Models of Strategic Reasoning Lecture 4

Transcription

1 Models of Strategic Reasoning Lecture 4 Eric Pacuit University of Maryland, College Park ai.stanford.edu/~epacuit August 9, 2012 Eric Pacuit: Models of Strategic Reasoning 1/55

2 Game Plan Lecture 4: Lecture 5: Introduction, Motivation and Background The Dynamics of Rational Deliberation Reasoning to a Solution: Common Modes of Reasoning in Games Reasoning to a Model: Iterated Belief Change as Deliberation Reasoning in Specific Games: Experimental Results Eric Pacuit: Models of Strategic Reasoning 2/55

3 Informational Context of a Game In any particular structure, certain beliefs, beliefs about belief,..., will be present and others won t be. So, there is an important implicit assumption behind the choice of a structure. This is that it is transparent to the players that the beliefs in the type structure and only those beliefs are possible...the idea is that there is a context to the strategic situation (eg., history, conventions, etc.) and this context causes the players to rule out certain beliefs. (pg. 810) Adam Brandenburger and Amanda Friedenberg. Self-Admissible Sets. Journal of Economic Theory, 145, , Eric Pacuit: Models of Strategic Reasoning 3/55

4 Informational Context of a Game Game G Strategy Space b s a Rat Rat Game Model Eric Pacuit: Models of Strategic Reasoning 3/55

5 Informational Context of a Game w s s v Epistemic Model: M = W, { i } i A, V w i v means i cannot rule out v according to her information. Language: ϕ := p ϕ ϕ ψ K i ϕ Truth: M, w = p iff w V (p) (p an atomic proposition) Boolean connectives as usual M, w = K i ϕ iff for all v W, if w i v then M, v = ϕ Eric Pacuit: Models of Strategic Reasoning 3/55

6 Informational Context of a Game w s s v Epistemic-Plausibility Model: M = W, { i } i A, { i } i A, V w i v means v is at least as plausibility as w for agent i. Language: ϕ := p ϕ ϕ ψ K i ϕ B ϕ ψ [ i ]ϕ Truth: [[ϕ]] M = {w M, w = ϕ} M, w = B ϕ i ψ iff for all v Min i ([[ϕ]] M [w] i ), M, v = ψ M, w = [ i ]ϕ iff for all v W, if v i w then M, v = ϕ Eric Pacuit: Models of Strategic Reasoning 3/55

7 Informational Context of a Game 1 r s w s r v Epistemic-Plausibility Model: M = W, { i } i A, {π i } i A, V π i : W [0, 1] is a probability measure Language: ϕ := p ϕ ϕ ψ K i ϕ B p ψ Truth: [[ϕ]] M = {w M, w = ϕ} M, w = B p ϕ iff π i ([[ϕ]] M [w] i ) = π i ([[ϕ]] M [w] i ) π i ([w] i ) p, M, v = ψ M, w = K i ϕ iff for all v W, if w i v then M, v = ϕ Eric Pacuit: Models of Strategic Reasoning 3/55

8 Reasoning to a context It is important to understand that we have two forms of irrationality in this paper...for us, a player is rational if he optimizes and also rules nothing out. So irrationality might mean not optimizing. But it can also mean optimizing while not considering everything possible. (pg. 314) A. Brandenburger, A. Friedenberg and H. J. Keisler. Admissibility in Games. Econometrica, 76:2, 2008, pgs Eric Pacuit: Models of Strategic Reasoning 4/55

9 Reasoning to a context It is important to understand that we have two forms of irrationality in this paper...for us, a player is rational if he optimizes and also rules nothing out. So irrationality might mean not optimizing. But it can also mean optimizing while not considering everything possible. (pg. 314) A. Brandenburger, A. Friedenberg and H. J. Keisler. Admissibility in Games. Econometrica, 76:2, 2008, pgs A player can be rationally criticized for Eric Pacuit: Models of Strategic Reasoning 4/55

10 Reasoning to a context It is important to understand that we have two forms of irrationality in this paper...for us, a player is rational if he optimizes and also rules nothing out. So irrationality might mean not optimizing. But it can also mean optimizing while not considering everything possible. (pg. 314) A. Brandenburger, A. Friedenberg and H. J. Keisler. Admissibility in Games. Econometrica, 76:2, 2008, pgs A player can be rationally criticized for 1. not choosing what is best or what is rationally permissible, given one s information. Eric Pacuit: Models of Strategic Reasoning 4/55

11 Reasoning to a context It is important to understand that we have two forms of irrationality in this paper...for us, a player is rational if he optimizes and also rules nothing out. So irrationality might mean not optimizing. But it can also mean optimizing while not considering everything possible. (pg. 314) A. Brandenburger, A. Friedenberg and H. J. Keisler. Admissibility in Games. Econometrica, 76:2, 2008, pgs A player can be rationally criticized for 1. not choosing what is best or what is rationally permissible, given one s information. 2. not reasoning to a proper informational context. Eric Pacuit: Models of Strategic Reasoning 4/55

12 Key Idea Informational contexts of a game arise as fixed points of iterated rationality announcements. Eric Pacuit: Models of Strategic Reasoning 5/55

13 Key Idea Informational contexts of a game arise as fixed points of iterated rationality announcements. J. van Benthem. Rational dynamics and epistemic logic in games. International Game Theory Review 9, 1 (2007), A. Baltag, S. Smets, and J. Zvesper. Keep hoping for rationality: a solution to the backwards induction paradox. Synthese 169 (2009), K. Apt and J. Zvesper. Public announcements in strategic games with arbitrary strategy sets. Proceedings of LOFT 2010 (2010). J. van Benthem, and A. Gheerbrant. Game solution, epistemic dynamics and fixedpoint logics. Fund. Inform. 100 (2010), Eric Pacuit: Models of Strategic Reasoning 5/55

14 Dynamic Epistemic/Doxastic Logic J. van Benthem. Logical Dynamics of Information and Interaction. Cambridge University Press, EP. Dynamic Epistemic Logic Part I: Modeling Knowledge and Belief. Compass, forthcoming. EP. Dynamic Epistemic Logic Part II: Logics of Information Change. Compass, forthcoming. Philosophy Philosophy Eric Pacuit: Models of Strategic Reasoning 6/55

15 Modeling Information Change: Two Methodologies 1. Change-based modeling : describe the effect a learning experience has on a model 2. Explicit-temporal modeling : explicitly describe different moments in the model Eric Pacuit: Models of Strategic Reasoning 7/55

