Logic and Artificial Intelligence Lecture 24
|
|
- Melina Fletcher
- 5 years ago
- Views:
Transcription
1 Logic and Artificial Intelligence Lecture 24 Eric Pacuit Currently Visiting the Center for Formal Epistemology, CMU Center for Logic and Philosophy of Science Tilburg University ai.stanford.edu/ epacuit November 30, 2011 Logic and Artificial Intelligence 1/36
2 Merging Logics of Rational Agency Entangling Knowledge/Beliefs and Preferences Epistemizing Logics of Action and Ability BDI (Belief + Desires + Intentions) Logics Logic and Artificial Intelligence 2/36
3 Logic and Game Theory Game theory is full of deep puzzles, and there is often disagreement about proposed solutions to them. Logic and Artificial Intelligence 3/36
4 Logic and Game Theory Game theory is full of deep puzzles, and there is often disagreement about proposed solutions to them. The puzzlement and disagreement are neither empirical nor mathematical but, rather, concern the meanings of fundamental concepts ( solution, rational, complete information ) and the soundness of certain arguments... Logic and Artificial Intelligence 3/36
5 Logic and Game Theory Game theory is full of deep puzzles, and there is often disagreement about proposed solutions to them. The puzzlement and disagreement are neither empirical nor mathematical but, rather, concern the meanings of fundamental concepts ( solution, rational, complete information ) and the soundness of certain arguments...logic appears to be an appropriate tool for game theory both because these conceptual obscurities involve notions such as reasoning, knowledge and counter-factuality which are part of the stock-in-trade of logic, and because it is a prime function of logic to establish the validity or invalidity of disputed arguments. M.O.L. Bacharach. Logic and the Epistemic Foundations of Game Theory Logic and Artificial Intelligence 3/36
6 (Modal) Logic in Games M. Pauly and W. van der Hoek. Modal Logic for Games and Information. Handbook of Modal Logic (2006). G. Bonanno. Modal Logic and Game Theory: Two Alternative Approaches. Risk Decision and Policy 7 (2002). J. van Benthem. Extensive Games as Process Models. Journal of Logic, Language and Information 11 (2002). J. Halpern. A Computer Scientist Looks at Game Theory. Games and Economic Behavior 45:1 (2003). R. Parikh. Social Software. Synthese 132: 3 (2002). Logic and Artificial Intelligence 4/36
7 Many topics... Social Procedures: Fair-Division Algorithms, Voting Procedures, Cake-Cutting Algorithms Logics of rational agency Logics of rational interaction Game Logics When are two games the same? Epistemic program in game theory Social Choice Theory and Logic (Formally) Verifying that a social procedure is correct Develop ( well-behaved ) logical languages that can express game theoretic concepts, such as the Nash equilibrium Logic and Artificial Intelligence 5/36
8 Games for Logic Logic and Artificial Intelligence 6/36
9 Games for Logic A E p E A A q r p q p r A can force {p}, {q, r}, E can force {p, q}, {p, r} Logic and Artificial Intelligence 6/36
10 Games for Logic A E p E A A q r p q p r A can force {p}, {q, r}, E can force {p, q}, {p, r} Logic and Artificial Intelligence 6/36
11 Games for Logic A E p E A A q r p q p r A can force {p}, {q, r}, E can force {p, q}, {p, r} Logic and Artificial Intelligence 6/36
12 Games for Logic A E p E A A q r p q p r A can force {p}, {q, r}, E can force {p, q}, {p, r} Logic and Artificial Intelligence 6/36
13 Games for Logic A E p E A A q r p q p r A can force {p}, {q, r}, E can force {p, q}, {p, r} Logic and Artificial Intelligence 6/36
14 Games for Logic A E p E A A q r p q p r A can force {p}, {q, r}, E can force {p, q}, {p, r} Logic and Artificial Intelligence 6/36
15 Games for Logic A E p E A A q r p q p r p (q r) (p q) (p r) Logic and Artificial Intelligence 6/36
16 A primer on game theoretic models (extensive/normal form games) Logic and Artificial Intelligence 7/36
17 Game Situations Bob U L R Ann U 1,1 0,0 U D 0,0 1,1 U 1. a group of self-interested agents (players) involved in some interdependent decision problem, and the players recognize that they are engaged in a game situation Logic and Artificial Intelligence 8/36
18 Game Situations Bob U L R Ann U 1,1 0,0 U D 0,0 1,1 U 1. a group of self-interested agents (players) involved in some interdependent decision problem, and the players recognize that they are engaged in a game situation Logic and Artificial Intelligence 8/36
19 Game Situations Bob U L R Ann U 1,1 0,0 U D 0,0 1,1 U 1. a group of self-interested agents (players) involved in some interdependent decision problem, and the players recognize that they are engaged in a game situation Logic and Artificial Intelligence 8/36
20 Game Situations Bob U L R Ann U 1,1 0,0 U D 0,0 1,1 U 1. a group of self-interested agents (players) involved in some interdependent decision problem, and the players recognize that they are engaged in a game situation Logic and Artificial Intelligence 8/36
21 Game Situations Bob U L R Ann U 1,1 0,0 U D 0,0 1,1 U 1. a group of self-interested agents (players) involved in some interdependent decision problem, and the players recognize that they are engaged in a game situation Logic and Artificial Intelligence 8/36
22 Game Situations Bob U L R Ann U 1,1 0,0 U D 0,0 1,1 U 1. a group of self-interested agents (players) involved in some interdependent decision problem, and the players recognize that they are engaged in a game situation Logic and Artificial Intelligence 8/36
23 Game Situations Bob U L R Ann U 1,1 0,0 U D 0,0 1,1 U 1. a group of self-interested agents (players) involved in some interdependent decision problem, and 2. the players recognize that they are engaged in a game situation Logic and Artificial Intelligence 8/36
24 Game Situations Bob U L R Ann U 1,1 0,0 U D 0,0 1,1 U What should Ann (Bob) do? asdfasdf asdf asdfjasdfasd f asdf asd f asd fasd It depends on what she expects Bob to do, but this depends on what she thinks Bob expects her to do, and so on... Logic and Artificial Intelligence 8/36
25 Game Situations Bob U L R Ann U 1,1 0,0 U D 0,0 1,1 U What should Ann (Bob) do? asdfasdf asdf asdfjasdfasd f asdf asd f asd fasd It depends on what she expects Bob to do, but this depends on what she thinks Bob expects her to do, and so on... Logic and Artificial Intelligence 8/36
26 Just Enough Game Theory Osborne and Rubinstein. Introduction to Game Theory. MIT Press. Logic and Artificial Intelligence 9/36
27 Just Enough Game Theory Osborne and Rubinstein. Introduction to Game Theory. MIT Press. A game is a description of strategic interaction that includes actions the players can take description of the players interests (i.e., preferences), description of the structure of the decision problem Logic and Artificial Intelligence 9/36
28 Just Enough Game Theory Osborne and Rubinstein. Introduction to Game Theory. MIT Press. A game is a description of strategic interaction that includes actions the players can take description of the players interests (i.e., preferences), description of the structure of the decision problem It does not specify the actions that the players do take. Logic and Artificial Intelligence 9/36
