allowance within it for the cost of thinking, or would that entail paradox, as I am inclined to believe but unable to demonstrate?

Eric Pacuit: Models of Strategic Reasoning 10/43

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The Cost of Thinking

“A person required to risk money on a remote digit of π would have to compute that digit in order to comply fully with the theory, though this would really be wasteful if the cost of computation were more than the prize involved. For the postulates of the theory imply that you should behave in accordance with the logical implications of all that you know. Is it possible to improve the theory in this respect, making allowance within it for the cost of thinking, or would that entail paradox, as I am inclined to believe but unable to demonstrate? If the remedy is not in changing the theory but rather in the way in which we attempt to use it, clarification is still to be desired.” (pg.308)

  • L. J. Savage. Difficulties in the theory of personal probability. Philosophy of Science,

34(4), pgs. 305 - 310, 1967.

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What are the players deliberating/reasoning about?

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What are the players deliberating/reasoning about? Their preferences?

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What are the players deliberating/reasoning about? Their preferences? The model?

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What are the players deliberating/reasoning about? Their preferences? The model? The other players?

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What are the players deliberating/reasoning about? Their preferences? The model? The other players? What to do?

Conclusion Eric Pacuit: Models of Strategic Reasoning 11/43

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  • I. Douven. Decision theory and the rationality of further deliberation. Economics and

Philosophy, 18, pgs. 303 - 328, 2002.

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Deliberation in Decision Theory

“deliberation crowds out prediction”

  • F. Schick.

Self-Knowledge, Uncertainty and Choice. The British Journal for the Philosophy of Science, 30:3, pgs. 235 - 252, 1979.

  • I. Levi. Feasibility. in Knowledge, belief and strategic interaction, C. Bicchieri and M.
  • L. D. Chiara (eds.), pgs. 1 - 20, 1992.
  • W. Rabinowicz. Does Practical deliberation Crowd Out Self-Prediction?. Erkenntnis,

57, 91-122, 2002.

Conclusion Eric Pacuit: Models of Strategic Reasoning 13/43

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Meno’s Paradox

  • 1. If you know what youre looking for, inquiry is unnecessary.
  • 2. If you do not know what youre looking for, inquiry is impossible.

Therefore, inquiry is either unnecessary or impossible.

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Meno’s Paradox

  • 1. If you know what youre looking for, inquiry is unnecessary.
  • 2. If you do not know what youre looking for, inquiry is impossible.

Therefore, inquiry is either unnecessary or impossible. Levi’s Argument

  • 1. If you have access to self-knowledge and logical omniscience to

apply the principles of rational choice to determine which options are admissible, then the principles of rational choice are vacuous for the purposes of deciding what to do.

  • 2. If you do not have access to self-knowledge and logical
  • mniscience in this sense, then the principles of rational choice are

inapplicable for the purposes of deciding what do. Therefore, the principles of rational choice are either unnecessary or impossible.

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If X takes the sentence “Sam behaves in manner R” to be an act description vis-´ a-vis a decision problem faced by Sam, then X is in a state of full belief that has the following contents:

Eric Pacuit: Models of Strategic Reasoning 15/43

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If X takes the sentence “Sam behaves in manner R” to be an act description vis-´ a-vis a decision problem faced by Sam, then X is in a state of full belief that has the following contents:

  • 1. Ability Condition: Sam has the ability to choose that Sam will R
  • n a trial of kind S, where the trial of kind S is a process of

deliberation eventuating in choice.

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If X takes the sentence “Sam behaves in manner R” to be an act description vis-´ a-vis a decision problem faced by Sam, then X is in a state of full belief that has the following contents:

  • 1. Ability Condition: Sam has the ability to choose that Sam will R
  • n a trial of kind S, where the trial of kind S is a process of

deliberation eventuating in choice.

  • 2. Deliberation Condition: Sam is is subject to a trial of kind S at

time t; that is Sam is deliberating at time t

Eric Pacuit: Models of Strategic Reasoning 15/43

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If X takes the sentence “Sam behaves in manner R” to be an act description vis-´ a-vis a decision problem faced by Sam, then X is in a state of full belief that has the following contents:

  • 1. Ability Condition: Sam has the ability to choose that Sam will R
  • n a trial of kind S, where the trial of kind S is a process of

deliberation eventuating in choice.

  • 2. Deliberation Condition: Sam is is subject to a trial of kind S at

time t; that is Sam is deliberating at time t

  • 3. Efficaciousness Condition: Adding the claim that Sam chooses that

he will R to X’s current body of full beliefs entails that Sam will R

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If X takes the sentence “Sam behaves in manner R” to be an act description vis-´ a-vis a decision problem faced by Sam, then X is in a state of full belief that has the following contents:

  • 1. Ability Condition: Sam has the ability to choose that Sam will R
  • n a trial of kind S, where the trial of kind S is a process of

deliberation eventuating in choice.

