Logical Foundations of Negotiation: Strategies and Preference

1. Logical Foundations of Negotiation:Strategies and Preference Thomas Meyer, Norman Foo Rex Kwok Dongmo Zhang Presented by Shiyan Li

2. Introduction • Quantitative Approacheseg. Utility Function; Game Theory…… • Qualitative Approaches

3. Introduction • Extend the logical framework introduced in a previous paper (Meyer, T.; Kwok, R.; and Zhang, D. 2003. Logical foundations of negotiation: Outcome, concession and adaptation. Norman Foo’s Festschrift, http://www.cse.unsw.edu.au/ ~ksg/Norman/) by considering scenarios in which the initial demands of agents may vary. • Results:(1) A negotiation strategy should be an AGM belief revision operation.(2) A negotiation strategy should be a preference relation on demands.

4. Introduction • In the framework, agents are assumed to truthful, rational and cooperative. • Truthfulness:Pose true and most preferred demands. • Rationality:Attempt to maximize their own gains without being concerned others’. • Cooperation:Accommodate the demands of others. They must make a deal after negotiation.

5. Logical Framework • Negotiation Process:Considers two TRC agents.Terminates when they strike a deal. • Aims of Negotiation:(1) Have as many of their initial demands in the outcome as possible.(2) Reach an permissible agreement as quickly as possible.

6. Logical Framework • Conession ModelParticipants are required to concede by retracting some of their initial demands. • Adaptation ModelThe final outcome consists of those demands common to the adaptation of the two agents.

7. Outcome of Negotiation • DemandDemand Pair: K = (K0, K1)Ki (i = 0, 1): is a consistent theory (i.e. Ki = Cn(Ki)), represents the initial demands of agent i • Postulates of Permissible Deal D(O1) O(D) = Cn(O(D))(O2) O(D) ⊭ ⊥(O3) If K0 ∪ K1 ⊭ ⊥ then O(D) = (K0 ∪ K1)(O4) (K0 ∩ K1) ⊆ O(D) or O(D) ∪ (K0 ∩ K1) ⊭ ⊥

8. M(K0) M(K1) M(K0) M(K1) M(O(D)) M(O(D)) 0-dominated deals 1-dominated deals M(K0) M(K1) M(K0) M(K1) M(O(D)) M(O(D)) cooperative deals neutral deals Outcome of Negotiation • Trivial DealA trivial deal is one for which the outcome is Cn(K0∪K1). • Non-trivial Deal

9. Concession Model • i-concession Ci(D)Represents the weakened demands of agent i. • Outcome of Concession Model(O1): (OC) O(D) = Cn(C0(D)∪ C1(D)) • Permissible Deals of Concession Model(O1): (C1) Ci(D) = Cn(Ci(D)) for i = 0, 1 (C2) Ci(D) ⊆ Ki for i = 0, 1(O3): (C3) If K0 ∪ K1⊭ ⊥ then Ci(D) = Ki for i = 0, 1(O2): (C4) C0(D)∪ C1(D) ⊭ ⊥(O4): (C5) If C0(D)∪ K1⊭ ⊥ or C1(D)∪ K0⊭ ⊥ then K0 ∩ K1⊆C0(D)∪ C1(D)(O4): (C6) If C0(D)∪ K1⊨ ⊥ and C1(D)∪ K0⊨ ⊥ then C0(D)∪ C1(D) ∪ (K0 ∩ K1) ⊨ ⊥

10. Adaptation Model • i-adaptation Ai(D)Represents the adapted demands of agent i. • Outcome of Adaptation Model(O1): (OA) O(D) = A0(D) ∩ A1(D) • Permissible Deals of Adaptation Model(O1): (A1) Ai(D) = Cn(Ai(D)) for i = 0, 1(O3): (A2) If K0∪ K1⊭ ⊥ then A0(D) = A1(D) = Cn(K0 ∪ K1)(O4): (A3) K0⊆ Ai(D), or K1⊆ Ai(D), or Ai(D) ∪ (K0 ∩ K1) ⊨ ⊥, for i = 0, 1(O4): (A4) For i = 0, 1, if Ki⊈ Ai(D) then A0(D) = A1(D)Note: Why is there no this postulate: A0(D) ∩ A1(D) ⊭ ⊥ being a correspondence of (O2)?

