From Axiomatic to Strategic Models of Bargaining with Logical Beliefs and Goals

From axiomatic to strategic models of bargaining with logical beliefs and goals

Quoc Bao Vo and Minyi Li Faculty of Information & Communication Technologies Swinburne University of Technology, Australia email: {bvo | myli}@swin.edu.au

ABSTRACT the strategic (non-cooperative) approach to provide the foundations 1 In this paper, we introduce axiomatic and strategic models for bar- for cooperative bargaining solution concepts. His approach was to gaining and investigate the link between the two. Bargaining situ- design a non-cooperative game, now known as the Nash demand ations are described in propositional logic while the agents’ pref- game, in which the only equilibrium outcome is exactly the one erences over the outcomes are expressed as ordinal preferences. suggested by Nash solution. Our main contribution is an axiomatic theory of bargaining. We We’ll now turn our attention to multiagent systems in which propose a bargaining solution based on the well-known egalitarian agents hold beliefs about the environment they are operating in and social welfare for bargaining problems in which the agents’ logical have goals they want to achieve or maintain. In many multiagent beliefs specify their bottom lines. We prove that the proposed so- frameworks, the agents beliefs and goals are represented as logical lution is uniquely identified by a set of axioms. We further present sentences. Moreover, the agents are required to interact, coordi- a model of bargaining based on argumentation frameworks with nate and in most cases, negotiate to reach agreements about who the view to develop a strategic model of bargaining using the con- do what and who get what. Given that Nash’s theories and most lit- cept of minimal concession strategy in argument-based negotiation erature on bargaining have been based exclusively on utility theory frameworks. (and possibly probability theory when uncertainty is present), it has been a challenge to apply these game-theoretic models to develop solutions for or analyse the agent negotiation problems when agents Categories and Subject Descriptors hold logical beliefs and goals. Attempts have been made to convert I.2.11 [Distributed Artificial Intelligence]: Multiagent Systems, agents’ goals to a form of utility through a cost function (see e.g., Intelligent Agents; J.4 [Computer Applications]: Social and Be- [13]). However, questions remain on how agents’ logical beliefs havioral Sciences can be integrated into such a framework. Other researchers have attempted to revive Nash’s approaches, particularly the axiomatic General Terms approach, by studying the properties that a bargaining solution (of a bargaining problem with logical goals) should satisfy (see e.g., Design, Economics [9, 8, 16, 15] and the reference therein). While [9] and [8] study a number of logical properties for negotiation, it’s not clear what Keywords bargaining solution they would suggest for self-interested bargain- Bargaining and negotiation, Collective decision making, Judgment ers. In [16], an interesting bargaining solution is proposed together aggregation and belief merging with a study on a number of game-theoretic properties the proposed solution satisfies. Zhang [15] introduces a framework that is perhaps closest to what Nash intended with his axiomatic approach. In 1. INTRODUCTION this paper, Zhang proposes a solution, which he calls simultaneous The formal theory of bargaining originated with John Nash’s concession solution, and shows that it is exactly characterised by a seminal papers [10, 11]. Nash’s 1950 paper establishes a frame- set of axioms. Zhang’s bargaining solution is, however, quite prob- work for bargaining analysis. In this paper, Nash initiated an ax- lematic because: (i) it’s syntax sensitive and, as a consequence, iomatic approach to bargaining, in which we abstract from the bar- prone to manipulation; and (ii) it actually removes the goals that gaining process itself and specify a list of properties (axioms) that a both agents agree on, the so-called “drowning effect”. bargaining solution should satisfy. Nash then proposed four axioms The present paper is an effort to reopen the Nash program, partic- and proved that they uniquely characterise what is now known as ularly for agent-based bargaining problems in which agents’ beliefs the Nash bargaining solution. In Nash’s 1953 paper, he then turned and goals are expressed as logical sentences. Towards that end, we to the question of how this solution might be obtained in bargain- introduce axiomatic and strategic models of bargaining. We pro- ing situations between self-interested agents; i.e., investigate the pose a set of axioms that a bargaining solution should satisfy and bargaining problem using a strategic approach. In this paper, Nash introduce a bargaining solution that is exactly characterised by the implicitly established a new research agenda, attempting to utilise proposed axioms. Our proposed bargaining solution is quite intuitive and based on the well-known egalitarian social welfare. Sub- Appears in: Proceedings of the 11th International Conference sequently, we also present a strategic model of bargaining based on Autonomous Agents and Multiagent Systems (AAMAS 2012), on the minimal concession strategy and shown that its equilibrium Conitzer, Winikoff, Padgham, and van der Hoek (eds.), June, 4–8, 2012, Valencia, Spain. 