  • 2. Deliberation Condition: Sam is is subject to a trial of kind S at

time t; that is Sam is deliberating at time t

  • 3. Efficaciousness Condition: Adding the claim that Sam chooses that

he will R to X’s current body of full beliefs entails that Sam will R

  • 4. Serious Possibility: For each feasible option for Sam, nothing in

X’s state of full belief is incompatible with Sam’s choosing that

  • ption

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If X takes the sentence “Sam behaves in manner R” to be an act description vis-´ a-vis a decision problem faced by Sam, then X is in a state of full belief that has the following contents:

  • 1. Ability Condition: Sam has the ability to choose that Sam will R
  • n a trial of kind S, where the trial of kind S is a process of

deliberation eventuating in choice.

  • 2. Deliberation Condition: Sam is is subject to a trial of kind S at

time t; that is Sam is deliberating at time t

  • 3. Efficaciousness Condition: Adding the claim that Sam chooses that

he will R to X’s current body of full beliefs entails that Sam will R

  • 4. Serious Possibility: For each feasible option for Sam, nothing in

X’s state of full belief is incompatible with Sam’s choosing that

  • ption

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Foreknowledge of Rationality

Let A be a set of feasible options and C(A) ⊆ A the admissible

  • ptions.

Eric Pacuit: Models of Strategic Reasoning 16/43

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Foreknowledge of Rationality

Let A be a set of feasible options and C(A) ⊆ A the admissible

  • ptions.
  • 1. Logical Omniscience: The agent must have enough logical
  • mniscience and computational capacity to use his principles of

choice to determine the set C(A) of admissible outcomes

Eric Pacuit: Models of Strategic Reasoning 16/43

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Foreknowledge of Rationality

Let A be a set of feasible options and C(A) ⊆ A the admissible

  • ptions.
  • 1. Logical Omniscience: The agent must have enough logical
  • mniscience and computational capacity to use his principles of

choice to determine the set C(A) of admissible outcomes

  • 2. Self-Knowledge: The agent must know “enough” about his own

values (goal, preferences, utilities) and beliefs (both full beliefs and probability judgements)

Eric Pacuit: Models of Strategic Reasoning 16/43

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Foreknowledge of Rationality

Let A be a set of feasible options and C(A) ⊆ A the admissible

  • ptions.
  • 1. Logical Omniscience: The agent must have enough logical
  • mniscience and computational capacity to use his principles of

choice to determine the set C(A) of admissible outcomes

  • 2. Self-Knowledge: The agent must know “enough” about his own

values (goal, preferences, utilities) and beliefs (both full beliefs and probability judgements)

  • 3. Smugness: The agent is certain that in the deliberation taking

place at time t, X will choose an admissible option

Eric Pacuit: Models of Strategic Reasoning 16/43

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Foreknowledge of Rationality

Let A be a set of feasible options and C(A) ⊆ A the admissible

  • ptions.
  • 1. Logical Omniscience: The agent must have enough logical
  • mniscience and computational capacity to use his principles of

choice to determine the set C(A) of admissible outcomes

  • 2. Self-Knowledge: The agent must know “enough” about his own

values (goal, preferences, utilities) and beliefs (both full beliefs and probability judgements)

  • 3. Smugness: The agent is certain that in the deliberation taking

place at time t, X will choose an admissible option If all the previous conditions are satisfied, then no inadmissible option is feasible from the deliberating agent’s point of view when deciding what to do: C(A) = A.

Eric Pacuit: Models of Strategic Reasoning 16/43

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“Though this result is not contradictory, it implies the vacuousness of principles of rational choice for the purpose of deciding what to do...If they are useless for this purpose, then by the argument of the previous section, they are useless for passing judgement on the rationality of choice as well.” (L, pg. 10)

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“Though this result is not contradictory, it implies the vacuousness of principles of rational choice for the purpose of deciding what to do...If they are useless for this purpose, then by the argument of the previous section, they are useless for passing judgement on the rationality of choice as well.” (L, pg. 10) (Earlier argument: “If X is merely giving advice, it is pointless to advise Sam to do something X is sure Sam will not do...The point I mean to belabor is that passing judgement on the rationality of Sam’s choices has little merit unless it gives advice to how one should choose in predicaments similar to Sam’s in relevant aspects”)

Eric Pacuit: Models of Strategic Reasoning 17/43

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Weak Thesis: In a situation of choice, the DM does not assign extreme probabilities to options among which his choice is being made. Strong Thesis: In a situation of choice, the DM does not assign any probabilities to options among which his choice is being made.

