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

Introduction Quantitative Approaches eg. Utility Function; Game Theory…… Qualitative Approaches

Introduction Extend the logical framework introduced in a previous paper (Meyer, T.; Kwok, R.; and Zhang, D Logical foundations of negotiation: Outcome, concession and adaptation. Norman Foo’s Festschrift, ~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.

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.

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.

Logical Framework Conession Model Participants are required to concede by retracting some of their initial demands. Adaptation Model The final outcome consists of those demands common to the adaptation of the two agents.

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

Outcome of Negotiation Trivial Deal A trivial deal is one for which the outcome is Cn(K 0 ∪ K 1 ). Non-trivial Deal M(K 0 )M(K 1 )M(K 0 )M(K 1 ) M(K 0 )M(K 1 )M(K 0 )M(K 1 ) M(O(D)) 0-dominated deals1-dominated deals cooperative dealsneutral deals

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

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

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

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

AGM Belief Revision ≾ is K-faithful iff the ≾ –minimal valations are exactly the models of K (i.e. M(K) = M ≾ (Cn( ⊤ )). Theorem 1 For 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.

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

Negotiation Strategies By conceding Agent 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.

Negotiation Strategies Mutual Belief Revision Whenever 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.

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

Rational Negotiation Strategies Cooperation: (A) (K i ⊗ C) ∩ K 1-i = C. Rationality: It is required the inclusion of as many demands as possible. K i ⊗ C should be the largest set of sentences subject to the restriction imposed on it in (A).

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

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

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

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

Determination of Deals Definition 4 Let K = (K 0, K 1 ) be a demand pair and i ∈ {0, 1}. A rational negotiation strategy ⊗ i-determines a permissible deal D iff K i ⊗ C 1-i (D) = A i (D) and C i (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.

Determination of Deals Proposition 2 If K 0 ∪ K 1 ⊭ ⊥ then every rational negotiation strategy uniqu ely i-determines the trivial deal, and only the trivial deal. Now suppose that K 0 ∪ K 1 ⊨ ⊥, and consider any rational negotiation strategy ⊗. Then 1. A deal D is i-determines by ⊗ iff C i (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 not i- 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 C i (D’) is i-compatible with ⊗, such that D and D’ are i-codetermined by ⊗. 6. For every cooperative deal D for which C i (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 ⊗.

Determination of Deals Theorem 2 1. For every pair of rational negotiation strategies ( ⊗ 0, ⊗ 1 ), there is a permissible deal that is i-determined by ⊗ i, for i = 0, 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.

Negotiation Strategies and Compound Deals Compound Deal D Let K be an enumeration of all demand pairs K = (K 0, K 1 ). 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 D j, is a permissible deal with respect to the jth demand pair K j = (K j 0, K j 1 ) in the enumeration K.

Negotiation Strategies and Compound Deals (O5) ∀ j, k ∈ {1, …, n}, i ∈ {0, 1}, if K j i = K k 1-i, K j 1-i ⊆ K k 1-i, K j 1-i ⊆ O(D j ), and O(D j ) ∪ K k 1-i ⊭ ⊥, then O(D k ) = Cn(O(D j ) ∪ K k 1-i ) (C7) ∀ j, k ∈ {1, …, n}, i ∈ {0, 1}, if K j i = K k 1-i, K j 1-i ⊆ K k 1-i, C i (D j ) ∪ K j 1-i ⊭ ⊥, and C i (D j ) ∪ K k 1-i ⊭ ⊥, then C i (D k ) = Cn(C i (D j ) ∪ K k 1-i ) ∩ K j i (A5) ∀ j, k ∈ {1, …, n}, i ∈ {0, 1}, if K j i = K k 1-i, K j 1-i ⊆ K k 1-i, K j 1-i ⊆ A i (D j ), and A i (D j ) ∪ K k 1-i ⊭ ⊥, then A i (D k ) = Cn(A i (D j ) ∪ K k 1-i )

Negotiation Strategies and Compound Deals Definition 5 For i ∈ {0, 1}, a negotiation strategy ⊗ i- determines a permissible compound deal D iff for every j ∈ {1, …, n}, K j i ⊗ C 1-i (D j ) = A i (D j ) and C i (D j ) is i-compatible with ⊗.

Negotiation Strategies and Compound Deals Theorem 3 1. For every pair of systematic negotiation strategies ( ⊗ 0, ⊗ 1 ), there is a permissible compound deal that is i-determined by ⊗ i, for i = 0, 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.

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.