Towards a Theory of Correctness of Social. Procedures. ILLC, University of Amsterdam. staff.science.uva.

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1 Towards a Theory of Correctness of Social Procedures Eric Pacuit April 2, 26 ILLC, University of Amsterdam staff.science.uva.nl/ epacuit epacuit@science.uva.nl

2 Introduction: Social Software Social software is an interdisciplinary research program that combines mathematical tools and techniues from game theory and computer science in order to analyze and design social procedures.

3 Introduction: Social Software Social software is an interdisciplinary research program that combines mathematical tools and techniues from game theory and computer science in order to analyze and design social procedures. Research in Social Software can be divided into three dierent but related categories:

4 Introduction: Social Software Social software is an interdisciplinary research program that combines mathematical tools and techniues from game theory and computer science in order to analyze and design social procedures. Research in Social Software can be divided into three dierent but related categories: Mathematical Models of Social Situations

5 Introduction: Social Software Social software is an interdisciplinary research program that combines mathematical tools and techniues from game theory and computer science in order to analyze and design social procedures. Research in Social Software can be divided into three dierent but related categories: Mathematical Models of Social Situations A Theory of Correctness of Social Procedures

6 Introduction: Social Software Social software is an interdisciplinary research program that combines mathematical tools and techniues from game theory and computer science in order to analyze and design social procedures. Research in Social Software can be divided into three dierent but related categories: Mathematical Models of Social Situations A Theory of Correctness of Social Procedures Designing Social Procedures

7 Introduction: Social Software Social software is an interdisciplinary research program that combines mathematical tools and techniues from game theory and computer science in order to analyze and design social procedures. Research in Social Software can be divided into three dierent but related categories: Mathematical Models of Social Situations A Theory of Correctness of Social Procedures Designing Social Procedures

8 Introduction: Social Software Social software is an interdisciplinary research program that combines mathematical tools and techniues from game theory and computer science in order to analyze and design social procedures. For more information see R. Parikh. Social Software. Synthese 132 (22). R. Parikh. Language as Social Software. in Future Pasts: The Analytic Tradition in the Twentieth Century (21). EP and R. Parikh. Social Interaction, Knowledge, and Social Software. in Interactive Computation: The New Paradigm (forthcoming).

9 Logic for Mechanism Design Computational aspects of computer science vs. using ideas from computer science (eg. program verication) in game theory.

10 Logic for Mechanism Design Computational aspects of computer science vs. using ideas from computer science (eg. program verication) in game theory. Formally verifying mechanisms: J. Halpern. A Computer Scientist Looks at Game Theory. Games and Economic Behavior 45 (23). J. van Benthem. Extensive Games as Process Models. JOLLI 11 (22). M. Pauly and M. Wooldridge. Logics for Mechanism Design A Manifesto. available at the author's websites. S. van Otterloo. Strategic Analysis of Multi-agent Protocols. Ph.D. Thesis, University of Liverpool (25).

11 Outline of the Talk Strategy Logics Coalitional Logic A Simple Example Alternating Time Temporal Logic From Hoare Logic to PDL Game Logic Banach-Knaster Cake Cutting Algorithm Pauly's Mechanism Programming Language Example Case Study: Adjusted Winner

12 From Temporal Logic to Strategy Logic

13 From Temporal Logic to Strategy Logic Linear Time Temporal Logic: Reasoning about computation paths: φ: φ is true some time in the future. A. Pnuelli. A Temporal Logic of Programs. in Proc. 18th IEEE Symposium on Foundations of Computer Science (1977).

14 From Temporal Logic to Strategy Logic Linear Time Temporal Logic: Reasoning about computation paths: φ: φ is true some time in the future. A. Pnuelli. A Temporal Logic of Programs. in Proc. 18th IEEE Symposium on Foundations of Computer Science (1977). Branching Time Temporal Logic: Allows uantication over paths: φ: there is a path in which φ is eventually true. E. M. Clarke and E. A. Emerson. Design and Synthesis of Synchronization Skeletons using Branching-time Temproal-logic Specications. In Proceedings Workshop on Logic of Programs, LNCS (1981).

15 From Temporal Logic to Strategy Logic Alternating-time Temporal Logic: Reasoning about (local and global) group power: A φ: The coalition A has a joint strategy to ensure that φ will remain true. R. Alur, T. Henzinger and O. Kupferman. Alternating-time Temproal Logic. Jouranl of the ACM (22).

16 From Temporal Logic to Strategy Logic Alternating-time Temporal Logic: Reasoning about (local and global) group power: A φ: The coalition A has a joint strategy to ensure that φ will remain true. R. Alur, T. Henzinger and O. Kupferman. Alternating-time Temproal Logic. Jouranl of the ACM (22). Coalitional Logic: Reasoning about (local) group power (fragment of ATL). [C]φ (euivalently C φ): coalition C has a joint strategy to bring about φ. M. Pauly. A Modal Logic for Coalition Powers in Games. Journal of Logic and Computation 12 (22).