Eric Pacuit: Models of Strategic Reasoning 18/43

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Weak Thesis: In a situation of choice, the DM does not assign extreme probabilities to options among which his choice is being made. Strong Thesis: In a situation of choice, the DM does not assign any probabilities to options among which his choice is being made. “...the probability assignment to A may still be available to the subject in his purely doxastic capacity but not in his capacity of an agent or practical deliberator. The agent qua agent must abstain from assessing the probability of his options.” (Rabinowicz, pg. 3)

Eric Pacuit: Models of Strategic Reasoning 18/43

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“(...) probabilities of acts play no role in decision making. (...) The decision maker chooses the act he likes most be its probability as it

  • may. But if this is so, there is no sense in imputing probabilities for acts

to the decision maker.” (Spohn (1977), pg. 115)

Eric Pacuit: Models of Strategic Reasoning 19/43

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“(...) probabilities of acts play no role in decision making. (...) The decision maker chooses the act he likes most be its probability as it

  • may. But if this is so, there is no sense in imputing probabilities for acts

to the decision maker.” (Spohn (1977), pg. 115)

◮ Levi: “I never deliberate about an option I am certain that I am

not going to choose”. If I have a low probability for doing some action A, then I may spend less time and effort in deliberation...

Eric Pacuit: Models of Strategic Reasoning 19/43

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“(...) probabilities of acts play no role in decision making. (...) The decision maker chooses the act he likes most be its probability as it

  • may. But if this is so, there is no sense in imputing probabilities for acts

to the decision maker.” (Spohn (1977), pg. 115)

◮ Levi: “I never deliberate about an option I am certain that I am

not going to choose”. If I have a low probability for doing some action A, then I may spend less time and effort in deliberation...

◮ Deliberation as a feedback process: change in inclinations causes a

change in probabilities assigned to various options, which in turn may change my inclinations towards particular options....

Eric Pacuit: Models of Strategic Reasoning 19/43

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Discussion

Eric Pacuit: Models of Strategic Reasoning 20/43

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Discussion

◮ Logical Omniscience/Self-Knowledge: “decision makers do not

know their preferences at the time of deliberation” (Schick):

Eric Pacuit: Models of Strategic Reasoning 20/43

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Discussion

◮ Logical Omniscience/Self-Knowledge: “decision makers do not

know their preferences at the time of deliberation” (Schick): “If decision makers never have the capacities to apply the principles of rational choice and cannot have their capacities improved by new technology and therapy, the principles are inapplicable. Inapplicability is no better a fate than vacuity.”

Eric Pacuit: Models of Strategic Reasoning 20/43

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Discussion

◮ Logical Omniscience/Self-Knowledge: “decision makers do not

know their preferences at the time of deliberation” (Schick): “If decision makers never have the capacities to apply the principles of rational choice and cannot have their capacities improved by new technology and therapy, the principles are inapplicable. Inapplicability is no better a fate than vacuity.”

◮ Drop smugness: “the agent need not assume he will choose

rationally...the agent should be in a state of suspense as to which

  • f the feasible options will be chosen” (Levi)

Eric Pacuit: Models of Strategic Reasoning 20/43

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Discussion

◮ Logical Omniscience/Self-Knowledge: “decision makers do not

know their preferences at the time of deliberation” (Schick): “If decision makers never have the capacities to apply the principles of rational choice and cannot have their capacities improved by new technology and therapy, the principles are inapplicable. Inapplicability is no better a fate than vacuity.”

◮ Drop smugness: “the agent need not assume he will choose

rationally...the agent should be in a state of suspense as to which

  • f the feasible options will be chosen” (Levi)

◮ Implications for game theory (common knowledge of rationality

implies, in particular, that agents satisfy Smugness).

Skip Eric Pacuit: Models of Strategic Reasoning 20/43

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Game Plan

  • Introduction, Motivation and Background
  • The Dynamics of Rational Deliberation

Lecture 3: Reasoning to a Solution: Common Modes of Reasoning in Games Lecture 4: Reasoning to a Model: Iterated Belief Change as Deliberation Lecture 5: Reasoning in Specific Games: Experimental Results

Eric Pacuit: Models of Strategic Reasoning 21/43

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Iterative Solution Concepts: Two Views

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Iterative Solution Concepts: Two Views

Eg., Iterated removal of weakly/strictly dominated strategies

Eric Pacuit: Models of Strategic Reasoning 22/43

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Iterative Solution Concepts: Two Views

Eg., Iterated removal of weakly/strictly dominated strategies

  • 1. iterative procedures narrow down or assist in the search for a

equilibria

  • 2. iterative procedures represent a rational deliberation process

Eric Pacuit: Models of Strategic Reasoning 22/43

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Iterative Solution Concepts: Two Views

Eg., Iterated removal of weakly/strictly dominated strategies

  • 1. iterative procedures narrow down or assist in the search for a

equilibria successive stages of strategy deletion may correspond to different levels of belief (in a lexicographic probability system)

  • 2. iterative procedures represent a rational deliberation process

successive stages of a strategy deletion can be interpreted as tracking successive steps of reasoning that players can perform

Eric Pacuit: Models of Strategic Reasoning 22/43

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Aumann “versus” Lewis on Common Knowledge

Aumann defines common knowledge to be the infinite conjunction of iterations of “everyone knows that” operators. Lewis offers an analysis of how common knowledge is achieved

  • R. Cubitt and R. Sugden. Common Knowledge, Salience and Convention: A Recon-

struction of David Lewis’ Game Theory. Economics and Philosophy, 19, pgs. 175 - 210, 2003.

