Agreement on Truth through Communication under robust messages

1. Agreement on Truth through Communication under robust messages LATD2008Siena Takashi Matsuhisa Ibaraki National College of Technology Ibaraki 312-8508, Japan E-mail: mathisa@ge.ibaraki-ct.ac.jp September 11, 2008

2. Purpose How dose communication affect individual decision making ? How we can reach consensus or agreement? B ● A● message D● C● Agreement theorem: all agents can agree on something (a sentence ) by communication through messages.

3. Outline • Background • Formal Model: Knowledge structure Aumman’s agreement theorem Communication under robust messages • Main theorem • Concluding remarks

4. Background 1 • Game theorists’ point of view: 1. Each agent chooses some definite his/her own action, and the other agents need not know which one. 2. The mixed strategy represents their uncertainty as their conjectures on his/her actions. 3. Each agent’s decision makings are determined by his/her own private information. 4. The private information comes from agents’ knowledge. • Theorem (Aumann, 1976): Common-knowledge of the conjectures implies consensus among agents; i.e., all the conjectures of something are equal.

5. Background 2 • By common-knowledge of something, we mean the infinite regress of mutual knowledge of players; ‘{every body knows it} and [every body knows that {everybody knows it}] and every body knows that [ ⋯⋯⋯] and so on.’ • It is not clear the process that yields the agreement in Aumann. • The aim fills this gap.

6. Information structure 〈W, m, (Pi)i∈N〉 • (W, m ):= a probability space • W∋w : a state • 2W∋E : an event • N : a finite set {1,2,..,i,..,n} of agents • i’s information partitionPi: W → 2W; { Pi(w) | wW }:= a partition of W • M = 〈W,(Pi)i∈N〉 := a model for the multi-modal logic S5n = 〈L,T,v,¬, →,..,(□i)i∈N〉 with L⊇PL with suitable inference rules andaxioms: e.g. MP and • N: □i TT: □iφ→φ 4: □iφ→ □i □iφ • 5: ¬□iφ→ □i¬ □iφ etc.

7. Knowledge structure 〈W, μ, (Pi)i∈N, (Ki)i∈N〉:= a knowledge structure: i’s knowledge operatorKi:2W → 2W • Ki(E) :＝ { w∈W｜Pi(w) ⊆ E} := the event that ‘iknows E’ • T: Ki(E)⊆E (equiv. ∖E⊆∖Ki(E) ); • 4: Ki(E)⊆Ki(Ki(E)) ; • 5: ∖Ki(E) Ki(∖Ki(E)) ; N.B. 〈W, (Ki)i∈N〉 is also a model for the logic S5n

8. Game theoretical context Let fbe a sentence in S5n, M = 〈W,(Pi)i∈N〉 a model and m a probability measure with m(w) ≠0. Set X = ||f ||M : thetruth set of f • i’s conjecture of f: qi(f ;ω) := μ(||f ||M ｜i(ω)) . • The event of the conjecture [qi(f) = qi(f;ω) ] = {ξ∈W ｜ qi(X;ξ) = qi(X;ω) } • The event of all the conjectures [q(f) = q(f ;ω) ] =∩i∈N[qi(f) = qi(f ;ω) ] • All agents agree ona sentence fif qi(f;ω) = qj(f;ω) for every i, jat any w ;

9. How the agents reach such agreement? • KE(E)= ∩i∈NKi(E) : The agents mutually know E. • KC(E)= ∩n=0,1,2,… (KE)n(E): They commonly know E. • E iscommon-knowledgeatwifw ∈KC(E). • E iscommon-knowledge everywhere ifKC(E)=W. `Agreeing to disagree’ theorem (Aumann,1964) : All the agents can agree on X if they commonly know their conjectures of X everywhere; i.e.; (1) If wKC ([q(f ) = q(f ;ω) ]) then qi(f ;ω)= qj(f ;ω) for all i, j, and hence (2)If wW;KC ([q(f) = q(f ;ω) ]) =W, then(X, qi)= (X, qj) for all i, j. ? How process they can reach such agreement?

10. Decision function Let Z be a set of decisions. A Z-valued decision function f = f(X ;·) :2W → Z • Disjoint union consistency (DUC):For any S, T  2Wwith S∩T=, if f (S) = f (T) = e then f (S∪T) = e • Preserving under difference (PUD): For any S, T  2Wwith S⊇T, if f(S) = f(T) = e then f (S∖T) = e • Convexity (COV): For a real-valued f = f(X ;·), if S, T  2Ware disjoint ( S∩T=,) there exist l1, l2[0,1] such that f (S∪T) = l1 f (S) + l2 f (T) with l1 + l1 = 1 . • Example: An [0, 1]-valued decision function of an eventX: f(X ;·) := μ(X｜·) satisfies (DUC), (PUD) and (COV)

11. Membership function An [0, 1]-valued decision function f (X ;·) of f:2W → [0, 1]. • i ’s membership value of X under private information according to f: di(X ;·): W → [0, 1], di(X ;w):= f (X;Pi(w)) • The event of di (X;ω): [di(X) = di (X;ω) ] = {ξ∈W ｜di (X;ξ) = di (X;ω) } • All agents agree onX at w if di (X;ω) = di (X;ω) for all i, j ; • All agents agree onX if (X, di) = (X, dj) for all i, j ; • Question: How the agents reach the agreement?

12. Epsilon-robust communication 〈, μ, Pr, (Pit) i∈N, (dit) i∈N |t = 0.1.2…..〉 • Each player i sends the messageMito the other player according to the protocol: • Pr : T → N×N, t → (s(t),r(t)) : a directed graph, s(t) is the sender at time t, r(t) is the recipient at t. • Fixε [0, 1) • (Pit)t∈T := the revision process ofi ’sinformation structure. • (dit)t∈T := the stochastic process of i’srevised membership values.