17 1 Background x=1 x= x=1 1 x= Background 1Reasoning Background April 29, 25 coalitions Reasoning about 1 xabout Background April 1 29,coalitions 25 = Background x= Rea April 29, 25 April 29, 25 x = x =Structures x=1 Computational vs. Behavioral 1 Background x= 1 Background x=1 Aprilabout 29, 25 Reasoning coalitions x = 1 x = x = 1 1 Background 1 Background x= April 29, x =25 1 about x = 1 coalitions 1 Background x Reasoning = about coalitions x= Reasoning x = 1 x = April 29, Background x1 = 1April about 1 x =25 1 April 29, 25Reasoning 29, 25 coalitions Reasoning abo 1 x 1 x = 29, 1 25 = 1 29, April 25 April 1 1 Background 1 Background 1 Backgroun x = x = 1 x =1 x= 1 1 Background 1 Background x = x=1 x= x = 1 x = =1 11 x=1 x Linear. Time Temporal Linear Time Temporal Logic (LTL) putations: putations: Logic (LTL) 1 1 Linear Time Temporal 1 Linear 1 Time Temporal 1 1 putations: φ: putations: som Logic [Pnuelli, Linear Time φ: (LTL) φ is true 1 1 Temporal 1 Linear Time Temporal Logic (LTL) [Pnu Branching-time φ: φ is true putations: so Temp putations: 1 Logic Branching-time Temporal (CT φ: Time (LTL) 1about Linear (LTL) [Pnuelli, 1977]: Reasoni Emerson andlogic Halpern, 1 Temporal 1977]: Linear Time Temporal Logic [Pnuelli, Reasoning com- Reasoning about coalit Reasoning about c 1 Background 1April Background 29, 25 April April 29, 25April 29, 2529, 25

18 x 1 Background Background x= Background April 29, 25 April 29, Rea April about 29, 25coalitionsabout coalitions Reasoning 1 xbackground April Reasoning = 129, 25 Background x= x= x=1 Background x =Behavioral 1 Background xstructures =1 x = 1Computational vs. Aprilabout 29, 25 Reasoning coalitions x = 1 x= x=1 1 Background = 1 Background x= April 29, x =25 1 about x = 1 coalitions 1 Background x Reasoning = about coalitions x = x = 1 Reasoning xapril = 129, x = Background x1 = 1April about 1 x =25 1 April 29, 25Reasoning 29, 25 coalitions Reasoning abo 1 1 x x = 29, 1 25 = 1 29, April 25 April 1 1 Background 1 Background 1 Backgroun x = x = 1 1 x = x= 1 1 Background1 Background x =11 1 x=1 x= x = 1 x = x=1 x = Linear. Time Temporal Linear Time Temporal Logic (LTL) putations: putations: Logic (LTL) Linear Time Temporal Linear 1 Time Temporal 1 1 putations: φ: putations: som 1 Logic [Pnuelli, 1 1 Linear Time φ: (LTL) φ is true 1 1 Temporal Linear Time Temporal Logic (LTL) [Pnu Branching-time P 1 1 φ: φ is true 1 x=1 putations: so Temp 1 putations: Logic Branching-time Temporal (CT φ: Time (LTL) 1about Linear (LTL) [Pnuelli, 1977]: Reasoni Emerson andlogic Halpern, 1 Temporal 1977]: Linear Time Temporal Logic [Pnuelli, Reasoning com and φlogic Halpern, 1986]: Allows 1 1Linear Time φ: istemporal true(ltl) some time in putations: Emerson Branching-time Logic (C Temporal [Pnu 11putations: φ: φ istime true Temporal some tim Linear putations: ]: 1 Emerson 1 Branching-time and Halpern, Allow Temp 1 φ: ther x=

19 x 1 Background Background x= Background April 29, 25 April 29, Rea April about 29, 25coalitionsabout coalitions Reasoning 1 xbackground April Reasoning = 129, 25 Background x= x= x=1 Background x =Behavioral 1 Background xstructures =1 x = 1Computational vs. Aprilabout 29, 25 Reasoning coalitions x = 1 x= x=1 1 Background = 1 Background x= April 29, x =25 1 about x = 1 coalitions 1 Background x Reasoning = about coalitions x = x = 1 Reasoning xapril = 129, x = Background x1 = 1April about 1 x =25 1 April 29, 25Reasoning 29, 25 coalitions Reasoning abo 1 1 x x = 29, 1 25 = 1 29, April 25 April 1 1 Background 1 Background 1 Backgroun x = x = 1 1 x = x= 1 1 Background1 Background x =11 1 x=1 x= x = 1 x = x=1 x = Linear. Time Temporal Linear Time Temporal Logic (LTL) putations: putations: Logic (LTL) Linear Time Temporal Linear 1 Time Temporal 1 1 putations: φ: putations: som 1 Logic [Pnuelli, 1 1 Linear Time φ: (LTL) φ is true 1 1 Temporal Linear Time Temporal Logic (LTL) [Pnu P Branching-time 1 x=1 putations: 1 1 φ: φ is true so Temp 1 putations: Logic Branching-time Temporal (CT φ: Time (LTL) 1about Linear (LTL) [Pnuelli, 1977]: Reasoni Emerson andlogic Halpern, 1 Temporal 1977]: Linear Time Temporal Logic [Pnuelli, Reasoning com and φlogic Halpern, 1986]: Allows 1 1Linear Time φ: istemporal true(ltl) some time in putations: Emerson Branching-time Logic (C Temporal [Pnu 11putations: φ: φ istime true Temporal some tim Linear putations: ]: 1 Emerson 1 Branching-time and Halpern, Allow Temp 1 φ: ther x=

20 Alternating Transition Systems The previous model assumes there is one agent that controls the transition system.

21 Alternating Transition Systems The previous model assumes there is one agent that controls the transition system. What if there is more than one agent?