Eric Pacuit: Models of Strategic Reasoning 23/43

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The Fixed-Point Definition

Separating the fixed-point/iteration definition of common knowledge/belief:

  • J. Barwise. Three views of Common Knowledge. TARK (1987).
  • J. van Benthem and D. Saraenac. The Geometry of Knowledge. Aspects of Universal

Logic (2004).

  • A. Heifetz. Iterative and Fixed Point Common Belief. Journal of Philosophical Logic

(1999).

Eric Pacuit: Models of Strategic Reasoning 24/43

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Reason to Believe

Biϕ: “i believes ϕ”

Eric Pacuit: Models of Strategic Reasoning 25/43

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Reason to Believe

Biϕ: “i believes ϕ” vs. Ri(ϕ): “i has a reason to believe ϕ”

Eric Pacuit: Models of Strategic Reasoning 25/43

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Reason to Believe

Biϕ: “i believes ϕ” vs. Ri(ϕ): “i has a reason to believe ϕ”

◮ “Although it is an essential part of Lewis’ theory that human

beings are to some degree rational, he does not want to make the strong rationality assumptions of conventional decision theory or game theory.” (CS, pg. 184).

Eric Pacuit: Models of Strategic Reasoning 25/43

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Reason to Believe

Biϕ: “i believes ϕ” vs. Ri(ϕ): “i has a reason to believe ϕ”

◮ “Although it is an essential part of Lewis’ theory that human

beings are to some degree rational, he does not want to make the strong rationality assumptions of conventional decision theory or game theory.” (CS, pg. 184).

◮ Anyone who accept the rules of arithmetic has a reason to believe

618 × 377 = 232, 986, but most of us do not hold have firm beliefs about this.

Eric Pacuit: Models of Strategic Reasoning 25/43

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Reason to Believe

Biϕ: “i believes ϕ” vs. Ri(ϕ): “i has a reason to believe ϕ”

◮ “Although it is an essential part of Lewis’ theory that human

beings are to some degree rational, he does not want to make the strong rationality assumptions of conventional decision theory or game theory.” (CS, pg. 184).

◮ Anyone who accept the rules of arithmetic has a reason to believe

618 × 377 = 232, 986, but most of us do not hold have firm beliefs about this.

◮ Definition: Ri(ϕ) means ϕ is true within some logic of reasoning

that is endorsed by (that is, accepted as a normative standard by) person i...ϕ must be either regarded as self-evident or derivable by rules of inference (deductive or inductive)

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SLIDE 68

A indicates to i that ϕ

A is a “state of affairs” A indi ϕ: i’s reason to believe that A holds provides i’s reason for believing that ϕ is true. (A1) For all i, for all A, for all ϕ: [Ri(A holds) ∧ (A indi ϕ)] ⇒ Ri(ϕ)

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SLIDE 69

Some Properties

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SLIDE 70

Some Properties

◮ [(A holds) entails (A′ holds)] ⇒ A indi(A′ holds)

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SLIDE 71

Some Properties

◮ [(A holds) entails (A′ holds)] ⇒ A indi(A′ holds) ◮ [(A indi ϕ) ∧ (A indiψ)] ⇒ A indi(ϕ ∧ ψ)

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SLIDE 72

Some Properties

◮ [(A holds) entails (A′ holds)] ⇒ A indi(A′ holds) ◮ [(A indi ϕ) ∧ (A indiψ)] ⇒ A indi(ϕ ∧ ψ) ◮ [(A indi[A′ holds]) ∧ (A′ indix)] ⇒ A indiϕ

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SLIDE 73

Some Properties

◮ [(A holds) entails (A′ holds)] ⇒ A indi(A′ holds) ◮ [(A indi ϕ) ∧ (A indiψ)] ⇒ A indi(ϕ ∧ ψ) ◮ [(A indi[A′ holds]) ∧ (A′ indix)] ⇒ A indiϕ ◮ [(A indiϕ) ∧ (ϕ entails ψ)] ⇒ A indiψ

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SLIDE 74

Some Properties

◮ [(A holds) entails (A′ holds)] ⇒ A indi(A′ holds) ◮ [(A indi ϕ) ∧ (A indiψ)] ⇒ A indi(ϕ ∧ ψ) ◮ [(A indi[A′ holds]) ∧ (A′ indix)] ⇒ A indiϕ ◮ [(A indiϕ) ∧ (ϕ entails ψ)] ⇒ A indiψ ◮ [(A indi Rj[A′ holds]) ∧ Ri(A′ indjϕ)] ⇒ A indiRj(ϕ)

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SLIDE 75

Reflexive Common Indicator

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SLIDE 76

Reflexive Common Indicator

◮ A holds ⇒ Ri(A holds)

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SLIDE 77

Reflexive Common Indicator

◮ A holds ⇒ Ri(A holds) ◮ A indi Rj(A holds)

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SLIDE 78

Reflexive Common Indicator

◮ A holds ⇒ Ri(A holds) ◮ A indi Rj(A holds) ◮ A indi ϕ

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SLIDE 79

Reflexive Common Indicator

◮ A holds ⇒ Ri(A holds) ◮ A indi Rj(A holds) ◮ A indi ϕ ◮ (A indi ψ) ⇒ Ri[A indj ψ]

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SLIDE 80

Let RG(ϕ): Riϕ, Rjϕ, . . ., Ri(Rjϕ), Rj(Ri(ϕ)), . . . iterated reason to believe ϕ.