1 Copyright c 2012, International Foundation for Autonomous Agents and This research agenda has been commonly referred to as the Nash Multiagent Systems (www.ifaamas.org). All rights reserved. program (see [2]). outcomes turns out to be the solution outcomes described by our A1. Invariance to equivalent utility representations. Let the bar- axiomatic theory. gaining problem (S′,d′) be obtained from (S,d) by the trans- ′ ′ The paper is organised as follows: we present some technical formations si = αisi + βi and di = αidi + βi, where ′ ′ definitions in Section 2. The axiomatic model of bargaining is in- αi > 0, then fi(S ,d ) = αifi(S,d) + βi, for i =1, 2. troduced in Section 3. In particular, we consider two cases: when the agents’ bottom lines are fixed and based entirely on the agents’ A2. Symmetry. If the bargaining problem (S,d) is symmetric initial beliefs; and when the agents’ bottom lines can be revised as (i.e., d1 = d2 and (s1,s2) ∈S ⇔ (s2,s1) ∈ S), then the negotiation progresses and the opponent can introduce convinc- f1(S,d) = f2(S,d). ing arguments to challenge the agent’s initial bottom line. A main ′ A3. Independence of irrelevant alternatives. If (S,d) and (S ,d) result is also introduced in Section 3. Section 4 presents a strategic ′ ′ are bargaining problems such that S⊆S and f(S ,d) ∈S bargaining model which is based on the minimal concession strat- ′ then f(S ,d) = f(S,d). egy with some substantial modification to allow the agents to hold their position without having to make a concession through the use A4. Pareto efficiency. If (S,d) is a bargaining problem with s,s′ ∈ ′ of sufficiently convincing arguments. We follow the anonymous S and si >si for i =1, 2, then f(S,d) 6= s. reviewers’ recommendations by: (i) omitting a number of prelim- inary results in Section 4, replacing them instead by a discussion Another important contribution in Nash’s seminal paper is that, about our model and solution; and (ii) providing the proofs for all he also tied the axiomatic theory of bargaining and his proposed of the theoretical results present in this paper. We agree with the bargaining solution up nicely by proving that the proposed axioms reviewers that the revised paper is more self-contained and thus, uniquely characterise the Nash bargaining solution. significantly improved. 3. A LOGICAL MODEL OF BARGAINING 2. BACKGROUND Consider a finite set N = {1, 2} of two agents who try to come 2.1 Logical Preliminaries to an agreement over the alternatives in O ⊆ PO. Each agent i has a preference relation i, which is assumed to be a total pre-order We consider a propositional language L defined from a finite (i.e., total, reflexive and transitive), defined over the set of alter- (and non-empty) alphabet P together with the standard logical con- natives O. Each agent i also maintains a set of beliefs Bi ⊆ L. nectives, including the Boolean constants ⊤ and ⊥. Furthermore, Essentially, Bi represents agent i’s beliefs about her available op- we also assume that PO ⊆ P is the non-empty alphabet for the tions outside of the negotiation or simply her reservation value. negotiation outcomes. That is, the propositional variables from PO constitute the issues to be settled by the negotiating agents. An REMARK: It’s important to note that the agents’ beliefs don’t en- interpretation ω is a total function from P to {⊤, ⊥}. An interpre- code their hard constraints. If an agent has a hard constraint that tation ω is a model of a set of sentences Φ ⊆L if and only if every can never be violated and that rules out an outcome o ∈ PO then sentence in Φ is satisfied by ω. JΦK denotes the set of models of the o∈ / O. Rather, the agent’s beliefs Bi encode her bottom line in the set of L-sentences Φ. Given a sentence φ ∈L, we’ll also write JφK bargaining, in the sense that, due to the requirement of individual instead of J{φ}K. An outcome (or alternative) o is a total function rationality, she will never agree on an outcome that is worse than from PO to {⊤, ⊥}. We denote by PO the set of all possible out- her bottom line. comes. We will identify each outcome o with the canonical term on We now define the bargaining problem to be