16 Modeling Information Change: Two Methodologies 1. Change-based modeling : describe the effect a learning experience has on a model 2. Explicit-temporal modeling : explicitly describe different moments in the model Eric Pacuit: Models of Strategic Reasoning 7/55

17 [ψ]ϕ: after everyone finds out that ψ is true, ϕ is true Eric Pacuit: Models of Strategic Reasoning 8/55

18 [ψ]ϕ: after everyone finds out that ψ is true, ϕ is true How did you find out that ψ? direct observation of ψ public announcement of ψ... Eric Pacuit: Models of Strategic Reasoning 8/55

19 Finding out that p is true w P P v w P M 0 M 1 Eric Pacuit: Models of Strategic Reasoning 9/55

20 Public Announcement Logic J. Plaza. Logics of Public Communications J. Gerbrandy. Bisimulations on Planet Kripke J. van Benthem. One is a lonely number Eric Pacuit: Models of Strategic Reasoning 10/55

21 Public Announcement Logic where p At and i A. p ϕ ϕ ϕ K i ϕ Cϕ [ψ]ϕ Eric Pacuit: Models of Strategic Reasoning 11/55

22 Public Announcement Logic where p At and i A. p ϕ ϕ ϕ K i ϕ Cϕ [ψ]ϕ [ψ]ϕ is intended to mean After ψ is publicly announced, ϕ is true. Eric Pacuit: Models of Strategic Reasoning 11/55

23 Public Announcement Logic where p At and i A. p ϕ ϕ ϕ K i ϕ Cϕ [ψ]ϕ [p]k i p: after publicly announcing P, agent i knows P Eric Pacuit: Models of Strategic Reasoning 11/55

24 Public Announcement Logic where p At and i A. p ϕ ϕ ϕ K i ϕ Cϕ [ψ]ϕ [ K i p]cp: after announcing that agent i does not know p, then p is common knowledge Eric Pacuit: Models of Strategic Reasoning 11/55

25 Public Announcement Logic where p At and i A. p ϕ ϕ ϕ K i ϕ Cϕ [ψ]ϕ [ K i p]k i p: after announcing i does not know p, then i knows p Eric Pacuit: Models of Strategic Reasoning 11/55

26 Public Announcement Logic Suppose M = W, { i } i A, V is a multi-agent epistemic models M, w = [ψ]ϕ iff M, w = ψ implies M ψ, w = ϕ where M ψ = W, { i } i A, V with W = W {w M, w = ψ} For each i, i = i (W W ) for all p At, V (p) = V (p) W Eric Pacuit: Models of Strategic Reasoning 12/55

27 Public Announcement Logic [ψ]p (ψ p) Eric Pacuit: Models of Strategic Reasoning 13/55

28 Public Announcement Logic [ψ]p (ψ p) [ψ] ϕ (ψ [ψ]ϕ) Eric Pacuit: Models of Strategic Reasoning 13/55

29 Public Announcement Logic [ψ]p (ψ p) [ψ] ϕ (ψ [ψ]ϕ) [ψ](ϕ χ) ([ψ]ϕ [ψ]χ) Eric Pacuit: Models of Strategic Reasoning 13/55

30 Public Announcement Logic [ψ]p (ψ p) [ψ] ϕ (ψ [ψ]ϕ) [ψ](ϕ χ) ([ψ]ϕ [ψ]χ) [ψ][ϕ]χ [ψ [ψ]ϕ]χ Eric Pacuit: Models of Strategic Reasoning 13/55

31 Public Announcement Logic [ψ]p (ψ p) [ψ] ϕ (ψ [ψ]ϕ) [ψ](ϕ χ) ([ψ]ϕ [ψ]χ) [ψ][ϕ]χ [ψ [ψ]ϕ]χ [ψ]k i ϕ (ψ K i (ψ [ψ]ϕ)) Eric Pacuit: Models of Strategic Reasoning 13/55

32 Public Announcement Logic [ψ]p (ψ p) [ψ] ϕ (ψ [ψ]ϕ) [ψ](ϕ χ) ([ψ]ϕ [ψ]χ) [ψ][ϕ]χ [ψ [ψ]ϕ]χ [ψ]k i ϕ (ψ K i (ψ [ψ]ϕ)) Theorem Every formula of Public Announcement Logic is equivalent to a formula of Epistemic Logic. Eric Pacuit: Models of Strategic Reasoning 13/55

33 Public Announcement Logic [ψ]p (ψ p) [ψ] ϕ (ψ [ψ]ϕ) [ψ](ϕ χ) ([ψ]ϕ [ψ]χ) [ψ][ϕ]χ [ψ [ψ]ϕ]χ [ψ]k i ϕ (ψ K i (ψ [ψ]ϕ)) The situation is more complicated with common knowledge. J. van Benthem, J. van Eijk, B. Kooi. Logics of Communication and Change Eric Pacuit: Models of Strategic Reasoning 13/55

34 Aspects of Informative Events 1. The agents observational powers. Agents may perceive the same event differently and this can be described in terms of what agents do or do not observe. Examples range from public announcements where everyone witnesses the same event to private communications between two or more agents with the other agents not even being aware that an event took place. Eric Pacuit: Models of Strategic Reasoning 14/55

35 Aspects of Informative Events 1. The agents observational powers. 2. The type of change triggered by the event. Agents may differ in precisely how they incorporate new information into their epistemic states. These differences are based, in part, on the agents perception of the source of the information. For example, an agent may consider a particular source of information infallible (not allowing for the possibility that the source is mistaken) or merely trustworthy (accepting the information as reliable though allowing for the possibility of a mistake). Eric Pacuit: Models of Strategic Reasoning 14/55

36 Aspects of Informative Events 1. The agents observational powers. 2. The type of change triggered by the event. 3. The underlying protocol specifying which events (observations, messages, actions) are available (or permitted) at any given moment. This is intended to represent the rules or conventions that govern many of our social interactions. For example, in a conversation, it is typically not polite to blurt everything out at the beginning, as we must speak in small chunks. Other natural conversational protocol rules include do not repeat yourself, let others speak in turn, and be honest. Imposing such rules restricts the legitimate sequences of possible statements or events. Eric Pacuit: Models of Strategic Reasoning 14/55

37 Aspects of Informative Events 1. The agents observational powers. 2. The type of change triggered by the event. 3. The underlying protocol specifying which events (observations, messages, actions) are available (or permitted) at any given moment. Eric Pacuit: Models of Strategic Reasoning 14/55

38 Aspects of Informative Events 1. The agents observational powers. 2. The type of change triggered by the event. 3. The underlying protocol specifying which events (observations, messages, actions) are available (or permitted) at any given moment. Eric Pacuit: Models of Strategic Reasoning 14/55