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SLIDE 81

Let RG(ϕ): Riϕ, Rjϕ, . . ., Ri(Rjϕ), Rj(Ri(ϕ)), . . . iterated reason to believe ϕ.

  • Theorem. (Lewis) For all states of affairs A, for all propositions ϕ, and

for all groups G: if A holds, and if A is a reflexive common indicator in G that ϕ, then RG(ϕ) is true.

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SLIDE 82

Lewis and Aumann

Lewis common knowledge that ϕ implies the iterated definition of common knowledge (“Aumann common knowledge”)

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SLIDE 83

Lewis and Aumann

Lewis common knowledge that ϕ implies the iterated definition of common knowledge (“Aumann common knowledge”), but the converse is not generally true....

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SLIDE 84

Lewis and Aumann

Lewis common knowledge that ϕ implies the iterated definition of common knowledge (“Aumann common knowledge”), but the converse is not generally true....

  • Example. Suppose there is an agent i ∈ G that is authoritative for

each member of G.

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SLIDE 85

Lewis and Aumann

Lewis common knowledge that ϕ implies the iterated definition of common knowledge (“Aumann common knowledge”), but the converse is not generally true....

  • Example. Suppose there is an agent i ∈ G that is authoritative for

each member of G. So, for j ∈ G, “i states to j that ϕ is true” indicates to j that ϕ.

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SLIDE 86

Lewis and Aumann

Lewis common knowledge that ϕ implies the iterated definition of common knowledge (“Aumann common knowledge”), but the converse is not generally true....

  • Example. Suppose there is an agent i ∈ G that is authoritative for

each member of G. So, for j ∈ G, “i states to j that ϕ is true” indicates to j that ϕ. Suppose that separately and privately to each member of G, i states that ϕ and RG(ϕ) are true.

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SLIDE 87

Lewis and Aumann

Lewis common knowledge that ϕ implies the iterated definition of common knowledge (“Aumann common knowledge”), but the converse is not generally true....

  • Example. Suppose there is an agent i ∈ G that is authoritative for

each member of G. So, for j ∈ G, “i states to j that ϕ is true” indicates to j that ϕ. Suppose that separately and privately to each member of G, i states that ϕ and RG(ϕ) are true.Then, we have Riϕ and Ri(RG(ϕ)) for each i ∈ G.

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SLIDE 88

Lewis and Aumann

Lewis common knowledge that ϕ implies the iterated definition of common knowledge (“Aumann common knowledge”), but the converse is not generally true....

  • Example. Suppose there is an agent i ∈ G that is authoritative for

each member of G. So, for j ∈ G, “i states to j that ϕ is true” indicates to j that ϕ. Suppose that separately and privately to each member of G, i states that ϕ and RG(ϕ) are true.Then, we have Riϕ and Ri(RG(ϕ)) for each i ∈ G. But there is no common indicator that ϕ is true.

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SLIDE 89

Lewis and Aumann

Lewis common knowledge that ϕ implies the iterated definition of common knowledge (“Aumann common knowledge”), but the converse is not generally true....

  • Example. Suppose there is an agent i ∈ G that is authoritative for

each member of G. So, for j ∈ G, “i states to j that ϕ is true” indicates to j that ϕ. Suppose that separately and privately to each member of G, i states that ϕ and RG(ϕ) are true.Then, we have Riϕ and Ri(RG(ϕ)) for each i ∈ G. But there is no common indicator that ϕ is true. The agents j ∈ G may have no reason to believe that everyone heard the statement from i or that all agents in G treat i as authoritative.

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SLIDE 90

Example

A and B are players in the same football team. A has the ball, but an

  • pposing player is converging on him.

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SLIDE 91

Example

A and B are players in the same football team. A has the ball, but an

  • pposing player is converging on him. He can pass the ball to B, who

has a chance to shoot.

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SLIDE 92

Example

A and B are players in the same football team. A has the ball, but an

  • pposing player is converging on him. He can pass the ball to B, who

has a chance to shoot. There are two directions in which A can move the ball, left and right, and correspondingly, two directions in which B can run to intercept the pass.

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SLIDE 93

Example

A and B are players in the same football team. A has the ball, but an

  • pposing player is converging on him. He can pass the ball to B, who

has a chance to shoot. There are two directions in which A can move the ball, left and right, and correspondingly, two directions in which B can run to intercept the pass. If both choose left there is a 10% chance that a goal will be scored.

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SLIDE 94

Example

A and B are players in the same football team. A has the ball, but an

  • pposing player is converging on him. He can pass the ball to B, who

has a chance to shoot. There are two directions in which A can move the ball, left and right, and correspondingly, two directions in which B can run to intercept the pass. If both choose left there is a 10% chance that a goal will be scored. If they both choose right, there is a 11% change.