29 Solution Concepts A solution concept is a systematic description of the outcomes that may emerge in a family of games. This is the starting point for most of game theory and includes many variants: Nash equilibrium, backwards inductions, or iterated dominance of various kinds. These are usually thought of as the embodiment of rational behavior in some way and used to analyze game situations. Logic and Artificial Intelligence 10/36
30 Strategic Games A strategic games is a tuple N, {A i } i N, { i } i N where N is a finite set of players Logic and Artificial Intelligence 11/36
31 Strategic Games A strategic games is a tuple N, {A i } i N, { i } i N where N is a finite set of players for each i N, A i is a nonempty set of actions Logic and Artificial Intelligence 11/36
32 Strategic Games A strategic games is a tuple N, {A i } i N, { i } i N where N is a finite set of players for each i N, A i is a nonempty set of actions for each i N, i is a preference relation on A = Π i N A i (Often i are represented by utility functions u i : A R) Logic and Artificial Intelligence 11/36
33 Strategic Games: Comments on Preferences Preferences may be over a set of consequences C. Assume g : A C and { i i N} a set of preferences on C. Then for a, b A, a i b iff g(a) i g(b) Consequences may be affected by exogenous random variable whose realization is not known before choosing actions. Let Ω be a set of states, then define g : A Ω C. Where g(a ) is interpreted as a lottery. Often i are represented by utility functions u i : A R Logic and Artificial Intelligence 12/36
34 Strategic Games: Example Row N = {Row, Column} r Column u (2,2) (0,0) d (0,0) (1,1) A Row = {u, d}, A Column = {r, l} (u, r) Row (d, l) Row (u, l) Row (d, r) (u, r) Column (d, l) Column (u, l) Column (d, r) l Logic and Artificial Intelligence 13/36
35 Strategic Games: Example Row r Column u (2,2) (0,0) d (0,0) (1,1) N = {Row, Column} A Row = {u, d}, A Column = {r, l} u Row : A Row A Column {0, 1, 2}, u Column : A Row A Column {0, 1, 2} with u Row (u, r) = u Column (u, r) = 2, u Row (d, l) = u Column (d, l) = 2, and u x (u, l) = u x (d, r) = 0 for x N. l Logic and Artificial Intelligence 13/36
36 Nash Equilibrium Let N, {A i } i N, { i } i N be a strategic game For a i A i, let B i (a i ) = {a i A i (a i, a i ) i (a i, a i) a i A i } B i is the best-response function. Logic and Artificial Intelligence 14/36
37 Nash Equilibrium Let N, {A i } i N, { i } i N be a strategic game For a i A i, let B i (a i ) = {a i A i (a i, a i ) i (a i, a i) a i A i } B i is the best-response function. a A is a Nash equilibrium iff ai B i (a i ) for all i N. Logic and Artificial Intelligence 14/36
38 Strategic Games Example: Bach or Stravinsky? b c s c b r 2,1 0,0 s r 0,0 1,2 N = {r, c} A r = {b r, s r }, A c = {b c, s c } B r (b c ) = {b r } B r (s c ) = {s r } B c (b r ) = {b c } B c (s r ) = {s c } (b r, b c ) is a Nash Equilibrium (s r, s c ) is a Nash Equilibrium Logic and Artificial Intelligence 15/36
39 Strategic Games Example: Bach or Stravinsky? b c s c b r 2,1 0,0 s r 0,0 1,2 N = {r, c} A r = {b r, s r }, A c = {b c, s c } B r (b c ) = {b r } B r (s c ) = {s r } B c (b r ) = {b c } B c (s r ) = {s c } (b r, b c ) is a Nash Equilibrium (s r, s c ) is a Nash Equilibrium Logic and Artificial Intelligence 15/36
40 Strategic Games Example: Bach or Stravinsky? b c s c b r 2,1 0,0 s r 0,0 1,2 N = {r, c} A r = {b r, s r }, A c = {b c, s c } B r (b c ) = {b r } B r (s c ) = {s r } B c (b r ) = {b c } B c (s r ) = {s c } (b r, b c ) is a Nash Equilibrium (s r, s c ) is a Nash Equilibrium Logic and Artificial Intelligence 15/36
41 Strategic Games Example: Bach or Stravinsky? b c s c b r 2,1 0,0 s r 0,0 1,2 N = {r, c} A r = {b r, s r }, A c = {b c, s c } B r (b c ) = {b r } B r (s c ) = {s r } B c (b r ) = {b c } B c (s r ) = {s c } (b r, b c ) is a Nash Equilibrium (s r, s c ) is a Nash Equilibrium Logic and Artificial Intelligence 15/36
42 Strategic Games Example: Bach or Stravinsky? b c s c b r 2,1 0,0 s r 0,0 1,2 N = {r, c} A r = {b r, s r }, A c = {b c, s c } B r (b c ) = {b r } B r (s c ) = {s r } B c (b r ) = {b c } B c (s r ) = {s c } (b r, b c ) is a Nash Equilibrium (s r, s c ) is a Nash Equilibrium Logic and Artificial Intelligence 15/36
43 Another Example: Pure Coordination Bob U L R Ann U 1,1 0,0 U D 0,0 1,1 U Logic and Artificial Intelligence 16/36
44 Another Example: Hi-Low Bob U L R Ann U 3,3 0,0 U D 0,0 1,1 U Logic and Artificial Intelligence 17/36
45 Reasoning about (strategic) games Logic and Artificial Intelligence 18/36
46 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) Logic and Artificial Intelligence 18/36
47 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) Logic and Artificial Intelligence 18/36
48 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) Logic and Artificial Intelligence 18/36
49 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) Logic and Artificial Intelligence 18/36
50 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 Logic and Artificial Intelligence 19/36
51 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 What is the complete logic of finite games? Logic and Artificial Intelligence 19/36
52 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 on full games Logic and Artificial Intelligence 19/36
53 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 Can we modally define the Nash Equilibrium? Logic and Artificial Intelligence 19/36
54 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 Can we modally define the Nash Equilibrium? Nash := i N Br i Logic and Artificial Intelligence 19/36
55 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 Can we modally define the best response for i? Logic and Artificial Intelligence 19/36
56 Reasoning about strategic games a b c d (2,3) (2,2) (1,1) e (0,2) (4,0) (1,0) f (0,1) (1,4) (2,0) Logic and Artificial Intelligence 19/36
57 Reasoning about strategic games (e, a) (f, a) (d, a) (d, b) (d, c) (f, b) (d, c) (e, c) (e, a) (e, b) (e, c) (d, a) (d, b) (f, c) (f, a) (f, b) (f, c) (e, b) σ = i : there is no outcome at least as good as σ Logic and Artificial Intelligence 19/36
58 Reasoning about strategic games (e, a) (f, a) (d, a) (d, b) (d, c) (f, b) (d, c) (e, c) (e, a) (e, b) (e, c) (d, a) (d, b) (f, c) (f, a) (f, b) (f, c) (e, b) σ = i : there is no outcome at least as good as σ Logic and Artificial Intelligence 19/36
59 Reasoning about strategic games (e, a) (f, a) (d, a) (d, b) (d, c) (f, b) (d, c) (e, c) (e, a) (e, b) (e, c) (d, a) (d, b) (f, c) (f, a) (f, b) (f, c) (e, b) (d, a) = i : there is no outcome at least as good as σ Logic and Artificial Intelligence 19/36
60 Reasoning about strategic games (e, a) (f, a) (d, a) (d, b) (d, c) (f, b) (d, c) (e, c) (e, a) (e, b) (e, c) (d, a) (d, b) (f, c) (f, a) (f, b) (f, c) (e, b) there is no outcome which i can choose that is at least as good Logic and Artificial Intelligence 19/36
61 Reasoning about strategic games (e, a) (f, a) (d, a) (d, b) (d, c) (f, b) (d, c) (e, c) (e, a) (e, b) (e, c) (d, a) (d, b) (f, c) (f, a) (f, b) (f, c) (e, b) σ = i i ϕ iff there is σ such that σ( i i )σ and σ = ϕ Logic and Artificial Intelligence 19/36