a tuple BP = (O, PO which has o as its unique model. For instance, if PO = {p,q} hB1, 1i, hB2, 2i), where O ⊆ PO is the set of outcomes and and o(p) = ⊤,o(q) = ⊥, then o is identified with the term p ∧¬q Bi and i are agent i’s beliefs and preference relation over O, (or pq¯, or {p, ¬q}). respectively. Following Nash [11] and other researchers in the lit- erature of (cooperative) game-theoretic bargaining (see e.g., [12]), 2.2 Nash bargaining theory we are also interested in a bargaining solution, by which we mean Nash [10] established a framework to study bargaining. In his a function f that specifies a unique outcome set f(BP ) ∈ 2O for framework, a set of bargainers N = {1, 2} tries to come to an every bargaining problem BP . The reason why we target an out- agreement over a set of possible alternatives A. If they fail to reach come set as the solution of the bargaining problem instead of an an agreement, a disagreement event D occurs. Each agent i ∈ outcome in O will become clear later. N has a von Neumann - Morgenstern utility function ui : A ∪ {D} → R from which the set of all utility pairs that result from EXAMPLE 1. A vendor agent of a house (agent 1) is negotiat- some agreement, ing with a prospective buyer (agent 2) over the sale of the house. The two issues they need to come to an agreement are: the price S = {(u (a),u (a)) ∈ R2 : a ∈ A}, 1 2 of the house (i.e. whether the buyer should pay the vendor’s ask- as well as the pair d = (d1,d2), where di = ui(D) can be con- ing price), a proposition denoted by P , and the settlement (i.e., structed. whether it should be an early settlement), denoted by E. Then Nash then defined the pair (S,d) to be a bargaining problem. O = {PE,P E,¯ PE,¯ P¯E¯} is the set of possible outcomes. In Nash subsequently defined the bargaining solution to be a function this example, an outcome, say P¯E¯, indicates that the buyer would f : B → R2 that specifies, for each bargaining problem (S,d), a pay a price lower than the vendor’s asking price such as the me- unique outcome f(S,d) ∈S. dian house price of the area and the settlement will be according In the same paper, Nash also introduced an axiomatic theory of to the standard settlement of three months after the date of pur- bargaining. Rather than specifying an explicit model of the bar- chase. Assume further that F denotes the proposition that the gaining procedure, the axiomatic approach aims to impose proper- vendor agent has an existing offer agreeing to pay him the ask- ties that one wants a bargaining solution to satisfy, and then look ing price, and A denotes the proposition that the buyer can get a for solutions with these properties. Nash proposed the following similar property at a price lower than the asking price. Given: four axioms: B1 = {F,F ⇒ P,A ⇒ ¬P } and B2 = {F ⇒ P,A ⇒ ¬P }, and P E ≻1 P E¯ ≻1 P¯ E ≻1 P¯E¯ and P¯E¯ ≻2 P E¯ ≻2 P¯ E ≻2 states that an outcome is only inefficient when there is some other P E, a bargaining problem BP = (O, hB1, 1i, hB2, 2i) can outcome that can improves for all agents. be defined. Next, the axiom of Independence of Irrelevant Alternatives will be formulated in our model as follows. REMARK: Our bargaining model defined above is quite similar ′ ′ to the ordinal bargaining model defined by Shubik [14]. In this IIA. If BP =(O, hB1, 1i, hB2, 2i) and BP =(O , hB1, 1 ′ model, a bargaining situation (between two players) can be rep- i, hB2, 2i) are bargaining problems with O ⊆ O , and ′ ′ resented as a tuple (A,D, ≥1, ≥2), where A is the set of possi- f(BP ) ⊆ O then f(BP ) = f(BP ). ble agreements, D is the disagreement (i.e., the outcome when the agents fail to reach an agreement), and ≥1 and ≥2 are the prefer- Finally, axiom Symmetry can be interpreted as imposing the ence orderings over the set A ∪{D} of the agents 1 and 2, respec- requirement of fairness on a bargaining solution. That is, when tively.2 In the ordinal bargaining model, a bargaining solution F the bargaining situation is symmetric for the two bargainers in the maps from every bargaining situation (A,D, ≥1, ≥2) to an agree- sense that they have similar bargaining power and that there is no ment f(A,D, ≥1, ≥2) ∈ A (see e.g., [12] for more details). possible agreement that can provide a particular payoff structure to the two bargainers without another possible agreement that can In the following, we’ll explore the cases when the agents’ beliefs provide the opposite payoff structure, then the solution should give define their bottom-lines (and thus, the disagreement position of the the bargainers the same payoffs. With a discrete set of alternatives, agents), and also when the agents’ beliefs are uncertain and can be it can not be guaranteed to have an alternative that satisfies this revised (in relation to the opponents’ beliefs and position). property.