39 Finding out that ϕ M = W, { i } i A, { i } i A, V Find out that ϕ M = W, { i } i A, { i } i A, V W Eric Pacuit: Models of Strategic Reasoning 15/55

40 Informative Actions w 1 w 2 w 3 w 1 w 2 and w 2 w 1 (w 1 and w 2 are equi-plausbile) w 1 w 3 (w 1 w 3 and w 3 w 1 ) E D B w 3 A ϕ w 2 w 3 (w 2 w 3 and w 3 w 2 ) {w 1, w 2 } Min ([w i ]) w 1 w 2 Eric Pacuit: Models of Strategic Reasoning 16/55

41 Informative Actions w 1 w 2 w 3 w 1 w 2 and w 2 w 1 (w 1 and w 2 are equi-plausbile) w 1 w 3 (w 1 w 3 and w 3 w 1 ) E D B w 3 A ϕ w 2 w 3 (w 2 w 3 and w 3 w 2 ) {w 1, w 2 } Min ([w i ]) w 1 w 2 Eric Pacuit: Models of Strategic Reasoning 16/55

42 Informative Actions w 1 w 2 w 3 w 1 w 2 and w 2 w 1 (w 1 and w 2 are equi-plausbile) w 1 w 3 (w 1 w 3 and w 3 w 1 ) E D B w 3 A ϕ w 2 w 3 (w 2 w 3 and w 3 w 2 ) {w 1, w 2 } Min ([w i ]) w 1 w 2 Eric Pacuit: Models of Strategic Reasoning 16/55

43 Informative Actions E D B A ϕ C Incorporate the new information ϕ(!ϕ): A i B Conservative Upgrade: Information from a trusted source ( ϕ): A i C i D i B E Radical Upgrade: Information from a strongly trusted source ( ϕ): A i B i C i D i E Eric Pacuit: Models of Strategic Reasoning 16/55

44 Informative Actions E D B A ϕ C Incorporate the information that ϕ Min (W [[ϕ]] M ) [[ψ]] M W!ϕ = [[ϕ]] M!ϕ i = i (W!ϕ W!ϕ )!ϕ i = i (W!ϕ W!ϕ ) Eric Pacuit:!ϕ Models of Strategic Reasoning 16/55

45 W!ϕ = [[ϕ]] M!ϕ i = i (W!ϕ W!ϕ )!ϕ i = i (W!ϕ W!ϕ )!ϕ Eric Pacuit: = Models i (W of Strategic!ϕ Reasoning W!ϕ ) 16/55 Informative Actions ψ E D B A ϕ C Conditional Belief: B ϕ ψ Min (W [[ϕ]] M ) [[ψ]] M

46 Informative Actions E D B A ϕ C Public Announcement: Information from an infallible source (!ϕ): A i B M!ϕ = W!ϕ, {!ϕ i } i A, V!ϕ W!ϕ = [[ϕ]] M!ϕ i = i (W!ϕ W!ϕ )!ϕ i = i (W!ϕ W!ϕ )!ϕ i = i (W!ϕ W!ϕ ) Eric Pacuit: Models of Strategic Reasoning 16/55

47 Informative Actions E D B A ϕ C Radical Upgrade: ( ϕ): A i B i C i D i E, M ϕ = W, { i } i A, { ϕ i } i A, V Let [[ϕ]] w i = {x M, x = ϕ} [w] i for all x [[ϕ]] w i and y [[ ϕ]] w i, set x ϕ i y, for all x, y [[ϕ]] w i, set x ϕ i y iff x i y, and for all x, y [[ ϕ]] w i, set x ϕ i y iff x i y. Eric Pacuit: Models of Strategic Reasoning 16/55

48 Informative Actions E D B A ϕ C Conservative Upgrade: ( ϕ): A i C i D i B E Conservative upgrade is radical upgrade with the formula best i (ϕ, w) := Min i ([w] i {x M, x = ϕ}) 1. If v best i (ϕ, w) then v ϕ i x for all x [w] i, and 2. for all x, y [w] i best i (ϕ, w), x ϕ i y iff x i y. Eric Pacuit: Models of Strategic Reasoning 16/55

49 [q]kq Eric Pacuit: Models of Strategic Reasoning 17/55

50 [q]kq Kp [q]kp Eric Pacuit: Models of Strategic Reasoning 17/55

51 [q]kq Kp [q]kp Bϕ [ψ]bϕ Eric Pacuit: Models of Strategic Reasoning 17/55

52 [q]kq Kp [q]kp Bϕ [ψ]bϕ p, q p, q p, q w 1 w 2 w 3 [ϕ]ϕ Eric Pacuit: Models of Strategic Reasoning 17/55

53 Public Announcement vs. Conditional Belief Are [ϕ]bψ and B ϕ ψ different? Eric Pacuit: Models of Strategic Reasoning 18/55

54 Public Announcement vs. Conditional Belief Are [ϕ]bψ and B ϕ ψ different? Yes! Eric Pacuit: Models of Strategic Reasoning 18/55

55 Public Announcement vs. Conditional Belief Are [ϕ]bψ and B ϕ ψ different? Yes! 1 2 p, q p, q w 1 w 2 w 3 p, q Eric Pacuit: Models of Strategic Reasoning 18/55

56 Public Announcement vs. Conditional Belief Are [ϕ]bψ and B ϕ ψ different? Yes! 1 2 p, q p, q w 1 w 2 w 3 p, q w 1 = B 1 B 2 q Eric Pacuit: Models of Strategic Reasoning 18/55

57 Public Announcement vs. Conditional Belief Are [ϕ]bψ and B ϕ ψ different? Yes! 1 2 p, q p, q w 1 w 2 w 3 p, q w 1 = B 1 B 2 q w 1 = B p 1 B 2q Eric Pacuit: Models of Strategic Reasoning 18/55

58 Public Announcement vs. Conditional Belief Are [ϕ]bψ and B ϕ ψ different? Yes! 1 2 p, q p, q w 1 w 2 w 3 p, q w 1 = B 1 B 2 q w 1 = B p 1 B 2q w 1 = [p] B 1 B 2 q Eric Pacuit: Models of Strategic Reasoning 18/55

59 Public Announcement vs. Conditional Belief Are [ϕ]bψ and B ϕ ψ different? Yes! 1 2 p, q p, q w 1 w 2 w 3 p, q w 1 = B 1 B 2 q w 1 = B p 1 B 2q w 1 = [p] B 1 B 2 q More generally, B p i (p K i p) is satisfiable but [p]b i (p K i p) is not. Eric Pacuit: Models of Strategic Reasoning 18/55