62 Reasoning about strategic games (e, a) (f, a) (d, a) (d, b) (d, c) (f, b) (d, c) (e, c) (e, a) (e, b) (e, c) (d, a) (d, b) (f, c) (f, a) (f, b) (f, c) (e, b) the best response for player i is defined as i i Logic and Artificial Intelligence 19/36
63 Reasoning about extensive games Logic and Artificial Intelligence 20/36
64 Reasoning about extensive games A B B (1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) Logic and Artificial Intelligence 20/36
65 Reasoning about extensive games Invented by Zermelo, Backwards Induction is an iterative algorithm for solving and extensive game. Logic and Artificial Intelligence 20/36
66 Reasoning about extensive games A B B (1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) Logic and Artificial Intelligence 20/36
67 Reasoning about extensive games A B B (1, 0) (2, 3) (1, 5) (4, 4) (3, 1) (4, 4) Logic and Artificial Intelligence 20/36
68 Reasoning about extensive games A B B (1, 0) (2, 3) (1, 5) (4, 4) (3, 1) (4, 4) Logic and Artificial Intelligence 20/36
69 Reasoning about extensive games A (2, 3) B (1, 0) (2, 3) (1, 5) (4, 4) (3, 1) (4, 4) Logic and Artificial Intelligence 20/36
70 Reasoning about extensive games A (2, 3) B (1, 0) (2, 3) (1, 5) (4, 4) (3, 1) (4, 4) Logic and Artificial Intelligence 20/36
71 Reasoning about extensive games A (2, 3) (1, 5) (1, 0) (2, 3) (1, 5) (4, 4) (3, 1) (4, 4) Logic and Artificial Intelligence 20/36
72 Reasoning about extensive games A (2, 3) (1, 5) (1, 0) (2, 3) (1, 5) (4, 4) (3, 1) (4, 4) Logic and Artificial Intelligence 20/36
73 Reasoning about extensive games (2, 3) (2, 3) (1, 5) (1, 0) (2, 3) (1, 5) (4, 4) (3, 1) (4, 4) Logic and Artificial Intelligence 20/36
74 Reasoning about extensive games A B B (1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) Logic and Artificial Intelligence 20/36
75 Reasoning about extensive games A B B (1, 0) (2, 3) (1, 5) A (3, 1) (4, 4) Logic and Artificial Intelligence 20/36
76 Characterizing Backwards Induction For each extensive game form, the strategy profile σ is a backward induction solution iff σ is played at the root of a tree satisfying the following modal axiom for all propositions p and players i: (turn i σ (end p)) [move i ] σ (end i p) move i = a is an i-move a, turn i is a propositional variable saying that it is i s turn to move, and end is a propositional variable true at only end nodes J. van Benthem, S. van Otterloo and O. Roy. Preference Logic, Conditionals, and Solution Concepts in Games. In Modality Matters: Twenty-Five Essays in Honour of Krister Segerberg, Logic and Artificial Intelligence 21/36
77 Characterizing Backwards Induction σ x y z via σ u via σ v Logic and Artificial Intelligence 21/36
78 Modal Languages for Games [a b] c d p: for each choice between a or b there is a choice between c or d ending in a p-state. Logic and Artificial Intelligence 22/36
79 Modal Languages for Games [a b] c d p: for each choice between a or b there is a choice between c or d ending in a p-state. Strategies as programs: [((turn E )?; σ) (turn A?; τ)) ](end p) Logic and Artificial Intelligence 22/36
80 Modal Languages for Games [a b] c d p: for each choice between a or b there is a choice between c or d ending in a p-state. Strategies as programs: [((turn E )?; σ) (turn A?; τ)) ](end p) A ϕ: A has a strategy to ensure that ϕ is true Logic and Artificial Intelligence 22/36
81 Modal Languages for Games [a b] c d p: for each choice between a or b there is a choice between c or d ending in a p-state. Strategies as programs: [((turn E )?; σ) (turn A?; τ)) ](end p) A ϕ: A has a strategy to ensure that ϕ is true Knowledge/beliefs: Logic and Artificial Intelligence 22/36
82 Modal Languages for Games [a b] c d p: for each choice between a or b there is a choice between c or d ending in a p-state. Strategies as programs: [((turn E )?; σ) (turn A?; τ)) ](end p) A ϕ: A has a strategy to ensure that ϕ is true Knowledge/beliefs: K E ( a p b p), K E a p K E b p Logic and Artificial Intelligence 22/36
83 Modal Languages for Games [a b] c d p: for each choice between a or b there is a choice between c or d ending in a p-state. Strategies as programs: [((turn E )?; σ) (turn A?; τ)) ](end p) A ϕ: A has a strategy to ensure that ϕ is true Knowledge/beliefs: K E ( a p b p), K E a p K E b p K A E ϕ vs. E K A ϕ Logic and Artificial Intelligence 22/36
84 Modal Languages for Games [a b] c d p: for each choice between a or b there is a choice between c or d ending in a p-state. Strategies as programs: [((turn E )?; σ) (turn A?; τ)) ](end p) A ϕ: A has a strategy to ensure that ϕ is true Knowledge/beliefs: K E ( a p b p), K E a p K E b p K A E ϕ vs. E K A ϕ preferences,... Logic and Artificial Intelligence 22/36
85 Reasoning with games Logic and Artificial Intelligence 23/36
86 Background: Propositional Dynamic Logic Let P be a set of atomic programs and At a set of atomic propositions. Formulas of PDL have the following syntactic form: where p At and a P. ϕ := p ϕ ϕ ψ [α]ϕ α := a α β α; β α ϕ? [α]ϕ is intended to mean after executing the program α, ϕ is true Logic and Artificial Intelligence 24/36
87 Background: Propositional Dynamic Logic Semantics: M = W, {R a a P}, V where for each a P, R a W W and V : At (W ) R α β := R α R β R α;β := R α R β R α := n 0 R n α R ϕ? = {(w, w) M, w = ϕ} M, w = [α]ϕ iff for each v, if wr α v then M, v = ϕ Logic and Artificial Intelligence 25/36
88 Background: Propositional Dynamic Logic 1. Axioms of propositional logic 2. [α](ϕ ψ) ([α]ϕ [α]ψ) 3. [α β]ϕ [α]ϕ [β]ϕ 4. [α; β]ϕ [α][β]ϕ 5. [ψ?]ϕ (ψ ϕ) 6. ϕ [α][α ]ϕ [α ]ϕ 7. ϕ [α ](ϕ [α]ϕ) [α ]ϕ 8. Modus Ponens and Necessitation (for each program α) Logic and Artificial Intelligence 26/36
89 Background: Propositional Dynamic Logic 1. Axioms of propositional logic 2. [α](ϕ ψ) ([α]ϕ [α]ψ) 3. [α β]ϕ [α]ϕ [β]ϕ 4. [α; β]ϕ [α][β]ϕ 5. [ψ?]ϕ (ψ ϕ) 6. ϕ [α][α ]ϕ [α ]ϕ (Fixed-Point Axiom) 7. ϕ [α ](ϕ [α]ϕ) [α ]ϕ (Induction Axiom) 8. Modus Ponens and Necessitation (for each program α) Logic and Artificial Intelligence 27/36
90 From PDL to Game Logic R. Parikh. The Logic of Games and its Applications.. Annals of Discrete Mathematics. (1985). Logic and Artificial Intelligence 28/36
91 From PDL to Game Logic R. Parikh. The Logic of Games and its Applications.. Annals of Discrete Mathematics. (1985). Main Idea: In PDL: w = π ϕ: there is a run of the program π starting in state w that ends in a state where ϕ is true. The programs in PDL can be thought of as single player games. Logic and Artificial Intelligence 28/36
92 From PDL to Game Logic R. Parikh. The Logic of Games and its Applications.. Annals of Discrete Mathematics. (1985). Main Idea: In PDL: w = π ϕ: there is a run of the program π starting in state w that ends in a state where ϕ is true. The programs in PDL can be thought of as single player games. Game Logic generalized PDL by considering two players: In GL: w = γ ϕ: Angel has a strategy in the game γ to ensure that the game ends in a state where ϕ is true. Logic and Artificial Intelligence 28/36
93 From PDL to Game Logic Consequences of two players: Logic and Artificial Intelligence 29/36
94 From PDL to Game Logic Consequences of two players: γ ϕ: Angel has a strategy in γ to ensure ϕ is true [γ]ϕ: Demon has a strategy in γ to ensure ϕ is true Logic and Artificial Intelligence 29/36