22 Alternating Transition Systems The previous model assumes there is one agent that controls the transition system. What if there is more than one agent? Example: Suppose that there are two agents: a server (s) and a client (c). The client asks to set the value of x and the server can either grant or deny the reuest. Assume the agents make simultaneous moves.

23 Alternating Transition Systems The previous model assumes there is one agent that controls the transition system. What if there is more than one agent? Example: Suppose that there are two agents: a server (s) and a client (c). The client asks to set the value of x and the server can either grant or deny the reuest. Assume the agents make simultaneous moves. set deny grant set1

24 Alternating Transition Systems The previous model assumes there is one agent that controls the transition system. What if there is more than one agent? Example: Suppose that there are two agents: a server (s) and a client (c). The client asks to set the value of x and the server can either grant or deny the reuest. Assume the agents make simultaneous moves. deny grant set, 1 set1 1, 1 1

25 Alternating Transition Systems The previous model assumes there is one agent that controls the transition system. What if there is more than one agent? Example: Suppose that there are two agents: a server (s) and a client (c). The client asks to set the value of x and the server can either grant or deny the reuest. Assume the agents make simultaneous moves. deny grant set, 1 set1 1, 1 1

26 Multi-agent Transition Systems grant, set deny, set1 deny, set x = grant, set grant, set1 1 x = 1 deny, set1 deny, set grant, set1

27 Multi-agent Transition Systems grant, set deny, set1 deny, set x = grant, set grant, set1 1 x = 1 deny, set1 deny, set grant, set1 (Px= [s]px=) (Px=1 [s]px=1)

28 Multi-agent Transition Systems grant, set deny, set1 deny, set x = grant, set grant, set1 1 x = 1 deny, set1 deny, set grant, set1 Px= [s]px=1

29 Multi-agent Transition Systems grant, set deny, set1 deny, set x = grant, set grant, set1 1 x = 1 deny, set1 deny, set grant, set1 Px= [s, c]px=1

30 An Example Two agents, A and B, must choose between two outcomes, p and. We want a mechanism that will allow them to choose, which will satisfy the following reuirements: 1. We denitely want an outcome to result, i.e., either p or must be selected 2. We want the agents to be able to collectively choose and outcome 3. We do not want them to be able to bring about both outcomes simultaneously 4. We want them both to have eual power

31 An Example Two agents, A and B, must choose between two outcomes, p and. We want a mechanism that will allow them to choose, which will satisfy the following reuirements: 1. We denitely want an outcome to result, i.e., either p or must be selected: [ ](p ) 2. We want the agents to be able to collectively choose and outcome 3. We do not want them to be able to bring about both outcomes simultaneously 4. We want them both to have eual power

32 An Example Two agents, A and B, must choose between two outcomes, p and. We want a mechanism that will allow them to choose, which will satisfy the following reuirements: 1. We denitely want an outcome to result, i.e., either p or must be selected: [ ](p ) 2. We want the agents to be able to collectively choose and outcome: [A, B]p [A, B] 3. We do not want them to be able to bring about both outcomes simultaneously 4. We want them both to have eual power

33 An Example Two agents, A and B, must choose between two outcomes, p and. We want a mechanism that will allow them to choose, which will satisfy the following reuirements: 1. We denitely want an outcome to result, i.e., either p or must be selected: [ ](p ) 2. We want the agents to be able to collectively choose and outcome: [A, B]p [A, B] 3. We do not want them to be able to bring about both outcomes simultaneously: [A, B](p ) 4. We want them both to have eual power

34 An Example Two agents, A and B, must choose between two outcomes, p and. We want a mechanism that will allow them to choose, which will satisfy the following reuirements: 1. We denitely want an outcome to result, i.e., either p or must be selected: [ ](p ) 2. We want the agents to be able to collectively choose and outcome: [A, B]p [A, B] 3. We do not want them to be able to bring about both outcomes simultaneously: [A, B](p ) 4. We want them both to have eual power: [x]p [x] where x {A, B}

35 An Example Consider the following mechanism: The two agents vote on the outcomes, i.e., they choose either p or. If there is a consensus, then the consensus is selected; if there is no consensus, then an outcome p or is selected non-deterministically. Pauly and Wooldridge use the MOCHA model checking system to verify that the above procedure satises the previous specications.

36 Pauly's Coalitional Logic: Syntax Given a nite non-empty set of agents N and a set of atomic propositions Φ, a formula φ can have the following syntactic form φ := p φ φ φ [C]φ where p Φ and C N. [C]φ is intended to mean coalition C can (locally) force φ to be true M. Pauly. Logics for Social Software. Ph.D. Thesis, ILLC (21).