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SLIDE 95

Example

A and B are players in the same football team. A has the ball, but an

  • pposing player is converging on him. He can pass the ball to B, who

has a chance to shoot. There are two directions in which A can move the ball, left and right, and correspondingly, two directions in which B can run to intercept the pass. If both choose left there is a 10% chance that a goal will be scored. If they both choose right, there is a 11%

  • change. Otherwise, the chance is zero.

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SLIDE 96

Example

A and B are players in the same football team. A has the ball, but an

  • pposing player is converging on him. He can pass the ball to B, who

has a chance to shoot. There are two directions in which A can move the ball, left and right, and correspondingly, two directions in which B can run to intercept the pass. If both choose left there is a 10% chance that a goal will be scored. If they both choose right, there is a 11%

  • change. Otherwise, the chance is zero. There is no time for

communication; the two players must act simultaneously.

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SLIDE 97

Example

A and B are players in the same football team. A has the ball, but an

  • pposing player is converging on him. He can pass the ball to B, who

has a chance to shoot. There are two directions in which A can move the ball, left and right, and correspondingly, two directions in which B can run to intercept the pass. If both choose left there is a 10% chance that a goal will be scored. If they both choose right, there is a 11%

  • change. Otherwise, the chance is zero. There is no time for

communication; the two players must act simultaneously. What should they do?

  • R. Sugden. The Logic of Team Reasoning. Philosophical Explorations (6)3, pgs. 165
  • 181 (2003).

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SLIDE 98

Example

l r l

10,10 00,00

r

00,00 11,11 A B A: What should I do? r if the probability of B choosing r is > 10

21 and l

if the probability of B choosing l is > 11

21

(symmetric reasoning for B)

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SLIDE 99

Example

l r l

10,10 0,0

r

0,0 11,11 A B A: What should I do? r if the probability of B choosing r is > 10

21 and l

if the probability of B choosing l is > 11

21

(symmetric reasoning for B)

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SLIDE 100

Example

l r l

10,10 0,0

r

0,0 11,11 A B A: What should I do? r if the probability of B choosing r is > 10

21 and l

if the probability of B choosing l is > 11

21

(symmetric reasoning for B)

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SLIDE 101

Example

l r l

10,10 0,0

r

0,0 11,11 A B A: What should I do? r if the probability of B choosing r is > 10

21 and l

if the probability of B choosing l is > 11

21

(symmetric reasoning for B)

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SLIDE 102

Example

l r l

10,10 0,0

r

0,0 11,11 A B A: What should we do?

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SLIDE 103

Example

l r l

10,10 0,0

r

0,0 11,11 A B A: What should we do? Team Reasoning: an escape from the infinite regress? why should this “mode of reasoning” be endorsed?

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SLIDE 104

Example

l r l

10,10 0,0

r

0,0 11,11 A B A: What should we do? Team Reasoning: why should this “mode of reasoning” be endorsed?

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SLIDE 105

Subject of the Proposition

Agent i is the subject of the proposition ϕi if ϕi makes an assertion about a current or future act of is will

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SLIDE 106

Subject of the Proposition

Agent i is the subject of the proposition ϕi if ϕi makes an assertion about a current or future act of is will:

◮ a prediction about what i will choose in a future decision problem; ◮ a deontic statement about what i ought to choose; ◮ assert that i endorses some inference rule; or ◮ assert that i has reason to believe some proposition

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SLIDE 107

Subject of the Proposition

Agent i is the subject of the proposition ϕi if ϕi makes an assertion about a current or future act of is will:

◮ a prediction about what i will choose in a future decision problem; ◮ a deontic statement about what i ought to choose; ◮ assert that i endorses some inference rule; or ◮ assert that i has reason to believe some proposition

Ri(ϕi) vs. Rj(ϕi): Suppose i reliable takes a bus every Monday.

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SLIDE 108

Subject of the Proposition

Agent i is the subject of the proposition ϕi if ϕi makes an assertion about a current or future act of is will:

◮ a prediction about what i will choose in a future decision problem; ◮ a deontic statement about what i ought to choose; ◮ assert that i endorses some inference rule; or ◮ assert that i has reason to believe some proposition

Ri(ϕi) vs. Rj(ϕi): Suppose i reliable takes a bus every Monday. The

  • ther commuters may all make the inductive inference that i will take

the bus next Monday (Mi).

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SLIDE 109

Subject of the Proposition

Agent i is the subject of the proposition ϕi if ϕi makes an assertion about a current or future act of is will:

◮ a prediction about what i will choose in a future decision problem; ◮ a deontic statement about what i ought to choose; ◮ assert that i endorses some inference rule; or ◮ assert that i has reason to believe some proposition

Ri(ϕi) vs. Rj(ϕi): Suppose i reliable takes a bus every Monday. The

  • ther commuters may all make the inductive inference that i will take

the bus next Monday (Mi). In fact, we may assume that this is a common mode of reasoning, so everyone reliably makes the inference that i will catch the bus next Monday.