95 From PDL to Game Logic Consequences of two players: γ ϕ: Angel has a strategy in γ to ensure ϕ is true [γ]ϕ: Demon has a strategy in γ to ensure ϕ is true Either Angel or Demon can win: γ ϕ [γ] ϕ Logic and Artificial Intelligence 29/36
96 From PDL to Game Logic Consequences of two players: γ ϕ: Angel has a strategy in γ to ensure ϕ is true [γ]ϕ: Demon has a strategy in γ to ensure ϕ is true Either Angel or Demon can win: γ ϕ [γ] ϕ But not both: ( γ ϕ [γ] ϕ) Logic and Artificial Intelligence 29/36
97 From PDL to Game Logic Consequences of two players: γ ϕ: Angel has a strategy in γ to ensure ϕ is true [γ]ϕ: Demon has a strategy in γ to ensure ϕ is true Either Angel or Demon can win: γ ϕ [γ] ϕ But not both: ( γ ϕ [γ] ϕ) Thus, [γ]ϕ γ ϕ is a valid principle Logic and Artificial Intelligence 29/36
98 From PDL to Game Logic Consequences of two players: γ ϕ: Angel has a strategy in γ to ensure ϕ is true [γ]ϕ: Demon has a strategy in γ to ensure ϕ is true Either Angel or Demon can win: γ ϕ [γ] ϕ But not both: ( γ ϕ [γ] ϕ) Thus, [γ]ϕ γ ϕ is a valid principle However, [γ]ϕ [γ]ψ [γ](ϕ ψ) is not a valid principle Logic and Artificial Intelligence 29/36
99 From PDL to Game Logic Reinterpret operations and invent new ones:?ϕ: Check whether ϕ currently holds γ 1 ; γ 2 : First play γ 1 then γ 2 γ 1 γ 2 : Angel choose between γ 1 and γ 2 γ : Angel can choose how often to play γ (possibly not at all); each time she has played γ, she can decide whether to play it again or not. γ d : Switch roles, then play γ γ 1 γ 2 := (γ d 1 γd 2 )d : Demon chooses between γ 1 and γ 2 γ x := ((γ d ) ) d : Demon can choose how often to play γ (possibly not at all); each time he has played γ, he can decide whether to play it again or not. Logic and Artificial Intelligence 30/36
100 From PDL to Game Logic Reinterpret operations and invent new ones:?ϕ: Check whether ϕ currently holds γ 1 ; γ 2 : First play γ 1 then γ 2 γ 1 γ 2 : Angel choose between γ 1 and γ 2 γ : Angel can choose how often to play γ (possibly not at all); each time she has played γ, she can decide whether to play it again or not. γ d : Switch roles, then play γ γ 1 γ 2 := (γ d 1 γd 2 )d : Demon chooses between γ 1 and γ 2 γ x := ((γ d ) ) d : Demon can choose how often to play γ (possibly not at all); each time he has played γ, he can decide whether to play it again or not. Logic and Artificial Intelligence 30/36
101 From PDL to Game Logic Reinterpret operations and invent new ones:?ϕ: Check whether ϕ currently holds γ 1 ; γ 2 : First play γ 1 then γ 2 γ 1 γ 2 : Angel choose between γ 1 and γ 2 γ : Angel can choose how often to play γ (possibly not at all); each time she has played γ, she can decide whether to play it again or not. γ d : Switch roles, then play γ γ 1 γ 2 := (γ d 1 γd 2 )d : Demon chooses between γ 1 and γ 2 γ x := ((γ d ) ) d : Demon can choose how often to play γ (possibly not at all); each time he has played γ, he can decide whether to play it again or not. Logic and Artificial Intelligence 30/36
102 From PDL to Game Logic Reinterpret operations and invent new ones:?ϕ: Check whether ϕ currently holds γ 1 ; γ 2 : First play γ 1 then γ 2 γ 1 γ 2 : Angel choose between γ 1 and γ 2 γ : Angel can choose how often to play γ (possibly not at all); each time she has played γ, she can decide whether to play it again or not. γ d : Switch roles, then play γ γ 1 γ 2 := (γ d 1 γd 2 )d : Demon chooses between γ 1 and γ 2 γ x := ((γ d ) ) d : Demon can choose how often to play γ (possibly not at all); each time he has played γ, he can decide whether to play it again or not. Logic and Artificial Intelligence 30/36
103 From PDL to Game Logic Reinterpret operations and invent new ones:?ϕ: Check whether ϕ currently holds γ 1 ; γ 2 : First play γ 1 then γ 2 γ 1 γ 2 : Angel choose between γ 1 and γ 2 γ : Angel can choose how often to play γ (possibly not at all); each time she has played γ, she can decide whether to play it again or not. γ d : Switch roles, then play γ γ 1 γ 2 := (γ d 1 γd 2 )d : Demon chooses between γ 1 and γ 2 γ x := ((γ d ) ) d : Demon can choose how often to play γ (possibly not at all); each time he has played γ, he can decide whether to play it again or not. Logic and Artificial Intelligence 30/36
104 From PDL to Game Logic Reinterpret operations and invent new ones:?ϕ: Check whether ϕ currently holds γ 1 ; γ 2 : First play γ 1 then γ 2 γ 1 γ 2 : Angel choose between γ 1 and γ 2 γ : Angel can choose how often to play γ (possibly not at all); each time she has played γ, she can decide whether to play it again or not. γ d : Switch roles, then play γ γ 1 γ 2 := (γ d 1 γd 2 )d : Demon chooses between γ 1 and γ 2 γ x := ((γ d ) ) d : Demon can choose how often to play γ (possibly not at all); each time he has played γ, he can decide whether to play it again or not. Logic and Artificial Intelligence 30/36
105 From PDL to Game Logic Reinterpret operations and invent new ones:?ϕ: Check whether ϕ currently holds γ 1 ; γ 2 : First play γ 1 then γ 2 γ 1 γ 2 : Angel choose between γ 1 and γ 2 γ : Angel can choose how often to play γ (possibly not at all); each time she has played γ, she can decide whether to play it again or not. γ d : Switch roles, then play γ γ 1 γ 2 := (γ d 1 γd 2 )d : Demon chooses between γ 1 and γ 2 γ x := ((γ d ) ) d : Demon can choose how often to play γ (possibly not at all); each time he has played γ, he can decide whether to play it again or not. Logic and Artificial Intelligence 30/36
106 From PDL to Game Logic Reinterpret operations and invent new ones:?ϕ: Check whether ϕ currently holds γ 1 ; γ 2 : First play γ 1 then γ 2 γ 1 γ 2 : Angel choose between γ 1 and γ 2 γ : Angel can choose how often to play γ (possibly not at all); each time she has played γ, she can decide whether to play it again or not. γ d : Switch roles, then play γ γ 1 γ 2 := (γ d 1 γd 2 )d : Demon chooses between γ 1 and γ 2 γ x := ((γ d ) ) d : Demon can choose how often to play γ (possibly not at all); each time he has played γ, he can decide whether to play it again or not. Logic and Artificial Intelligence 30/36
107 Game Logic Syntax Let Γ 0 be a set of atomic games and At a set of atomic propositions. Then formulas of Game Logic are defined inductively as follows: where p At, g Γ 0. γ := g ϕ? γ; γ γ γ γ γ d ϕ := p ϕ ϕ ϕ γ ϕ [γ]ϕ Logic and Artificial Intelligence 31/36
108 Game Logic A neighborhood game model is a tuple M = W, {E g g Γ 0 }, V where W is a nonempty set of states For each g Γ 0, E g : W ( (W )) is a monotonic neighborhood function. X E g (w) means in state s, Angel has a strategy to force the game to end in some state in X (we may write we g X ) V : At (W ) is a valuation function. Logic and Artificial Intelligence 32/36
109 Game Logic Propositional letters and boolean connectives are as usual. M, w = γ ϕ iff (ϕ) M E γ (w) Logic and Artificial Intelligence 33/36
110 Game Logic Propositional letters and boolean connectives are as usual. M, w = γ ϕ iff (ϕ) M E γ (w) Suppose E γ (Y ) := {s Y E g (s)} E γ1 ;γ 2 (Y ) := E γ1 (E γ2 (Y )) E γ1 γ 2 (Y ) := E γ1 (Y ) E γ2 (Y ) E ϕ? (Y ) := (ϕ) M Y E γ d (Y ) := E γ (Y ) E γ (Y ) := µx.y E γ (X ) Logic and Artificial Intelligence 33/36