3.1 Bargaining with fixed bottom lines EXAMPLE 2. Continue with our running example, consider the When the agents’ beliefs are not revisable, they define the agents’ following bargaining situation: BP = (O, hB1, 1i, hB2, 2i) disagreement points. We can now discuss a reformulation of Nash’s where B1 = {E} and B2 = {E} (i.e., both bargainers insist in an axioms for the logical bargaining model by associating agent i’s early settlement), and P E ≻1 P¯ E ≻1 P E¯ ≻1 P¯E¯ and P¯ E ≻2 disagreement point with his beliefs. Firstly, we will define the dis- P E ≻2 P¯E¯ ≻2 P E¯. It’s easy to see that the two agents have agreement point for a logical bargaining problem. Intuitively, the similar bargaining power in which they would both rule out the two notion of disagreement point (or threat point) has been used to en- least preferred outcomes and it happens that, in this bargaining code a bargainer’s bargaining power. That is, the higher the utility situation, they share the same set of outcomes they are willing to of the disagreement point to a bargainer, the more power she has as agree on, namely the set {PE, P¯ E}. However, they have opposite she can walk away from the negotiation and obtain that high utility preferences over the outcomes in this set. Thus, neither outcome outside of the negotiation. This matches with our designation of would be an attractive solution in this bargaining situation. By the agent’s beliefs. It’s her beliefs that define what she thinks she allowing this set of outcomes to be the solution in this situation, and her opponent can get outside of the negotiation, which subse- we are open to any resolution, including Nash’s suggestion of non- quently defines her relative bargaining power and the negotiation physical agreements such as the lotteries over these outcomes. disagreement point. To ensure fairness, we will target outcomes that aim at maximis- DEFINITION 1. Let BP = (O, hB1, 1i, hB2, 2i) be a bar- ing the payoffs for agents with the smallest gains (in utility). We de- gaining problem, agent i’s disagreement point is the outcome Di ∈ fine a cardinal gain of an outcome o for an agent i to be the number such that (i) is least preferred to , and (ii) is consistent O Di i Di of outcomes between o and the disagreement point Di. Formally, with Bi. Then, agent i’s bargaining power is defined to be the number of outcomes ruled out by Di, according to agent i’s prefer- DEFINITION 2. Let BP = (O, hB1, 1i, hB2, 2i) be a bar- ence ordering: gaining problem and an agreement-feasible outcome σ ∈ O, the (cardinal) gain of outcome σ for agent i is defined as follows: Ui(Di)=#{o ∈ O : Di ≻i o}. Gi(σ)=#{o ∈ O : σ ≻i o & o i Di}. An outcome o ∈ O is agreement-feasible if o i Di for i =1, 2.

For bargaining with ordinal preference, it has been shown by Basically, the axiom for fairness, to be presented in the follow- Osborne and Rubinstein [12] that a reformulation of A1 (i.e., an ing, ensures that the difference between the bargainers’ gains would axiom expressing Invariance of Equivalent Preference Representa- be minimal. However, to avoid the bargainers to settle for fair but tions) would result in unattractive bargaining solutions. suboptimal outcomes, we require only that fairness be subject to In the rest of this paper, we’ll denote the agent other than agent the optimality of the outcome. We will introduce a concept to allow efficiency to be formulated in unanimity, namely Unanimous i by −i. Furthermore, for convenience of presentation, given two ′ ′ ′ ′ Efficiency (UE). Intuitively, if o can improve for both the worst-off outcomes o,o ∈ O, we’ll say that o o if and only if o i o for ′ ′ ′ agent and the better-off agent in comparison to o then o is con- i =1, 2; similarly, o ≈ o if and only if o o and o o . ′ Pareto Efficiency axiom can be formulated in our model as fol- sidered to be UE-dominated by o and should not be selected as a lows. bargaining agreement. Formally,