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SLIDE 110

Subject of the Proposition

Agent i is the subject of the proposition ϕi if ϕi makes an assertion about a current or future act of is will:

◮ a prediction about what i will choose in a future decision problem; ◮ a deontic statement about what i ought to choose; ◮ assert that i endorses some inference rule; or ◮ assert that i has reason to believe some proposition

Ri(ϕi) vs. Rj(ϕi): Suppose i reliable takes a bus every Monday. The

  • ther commuters may all make the inductive inference that i will take

the bus next Monday (Mi). In fact, we may assume that this is a common mode of reasoning, so everyone reliably makes the inference that i will catch the bus next Monday. So, Rj(Mi), RiRj(Mi)

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SLIDE 111

Subject of the Proposition

Agent i is the subject of the proposition ϕi if ϕi makes an assertion about a current or future act of is will:

◮ a prediction about what i will choose in a future decision problem; ◮ a deontic statement about what i ought to choose; ◮ assert that i endorses some inference rule; or ◮ assert that i has reason to believe some proposition

Ri(ϕi) vs. Rj(ϕi): Suppose i reliable takes a bus every Monday. The

  • ther commuters may all make the inductive inference that i will take

the bus next Monday (Mi). In fact, we may assume that this is a common mode of reasoning, so everyone reliably makes the inference that i will catch the bus next Monday. So, Rj(Mi), RiRj(Mi), but i should still be free to choose whether he wants to take the bus on Monday, so ¬Ri(Mi) and ¬Rj(Ri(Mi)), etc.

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SLIDE 112

Common Reason to Believe

Awareness of Common Reason: for all i ∈ G and all propositions ϕ, RG(ϕ) ⇒ Ri[RG(ϕ)]

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SLIDE 113

Common Reason to Believe

Awareness of Common Reason: for all i ∈ G and all propositions ϕ, RG(ϕ) ⇒ Ri[RG(ϕ)] Authority of Common Reason: for all i ∈ G and all propositions ϕ for which i is not the subject inf (Ri) : RG(ϕ) → ϕ

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SLIDE 114

Common Reason to Believe

Awareness of Common Reason: for all i ∈ G and all propositions ϕ, RG(ϕ) ⇒ Ri[RG(ϕ)] Authority of Common Reason: for all i ∈ G and all propositions ϕ for which i is not the subject inf (Ri) : RG(ϕ) → ϕ Common Attribution of Common Reason: for all i ∈ G, for all propositions ϕ for which i is not the subject inf (RG) : ϕ → Ri(ϕ)

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SLIDE 115

Team Maximising

inf (Ri) : RN[opt(v, N, sN)], RN[ each i ∈ N endorses team maximising with respect to N and v ], RN[ each member of N acts on reasons ] → ought(i, si) Ri[ought(i, si)]: i has reason to choose si

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SLIDE 116

Team Maximising

inf (Ri) : RN[opt(v, N, sN)], RN[ each i ∈ N endorses team maximising with respect to N and v ], RN[ each member of N acts on reasons ] → ought(i, si) Ri[ought(i, si)]: i has reason to choose si

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SLIDE 117

Team Maximising

inf (Ri) : RN[opt(v, N, sN)], RN[ each i ∈ N endorses team maximising with respect to N and v ], RN[ each member of N acts on reasons ] → ought(i, si) i acts on reasons if for all si, Ri[ought(i, si)] ⇒ choice(i, si)

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SLIDE 118

Team Maximising

inf (Ri) : RN[opt(v, N, sN)], RN[ each i ∈ N endorses team maximising with respect to N and v ], RN[ each member of N acts on reasons ] → ought(i, si)

  • pt(v, N, sN): sN is maximal for the group N w.r.t. v

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SLIDE 119

Team Maximising

inf (Ri) : RN[opt(v, N, sN)], RN[ each i ∈ N endorses team maximising with respect to N and v ], RN[ each member of N acts on reasons ] → ought(i, si) Recursive definition: i’s endorsement of the rule depends on i having a reason to believe everyone else endorses the rule...

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SLIDE 120

Team modes of reasoning, group identification, frames and team preferences, ...

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SLIDE 121

Reasoning Based Expected Utility Procedure

  • R. Cubitt and R. Sugden. The reasoning-based expected utility procedure. Games

and Economic Behavior, 2010.

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SLIDE 122

Reasoning-Based Solution Concepts

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SLIDE 123

Reasoning-Based Solution Concepts

A categorization is a ternary partition of the players choices (rather than a binary partition of what is in and what is out):

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SLIDE 124

Reasoning-Based Solution Concepts

A categorization is a ternary partition of the players choices (rather than a binary partition of what is in and what is out): strategies are accumulated, deleted or neither.