111 Game Logic: Axioms 1. All propositional tautologies 2. α; β ϕ α β ϕ Composition 3. α β ϕ α ϕ β ϕ Union 4. ψ? ϕ (ψ ϕ) Test 5. α d ϕ α ϕ Dual 6. (ϕ α α ϕ) α ϕ Mix and the rules, ϕ ϕ ψ ψ ϕ ψ α ϕ α ψ (ϕ α ψ) ψ α ϕ ψ Logic and Artificial Intelligence 34/36
112 Game Logic Game Logic is more expressive than PDL Logic and Artificial Intelligence 35/36
113 Game Logic Game Logic is more expressive than PDL (g d ) Logic and Artificial Intelligence 35/36
114 Game Logic Game Logic is more expressive than PDL (g d ) All GL games are determined. Logic and Artificial Intelligence 35/36
115 Game Logic Theorem Dual-free game logic is sound and complete with respect to the class of all game models. Logic and Artificial Intelligence 36/36
116 Game Logic Theorem Dual-free game logic is sound and complete with respect to the class of all game models. Theorem Iteration-free game logic is sound and complete with respect to the class of all game models. Logic and Artificial Intelligence 36/36
117 Game Logic Theorem Dual-free game logic is sound and complete with respect to the class of all game models. Theorem Iteration-free game logic is sound and complete with respect to the class of all game models. Open Question Is (full) game logic complete with respect to the class of all game models? R. Parikh. The Logic of Games and its Applications.. Annals of Discrete Mathematics. (1985). M. Pauly. Logic for Social Software. Ph.D. Thesis, University of Amsterdam (2001). Logic and Artificial Intelligence 36/36
Logic and Artificial Intelligence Lecture 25
Logic and Artificial Intelligence Lecture 25 Eric Pacuit Currently Visiting the Center for Formal Epistemology, CMU Center for Logic and Philosophy of Science Tilburg University ai.stanford.edu/ epacuit
More informationEpistemic Game Theory
Epistemic Game Theory Lecture 1 ESSLLI 12, Opole Eric Pacuit Olivier Roy TiLPS, Tilburg University MCMP, LMU Munich ai.stanford.edu/~epacuit http://olivier.amonbofis.net August 6, 2012 Eric Pacuit and
More informationTR : Knowledge-Based Rational Decisions
City University of New York (CUNY) CUNY Academic Works Computer Science Technical Reports Graduate Center 2009 TR-2009011: Knowledge-Based Rational Decisions Sergei Artemov Follow this and additional works
More informationTR : Knowledge-Based Rational Decisions and Nash Paths
City University of New York (CUNY) CUNY Academic Works Computer Science Technical Reports Graduate Center 2009 TR-2009015: Knowledge-Based Rational Decisions and Nash Paths Sergei Artemov Follow this and
More information6.254 : Game Theory with Engineering Applications Lecture 3: Strategic Form Games - Solution Concepts
6.254 : Game Theory with Engineering Applications Lecture 3: Strategic Form Games - Solution Concepts Asu Ozdaglar MIT February 9, 2010 1 Introduction Outline Review Examples of Pure Strategy Nash Equilibria
More informationReasoning About Others: Representing and Processing Infinite Belief Hierarchies
Reasoning About Others: Representing and Processing Infinite Belief Hierarchies Sviatoslav Brainov and Tuomas Sandholm Department of Computer Science Washington University St Louis, MO 63130 {brainov,
More informationOutline Introduction Game Representations Reductions Solution Concepts. Game Theory. Enrico Franchi. May 19, 2010
May 19, 2010 1 Introduction Scope of Agent preferences Utility Functions 2 Game Representations Example: Game-1 Extended Form Strategic Form Equivalences 3 Reductions Best Response Domination 4 Solution
More informationNotes on Natural Logic
Notes on Natural Logic Notes for PHIL370 Eric Pacuit November 16, 2012 1 Preliminaries: Trees A tree is a structure T = (T, E), where T is a nonempty set whose elements are called nodes and E is a relation
More informationAlgorithmic Game Theory and Applications. Lecture 11: Games of Perfect Information
Algorithmic Game Theory and Applications Lecture 11: Games of Perfect Information Kousha Etessami finite games of perfect information Recall, a perfect information (PI) game has only 1 node per information
More informationOn Existence of Equilibria. Bayesian Allocation-Mechanisms
On Existence of Equilibria in Bayesian Allocation Mechanisms Northwestern University April 23, 2014 Bayesian Allocation Mechanisms In allocation mechanisms, agents choose messages. The messages determine
More informationA Knowledge-Theoretic Approach to Distributed Problem Solving
A Knowledge-Theoretic Approach to Distributed Problem Solving Michael Wooldridge Department of Electronic Engineering, Queen Mary & Westfield College University of London, London E 4NS, United Kingdom
More informationEpistemic Planning With Implicit Coordination
Epistemic Planning With Implicit Coordination Thomas Bolander, DTU Compute, Technical University of Denmark Joint work with Thorsten Engesser, Robert Mattmüller and Bernhard Nebel from Uni Freiburg Thomas
More information6.207/14.15: Networks Lecture 10: Introduction to Game Theory 2
6.207/14.15: Networks Lecture 10: Introduction to Game Theory 2 Daron Acemoglu and Asu Ozdaglar MIT October 14, 2009 1 Introduction Outline Review Examples of Pure Strategy Nash Equilibria Mixed Strategies
More informationStrategy composition in dynamic games with simultaneous moves
Strategy composition in dynamic games with simultaneous moves Sujata Ghosh 1, Neethi Konar 1 and R. Ramanujam 2 1 Indian Statistical Institute, Chennai, India 2 Institute of Mathematical Sciences, Chennai,
More informationPAULI MURTO, ANDREY ZHUKOV. If any mistakes or typos are spotted, kindly communicate them to
GAME THEORY PROBLEM SET 1 WINTER 2018 PAULI MURTO, ANDREY ZHUKOV Introduction If any mistakes or typos are spotted, kindly communicate them to andrey.zhukov@aalto.fi. Materials from Osborne and Rubinstein
More informationLecture 5 Leadership and Reputation
Lecture 5 Leadership and Reputation Reputations arise in situations where there is an element of repetition, and also where coordination between players is possible. One definition of leadership is that
More informationIntroduction to game theory LECTURE 2
Introduction to game theory LECTURE 2 Jörgen Weibull February 4, 2010 Two topics today: 1. Existence of Nash equilibria (Lecture notes Chapter 10 and Appendix A) 2. Relations between equilibrium and rationality
More informationAdvanced Microeconomics
Advanced Microeconomics ECON5200 - Fall 2014 Introduction What you have done: - consumers maximize their utility subject to budget constraints and firms maximize their profits given technology and market
More informationCut-free sequent calculi for algebras with adjoint modalities
Cut-free sequent calculi for algebras with adjoint modalities Roy Dyckhoff (University of St Andrews) and Mehrnoosh Sadrzadeh (Universities of Oxford & Southampton) TANCL Conference, Oxford, 8 August 2007
More informationMicroeconomics II Lecture 8: Bargaining + Theory of the Firm 1 Karl Wärneryd Stockholm School of Economics December 2016
Microeconomics II Lecture 8: Bargaining + Theory of the Firm 1 Karl Wärneryd Stockholm School of Economics December 2016 1 Axiomatic bargaining theory Before noncooperative bargaining theory, there was
More informationGAME THEORY. Department of Economics, MIT, Follow Muhamet s slides. We need the following result for future reference.