60 Hard and Soft Updates M = W, { i } i A, { i } i A, V Find out that ϕ M = W, { i } i A, { i } i A, V W Eric Pacuit: Models of Strategic Reasoning 19/55

61 T 1, H 2 w 2 H 1, H 2 T 1, T 2 w 1 H 1, T 2 w 4 w 3 Min ([w 1 ]) = {w 4 }, so w 1 = B(H 1 H 2 ) Min ([w 1 ] [[T 1 ]] M ) = {w 2 }, so w 1 = B T 1 H 2 Min ([w 1 ] [[T 1 ]] M ) = {w 3 }, so w 1 = B T 2 H 1 Eric Pacuit: Models of Strategic Reasoning 20/55

62 T 1, H 2 w 2 H 1, H 2 T 1, T 2 w 1 H 1, T 2 w 4 w 3 Suppose the agent finds out that T 1 is/may be true. Eric Pacuit: Models of Strategic Reasoning 20/55

63 T1, H2 w 1 T1, T2 w 2 H1, T2 w 3 w 4 H1, H2!(T 1 ) = T 1, T 2 w 1 T 1, H 2 w 2 Suppose the agent finds out that T 1 is/may be true. (T 1 ) = T 1, T 2 H 1, T 2 Logic and Artificial Intelligence 6/16 w 1 B T 2 H 1 w 3 w 4 H 1, H 2 w 2 T 1, H 2 (T 1 ) = H 1, T 2 w 3 B T 2 T 1 w 4 H 1, H 2 w 1 T 1, T 2 w 2 T 1, H 2 Eric Pacuit: Models of Strategic Reasoning 20/55

64 What happens as beliefs change over time (iterated belief revision)? Eric Pacuit: Models of Strategic Reasoning 21/55

65 O i (S) P j (S ) O j (T ) P j (T ) O i (S) P j (S ) nothing new!ϕ 1!ϕ 2!ϕ 3!ϕ n M 0 M 1 M 2 M f initial model fixed-point Where do the ϕ k come from? from the players practical reasoning/rational requirements Eric Pacuit: Models of Strategic Reasoning 22/55

66 O i (S) P j (S ) O j (T ) P j (T ) O i (S) P j (S ) nothing new ϕ 1 ϕ 2 ϕ 3 ϕ n M 0 M 1 M 2 M f initial model fixed-point Where do the ϕ k come from? from the players practical reasoning/rational requirements Eric Pacuit: Models of Strategic Reasoning 22/55

67 O i (S) P j (S ) O j (T ) P j (T ) O i (S) P j (S ) nothing new!ϕ 1 ϕ 2 ϕ 3 ϕ n M 0 M 1 M 2 M f initial model fixed-point Where do the ϕ k come from? from the players practical reasoning/rational requirements Eric Pacuit: Models of Strategic Reasoning 22/55

68 O i (S) P j (S ) O j (T ) P j (T ) O i (S) P j (S ) nothing new τ(ϕ 1 ) τ(ϕ 2 ) τ(ϕ 3 ) τ(ϕ n ) M 0 M 1 M 2 M f initial model fixed-point Where do the ϕ k come from? from the players practical reasoning/rational requirements Eric Pacuit: Models of Strategic Reasoning 22/55

69 Iterated Updates!ϕ 1,!ϕ 2,!ϕ 3,...,!ϕ n always reaches a fixed-point Eric Pacuit: Models of Strategic Reasoning 23/55

70 Iterated Updates!ϕ 1,!ϕ 2,!ϕ 3,...,!ϕ n always reaches a fixed-point p p p Contradictory beliefs leads to oscillations Eric Pacuit: Models of Strategic Reasoning 23/55

71 Iterated Updates!ϕ 1,!ϕ 2,!ϕ 3,...,!ϕ n always reaches a fixed-point p p p Contradictory beliefs leads to oscillations ϕ, ϕ,... Simple beliefs may never stabilize Eric Pacuit: Models of Strategic Reasoning 23/55

72 Iterated Updates!ϕ 1,!ϕ 2,!ϕ 3,...,!ϕ n always reaches a fixed-point p p p Contradictory beliefs leads to oscillations ϕ, ϕ,... Simple beliefs may never stabilize ϕ, ϕ,... Simple beliefs stabilize, but conditional beliefs do not A. Baltag and S. Smets. Group Belief Dynamics under Iterated Revision: Fixed Points and Cycles of Joint Upgrades. TARK, Eric Pacuit: Models of Strategic Reasoning 23/55

73 r n d w 1 w 2 w 3 (r (d Bd) ( d Bd) r d n w 1 w 3 w 2 (r (d Bd) ( d Bd) r n d w 1 w 2 w 3 Eric Pacuit: Models of Strategic Reasoning 24/55

74 Let ϕ be (r (B r q p) (B r p q)) w 3 p w 2 q w 3 p w 2 q ϕ = w 3 p ϕ = w 2 q ϕ = w 1 r w 1 r w 1 r M 1 M 2 M 3 Eric Pacuit: Models of Strategic Reasoning 25/55

75 Iterated Updates!ϕ 1,!ϕ 2,!ϕ 3,...,!ϕ n always reaches a fixed-point p p p Contradictory beliefs leads to oscillations ϕ, ϕ,... Simple beliefs may never stabilize ϕ, ϕ,... Simple beliefs stabilize, but conditional beliefs do not A. Baltag and S. Smets. Group Belief Dynamics under Iterated Revision: Fixed Points and Cycles of Joint Upgrades. TARK, Eric Pacuit: Models of Strategic Reasoning 26/55

76 Iterated belief revision: two issues C1: If α ϕ then Ψ(β 1,..., β n, ϕ, α) = Ψ(β 1,..., β n, α) C2: If α ϕ then Ψ(β 1,..., β n, ϕ, α) = Ψ(β 1,..., β n, α) R. Stalnaker. Iterated Belief Revision. Erkentnis 70, pgs , Eric Pacuit: Models of Strategic Reasoning 27/55

77 Suppose that you are in the forest and happen to a see strange-looking animal. Eric Pacuit: Models of Strategic Reasoning 28/55

78 Suppose that you are in the forest and happen to a see strange-looking animal. You consult your animal guidebook and find a picture that seems to match the animal you see. Eric Pacuit: Models of Strategic Reasoning 28/55

79 Suppose that you are in the forest and happen to a see strange-looking animal. You consult your animal guidebook and find a picture that seems to match the animal you see. The guidebook says that the animal is a type of bird, so that is what you conclude: The animal before you is a bird. After looking more closely, you also notice that the animal is also red. Eric Pacuit: Models of Strategic Reasoning 28/55