PE. If BP = (O, hB1, 1i, hB2, 2i) is a bargaining problem DEFINITION 3. Let BP = (O, hB1, 1i, hB2, 2i) be a bar- ′ ′ ′ ′ with o,o ∈ O and o o and o ≻j o for some j ∈{1, 2}, gaining problem with the agreement-feasible outcomes o,o ∈ O, o ′ ′ then o∈ / f(BP ). is UE-dominated by o if mini=1,2(Gi(o )) ≥ mini=1,2(Gi(o)) ′ and maxi=1,2(Gi(o )) ≥ maxi=1,2(Gi(o)), and at least one of Note that in this paper, by Pareto-efficiency we mean the Strong them has to be a strict inequality. Pareto Efficiency, rather than the Weak Pareto Efficiency which Moreover, we’ll also say that two outcomes o and o′ are UE- 2 ′ The preference orderings are required to be total pre-orders. That equivalent if and only if mini=1,2(Gi(o )) = mini=1,2(Gi(o)) ′ is, a complete transitive reflexive binary relation. and maxi=1,2(Gi(o )) = maxi=1,2(Gi(o)). An outcome is unanimously efficient if it is not UE-dominated UE. If BP = (O, hB1, 1i, hB2, 2i) is a bargaining problem by any other outcome.3 with the agreement-feasible outcomes o,o′ ∈ O and o is UE-dominated by o′ then o∈ / f(BP ). LEMMA 1. Let BP = (O, hB1, 1i, hB2, 2i) be a bargaining problem with an agreement-feasible outcome o ∈ O, if o is Obviously, by Lemma 1, if the bargaining solution f satisfies UE unanimously efficient then it is also Pareto efficient. then it also satisfies PE. In the following, we’ll develop a bargain- Before proving Lemma 1, we state a convention to be used through- ing solution that is uniquely identified by the axioms UE, FR and UB. The developed solution is based on the well-known egalitarian out the rest of the paper: Given an outcome o, if G1(o) = G2(o), solution: we assume that arg mini=1,2(Gi(o)) will pick out a single value, either 1 or 2 (which one would be picked doesn’t matter). Further- O THEOREM 1. There is a bargaining solution f E : BP → 2 more, if j = argmini , (Gi(o)) then arg maxi , (Gi(o)) = =1 2 =1 2 given by: −j. That is, arg mini=1,2(Gi(o)) (resp. arg maxi=1,2(Gi(o))) is E guaranteed to deterministically return a single agent j (resp. −j) f (O, hB1, 1i, hB2, 2i) = arg max(max Gi(o)), where o∈ES i=1,2 whose cardinal gain Gj (o) (resp. G−j (o)) is smallest (resp. largest).

Proof: Suppose, to the contrary, that o is not Pareto efficient. That ES = arg max (min Gi(o)) ′ ′ ′ o∈AF i=1,2 is, there are o ∈ O and j ∈{1, 2} such that o ≻j o and o −j o. ′ Obviously, o is agreement-feasible. We’ll consider two cases: where AF ⊆ O denotes the set of agreement-feasible outcomes of ′ O Case 1: arg mini=1,2(Gi(o )) = argmini=1,2(Gi(o)) = k. If 5 ′ ′ BP . Then, a bargaining solution f : BP → 2 satisfies UE, FR, k = j then min (Gi(o )) > min (Gi(o)) and max(Gi(o )) ≥ E i=1,2 i=1,2 i=1,2 and UB if and only if f = f . ′ max(Gi(o)). If k 6= j then max(Gi(o )) > max(Gi(o)) and i=1,2 i=1,2 i=1,2 Before proving Theorem 1, we’ll discuss a relevant result. Note ′ min (Gi(o )) ≥ min (Gi(o)). Either way, we have o is UE- that, ≈ is an equivalence relation on the set O. Moreover, if we’ll i=1,2 i=1,2 ′ only focus on the set AF ⊆ O of agreement-feasible outcomes dominated by o , and thus can not be unanimously efficient. ′ of BP , then AF can be reduced to a collection of equivalence Case 2: arg mini=1,2(Gi(o )) 6= argmini=1,2(Gi(o)). With- ′ classes, such that each equivalence class can be represented by a out loss of generality, we can assume that arg mini=1,2(Gi(o )) = ′ ′ pair (G1(o),G2(o)), where o is a member of the equivalence class. 