14.126 GAME THEORY MIHAI MANEA Department of Economics, MIT, 1. Existence and Continuity of Nash Equilibria Follow Muhamet s slides. We need the following result for future reference. Theorem 1. Suppose
More informationGames of Incomplete Information
Games of Incomplete Information EC202 Lectures V & VI Francesco Nava London School of Economics January 2011 Nava (LSE) EC202 Lectures V & VI Jan 2011 1 / 22 Summary Games of Incomplete Information: Definitions:
More informationStochastic Games and Bayesian Games
Stochastic Games and Bayesian Games CPSC 532l Lecture 10 Stochastic Games and Bayesian Games CPSC 532l Lecture 10, Slide 1 Lecture Overview 1 Recap 2 Stochastic Games 3 Bayesian Games 4 Analyzing Bayesian
More informationMicroeconomics of Banking: Lecture 5
Microeconomics of Banking: Lecture 5 Prof. Ronaldo CARPIO Oct. 23, 2015 Administrative Stuff Homework 2 is due next week. Due to the change in material covered, I have decided to change the grading system
More informationBest response cycles in perfect information games
P. Jean-Jacques Herings, Arkadi Predtetchinski Best response cycles in perfect information games RM/15/017 Best response cycles in perfect information games P. Jean Jacques Herings and Arkadi Predtetchinski
More informationIn the Name of God. Sharif University of Technology. Microeconomics 2. Graduate School of Management and Economics. Dr. S.
In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics 2 44706 (1394-95 2 nd term) - Group 2 Dr. S. Farshad Fatemi Chapter 8: Simultaneous-Move Games
More informationECON322 Game Theory Half II
ECON322 Game Theory Half II Part 1: Reasoning Foundations Rationality Christian W. Bach University of Liverpool & EPICENTER Agenda Introduction Rational Choice Strict Dominance Characterization of Rationality
More informationFDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.
FDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.) Hints for Problem Set 2 1. Consider a zero-sum game, where
More informationG5212: Game Theory. Mark Dean. Spring 2017
G5212: Game Theory Mark Dean Spring 2017 Modelling Dynamics Up until now, our games have lacked any sort of dynamic aspect We have assumed that all players make decisions at the same time Or at least no
More informationCharacterizing Solution Concepts in Terms of Common Knowledge of Rationality
Characterizing Solution Concepts in Terms of Common Knowledge of Rationality Joseph Y. Halpern Computer Science Department Cornell University, U.S.A. e-mail: halpern@cs.cornell.edu Yoram Moses Department
More information6.207/14.15: Networks Lecture 9: Introduction to Game Theory 1
6.207/14.15: Networks Lecture 9: Introduction to Game Theory 1 Daron Acemoglu and Asu Ozdaglar MIT October 13, 2009 1 Introduction Outline Decisions, Utility Maximization Games and Strategies Best Responses
More informationIntroduction to Industrial Organization Professor: Caixia Shen Fall 2014 Lecture Note 5 Games and Strategy (Ch. 4)
Introduction to Industrial Organization Professor: Caixia Shen Fall 2014 Lecture Note 5 Games and Strategy (Ch. 4) Outline: Modeling by means of games Normal form games Dominant strategies; dominated strategies,
More informationEpistemic Experiments: Utilities, Beliefs, and Irrational Play
Epistemic Experiments: Utilities, Beliefs, and Irrational Play P.J. Healy PJ Healy (OSU) Epistemics 2017 1 / 62 Motivation Question: How do people play games?? E.g.: Do people play equilibrium? If not,
More informationGame Theory: Normal Form Games
Game Theory: Normal Form Games Michael Levet June 23, 2016 1 Introduction Game Theory is a mathematical field that studies how rational agents make decisions in both competitive and cooperative situations.
More informationSolutions of Bimatrix Coalitional Games
Applied Mathematical Sciences, Vol. 8, 2014, no. 169, 8435-8441 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.410880 Solutions of Bimatrix Coalitional Games Xeniya Grigorieva St.Petersburg
More informationStrategies and Nash Equilibrium. A Whirlwind Tour of Game Theory
Strategies and Nash Equilibrium A Whirlwind Tour of Game Theory (Mostly from Fudenberg & Tirole) Players choose actions, receive rewards based on their own actions and those of the other players. Example,
More informationA reinforcement learning process in extensive form games
A reinforcement learning process in extensive form games Jean-François Laslier CNRS and Laboratoire d Econométrie de l Ecole Polytechnique, Paris. Bernard Walliser CERAS, Ecole Nationale des Ponts et Chaussées,
More informationIntroduction to Multi-Agent Programming
Introduction to Multi-Agent Programming 10. Game Theory Strategic Reasoning and Acting Alexander Kleiner and Bernhard Nebel Strategic Game A strategic game G consists of a finite set N (the set of players)
More informationarxiv: v1 [cs.gt] 12 Jul 2007
Generalized Solution Concepts in Games with Possibly Unaware Players arxiv:0707.1904v1 [cs.gt] 12 Jul 2007 Leandro C. Rêgo Statistics Department Federal University of Pernambuco Recife-PE, Brazil e-mail:
More informationStrategic Trading of Informed Trader with Monopoly on Shortand Long-Lived Information
ANNALS OF ECONOMICS AND FINANCE 10-, 351 365 (009) Strategic Trading of Informed Trader with Monopoly on Shortand Long-Lived Information Chanwoo Noh Department of Mathematics, Pohang University of Science
More informationEconomics and Computation
Economics and Computation ECON 425/56 and CPSC 455/555 Professor Dirk Bergemann and Professor Joan Feigenbaum Lecture I In case of any questions and/or remarks on these lecture notes, please contact Oliver
More informationReasoning with Probabilities. Compactness. Dutch Book. Dutch Book. Synchronic Dutch Book Diachronic Dutch Book. Some Puzzles.