11. AGM Belief Change • By Alchourron, Gardenfors, & Makinson in 1985 • Belief ChangeRevision: An agent has to incorporate new information while maintaining consistency.Contraction: An agent has to remove information from its current beliefs.

12. AGM Belief Revision • In Nayak’s Approach:In which C is a theory.(K*1) K * C = Cn(K * C)(K*2) K * C ⊆ Cn(K ∪ C)(K*3) If K ∪ C ⊭ ⊥ then K * C =Cn(K ∪ C)(K*4) C ⊆ K * C(K*5) C = Cn(C)(K*6) K * C ⊨ ⊥ iff C ⊨ ⊥(K*7) K * Cn(A ∪ B) ⊆ Cn((K * A) ∪ B)(K*8) If (K * A) ∪ B ⊭ ⊥ then Cn((K * A) ∪ B) ⊆ K * Cn(A ∪ B) • (K * 1)-(K * 6): Basic AGM Revision(K * 1)-(K * 8): Full AGM Revision

13. AGM Belief Revision • ≾ is K-faithfuliff the ≾–minimal valations are exactly the models of K (i.e. M(K) = M≾(Cn(⊤)). • Theorem 1For every K-faithful total preorder on valuations ≾, there is a full AGM revision operation * such that K * C =Th(M≾(C)) for every theory C. Conversely, for every full AGM revision operation *, there is a K-faithful total order on valuations ≾ such that K * C = Th(M≾(C)) for every theory C.

14. AGM Belief Revision • Result:Every full AGM revision operation is a representation of the preferences of an agent with regard to its beliefs.

15. Negotiation Strategies • By concedingAgent view: weaken its current demands to some acceptable level.AGM view: contract belief by a theory C to the belief not containing C. • By adapting(1)Agent view: strengthen its current demands, adopt a set which includes the demands of its adversary.AGM view: revise belief by a theory C to a strengthened the belief.(2)Agent view: settle on a set that is inconsistent with the initial commonly held demands.AGM view: revise belief by a theory C to a belief which includes C, and is inconsistent with the originally held belief.

16. Negotiation Strategies • Mutual Belief RevisionWhenever the initial demands of the agents are conflicting, each agent will be required to present a weakened version of their demands to the other which is obliged to accept this weaker set of demands. • The process of accepting weakened demands can be modeled by AGM Belief Revision.

17. Rational Negotiation Strategies • ⊗: negotiation strategyC: a set of weakened demands of agent 1-i which i has to accept • If Ki∪ C ⊭ ⊥:every basic AGM revision operation will produce Cn(Ki ∪ C) • If Ki∪ C ⊨ ⊥:(1) C = K1-iAgent 1-i regards all demands in as equally preferable(2) C is strictly weaker than K1-IAgent 1-i expresses a preference for the demands in C over the remaining demands in K1-I .

18. Rational Negotiation Strategies • Cooperation:(A) (Ki⊗ C) ∩ K1-i = C. • Rationality:It is required the inclusion of as many demands as possible. • Ki⊗ C should be the largest set of sentences subject to the restriction imposed on it in (A).

19. Rational Negotiation Strategies • Definition 1The negotiation strategy ⊗ for agent i (i = 0, 1) with demand set Ki is rational iff for every input C such that Ki∪ C ⊨ ⊥ and C ⊂ K1-i, it is the case that Ki ⊗ C is the largest set satisfying (A).

20. Permissible and Compatible Inputs • Definition 2A theory C is an i-permissible input (i = 0, 1) for a negotiation strategy iff there is a permissible deal D such that C1-i(D) = C.

21. Permissible and Compatible Inputs • Proposition 1Let K = (K0, K1) be any demand pair and let i ∈ {0, 1}. For every A ⊇ K1-i, Ki∩ A is an (1-i)-permissible input and is consistent with K1-i.

22. Permissible and Compatible Inputs • Definition 3Let K = (K0, K1) be any demand pair, let i ∈ {0, 1}, and ⊗ a rational negotiation strategy for agent i. The i-concession Ci(D) of a permissible deal D is i-compatible with ⊗ iff Ci(D) ∪ K1-i⊨ ⊥ or Ci(D) = Ki⊗ K1-i.