1 and arg mini=1,2(Gi(o))=2. That is, G1(o ) = mini=1,2(Gi(o )) ′ ′ and G2(o) = mini=1,2(Gi(o)) and G2(o ) = maxi=1,2(Gi(o )) LEMMA 2. If BP = (O, hB1, 1i, hB2, 2i) is a bargaining and G1(o) = maxi=1,2(Gi(o)). Therefore, G1(o) ≥ G2(o) and E ′ ′ problem then f (BP ) contains at most two equivalence classes G2(o ) ≥ G2(o). Also, because o ≻j o for some j ∈ {1, 2}, at E ′ of the agreement-feasible outcomes of BP . Moreover, if f (BP ) least one of the above inequality has to be strict. Since o ≻j o for ′ 4 ′ contains exactly one equivalence class then it has to be of the form some j ∈{1, 2} and o −j o, G1(o ) ≥ G1(o). E ′ ′ ′ (g, g) (i.e., it gives both agents the same gain); if f (BP ) contains Hence, G1(o ) ≥ G1(o) ≥ G2(o) and G2(o ) ≥ G1(o ) ≥ exactly two equivalence classes then they have to be of the form G1(o), and at least one of the inequalities G1(o) ≥ G2(o) and ′ ′ ′ (g,h) and (h, g). G2(o ) ≥ G1(o ) has to be strict. In other words, mini=1,2(Gi(o )) ≥ ′ mini=1,2(Gi(o)) and maxi=1,2(Gi(o )) ≥ maxi=1,2(Gi(o)), and Proof (of the Lemma): Obviously, from the definition of the func- at least one of these inequalities has to be strict. E tion f , there exist kmin, kmax ≥ 0 such that for every outcome Therefore, o is UE-dominated by o′, and thus can not be unani- o ∈ ES, mini=1,2 Gi(o) = kmin and for every outcome σ ∈ mously efficient. E f (BP ), maxi=1,2 Gi(σ) = kmax. Thus, when kmin = kmax, f E (BP ) contains exactly one equivalence class that is of the form The following Fairness axiom can now be formulated. (g, g), where g = kmin = kmax. E FR. If BP = (O, hB1, 1i, hB2, 2i) is a bargaining problem Otherwise, kmin 6= kmax and f (BP ) contains exactly two with an agreement-feasible outcome o ∈ O. If there is an equivalence classes of the form (g,h) and (h, g) where g = kmin ′ agreement-feasible and unanimously efficient outcome o ∈ and h = kmax. ′ ′ O such that |G1(o) − G2(o)| > |G1(o ) − G2(o )| then Proof (of the Theorem): First we prove that, f E satisfies UE, FR, o∈ / f(BP ). and UB. E In other words, axiom FR allows us to select the fairer outcomes That f satisfies UE: Let BP = (O, hB1, 1i, hB2, 2i) be among those that are unanimously efficient. a bargaining problem. Let σ ∈ f E (BP ), we’ll prove that σ is In addition to the above Fairness axiom, we will also require that unanimously efficient. Suppose, to the contrary, that there is an when outcomes are UE-equivalent, the bargaining solution will not agreement-feasible outcome o ∈ AF such that mini=1,2 Gi(o) ≥ be biased towards a particular one. The following axiom formulates mini=1,2 Gi(σ) and maxi=1,2 Gi(o) ≥ maxi=1,2 Gi(σ) and at this requirement for unbiasedness. least one of these inequalities is strict. Moreover, there exists kmin ≥ 0 such that ∀a ∈ AF. mini=1,2 Gi(a) ≤ kmin and kmin = UB. If BP = (O, hB1, 1i, hB2, 2i) is a bargaining problem ′ mini , Gi(σ). Thus, kmin = mini , Gi(o) = mini , Gi(σ). with agreement-feasible and UE-equivalent outcomes o,o ∈ =1 2 =1 2 =1 2 ′ In other words, o ∈ ES. However, according to Lemma 2, there O. Then, o ∈ f(BP ) if and only if o ∈ f(BP ). exists kmax ≥ 0 that, together with kmin, characterises the equiva- E Finally, we will replace the Pareto Efficiency (PE) axiom by a lence classes defining f (BP ) and ∀a ∈ ES. maxi=1,2 Gi(a) ≤ stronger one, requiring that the bargaining solution be unanimously kmax and kmax = maxi=1,2 Gi(σ). Thus, maxi=1,2 Gi(a) ≤ efficient, rather than only Pareto-efficient. maxi=1,2 Gi(σ). Contradiction. 