80 Suppose that you are in the forest and happen to a see strange-looking animal. You consult your animal guidebook and find a picture that seems to match the animal you see. The guidebook says that the animal is a type of bird, so that is what you conclude: The animal before you is a bird. After looking more closely, you also notice that the animal is also red. So, you also update your beliefs with that fact. Eric Pacuit: Models of Strategic Reasoning 28/55

81 Suppose that you are in the forest and happen to a see strange-looking animal. You consult your animal guidebook and find a picture that seems to match the animal you see. The guidebook says that the animal is a type of bird, so that is what you conclude: The animal before you is a bird. After looking more closely, you also notice that the animal is also red. So, you also update your beliefs with that fact. Now, suppose that an expert (whom you trust) happens to walk by and tells you that the animal is, in fact, not a bird. Eric Pacuit: Models of Strategic Reasoning 28/55

82 b, r b, r b, r b, r b b, r b, r b, r b, r r b, r b, r b, r b, r M 0 M 1 b M 2 b, r b, r b, r b, r M 3 Eric Pacuit: Models of Strategic Reasoning 29/55

83 Note that in the last model, M 3, the agent does not believe that the bird is red. Eric Pacuit: Models of Strategic Reasoning 30/55

84 Note that in the last model, M 3, the agent does not believe that the bird is red. The problem is that there does not seem to be any justification for why the agent drops her belief that the bird is red. This seems to result from the accidental fact that the agent started by updating with the information that the animal is a bird. Eric Pacuit: Models of Strategic Reasoning 30/55

85 Note that in the last model, M 3, the agent does not believe that the bird is red. The problem is that there does not seem to be any justification for why the agent drops her belief that the bird is red. This seems to result from the accidental fact that the agent started by updating with the information that the animal is a bird. In particular, note that the following sequence of updates is not problematic: Eric Pacuit: Models of Strategic Reasoning 30/55

86 b, r b, r b, r b, r r b, r b, r b, r b, r b b, r b, r b, r b, r M 0 M 1 b M 2 b, r b, r b, r b, r M 3 Eric Pacuit: Models of Strategic Reasoning 31/55

87 C1: If α ϕ then Ψ(β 1,..., β n, ϕ, α) = Ψ(β 1,..., β n, α) Eric Pacuit: Models of Strategic Reasoning 32/55

88 UUU UUD UDU UDD DDD DDU DUD DUU Three switches wired such that a light is on iff all three switches are up or all three are down. Three independent (reliable) observers report on the switches: Alice says switch 1 is U, Bob says switch 2 is D and Carla says switch 3 is U. I receive the information that the light is on. What should I believe? Cautious: UUU, DDD; Bold: UUU Eric Pacuit: Models of Strategic Reasoning 33/55

89 UUU UUD UDU UDD DDD DDU DUD DUU Three switches wired such that a light is on iff all three switches are up or all three are down. Three independent (reliable) observers report on the switches: Alice says switch 1 is U, Bob says switch 2 is D and Carla says switch 3 is U. I receive the information that the light is on. What should I believe? Cautious: UUU, DDD; Bold: UUU Eric Pacuit: Models of Strategic Reasoning 33/55

90 UUU UUD UDU UDD DDD DDU DUD DUU Three switches wired such that a light is on iff all three switches are up or all three are down. Three independent (reliable) observers report on the switches: Alice says switch 1 is U, Bob says switch 2 is D and Carla says switch 3 is U. I receive the information that the light is on. What should I believe? Cautious: UUU, DDD; Bold: UUU Eric Pacuit: Models of Strategic Reasoning 33/55

91 UUU UUD UDU UDD DDD DDU DUD DUU Three switches wired such that a light is on iff all three switches are up or all three are down. Three independent (reliable) observers report on the switches: Alice says switch 1 is U, Bob says switch 2 is D and Carla says switch 3 is U. I receive the information that the light is on. What should I believe? Cautious: UUU, DDD; Bold: UUU Eric Pacuit: Models of Strategic Reasoning 33/55

92 UUU UUD UDU UDD DDD DDU DUD DUU Suppose there are two switches: L 1 is the main switch and L 2 is a secondary switch controlled by the first two lights. (So L 1 L 2, but not the converse) Suppose I receive L 1 L 2, this does not change the story. Suppose I learn that L 2. This is irrelevant to Carla s report, but it means either Ann or Bob is wrong. Now, after learning L 1, the only rational thing to believe is that all three switches are up. Eric Pacuit: Models of Strategic Reasoning 33/55

93 UUU UUD UDU UDD DDD DDU DUD DUU Suppose there are two switches: L 1 is the main switch and L 2 is a secondary switch controlled by the first two lights. (So L 1 L 2, but not the converse) Suppose I receive L 1 L 2, this does not change the story. Suppose I learn that L 2. This is irrelevant to Carla s report, but it means either Ann or Bob is wrong. Now, after learning L 1, the only rational thing to believe is that all three switches are up. Eric Pacuit: Models of Strategic Reasoning 33/55

94 UUU UUD UDU UDD DDD DDU DUD DUU Suppose there are two switches: L 1 is the main switch and L 2 is a secondary switch controlled by the first two lights. (So L 1 L 2, but not the converse) Suppose I receive L 1 L 2, this does not change the story. Suppose I learn that L 2. This is irrelevant to Carla s report, but it means either Ann or Bob is wrong. Now, after learning L 1, the only rational thing to believe is that all three switches are up. Eric Pacuit: Models of Strategic Reasoning 33/55

95 UUU UUD UDU UDD DDD DDU DUD DUU Suppose there are two switches: L 1 is the main switch and L 2 is a secondary switch controlled by the first two lights. (So L 1 L 2, but not the converse) Suppose I receive L 1 L 2, this does not change the story. Suppose I learn that L 2. This is irrelevant to Carla s report, but it means either Ann or Bob is wrong. Now, after learning L 1, the only rational thing to believe is that all three switches are up. Eric Pacuit: Models of Strategic Reasoning 33/55

96 UUU UUD UDU UDD DDD DDU DUD DUU Suppose there are two switches: L 1 is the main switch and L 2 is a secondary switch controlled by the first two lights. (So L 1 L 2, but not the converse) Suppose I receive L 1 L 2, this does not change the story. Suppose I learn that L 2. This is irrelevant to Carla s report, but it means either Ann or Bob is wrong. Now, after learning L 1, the only rational thing to believe is that all three switches are up. Eric Pacuit: Models of Strategic Reasoning 33/55