Reasoning with July 31, 2009 Plan for the Course Introduction and Background Probabilistic Epistemic Logics : Dynamic Probabilistic Epistemic Logics : Reasoning with Day 5: Conclusions and General Issues
More informationComparative Study between Linear and Graphical Methods in Solving Optimization Problems
Comparative Study between Linear and Graphical Methods in Solving Optimization Problems Mona M Abd El-Kareem Abstract The main target of this paper is to establish a comparative study between the performance
More information6.207/14.15: Networks Lecture 9: Introduction to Game Theory 1
6.207/14.15: Networks Lecture 9: Introduction to Game Theory 1 Daron Acemoglu and Asu Ozdaglar MIT October 13, 2009 1 Introduction Outline Decisions, Utility Maximization Games and Strategies Best Responses
More informationPrisoner s dilemma with T = 1
REPEATED GAMES Overview Context: players (e.g., firms) interact with each other on an ongoing basis Concepts: repeated games, grim strategies Economic principle: repetition helps enforcing otherwise unenforceable
More informationMicroeconomics Comprehensive Exam
Microeconomics Comprehensive Exam June 2009 Instructions: (1) Please answer each of the four questions on separate pieces of paper. (2) When finished, please arrange your answers alphabetically (in the
More informationAsynchronous Announcements in a Public Channel
Asynchronous Announcements in a Public Channel Sophia Knight 1, Bastien Maubert 1, and François Schwarzentruber 2 1 LORIA - CNRS / Université de Lorraine, sophia.knight@gmail.com, bastien.maubert@gmail.com
More informationBehavioral Equilibrium and Evolutionary Dynamics
Financial Markets: Behavioral Equilibrium and Evolutionary Dynamics Thorsten Hens 1, 5 joint work with Rabah Amir 2 Igor Evstigneev 3 Klaus R. Schenk-Hoppé 4, 5 1 University of Zurich, 2 University of
More information6.896 Topics in Algorithmic Game Theory February 10, Lecture 3
6.896 Topics in Algorithmic Game Theory February 0, 200 Lecture 3 Lecturer: Constantinos Daskalakis Scribe: Pablo Azar, Anthony Kim In the previous lecture we saw that there always exists a Nash equilibrium
More informationAn introduction on game theory for wireless networking [1]
An introduction on game theory for wireless networking [1] Ning Zhang 14 May, 2012 [1] Game Theory in Wireless Networks: A Tutorial 1 Roadmap 1 Introduction 2 Static games 3 Extensive-form games 4 Summary
More informationOn Forchheimer s Model of Dominant Firm Price Leadership
On Forchheimer s Model of Dominant Firm Price Leadership Attila Tasnádi Department of Mathematics, Budapest University of Economic Sciences and Public Administration, H-1093 Budapest, Fővám tér 8, Hungary
More informationStochastic Games and Bayesian Games
Stochastic Games and Bayesian Games CPSC 532L Lecture 10 Stochastic Games and Bayesian Games CPSC 532L Lecture 10, Slide 1 Lecture Overview 1 Recap 2 Stochastic Games 3 Bayesian Games Stochastic Games
More informationM.Phil. Game theory: Problem set II. These problems are designed for discussions in the classes of Week 8 of Michaelmas term. 1
M.Phil. Game theory: Problem set II These problems are designed for discussions in the classes of Week 8 of Michaelmas term.. Private Provision of Public Good. Consider the following public good game:
More informationECO 5341 (Section 2) Spring 2016 Midterm March 24th 2016 Total Points: 100
Name:... ECO 5341 (Section 2) Spring 2016 Midterm March 24th 2016 Total Points: 100 For full credit, please be formal, precise, concise and tidy. If your answer is illegible and not well organized, if
More information2 Deduction in Sentential Logic
2 Deduction in Sentential Logic Though we have not yet introduced any formal notion of deductions (i.e., of derivations or proofs), we can easily give a formal method for showing that formulas are tautologies:
More informationEvolution & Learning in Games
1 / 27 Evolution & Learning in Games Econ 243B Jean-Paul Carvalho Lecture 1: Foundations of Evolution & Learning in Games I 2 / 27 Classical Game Theory We repeat most emphatically that our theory is thoroughly
More informationGame-Theoretic Risk Analysis in Decision-Theoretic Rough Sets
Game-Theoretic Risk Analysis in Decision-Theoretic Rough Sets Joseph P. Herbert JingTao Yao Department of Computer Science, University of Regina Regina, Saskatchewan, Canada S4S 0A2 E-mail: [herbertj,jtyao]@cs.uregina.ca
More informationCUR 412: Game Theory and its Applications, Lecture 9
CUR 412: Game Theory and its Applications, Lecture 9 Prof. Ronaldo CARPIO May 22, 2015 Announcements HW #3 is due next week. Ch. 6.1: Ultimatum Game This is a simple game that can model a very simplified
More informationFinding Equilibria in Games of No Chance
Finding Equilibria in Games of No Chance Kristoffer Arnsfelt Hansen, Peter Bro Miltersen, and Troels Bjerre Sørensen Department of Computer Science, University of Aarhus, Denmark {arnsfelt,bromille,trold}@daimi.au.dk
More informationHarvard School of Engineering and Applied Sciences CS 152: Programming Languages
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Lecture 3 Tuesday, January 30, 2018 1 Inductive sets Induction is an important concept in the theory of programming language.
More informationPhD Qualifier Examination
PhD Qualifier Examination Department of Agricultural Economics May 29, 2014 Instructions This exam consists of six questions. You must answer all questions. If you need an assumption to complete a question,
More informationCTL Model Checking. Goal Method for proving M sat σ, where M is a Kripke structure and σ is a CTL formula. Approach Model checking!
CMSC 630 March 13, 2007 1 CTL Model Checking Goal Method for proving M sat σ, where M is a Kripke structure and σ is a CTL formula. Approach Model checking! Mathematically, M is a model of σ if s I = M
More informationGame Theory. Lecture Notes By Y. Narahari. Department of Computer Science and Automation Indian Institute of Science Bangalore, India August 2012
Game Theory Lecture Notes By Y. Narahari Department of Computer Science and Automation Indian Institute of Science Bangalore, India August 01 Chapter 5: Pure Strategy Nash Equilibrium Note: This is a only
More informationYao s Minimax Principle
Complexity of algorithms The complexity of an algorithm is usually measured with respect to the size of the input, where size may for example refer to the length of a binary word describing the input,
More informationAuctions. Michal Jakob Agent Technology Center, Dept. of Computer Science and Engineering, FEE, Czech Technical University
Auctions Michal Jakob Agent Technology Center, Dept. of Computer Science and Engineering, FEE, Czech Technical University AE4M36MAS Autumn 2014 - Lecture 12 Where are We? Agent architectures (inc. BDI
More informationMarkov Decision Processes
Markov Decision Processes Ryan P. Adams COS 324 Elements of Machine Learning Princeton University We now turn to a new aspect of machine learning, in which agents take actions and become active in their
More informationPh.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program August 2017
Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program August 2017 The time limit for this exam is four hours. The exam has four sections. Each section includes two questions.
More informationBAYESIAN GAMES: GAMES OF INCOMPLETE INFORMATION
BAYESIAN GAMES: GAMES OF INCOMPLETE INFORMATION MERYL SEAH Abstract. This paper is on Bayesian Games, which are games with incomplete information. We will start with a brief introduction into game theory,
More informationMicroeconomic Theory May 2013 Applied Economics. Ph.D. PRELIMINARY EXAMINATION MICROECONOMIC THEORY. Applied Economics Graduate Program.
Ph.D. PRELIMINARY EXAMINATION MICROECONOMIC THEORY Applied Economics Graduate Program May 2013 *********************************************** COVER SHEET ***********************************************
More informationMicroeconomic Theory III Final Exam March 18, 2010 (80 Minutes)
4. Microeconomic Theory III Final Exam March 8, (8 Minutes). ( points) This question assesses your understanding of expected utility theory. (a) In the following pair of games, check whether the players
More informationCMSC 474, Introduction to Game Theory 16. Behavioral vs. Mixed Strategies
CMSC 474, Introduction to Game Theory 16. Behavioral vs. Mixed Strategies Mohammad T. Hajiaghayi University of Maryland Behavioral Strategies In imperfect-information extensive-form games, we can define
More informationRationalizable Strategies
Rationalizable Strategies Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu Jun 1st, 2015 C. Hurtado (UIUC - Economics) Game Theory On the Agenda 1
More informationPre-vote negotiations and the outcome of collective decisions
and the outcome of collective decisions Department of Computing, Imperial College London joint work with Umberto Grandi (Padova) & Davide Grossi (Liverpool) Bits of Roman politics Cicero used to say that
More informationElements of Economic Analysis II Lecture X: Introduction to Game Theory
Elements of Economic Analysis II Lecture X: Introduction to Game Theory Kai Hao Yang 11/14/2017 1 Introduction and Basic Definition of Game So far we have been studying environments where the economic
More informationPh.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2017
Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2017 The time limit for this exam is four hours. The exam has four sections. Each section includes two questions.