37 Multi-player Game Models A Strategic Game Form is a tuple N, {Σi i N}, Q, o where N is a set of agents Σi is a set of actions Q is a set of states o : Πi N Σi Q assigns an outcome to each choice of aciton. Let Γ N Q be the set of all such strategic game forms. A Multi-Player Game Model is a tuple Q, γ, π where Q is a set of states and γ : Q Γ N Q associates strategic games form to each state = [C]φi σc σn C, o(σc, σn C) = φ

38 Eectivity Functions Let G be a strategic game. X E α G(C)i σc σ C o(σc, σ C ) X X E β G (C)i σ C σ C o(σc, σ C ) X E α G Eβ G

39 E β G Eα G Player 1 chooses the row, Player 2 chooses the column, Player 3 chooses the table l m r l m r l s1 s2 s1 l s3 s1 s2 r s2 s1 s3 r s2 s3 s3 {s2} E β G ({2}) but {s 2} E α G ({2})

40 Coalition Eectivity Models An eectivity function is playable i 1. For each C N, E(C) 2. For each C N, Q E(C) 3. If X E(N), then Q X E( ) 4. If X Y and X E(C) then Y E(C) 5. for all C1, C2 N and X1, X2 Q, if C1 C2 =, X1 E(C1) and X2 E(C2) then X1 X2 E(C1 C2) Characterization Theorem: An α-eectivity function E is playable i it is the eectivity function of some strategic game. M. Pauly. A Modal Logic for Coalition Powers in Games. Journal of Logic and Computation 12 (22).

41 Coalitional Logic: Coalition Eectivity Models A coalitional eectivity model is a tuple Q, E, V where E : Q (2 N 2 2Q ) assigns a playable eectivity function to each state and V is a valuation function. = [C]φ i (φ) M E(C)

42 Main Results Theorem Coalitional Logic is sound and strongly complete with respect to the class of eectivity models. Theorem The complexity of the satisability problem of coalitional logic is PSPACE-complete. M. Pauly. A Modal Logic for Coalitional Powers in Games. Journal of Logic and Computation (22). M. Pauly. On the Complexity of Coalitional Reasoning. International Game Theory Review (22).

43 ATL: Syntax Let A be a set of agents, Π a set of propositional variables and A A. 1. p where p Π 2. φ 3. φ ψ 4. A φ meaning `The coalition A can force in the next move an outcome satisfying φ' 5. A φ meaning `The coalition A can maintain forever outcomes satisfying φ' 6. A φu ψ meaning `The coalition A can eventually force an outcome satisfying ψ while meanwhile maintaining the truth of φ

44 Coalition Logic is a Fragment of ATL Dene [A]φ to be A φ Multi-player Game Models and Concurrent-game Models only dier in notation Coalitional Eectivity Models can be used as a semantics for ATL Goranko and Jamroga. Comparing Semantics of Logics fro Multi-Agent Systems. See the website.

45 Results Theorem All of the semantics (concurrent game structures, alternating transitions systems and coalitional eectivity models) are euivalent. Goranko and Jamroga. Comparing Semantics of Logics for Multi-Agent Systems. See the website. Theorem ATL is sound and (weakly) complete. Theorem Given a nite set of players, the satisability problem for ATL-formulas over N with respect to concurrent game structures over N is EXPTIME-complete. Goranko and van Drimmelen. Complete Axiomatization and Decidability of the Alternating-Time Temporal Logic. Theoretical Computer Science (25).

46 From Hoare Logic to Game Logic Hoare Logic Partial Correctness of Procedures {φ}α{ψ}: If the program α begins in a state in which φ is true, then after α terminates (!), ψ will be true. C. A. R. Hoare. An Axiomatic Basis for Computer Programming.. Comm. Assoc. Comput. Mach Propositional Dynamic Logic (PDL) [Pratt, 1976]: Reason about programs explicitly: [α]φ: after executing α, φ is true. C. A. R. Hoare. An Axiomatic Basis for Computer Programming.. Comm. Assoc. Comput. Mach

47 From Hoare Logic to Game Logic Game Logic (GL) [Parikh, 1985]: Reasoning about games: (γ)φ: Agent I has a strategy to bring about φ in game γ. R. Parikh. The Logic of Games and its Applications.. Annals of Discrete Mathematics. (1985). More information: K.R. Apt and E.R. Olderog. Verication of Seuential and Concurrent Programs, Second Edition. Springer-Verlag (1997). D. Harel, D. Kozen and J. Tiuryn. Dynamic Logic. MIT Press (2).

48 Background: Hoare Logic

49 Background: Hoare Logic Motivation: Formally verify the correctness of a program via partial correctness assertions: {φ}α{ψ}

50 Background: Hoare Logic Motivation: Formally verify the correctness of a program via partial correctness assertions: {φ}α{ψ} Intended Interpretation: If the program α begins in a state in which φ is true, then after α terminates (!), ψ will be true.

51 Background: Hoare Logic Motivation: Formally verify the correctness of a program via partial correctness assertions: {φ}α{ψ} Intended Interpretation: If the program α begins in a state in which φ is true, then after α terminates (!), ψ will be true. C. A. R. Hoare. An Axiomatic Basis for Computer Programming.. Comm. Assoc. Comput. Mach

52 Main Rules: Background: Hoare Logic

53 Background: Hoare Logic Main Rules: Assignment Rule: {φ[x/e]} x := e {φ}

54 Background: Hoare Logic Main Rules: Assignment Rule: {φ[x/e]} x := e {φ} Composition Rule: {φ} α {σ} {σ} β {ψ} {φ} α; β {ψ}

55 Background: Hoare Logic Main Rules: Assignment Rule: {φ[x/e]} x := e {φ} Composition Rule: {φ} α {σ} {σ} β {ψ} {φ} α; β {ψ} Conditional Rule: {φ σ} α {ψ} {φ σ} β {ψ} {φ} if σ then α else β {ψ}

56 Background: Hoare Logic Main Rules: Assignment Rule: {φ[x/e]} x := e {φ} Composition Rule: {φ} α {σ} {σ} β {ψ} {φ} α; β {ψ} Conditional Rule: {φ σ} α {ψ} {φ σ} β {ψ} {φ} if σ then α else β {ψ} While Rule: {φ σ} α {φ} {φ} while σ do α {φ σ}

57 Example: Euclid's Algorithm x := u; y := v; while x y do if x < y then else y := y x; x := x y; Let φ := gcd(x, y) = gcd(u, v)

58 Example: Euclid's Algorithm x := u; y := v; while x y do if x < y then else y := y x; x := x y; Let α be the inner if statement.