97 UUU UUD UDU UDD DDD DDU DUD DUU Suppose there are two switches: L 1 is the main switch and L 2 is a secondary switch controlled by the first two lights. (So L 1 L 2, but not the converse) Suppose I receive L 1 L 2, this does not change the story. Suppose I learn that L 2. This is irrelevant to Carla s report, but it means either Ann or Bob is wrong. Now, after learning L 1, the only rational thing to believe is that all three switches are up. Eric Pacuit: Models of Strategic Reasoning 33/55

98 C2: If α ϕ then Ψ(β 1,..., β n, ϕ, α) = Ψ(β 1,..., β n, α) Eric Pacuit: Models of Strategic Reasoning 34/55

99 1. Alice and Bert (who I initially take to be reliable) report to me, independently, about the results: Alice tells me that the coin in room A came up heads (H A ), while Bert tells me that the coin in room B came up heads (H B ). Eric Pacuit: Models of Strategic Reasoning 35/55

100 1. Alice and Bert (who I initially take to be reliable) report to me, independently, about the results: Alice tells me that the coin in room A came up heads (H A ), while Bert tells me that the coin in room B came up heads (H B ). 2. Carla and Dora, also two independent witnesses whose reliability, in my view, trumps that of Alice and Bert, give me conflicting information: Carla tells me that the coin in room A came up tails (T A ), and Dora tells me the same about the coin in room B (T B ) Eric Pacuit: Models of Strategic Reasoning 35/55

101 1. Alice and Bert (who I initially take to be reliable) report to me, independently, about the results: Alice tells me that the coin in room A came up heads (H A ), while Bert tells me that the coin in room B came up heads (H B ). 2. Carla and Dora, also two independent witnesses whose reliability, in my view, trumps that of Alice and Bert, give me conflicting information: Carla tells me that the coin in room A came up tails (T A ), and Dora tells me the same about the coin in room B (T B ) 3. Elmer, whose reliability trumps everyone else, tells me that that the coin in room A in fact landed heads (H A ) Eric Pacuit: Models of Strategic Reasoning 35/55

102 1. Alice and Bert (who I initially take to be reliable) report to me, independently, about the results: Alice tells me that the coin in room A came up heads (H A ), while Bert tells me that the coin in room B came up heads (H B ). 2. Carla and Dora, also two independent witnesses whose reliability, in my view, trumps that of Alice and Bert, give me conflicting information: Carla tells me that the coin in room A came up tails (T A ), and Dora tells me the same about the coin in room B (T B ) 3. Elmer, whose reliability trumps everyone else, tells me that that the coin in room A in fact landed heads (H A ) What should I now believe about the coin in room B? Eric Pacuit: Models of Strategic Reasoning 35/55

103 Many of the recent developments in this area have been driven by analyzing concrete examples. This raises an important methodological issue: Implicit assumptions about what the actors know and believe about the situation being modeled often guide the analyst s intuitions. In many cases, it is crucial to make these underlying assumptions explicit. The general point is that how the agent(s) come to know or believe that some proposition p is true is as important (or, perhaps, more important) than the fact that the agent(s) knows or believes that p is the case Eric Pacuit: Models of Strategic Reasoning 36/55

104 Discussion A key aspect of any formal model of a (social) interactive situation or situation of rational inquiry is the way it accounts for the...information about how I learn some of the things I learn, about the sources of my information, or about what I believe about what I believe and don t believe. If the story we tell in an example makes certain information about any of these things relevant, then it needs to be included in a proper model of the story, if it is to play the right role in the evaluation of the abstract principles of the model. (Stalnaker, pg. 203) R. Stalnaker. Iterated Belief Revision. Erkentnis 70, pgs , Eric Pacuit: Models of Strategic Reasoning 37/55

105 Iterated belief revision as a model of deliberation Eric Pacuit: Models of Strategic Reasoning 38/55

106 Reasoning about (strategic) games Eric Pacuit: Models of Strategic Reasoning 39/55

107 Reasoning about (strategic) games There is Kripke structure built in a strategic game. W = {σ σ is a strategy profile: σ Π i N S i } (d, a) (d, b) (d, c) a b c d (2,3) (2,2) (1,1) e (0,2) (4,0) (1,0) f (0,1) (1,4) (2,0) (e, a) (e, b) (e, c) (f, a) (f, b) (f, c) Eric Pacuit: Models of Strategic Reasoning 39/55

108 Reasoning about (strategic) games σ i σ iff σ i = σ i : this epistemic relation represents player i s view of the game at the ex interim stage where i s choice is fixed but the choices of the other players are unknown (d, a) (d, b) (d, c) a b c d (2,3) (2,2) (1,1) e (0,2) (4,0) (1,0) f (0,1) (1,4) (2,0) (e, a) (e, b) (e, c) (f, a) (f, b) (f, c) Eric Pacuit: Models of Strategic Reasoning 39/55

109 Reasoning about (strategic) games σ i σ iff σ i = σ i : this relation of action freedom gives the alternative choices for player i when the other players choices are fixed. (d, a) (d, b) (d, c) a b c d (2,3) (2,2) (1,1) e (0,2) (4,0) (1,0) f (0,1) (1,4) (2,0) (e, a) (e, b) (e, c) (f, a) (f, b) (f, c) Eric Pacuit: Models of Strategic Reasoning 39/55

110 Reasoning about (strategic) games σ i σ iff player i prefers the outcome σ at least as much as outcome σ (e, a) (f, a) a b c d (2,3) (2,2) (1,1) e (0,2) (4,0) (1,0) f (0,1) (1,4) (2,0) (f, b) (d, c) (e, c) (d, a) (d, b) (f, c) (e, b) Eric Pacuit: Models of Strategic Reasoning 39/55

111 Reasoning about (strategic) games M = W, { i } i N, { i } i N, { i } i N σ = [ i ]ϕ iff for all σ, if σ i σ then σ = ϕ. σ = [ i ]ϕ iff for all σ, if σ i σ then σ = ϕ. σ = i ϕ iff there exists σ such that σ i σ and σ = ϕ. σ = i ϕ iff there is a σ with σ i σ, σ i σ, and σ = ϕ the equivalence [ i ][ i ]ϕ [ i ][ i ]ϕ is valid Eric Pacuit: Models of Strategic Reasoning 39/55

112 Rationality Announcements: Theorem Weak Rationality: w = WR j means a w(j) j thinks that j s current action is at least as good for j as a., where the a s run over the current model. Theorem The following are equivalent for all states s in a full game model 1. s survives iterated removal of strongly dominated strategies 2. repeated successive public announcements of WR for the players stabilizes at a submodel whose domain contains s. J. van Benthem. Rational dynamics and epistemic logic in games. International Game Theory Review 9, 1 (2007), Eric Pacuit: Models of Strategic Reasoning 40/55