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SLIDE 125

Reasoning-Based Solution Concepts

A categorization is a ternary partition of the players choices (rather than a binary partition of what is in and what is out): strategies are accumulated, deleted or neither. Example: RBEU (reasoning based expected utility):

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SLIDE 126

Reasoning-Based Solution Concepts

A categorization is a ternary partition of the players choices (rather than a binary partition of what is in and what is out): strategies are accumulated, deleted or neither. Example: RBEU (reasoning based expected utility):

◮ accumulate strategies that maximize expected utility for every

possibly probability distribution

◮ delete strategies that do not maximize probability against any

probability distribution

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SLIDE 127

Reasoning-Based Solution Concepts

A categorization is a ternary partition of the players choices (rather than a binary partition of what is in and what is out): strategies are accumulated, deleted or neither. Example: RBEU (reasoning based expected utility):

◮ accumulate strategies that maximize expected utility for every

possibly probability distribution

◮ delete strategies that do not maximize probability against any

probability distribution

◮ accumulated strategies must receive positive probability, deleted

strategies must receive zero probability

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SLIDE 128

RBEU: Example

L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 S+ = {L} S− = {B} S+ = {L, R} S− = {B, M1} S+ = {L, R} S− = {B, M1}

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SLIDE 129

RBEU: Example

L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 S+ = {L} S− = {B} S+ = {L, R} S− = {B, M1} S+ = {L, R} S− = {B, M1}

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SLIDE 130

RBEU: Example

L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 S+ = {L} S− = {B} S+ = {L, R} S− = {B, M1} S+ = {L, R} S− = {B, M1}

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SLIDE 131

RBEU: Example

L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 S+ = {L} S− = {B} S+ = {L, R} S− = {B, M1} S+ = {L, R} S− = {B, M1}

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SLIDE 132

RBEU: Example

L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 S+ = {L} S− = {B} S+ = {L, R} S− = {B, M1} S+ = {L, R} S− = {B, M1}

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SLIDE 133

RBEU: Example

L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 S+ = {L} S− = {B} S+ = {L, R} S− = {B, M1} S+ = {L, R} S− = {B, M1}

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SLIDE 134

RBEU: Example

L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 S+ = {L} S− = {B} S+ = {L, R} S− = {B, M1} S+ = {L, R} S− = {B, M1}

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SLIDE 135

RBEU: Example

L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 L R U 1,1 1,1 M1 0,0 1,0 M2 2,0 0,0 B 0,2 0,0 S+ = {L} S− = {B} S+ = {L, R} S− = {B, M1} S+ = {L, R} S− = {B, M1}

Eric Pacuit: Models of Strategic Reasoning 39/43

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SLIDE 136

RBEU Example 2

l r u 1,1 0,0 d 0,0 0,0 l r u 1,1 0,0 d 0,0 0,0 S+ = {L} S− = {B} S+ = {L, R} S− = {B, M1}

Eric Pacuit: Models of Strategic Reasoning 40/43

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SLIDE 137

RBEU Example 2

l r u 1,1 0,0 d 0,0 0,0 l r u 1,1 1,0 d 1,0 0,1 S+ = {u, l} S− = ∅ S+ = {L, R} S− = {B, M1}

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SLIDE 138

RBEU Example 2

l r u 1,1 0,0 d 0,0 0,0 l r u 1,1 0,0 d 0,0 0,0 S+ = {u, l} S− = ∅ S+ = {u, l} S− = {d, r}

Eric Pacuit: Models of Strategic Reasoning 40/43

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SLIDE 139

RBEU Example 3

L R u 1,1 1,0 d 1,0 0,1 L R u 1,1 1,0 d 1,0 0,1 S+ = {L} S− = {B} S+ = {L, R} S− = {B, M1}

Eric Pacuit: Models of Strategic Reasoning 41/43

slide-140
SLIDE 140

RBEU Example 3

L R u 1,1 1,0 d 1,0 0,1 L R u 1,1 1,0 d 1,0 0,1 S+ = {u} S− = ∅ S+ = {L, R} S− = {B, M1}

Eric Pacuit: Models of Strategic Reasoning 41/43

slide-141
SLIDE 141

RBEU Example 3

L R u 1,1 1,0 d 1,0 0,1 L R u 1,1 1,0 d 1,0 0,1 S+ = {u} S− = ∅ S+ = {L, R} S− = {B, M1}

Eric Pacuit: Models of Strategic Reasoning 41/43

slide-142
SLIDE 142

RBEU Example 3

L R u 1,1 1,0 d 1,0 0,1 L R u 1,1 1,0 d 1,0 0,1 S+ = {u} S− = ∅ S+ = {u} S− = ∅

Eric Pacuit: Models of Strategic Reasoning 41/43

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SLIDE 143
  • R. Cubitt and R. Sugden.

Common reasoning in games: A Lewisian analysis of common knowledge of rationality. Discussion paper, 2011.

Eric Pacuit: Models of Strategic Reasoning 42/43

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SLIDE 144

Today: foundational issues (value of information, deliberation in decision theory), Lewisian common knowledge, common modes of reasoning Tomorrow: Dynamic logic perspective on games.

Eric Pacuit: Models of Strategic Reasoning 43/43