13. ● ● ● ● ● ● 〈 , μ, Pr, (Pit) i∈N , (dit) i∈N | t=1,2,…..〉 T ={ 0,1,2,3,…,t,.....} : line of times Communication graph Pr: T → N×N, t → (s(t), r(t)) is a directed graph with N := the set of edges,and i →j if and only if # { tT | i =s(t), j = r(t) } = ∞ contains no cycle contains a cycle

14. 〈 , μ,Pr,(Pit) i∈N, (dit) i∈N ,ε |t=1,2,…..〉 • fair： the graph is strongly-connected. • rounds：∃m such that ∀t ∈T, s(t)＝s(t＋m) • may contain cycles：There may be at least one group of agents {i1,i2,…,ik }(k≧3)such that ∀m＜k , s(t)＝im , r(t)＝im＋1 , s(t)＝ik and r(t) ＝ i1 Pr : T → N×N : as a directed graph

15. message 〈 , μ,Pr,(Pit) i∈N, (dit) i∈N,ε |t=1,2,…..〉 • (Pit) t=0,1,2,..is inductively defined as follows: • For t = 0, Pi0(ω):=Pi (ω) • For t ≥1, Pit+1(ω):=Pit(ω)ifir(t+1); Pit(ω)∩Ms(t)t(f ;ω)ifi = r(t+1). Mit(f ; ·) :  → 2 is the ε-robust message abouta sentence f to accuracy ε: Mit (f ;ω) :=｛ξ∈ ｜| dit (f;ξ) -dit (f;ω) | < ε} NB: We call it exact if ε= 0

16. Example: N = { I, II, Ⅲ}, ε=1/2 W= {w1, w2,….., w8} , m(w)= 1/8, Pr II● I● III●

17. X={w1,w2,w3. w4}, di(X ;ω) := μ(X | i (ω)) , ,ε=1/2 Player II Player I Robust information on dI (X;5) w. accuracy to ε = ½ MI0 (X;5) :=｛ξ∈ ｜ | dI(X;ξ) - dI (X;5) | < ½} Player Ⅲ

18. 〈W, μ, (Pti)i∈N〉 with ε= 1/2 Player II MI0 Player I PII1(w) = PII1(w)∩ MI0() dII1(X;ω):=μ(X｜II1(ω)) Player III

19. 〈W, μ, (Pti)i∈N〉 with ε= 1/2 Player II Player I MII1() :=｛ξ∈ ｜ | dII1(X:ξ) – dII1 (X;) | < ½ } Player III

20. 〈W, μ, (Pti)i∈N〉 with ε= 1/2 Player II MII1 Player I PⅢ2(w) = PⅢ(w)∩ MII1() dⅢ2(aj)(ω):=μ(X｜PⅢ2(w)) Player III

21. 〈W, μ, (Pti)i∈N〉 with ε= 1/2 Player II Player I MⅢ2(5) :=｛ξ∈ ｜ | dⅢ2(ξ) - dⅢ2 (5)| < ½ } Player III

22. 〈W, μ, (Pti)i∈N〉 with ε= 1/2 Player II Player I MIII2 Player III

23. 〈W, μ, (Pti)i∈N〉 with ε= 1/2 Player II Player I Player III

24. 〈W, μ, (Pti)i∈N〉 with ε= 1/2 Player II MI3 Player I Player III

25. 〈W, μ, (Pti)i∈N〉 with ε= 1/2 Player II Player I Player III

26. 〈W, μ, (Pti)i∈N〉 with ε= 1/2 Player II Player I MII4 Player III

27. 〈W, μ, (Pti)i∈N〉 with ε= 1/2 Player II Player I Player III

28. 〈W, μ, (Pti)i∈N〉 with ε= 1/2 Player II Player I MIII5 Player III

29. Stationary Information Structure (Pi )i=I,II,III Player II Player I Player III

30. The agents agree on X={w1, w2,w3. w4}by communication with accuracy toε= 1/2 Player II Player I dⅠ∞(X;ω)=dⅡ ∞(X;ω)=dⅢ ∞(X;ω) = 1 (ω=ω1, ω2, ω3, ω4) 0 (ω=ω5, ω6, ω7, ω8) Player III

31. Main theorem (acyclic case) • Set the stochastic process (dit)t=0,1,2,.. according to the decision function f and sentence f of S5n: dit(f,ω):= f ( ||f ||M｜it(ω)) Proposition 1: The stochastic process(dit )t=0,1,2… converges toa distributiondi∞ Theorem (Matsuhisa, 2007): Let fsentence of S5n and ε[0, ½). Assume the decision function f of X satisfies (DUC) and (PUD). In the communication under ε-robust messages, if the communication graph Pr has no cycle, the agents can agree on f: (||f ||M, di) = (||f ||M, dj) (fuzzy sets)

32. Concluding remarks Each player receives not the exact information but the robust information accuracy to εabout the other players’ actions as messages. • The communication-graph contains no cycles. • Theorem is still valid for models for S4n. Non acyclic case: • Question: Theorem is still valid when the communication-graph contains cycles? • Proposition 2: In the communication without acyclic condition, assume the decision function f of X satisfies (UCD), (PUD) and (COV). Then for anywW, there exists{w2, w3,….., wn} such that d1∞(f,ω)= d2∞(f, ω2) =…= dn∞(f, ωn) . • Unknown whether (||f ||M, di) = (||f ||M, dj) or not ?

33. Chronicle of Agreement theorem

34. Grazie! Thank you! Спасибо! 謝謝 御静聴有難うございました