113 Dynamics for the tree Where do the models satisfying common knowledge/belief of rationality come from? J. van Benthem and A. Gheerbrant. Game solution, epistemic dynamics and fixedpoint logics. Fund. Inform., 100 (2010) Eric Pacuit: Models of Strategic Reasoning 41/55

114 Dynamics for the tree x 1, 0 A y 0, 5 E A z u 6, 4 5, 5 Eric Pacuit: Models of Strategic Reasoning 41/55

115 Dynamics for the tree A A A x 1, 0 E Rat = x 1, 0 E Rat = x 1, 0 y 0, 5 A y 0, 5 A z 6, 4 Eric Pacuit: Models of Strategic Reasoning 41/55

116 Dynamics for the tree A A A x 1, 0 E x 1, 0 E x 1, 0 E y z 0, , 99 y z 0, , 99 y z 0, , 99 x y z rat = x y > z rat = x > y > z Eric Pacuit: Models of Strategic Reasoning 41/55

117 The Dynamics of Rational Play A. Baltag, S. Smets and J. Zvesper. Keep hoping for rationality: a solution to the backward induction paradox. Synthese, 169, pgs , Eric Pacuit: Models of Strategic Reasoning 42/55

118 Hard vs. Soft Information in a Game The structure of the game and past moves are hard information : irrevocably known Eric Pacuit: Models of Strategic Reasoning 43/55

119 Hard vs. Soft Information in a Game The structure of the game and past moves are hard information : irrevocably known Players knowledge of other players rationality and knowledge of her own future moves at nodes not yet reached are not of the same degree of certainty. Eric Pacuit: Models of Strategic Reasoning 43/55

120 Hard vs. Soft Information in a Game The structure of the game and past moves are hard information : irrevocably known Players knowledge of other players rationality and knowledge of her own future moves at nodes not yet reached are not of the same degree of certainty. Eric Pacuit: Models of Strategic Reasoning 43/55

121 I i I 2 I v 1 v 2 v 3 o 4 4, 4 O1 O2 O3 1, 1 0, 3 5, 2 o 1 o 2 o 3 o 4 o 3 2 o w w o 1 o 1 w w 4 o w o 2 1 w Eric Pacuit: Models of Strategic Reasoning 44/45 w 3 2 o 2 w 3 Eric Pacuit: Models of Strategic Reasoning 44/55

122 There are atomic propositions for each possible outcome o i is true only at state o i ). Eric Pacuit: Models of Strategic Reasoning 45/55

123 There are atomic propositions for each possible outcome o i is true only at state o i ). The non-terminal nodes v V are then identified with the set of outcomes reachable from that node: v := o v o Eric Pacuit: Models of Strategic Reasoning 45/55

124 There are atomic propositions for each possible outcome o i is true only at state o i ). The non-terminal nodes v V are then identified with the set of outcomes reachable from that node: v := o v o Open future: none of the players have hard information that an outcome is ruled out Eric Pacuit: Models of Strategic Reasoning 45/55

125 2 1 w 2 o 3 1 w 1 o w 4 o 1 2 o 2 w 3 Eric Pacuit: Models of Strategic Reasoning 46/55

126 2 1 w 2 o 3 1 w 1 o w 4 o 1 2 o 2 w 3 Player 1 is committed to the BI strategy is encoded in the conditional beliefs of the player: both B v 1 1 o 1 and B v 3 1 o 3 are true in the previous model. Eric Pacuit: Models of Strategic Reasoning 46/55

127 2 1 w 2 o 3 1 w 1 o w 4 o 1 2 o 2 w 3 Player 1 is committed to the BI strategy is encoded in the conditional beliefs of the player: both B v 1 1 o 1 and B v 3 1 o 3 are true in the previous model. For player 2, B v 2 2 (o 3 o 4 ) is true in the above model, which implies player 2 plans on choosing action I 2 at node v 2. Eric Pacuit: Models of Strategic Reasoning 46/55

128 The players belief change as they learn (irrevocably) which of the nodes in the game are reached: M = M!v 1 ; M!v 2 ; M!v 3 ; M!o 4 Eric Pacuit: Models of Strategic Reasoning 47/55

129 The players belief change as they learn (irrevocably) which of the nodes in the game are reached: M = M!v 1 ; M!v 2 ; M!v 3 ; M!o 4 The assumption that the players are incurably optimistic is represented as follows: no matter what true formula is publicly announced (i.e., no matter how the game proceeds), there is common belief that the players will make a rational choice (when it is their turn to move). Eric Pacuit: Models of Strategic Reasoning 47/55

130 The players belief change as they learn (irrevocably) which of the nodes in the game are reached: M = M!v 1 ; M!v 2 ; M!v 3 ; M!o 4 The assumption that the players are incurably optimistic is represented as follows: no matter what true formula is publicly announced (i.e., no matter how the game proceeds), there is common belief that the players will make a rational choice (when it is their turn to move). M, w = [! ]ϕ provided for all formulas ψ if M, w = ψ then M, w = [!ψ]ϕ. Eric Pacuit: Models of Strategic Reasoning 47/55

131 Theorem (Baltag, Smets and Zvesper). Common knowledge of the game structure, of open future and common stable belief in dynamic rationality implies common belief in the backward induction outcome. Ck(Struct G F G [! ]CbRat) Cb(BI G ) Eric Pacuit: Models of Strategic Reasoning 48/55

132 l r u 3, 3 0, 0 d 0, 0 1, 1 b a w 2 d, l a w 1 d, r a a, b b a w 4 u, l b u, r w 3 Eric Pacuit: Models of Strategic Reasoning 49/55

133 EP and O. Roy. A Dynamic Analysis of Interactive Rationality. Proceedings of LORI- III, Eric Pacuit: Models of Strategic Reasoning 50/55

134 a, b R i c, d R i e R i R i a, b R i c R i f R i R i a R i c, d R i e R i R i nothing new τ(ϕ 1 ) τ(ϕ 2 ) τ(ϕ 3 ) τ(ϕ n ) M 0 M 1 M 2 M f initial model fixed-point Where do the ϕ k come from? from the players categorization (e.g., practical reasoning) Eric Pacuit: Models of Strategic Reasoning 51/55