More informationRandom Search Techniques for Optimal Bidding in Auction Markets
Random Search Techniques for Optimal Bidding in Auction Markets Shahram Tabandeh and Hannah Michalska Abstract Evolutionary algorithms based on stochastic programming are proposed for learning of the optimum
More informationChair of Communications Theory, Prof. Dr.-Ing. E. Jorswieck. Übung 5: Supermodular Games
Chair of Communications Theory, Prof. Dr.-Ing. E. Jorswieck Übung 5: Supermodular Games Introduction Supermodular games are a class of non-cooperative games characterized by strategic complemetariteis
More informationA Decidable Logic for Time Intervals: Propositional Neighborhood Logic
From: AAAI Technical Report WS-02-17 Compilation copyright 2002, AAAI (wwwaaaiorg) All rights reserved A Decidable Logic for Time Intervals: Propositional Neighborhood Logic Angelo Montanari University
More informationStrong normalisation and the typed lambda calculus
CHAPTER 9 Strong normalisation and the typed lambda calculus In the previous chapter we looked at some reduction rules for intuitionistic natural deduction proofs and we have seen that by applying these
More informationGame Theory with Translucent Players
Game Theory with Translucent Players Joseph Y. Halpern Cornell University Dept. Computer Science Ithaca, NY 14853, USA halpern@cs.cornell.edu Rafael Pass Cornell University Dept. Computer Science Ithaca,
More informationAnswers to Problem Set 4
Answers to Problem Set 4 Economics 703 Spring 016 1. a) The monopolist facing no threat of entry will pick the first cost function. To see this, calculate profits with each one. With the first cost function,
More informationCredibilistic Equilibria in Extensive Game with Fuzzy Payoffs
Credibilistic Equilibria in Extensive Game with Fuzzy Payoffs Yueshan Yu Department of Mathematical Sciences Tsinghua University Beijing 100084, China yuyueshan@tsinghua.org.cn Jinwu Gao School of Information
More informationThe Turing Definability of the Relation of Computably Enumerable In. S. Barry Cooper
The Turing Definability of the Relation of Computably Enumerable In S. Barry Cooper Computability Theory Seminar University of Leeds Winter, 1999 2000 1. The big picture Turing definability/invariance
More informationAuctions That Implement Efficient Investments
Auctions That Implement Efficient Investments Kentaro Tomoeda October 31, 215 Abstract This article analyzes the implementability of efficient investments for two commonly used mechanisms in single-item
More informationCS 331: Artificial Intelligence Game Theory I. Prisoner s Dilemma
CS 331: Artificial Intelligence Game Theory I 1 Prisoner s Dilemma You and your partner have both been caught red handed near the scene of a burglary. Both of you have been brought to the police station,
More informationKuhn s Theorem for Extensive Games with Unawareness
Kuhn s Theorem for Extensive Games with Unawareness Burkhard C. Schipper November 1, 2017 Abstract We extend Kuhn s Theorem to extensive games with unawareness. This extension is not entirely obvious:
More informationTHE NUMBER OF UNARY CLONES CONTAINING THE PERMUTATIONS ON AN INFINITE SET
THE NUMBER OF UNARY CLONES CONTAINING THE PERMUTATIONS ON AN INFINITE SET MICHAEL PINSKER Abstract. We calculate the number of unary clones (submonoids of the full transformation monoid) containing the
More informationTHE UNIVERSITY OF NEW SOUTH WALES
THE UNIVERSITY OF NEW SOUTH WALES FINS 5574 FINANCIAL DECISION-MAKING UNDER UNCERTAINTY Instructor Dr. Pascal Nguyen Office: #3071 Email: pascal@unsw.edu.au Consultation hours: Friday 14:00 17:00 Appointments
More informationMA200.2 Game Theory II, LSE
MA200.2 Game Theory II, LSE Problem Set 1 These questions will go over basic game-theoretic concepts and some applications. homework is due during class on week 4. This [1] In this problem (see Fudenberg-Tirole
More informationA brief introduction to evolutionary game theory
A brief introduction to evolutionary game theory Thomas Brihaye UMONS 27 October 2015 Outline 1 An example, three points of view 2 A brief review of strategic games Nash equilibrium et al Symmetric two-player
More informationTABLEAU-BASED DECISION PROCEDURES FOR HYBRID LOGIC
TABLEAU-BASED DECISION PROCEDURES FOR HYBRID LOGIC THOMAS BOLANDER AND TORBEN BRAÜNER Abstract. Hybrid logics are a principled generalization of both modal logics and description logics. It is well-known
More informationTopics in Contract Theory Lecture 1
Leonardo Felli 7 January, 2002 Topics in Contract Theory Lecture 1 Contract Theory has become only recently a subfield of Economics. As the name suggest the main object of the analysis is a contract. Therefore
More informationChapter 3. Dynamic discrete games and auctions: an introduction
Chapter 3. Dynamic discrete games and auctions: an introduction Joan Llull Structural Micro. IDEA PhD Program I. Dynamic Discrete Games with Imperfect Information A. Motivating example: firm entry and
More informationAll Equilibrium Revenues in Buy Price Auctions
All Equilibrium Revenues in Buy Price Auctions Yusuke Inami Graduate School of Economics, Kyoto University This version: January 009 Abstract This note considers second-price, sealed-bid auctions with
More informationMicroeconomic Theory II Preliminary Examination Solutions
Microeconomic Theory II Preliminary Examination Solutions 1. (45 points) Consider the following normal form game played by Bruce and Sheila: L Sheila R T 1, 0 3, 3 Bruce M 1, x 0, 0 B 0, 0 4, 1 (a) Suppose
More informationGame Theory. Lecture Notes By Y. Narahari. Department of Computer Science and Automation Indian Institute of Science Bangalore, India October 2012
Game Theory Lecture Notes By Y. Narahari Department of Computer Science and Automation Indian Institute of Science Bangalore, India October 22 COOPERATIVE GAME THEORY Correlated Strategies and Correlated
More informationECON FINANCIAL ECONOMICS
ECON 337901 FINANCIAL ECONOMICS Peter Ireland Boston College Spring 2018 These lecture notes by Peter Ireland are licensed under a Creative Commons Attribution-NonCommerical-ShareAlike 4.0 International
More informationProblem 3 Solutions. l 3 r, 1
. Economic Applications of Game Theory Fall 00 TA: Youngjin Hwang Problem 3 Solutions. (a) There are three subgames: [A] the subgame starting from Player s decision node after Player s choice of P; [B]
More informationSubgame Perfect Cooperation in an Extensive Game
Subgame Perfect Cooperation in an Extensive Game Parkash Chander * and Myrna Wooders May 1, 2011 Abstract We propose a new concept of core for games in extensive form and label it the γ-core of an extensive
More informationGame Theory I 1 / 38
Game Theory I 1 / 38 A Strategic Situation (due to Ben Polak) Player 2 α β Player 1 α B-, B- A, C β C, A A-, A- 2 / 38 Selfish Students Selfish 2 α β Selfish 1 α 1, 1 3, 0 β 0, 3 2, 2 3 / 38 Selfish Students
More information