59 Example: Euclid's Algorithm x := u; y := v; while x y do if x < y then else y := y x; x := x y; Let α be the inner if statement. Then {gcd(x, y) = gcd(u, v)} α {gcd(x, y) = gcd(u, v)}

60 Example: Euclid's Algorithm x := u; y := v; while x y do if x < y then else y := y x; x := x y; Hence by the while-rule (using a weakening rule) {(gcd(x, y) = gcd(u, v)) (x y)} α {gcd(x, y) = gcd(u, v))} {gcd(x, y) = gcd(u, v)} while σ do α {(gcd(x, y) = gcd(u, v)) (x y)}

61 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: φ := p φ φ ψ [α]φ α := a α β α; β α φ? where p At and a P.

62 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: φ := p φ φ ψ [α]φ α := a α β α; β α φ? where p At and a P. {φ} α {ψ} is replaced with φ [α]ψ

63 From PDL to Game Logic Game Logic (GL) was introduced by Rohit Parikh in R. Parikh. The Logic of Games and its Applications.. Annals of Discrete Mathematics. (1985).

64 From PDL to Game Logic Game Logic (GL) was introduced by Rohit Parikh in 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.

65 From PDL to Game Logic Game Logic (GL) was introduced by Rohit Parikh in 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.

66 Conseuences of two players: From PDL to Game Logic

67 From PDL to Game Logic Conseuences 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

68 From PDL to Game Logic Conseuences 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

69 From PDL to Game Logic Conseuences 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)

70 From PDL to Game Logic Conseuences 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

71 From PDL to Game Logic Conseuences 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

72 From PDL to Game Logic Reinterpret operations and invent new ones:?φ: Check whether φ currently holds

73 From PDL to Game Logic Reinterpret operations and invent new ones:?φ: Check whether φ currently holds γ1; γ2: First play γ1 then γ2

74 From PDL to Game Logic Reinterpret operations and invent new ones:?φ: Check whether φ currently holds γ1; γ2: First play γ1 then γ2 γ1 γ2: Angel choose between γ1 and γ2

75 From PDL to Game Logic Reinterpret operations and invent new ones:?φ: Check whether φ currently holds γ1; γ2: First play γ1 then γ2 γ1 γ2: Angel choose between γ1 and γ2 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.

76 From PDL to Game Logic Reinterpret operations and invent new ones:?φ: Check whether φ currently holds γ1; γ2: First play γ1 then γ2 γ1 γ2: Angel choose between γ1 and γ2 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

77 From PDL to Game Logic Reinterpret operations and invent new ones:?φ: Check whether φ currently holds γ1; γ2: First play γ1 then γ2 γ1 γ2: Angel choose between γ1 and γ2 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 γ1 γ2 := (γ d 1 γ d 2) d : Demon chooses between γ1 and γ2

78 From PDL to Game Logic Reinterpret operations and invent new ones:?φ: Check whether φ currently holds γ1; γ2: First play γ1 then γ2 γ1 γ2: Angel choose between γ1 and γ2 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 γ1 γ2 := (γ d 1 γ d 2) d : Demon chooses between γ1 and γ2 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.

79 Game Logic: Syntax Syntax Let Γ be a set of atomic games and At a set of atomic propositions. Then formulas of Game Logic are dened inductively as follows: γ := g φ? γ; γ γ γ γ γ d φ := p φ φ φ γ φ [γ]φ where p At, g Γ.

80 Game Logic: Semantics I A neighborhood game model is a tuple M = W, {Eg g Γ}, V where W is a nonempty set of states For each g Γ, Eg : W 2 2W is an eectivity function such that if X X and X Eg(w) then X Eg(w). X Eg(w) means in state s, Angel has a strategy to force the game to end in some state in X (we may write wegx) V : At 2 W is a valuation function.

81 Game Logic: Semantics A neighborhood game model is a tuple M = W, {Eg g Γ}, V where Propositional letters and boolean connectives are as usual. M, w = γ φ i (φ) M Eγ(w)

82 Game Logic: Semantics A neighborhood game model is a tuple M = W, {Eg g Γ}, V where Propositional letters and boolean connectives are as usual. M, w = γ φ i (φ) M Eγ(w) Suppose Eγ(Y ) = {s Y Eg(s)} Eγ1;γ2 (Y ) := E γ1 (E γ2 (Y )) Eγ1 γ2(y ) := Eγ1(Y ) Eγ2(Y ) Eφ?(Y ) := (φ) M Y E γ d(y ) := Eγ(Y ) Eγ (Y ) := µx.y Eγ(X)

83 Some Results Fact Game Logic is more expressive than PDL

84 Some Results Fact Game Logic is more expressive than PDL (g d )

85 Some Results Fact Game Logic is more expressive than PDL (g d ) Theorem Game Logic x, where x {, d} 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 (21)..