135 a, b R i c, d R i e R i R i a, b R i c R i f R i R i a R i c, d R i e R i R i nothing new τ(ϕ 1 ) τ(ϕ 2 ) τ(ϕ 3 ) τ(ϕ n ) M 0 M 1 M 2 M f initial model fixed-point Where do the ϕ k come from? from the players practical reasoning (i.e., their categorization of their feasible moves) Eric Pacuit: Models of Strategic Reasoning 51/55

136 Iterated Admissibility Bob T L R Ann T 1,1 1,0 U B 1,0 0,1 U T weakly dominates B Eric Pacuit: Models of Strategic Reasoning 52/55

137 Iterated Admissibility Bob T L R Ann T 1,1 1,0 U B 1,0 0,1 U T weakly dominates B Eric Pacuit: Models of Strategic Reasoning 52/55

138 Iterated Admissibility Bob T L R Ann T 1,1 1,0 U B 1,0 0,1 U Then L strictly dominates R. Eric Pacuit: Models of Strategic Reasoning 52/55

139 Iterated Admissibility Bob T L R Ann T 1,1 1,0 U B 1,0 0,1 U The IA set Eric Pacuit: Models of Strategic Reasoning 52/55

140 Iterated Admissibility Bob T L R Ann T 1,1 1,0 U B 1,0 0,1 U But, now what is the reason for not playing B? Eric Pacuit: Models of Strategic Reasoning 52/55

141 A Dynamic Analysis of Iterated Admissibility L R u 1,1 1,0 d 1,0 0,1 u, L u, R d, L d, R D 0 d, L d, R u, L u, R M 0 D 1 M 1 d, L d, R u, R u, L D 2 d, R u, R D 3 u, L d, L u, R d, R D 0 u, L d, L M 2 M 3 M 4 = M 0 Eric Pacuit: Models of Strategic Reasoning 53/55

142 A Dynamic Analysis of Iterated Admissibility L R u 1,1 1,0 d 1,0 0,1 u, L u, R d, L d, R D 0 d, L d, R u, L u, R M 0 D 1 M 1 d, L d, R u, R u, L D 2 d, R u, R D 3 u, L d, L u, R d, R D 0 u, L d, L M 2 M 3 M 4 = M 0 Eric Pacuit: Models of Strategic Reasoning 53/55

143 A Dynamic Analysis of Iterated Admissibility L R u 1,1 1,0 d 1,0 0,1 u, L u, R d, L d, R D 0 d, L d, R u, L u, R M 0 D 1 M 1 d, L d, R u, R u, L D 2 d, R u, R D 3 u, L d, L u, R d, R D 0 u, L d, L M 2 M 3 M 4 = M 0 Eric Pacuit: Models of Strategic Reasoning 53/55

144 A Dynamic Analysis of Iterated Admissibility L R u 1,1 1,0 d 1,0 0,1 u, L u, R d, L d, R D 0 d, L d, R u, L u, R M 0 D 1 M 1 d, L d, R u, R u, L D 2 d, R u, R D 3 u, L d, L u, R d, R D 0 u, L d, L M 2 M 3 M 4 = M 0 Eric Pacuit: Models of Strategic Reasoning 53/55

145 A Dynamic Analysis of Iterated Admissibility L R u 1,1 1,0 d 1,0 0,1 u, L u, R d, L d, R D 0 d, L d, R u, L u, R M 0 D 1 M 1 d, L d, R u, R u, L D 2 d, R u, R D 3 u, L d, L u, R d, R D 0 u, L d, L M 2 M 3 M 4 = M 0 Eric Pacuit: Models of Strategic Reasoning 53/55

146 A Dynamic Analysis of Iterated Admissibility L R u 1,1 1,0 d 1,0 0,1 u, L u, R d, L d, R D 0 d, L d, R u, L u, R M 0 D 1 M 1 d, L d, R u, R u, L D 2 d, R u, R D 3 u, L d, L u, R d, R D 0 u, L d, L M 2 M 3 M 4 = M 0 Eric Pacuit: Models of Strategic Reasoning 53/55

147 A Dynamic Analysis of Iterated Admissibility L R u 1,1 1,0 d 1,0 0,1 u, L u, R d, L d, R D 0 d, L d, R u, L u, R M 0 D 1 M 1 d, L d, R u, R u, L D 2 d, R u, R D 3 u, L d, L u, R d, R D 0 u, L d, L M 2 M 3 M 4 = M 0 Eric Pacuit: Models of Strategic Reasoning 53/55

148 A Dynamic Analysis of Iterated Admissibility L R u 1,1 1,0 d 1,0 0,1 u, L u, R d, L d, R D 0 d, L d, R u, L u, R M 0 D 1 M 1 d, L d, R u, R u, L D 2 d, R u, R D 3 u, L d, L u, R d, R D 0 u, L d, L M 2 M 3 M 4 = M 0 Eric Pacuit: Models of Strategic Reasoning 53/55

149 Suspending Judgement Both ϕ 1 and ϕ 2 describe good options... F G C D E ϕ 2 ϕ 1 A B {ϕ 1, ϕ 2 } : A E B C D F G {ϕ 1, ϕ 2 } : A E B C D F G Eric Pacuit: Models of Strategic Reasoning 54/55

150 Suspending Judgement Both ϕ 1 and ϕ 2 describe good options... F G C D E ϕ 2 ϕ 1 A B {ϕ 1, ϕ 2 } : A E B C D F G {ϕ 1, ϕ 2 } : A E B C D F G Eric Pacuit: Models of Strategic Reasoning 54/55

151 Suspending Judgement Both ϕ 1 and ϕ 2 describe good options... F G C D E ϕ 2 ϕ 1 A B {ϕ 1, ϕ 2 } : A E B C D F G {ϕ 1, ϕ 2 } : A E B C D F G Eric Pacuit: Models of Strategic Reasoning 54/55

152 Suspending Judgement Both ϕ 1 and ϕ 2 describe good options... F G C D E ϕ 2 ϕ 1 A B {ϕ 1, ϕ 2 } : A E B C D F G {ϕ 1, ϕ 2 } : A E B C D F G Eric Pacuit: Models of Strategic Reasoning 54/55

153 Suspending Judgement Both ϕ 1 and ϕ 2 describe good options... F G C D E ϕ 2 ϕ 1 A B {ϕ 1, ϕ 2 } : A E B C D F G {ϕ 1, ϕ 2 } : A E B C D F G Eric Pacuit: Models of Strategic Reasoning 54/55

154 Remembering Reasons L R u 1,1 1,0 d 1,0 0,1 u, L u, R D 0 d, L d, R D 1 d, R u, R d, L d, R u, L u, R d, L u, L M 0 M 1 M 2 Eric Pacuit: Models of Strategic Reasoning 55/55