23. Determination of Deals • Definition 4Let K = (K0, K1) be a demand pair and i ∈ {0, 1}.A rational negotiation strategy ⊗ i-determines a permissible deal D iff Ki⊗ C1-i(D) = Ai(D) and Ci(D) is i-compatible with ⊗.Two permissible deals D and D’ are i-codetermined by ⊗ iff they are both i-determined by it.A set of permissible deals D is i-codetermined by ⊗ iff the elements of D are pairwise i-codetermined by ⊗.A permissible deal D is uniquely i-determined by ⊗ iff it is i-determines by ⊗ and i-codetermined only by itself.

24. Determination of Deals • Proposition 2If K0∪ K1⊭ ⊥ then every rational negotiation strategy uniquely i-determines the trivial deal, and only the trivial deal. Now suppose that K0∪ K1⊨ ⊥, and consider any rational negotiation strategy ⊗. Then1. A deal D is i-determines by ⊗ iff Ci(D) is i-compatible with ⊗.2. ⊗ uniquely i-determines every neutral deal.3. ⊗ uniquely i-determines exactly one (1-i)-dominated deal D and does noti-determines any other (1-i)-dominated deal.4. ⊗ i-determines every i-dominated deal.5. For every i-dominated deal D there is a single cooperative deal D’ for which Ci(D’) is i-compatible with ⊗, such that D and D’ are i-codetermined by ⊗.6. For every cooperative deal D for which Ci(D) is i-compatible with ⊗, there is an i-dominated deal D’ such that D and D’ are i-codetermined by ⊗.7. No two i-dominated deals are i-codetermined by ⊗.

25. Determination of Deals • Theorem 21. For every pair of rational negotiation strategies (⊗0, ⊗1), there is a permissible deal that is i-determined by ⊗i, for i = 0, 1.2. For every permissible deal D there is a pair of rational negotiation strategies (⊗0, ⊗1) such that D is i-determined by ⊗i, for i = 0, 1.

26. Negotiation Strategies and Compound Deals • Compound Deal DLet K be an enumeration of all demand pairs K = (K0, K1). A compound deal D is a n-tuple, where n is the number of demand pairs, and for every j ∈ {1, …, n}, entry j in D, denoted by Dj, is a permissible deal with respect to the jth demand pair Kj = (Kj0, Kj1) in the enumeration K.

27. Negotiation Strategies and Compound Deals • (O5) ∀ j, k ∈ {1, …, n}, i ∈ {0, 1}, if Kji = Kk1-i, Kj1-i⊆ Kk1-i, Kj1-i⊆ O(Dj), and O(Dj) ∪ Kk1-i ⊭ ⊥, then O(Dk) = Cn(O(Dj) ∪ Kk1-i) • (C7) ∀ j, k ∈ {1, …, n}, i ∈ {0, 1}, if Kji = Kk1-i, Kj1-i⊆ Kk1-i, Ci(Dj) ∪ Kj1-i ⊭ ⊥, and Ci(Dj) ∪ Kk1-i ⊭ ⊥, then Ci(Dk) = Cn(Ci(Dj) ∪ Kk1-i) ∩ Kji • (A5) ∀ j, k ∈ {1, …, n}, i ∈ {0, 1}, if Kji = Kk1-i, Kj1-i⊆ Kk1-i, Kj1-i⊆ Ai(Dj), and Ai(Dj) ∪ Kk1-i ⊭ ⊥, then Ai(Dk) = Cn(Ai(Dj) ∪ Kk1-i)

28. Negotiation Strategies and Compound Deals • Definition 5For i ∈ {0, 1}, a negotiation strategy ⊗ i-determines a permissible compound dealD iff for every j ∈ {1, …, n}, Kji⊗C1-i(Dj) = Ai(Dj) and Ci(Dj) is i-compatible with ⊗.

29. Negotiation Strategies and Compound Deals • Theorem 31. For every pair of systematic negotiation strategies (⊗0, ⊗1), there is a permissible compound deal that is i-determined by ⊗i, for i = 0, 1.2. For every permissible compound deal D there is a pair of systematic negotiation strategies (⊗0, ⊗1) such that D is i-determined by ⊗i, for i = 0, 1.

30. Negotiation Strategies and Compound Deals • Result:By T1 and T3 it follows that each systematic negotiation strategy can be redefined as a total preorder on valutions.