86 Some Results Theorem [2] Given a game logic formula φ and a nite game model M, model checking can be done in time O( M ad(φ)+1 φ ) Theorem [1,2] The satisability problem for game logic is in EXPTIME. Theorem [1] Game logic can be translated into the modal µ-calculus [1] R. Parikh. The Logic of Games and its Applications.. Annals of Discrete Mathematics. (1985). [2] M. Pauly. Logic for Social Software. Ph.D. Thesis, University of Amsterdam (21)..

87 More Information Editors: M. Pauly and R. Parikh. Special Issue on Game Logic. Studia Logica 75, 23. M. Pauly and R. Parikh. Game Logic An Overview. Studia Logica 75, 23. R. Parikh. The Logic of Games and its Applications.. Annals of Discrete Mathematics. (1985). M. Pauly. Game Logic for Game Theorists. Available at pianoman/.

88 Example: Banach-Knaster Cake Cutting Algorithm The Algorithm:

89 Example: Banach-Knaster Cake Cutting Algorithm The Algorithm: The rst person cuts out a piece which he claims is his fair share.

90 Example: Banach-Knaster Cake Cutting Algorithm The Algorithm: The rst person cuts out a piece which he claims is his fair share. The piece goes around being inspected by each agent.

91 Example: Banach-Knaster Cake Cutting Algorithm The Algorithm: The rst person cuts out a piece which he claims is his fair share. The piece goes around being inspected by each agent. Each agent, in turn, can either reduce the piece, putting some back to the main part, or just pass it.

92 Example: Banach-Knaster Cake Cutting Algorithm The Algorithm: The rst person cuts out a piece which he claims is his fair share. The piece goes around being inspected by each agent. Each agent, in turn, can either reduce the piece, putting some back to the main part, or just pass it. After the piece has been inspected by pn, the last person who reduced the piece, takes it. If there is no such person, then the piece is taken by p1.

93 Example: Banach-Knaster Cake Cutting Algorithm The Algorithm: The rst person cuts out a piece which he claims is his fair share. The piece goes around being inspected by each agent. Each agent, in turn, can either reduce the piece, putting some back to the main part, or just pass it. After the piece has been inspected by pn, the last person who reduced the piece, takes it. If there is no such person, then the piece is taken by p1. The algorithm continues with n 1 participants.

94 Example: Banach-Knaster Cake Cutting Algorithm Correctness: The algorithm is correct i each player has a winning strategy for achieving a fair outcome (1/n of the pie according to pi's own valuation).

95 Example: Banach-Knaster Cake Cutting Algorithm Correctness: The algorithm is correct i each player has a winning strategy for achieving a fair outcome (1/n of the pie according to pi's own valuation). Towards a Formal Proof: F (m, k): the piece m is big enough for k people. F (m, k) (c, i)(f (m, k 1) H(x))

96 Example: Banach-Knaster Cake Cutting Algorithm Correctness: The algorithm is correct i each player has a winning strategy for achieving a fair outcome (1/n of the pie according to pi's own valuation). Towards a Formal Proof: F (m, k): the piece m is big enough for k people. F (m, k) (c, i)(f (m, k 1) H(x)) Goal: Derive a formula expressing that every individual has a strategy that guarantees her fair share. M. Pauly and R. Parikh. Game Logic An Overview. Studia Logica 75, 23. R. Parikh. The Logic of Games and its Applications.. Annals of Discrete Mathematics. (1985).

97 A Hoare-style Logic for Reasoning about Mechanisms

98 A Hoare-style Logic for Reasoning about Mechanisms Add a (simultaneous) choice construct to the WHILE-language: cha({xa a A})

99 A Hoare-style Logic for Reasoning about Mechanisms Add a (simultaneous) choice construct to the WHILE-language: cha({xa a A}) A state is a function s that assigns element of some domain D to variables

100 A Hoare-style Logic for Reasoning about Mechanisms Add a (simultaneous) choice construct to the WHILE-language: cha({xa a A}) A state is a function s that assigns element of some domain D to variables An interpretation I is a rst order structure (a domain DI and an interpretation of function and relation symbols) and preference relations I a on DI for each agent a.

101 A Hoare-style Logic for Reasoning about Mechanisms Add a (simultaneous) choice construct to the WHILE-language: cha({xa a A}) A state is a function s that assigns element of some domain D to variables An interpretation I is a rst order structure (a domain DI and an interpretation of function and relation symbols) and preference relations I a on DI for each agent a. Associate with each expression γ and state s a semi-game G(γ, s, I)

102 A Hoare-style Logic for Reasoning about Mechanisms A semi-game G(γ, s, I) can be turned into a game by adding an outcome function ô that assigns an element of DI to terminal histories.

103 A Hoare-style Logic for Reasoning about Mechanisms A semi-game G(γ, s, I) can be turned into a game by adding an outcome function ô that assigns an element of DI to terminal histories. A predicate is any set of states P SI An e-predicate is any subset P SI DI

104 A Hoare-style Logic for Reasoning about Mechanisms A semi-game G(γ, s, I) can be turned into a game by adding an outcome function ô that assigns an element of DI to terminal histories. A predicate is any set of states P SI An e-predicate is any subset P SI DI Dene strategies and strategy proles (σ) as usual. Each strategy prole corresponds to a run run(σ). Let sσ denote the last state of (a nite) run(σ). Given an e-predicate Q, let ÔQ = {ô Ô for each terminal run σ, if σ is nite, then (last(σ), ô(σ)) Q}

105 Digression: Subgame Perfect Euilibrium A Nash Euilibrium is a strategy prole in which no agent has an incentive to (unilaterally) deviate from their chosen strategy. A Subgame Perfect Euilibrium is a strategy prole that is a Nash euilibrium in every subgame.

106 Digression: Subgame Perfect Euilibrium 1 A B 2 1,2 L R, 2,1

107 Digression: Subgame Perfect Euilibrium 1 A B 2 1,2 L R, 2,1

108 Digression: Subgame Perfect Euilibrium 1 A B 2 1,2 L R, 2,1

109 A Hoare-style Logic for Reasoning about Mechanisms A correctness assertion is an expression of the form {P }γ{q} where P, Q are e-predicate

110 A Hoare-style Logic for Reasoning about Mechanisms A correctness assertion is an expression of the form {P }γ{q} where P, Q are e-predicate We say I = {P }γ{q} provided For each (s, o) P there is a outcome function ˆf ÔQ and a strategy prole σ such that σ is a subgame perfect euilibrium in G(γ, s, I, ˆf) and ˆ(f)(sσ) = o.

111 Mechanism Design Problem A social choice correspondence f maps a preference prole ( i)i A to a set of outcomes X DI.

112 Mechanism Design Problem A social choice correspondence f maps a preference prole ( i)i A to a set of outcomes X DI. Mechanism Design Problem: nd a mechanism which implements the social choice correspondence such that no matter what the preferences of the agents are, self-interested agents will have an incentive to play so that the outcome intended by the designer will obtain. M. Osborne and A. Rubinstein. A Course in Game Theory. Chapter 1.

113 Mechanism Design Problem Give a social choice correspondence f, let f (x) = {(s, o) SI DI o f(x)} and let Q be any functional e-predicate.

114 Mechanism Design Problem Give a social choice correspondence f, let f (x) = {(s, o) SI DI o f(x)} and let Q be any functional e-predicate. Then we say that (γ, Q) SPE-implements a social choice correspondence f i for all preference proles ( i)i A we have I[( i)i A] = {f (( i)i A)} γ {Q}

115 Solomon's Dilema Two women have come before him with a small child, both claiming to be the mother of the child.

116 Solomon's Dilema Two women have come before him with a small child, both claiming to be the mother of the child. Suppose that a is `give the baby to A', b is `give the baby to B' and c is `cut the baby in half'. Suppose Θ1 : a >1 b >1 c and b >2 c >2 a Θ2 : a >1 c >1 b and b >2 a >2 c

117 Solomon's Dilema Two women have come before him with a small child, both claiming to be the mother of the child. Suppose that a is `give the baby to A', b is `give the baby to B' and c is `cut the baby in half'. Suppose Θ1 : a >1 b >1 c and b >2 c >2 a Θ2 : a >1 c >1 b and b >2 a >2 c Solomon must nd a mechanism which implements the social choice rule f(θ1) = {a} and f(θ2) = {b}. It is well-known that f is not Nash-implementable M. Osborne and A. Rubinstein. A Course on Game Theory..

118 Solomon's Dilema However, there is a solution involving money. Allow Solomon to impose nes on the women, so outcomes are of the form: (x, m1, m2) where x {, 1, 2} Suppose the legitimate owner has valuation vh and the other women has valuation vl where vh > vl >

119 Solomon's Dilema If i does not get the painting then i's payo is mi If i gets the painting and i is the legitimate owner then i's payo is vh mi If i gets the painting and i is not the legitimate owner then i's payo is vl mi Solomon wishes to nd a γ and Q such that f(θi) = (i,, ). Let ɛ > and M be such that vl < M < vh.

120 Solomon's Dilema 1 2 mine mine hers hers (2,,) (1,,) (2, ɛ, M)

121 Solomon's Dilema: A Formal Approach ch {1} ({x1}); if x1 > then owner := 2 else ch {2} ({x2}); if x2 > then owner := 1 else owner := ;

122 Solomon's Dilema: A Formal Approach ch {1} ({x1}); if x1 > then owner := 2 else ch {2} ({x2}); if x2 > then owner := 1 else owner := ; Q is the conjunction of owner = 1 o = (1,, ) owner = 2 o = (1,, ) owner = o = (2, ɛ, M)

123 Solomon's Dilema: A Formal Approach ch {1} ({x1}); if x1 > then owner := 2 else ch {2} ({x2}); if x2 > then owner := 1 else owner := ; Q is the conjunction of owner = 1 o = (1,, ) owner = 2 o = (1,, ) owner = o = (2, ɛ, M) I[Θ1] = {o = (1,, )}γ{q} and I[Θ2] = {o = (2,, )}γ{q}

124 Adjusted Winner Adjusted winner (AW ) is an algorithm for dividing n divisible goods among two people (invented by Steven Brams and Alan Taylor). For more information see Fair Division: From cake-cutting to dispute resolution by Brams and Taylor, 1998 The Win-Win Solution by Brams and Taylor, 2

125 Adjusted Winner: Example

126 Adjusted Winner: Example Suppose Ann and Bob are dividing three goods: A, B, and C.

127 Adjusted Winner: Example Suppose Ann and Bob are dividing three goods: A, B, and C. Step 1. Both Ann and Bob divide 1 points among the three goods.

128 Adjusted Winner: Example Suppose Ann and Bob are dividing three goods: A, B, and C. Step 1. Both Ann and Bob divide 1 points among the three goods. Item Ann Bob A 5 4 B C 3 5 Total 1 1

129 Adjusted Winner: Example Suppose Ann and Bob are dividing three goods: A, B, and C. Step 2. The agent who assigns the most points receives the item.

130 Adjusted Winner: Example Suppose Ann and Bob are dividing three goods: A, B, and C. Step 2. The agent who assigns the most points receives the item. Item Ann Bob A 5 4 B C 3 5 Total 1 1

131 Adjusted Winner: Example Suppose Ann and Bob are dividing three goods: A, B, and C. Step 2. The agent who assigns the most points receives the item. Item Ann Bob A 5 B 65 C 5 Total 7 5

132 Adjusted Winner: Example Suppose Ann and Bob are dividing three goods: A, B, and C. Step 3. Euitability adjustment:

133 Adjusted Winner: Example Suppose Ann and Bob are dividing three goods: A, B, and C. Step 3. Euitability adjustment: Notice that 65/46 5/4 1 3/5 Item Ann Bob A 5 4 B C 3 5 Total 1 1

134 Adjusted Winner: Example Suppose Ann and Bob are dividing three goods: A, B, and C. Step 3. Euitability adjustment:

135 Adjusted Winner: Example Suppose Ann and Bob are dividing three goods: A, B, and C. Step 3. Euitability adjustment: Give A to Bob (the item whose ratio is closest to 1)

136 Adjusted Winner: Example Suppose Ann and Bob are dividing three goods: A, B, and C. Step 3. Euitability adjustment: Give A to Bob (the item whose ratio is closest to 1) Item Ann Bob A 5 B 65 C 5 Total 7 5

137 Adjusted Winner: Example Suppose Ann and Bob are dividing three goods: A, B, and C. Step 3. Euitability adjustment: Give A to Bob (the item whose ratio is closest to 1) Item Ann Bob A 4 B 65 C 5 Total 65 54

138 Adjusted Winner: Example Suppose Ann and Bob are dividing three goods: A, B, and C. Step 3. Euitability adjustment: Still not eual, so give (some of) B to Bob: 65p = 1 46p. Item Ann Bob A 4 B 65 C 5 Total 65 54

139 Adjusted Winner: Example Suppose Ann and Bob are dividing three goods: A, B, and C. Step 3. Euitability adjustment: yielding p = 1/111 =.99 Item Ann Bob A 4 B 65 C 5 Total 65 54

140 Adjusted Winner: Example Suppose Ann and Bob are dividing three goods: A, B, and C. Step 3. Euitability adjustment: yielding p = 1/111 =.99 Item Ann Bob A 4 B C 5 Total

141 Adjusted Winner: Formal Denition Suppose that G1,..., Gn is a xed set of goods.

142 Adjusted Winner: Formal Denition Suppose that G1,..., Gn is a xed set of goods. A valuation of these goods is a vector of natural numbers a1,..., an whose sum is 1. Let α, α, α,... denote possible valuations for Ann and β, β, β,... denote possible valuations for Bob.

143 Adjusted Winner: Formal Denition Suppose that G1,..., Gn is a xed set of goods.

144 Adjusted Winner: Formal Denition Suppose that G1,..., Gn is a xed set of goods. An allocation is a vector of n real numbers where each component is between and 1 (inclusive). An allocation σ = s1,..., sn is interpreted as follows. For each i = 1,..., n, si is the proportion of Gi given to Ann. Thus if there are three goods, then 1,.5, means, Give all of item 1 and half of item 2 to Ann and all of item 3 and half of item 2 to Bob."

145 Adjusted Winner: Formal Denition Suppose that G1,..., Gn is a xed set of goods.

146 Adjusted Winner: Formal Denition Suppose that G1,..., Gn is a xed set of goods. VA(α, σ) = n i=1 a isi is the total number of points that Ann receives. VB(β, σ) = n i=1 b i(1 si) is the total number of points that Bob receives. Thus AW can be viewed as a function from pairs of valuations to allocations: AW(α, β) = σ if σ is the allocation produced by the AW algorithm.

147 Adjusted Winner is Fair Theorem AW produces allocations that are ecient, euitable and envy-free (with respect to the announced valuations) S. Brams and A. Taylor. Fair Division. Cambridge University Press.

148 Adjusted Winner is Fair Theorem AW produces allocations that are ecient, euitable and envy-free (with respect to the announced valuations) S. Brams and A. Taylor. Fair Division. Cambridge University Press. cha({x1, x2}); s := wta(x1, x2); while E(s, x1, x2) do s := t(s, x1, x2);

149 Adjusted Winner: Strategizing Item Ann Bob Matisse Picasso Ann will get the Matisse and Bob will get the Picasso and each gets 75 of his or her points.

150 Adjusted Winner: Strategizing Suppose Ann knows Bob's preferences, but Bob does not know Ann's. Item Ann Bob M P Item Ann Bob M P So Ann will get M plus a portion of P. According to Ann's announced allocation, she receives 5 points According to Ann's actual allocation, she receives = points.

151 Conclusion How should we deal with strategizing? EP. Towards a Logical Analysis of Adjusted Winner. working paper. Expressivity issues. Other euilibrium notions. Apply these ideas to more sophisticated mechanisms

152 Thank you.