3 5 Note that, similar to our remark about Pareto-efficiency, our defi- Note that more precise notations would be AFBP and ESBP . But nition of Unanimous Efficiency is also a strong one. since the bargaining problem BP is always clear from the context, 4We use −j to denote the agent other than j. we’ll drop these subscripts. That f E satisfies FR: Suppose, to the contrary, that there is then ¬P doesn’t necessarily define the buyer’s disagreement point. a bargaining problem BP = (O, hB1, 1i, hB2, 2i) and σ ∈ It could be the case that the buyer believes in ¬P , but is also willing f E (BP ) such that there is a unanimously efficient outcome o ∈ to retract this belief when learning about the scarcity of houses as AF and |G1(σ) − G2(σ)| > |G1(o) − G2(o)|. well as the high demand for houses in the area. According to Lemma 2, σ belongs to the equivalence classes Given that the agents’ beliefs (and constraints) will play a cru- characterised by the two non-negative numbers kmin and kmax cial role in this model of bargaining, we will make the agents’ hard (possibly equal to each other). constraints explicit in our bargaining model. In the rest of the pa- We have kmin = mini=1,2 Gi(σ) and kmax = maxi=1,2 Gi(σ). per, we will assume that the set of feasible negotiation outcomes is Case 1: mini=1,2 Gi(o) = kmin. Since |G1(σ) − G2(σ)| > defined by the hard constraints C. We denote by OC the set of fea- |G1(o) − G2(o)|, clearly maxi=1,2 Gi(o) < kmax, which is a sible outcomes that satisfy the hard constraints C. That is, o ∈ OC contradiction because o is UE-dominated by σ. if and only if JoK ∩ JCK 6= ∅. Case 2: mini=1,2 Gi(o) < kmin. Since o is unanimously ef- Given the propositional language L and the non-empty alphabet ficient, it is the case that maxi=1,2 Gi(o) > kmax. But then PO for the negotiation outcomes, a bargaining problem is defined |G1(σ) − G2(σ)| = kmax − kmin < max Gi(o) − min Gi(o) = to be a tuple BP = (C, hB1, 1i, hB2, 2i), where C⊆L is the i=1,2 i=1,2 set of hard constraints shared by all agents, Bi ⊆ L is the set of |G1(o) − G2(o)|. Contradiction. agent i’s beliefs, and i is agent i’s preference relation over the set It’s obvious that f E satisfies UB. OC. Subsequently, the (movable) disagreement points Di ∈ OC We are now proving that a solution f satisfying UE, FR, and UB are defined such that Di is least preferred to agent i and Di is con- necessarily obtains f E . sistent with Bi. Moreover, the set of agreement-feasible outcomes Suppose, to the contrary that, there exists a solution f satisfying AFBP is defined to be {o ∈ OC : o 1 D1}∩{o ∈ OC : o 2 UE, FR, and UB and a bargaining problem BP = (O, hB1, 1 6 E D2}. We will first state a trivial lemma: i, hB2, 2i) such that f(BP ) 6= f (BP ). First, we’ll show that it’s not possible for f(BP ) \ f E (BP ) 6= ∅. Assume by LEMMA 3. Let BP = (C, hB1, 1i, hB2, 2i) be a bargain- way of contradiction that there exists an outcome o ∈ AF such ing problem, if the set C∪ B1 ∪ B2 is consistent then AFBP 6= ∅. that o ∈ f(BP ) \ f E (BP ). Then o is unanimously efficient and is not ruled out by axiom FR. According to Lemma 2, the set Proof: Let ω be a model of C∪ B1 ∪ B2. Let o ∈ PO be such that E of outcomes f (BP ) can be partitioned into equivalence classes ω ∈ JoK. Clearly, o ∈ OC and o is consistent with both B1 and B2. characterised by the non-negative numbers kmin and kmax (pos- Thus, o 1 D1 and o 2 D2. Thus, o ∈ AFBP . sibly equal to each other). Clearly, mini=1,2 Gi(o) ≤ kmin. If On the other hand, when the set C∪ B1 ∪ B2 is inconsistent, it mini=1,2 Gi(o) = kmin then, for o to be unanimously efficient, doesn’t always mean that AFBP = ∅. maxi=1,2 Gi(o) ≥ kmax. Thus, maxi=1,2 Gi(o) = kmax In other E words, o ∈ f (BP ). Contradiction. If mini=1,2 Gi(o) < kmin EXAMPLE 3. Continue with our running example and consider then, for o to be unanimously efficient, maxi=1,2 Gi(o) > kmax. the following bargaining situation E BP = (C, hB1, 1i, hB2, 2 But then, there is a unanimously efficient outcome σ ∈ f (BP ) i), where C = {P } (e.g., the agent receives the instruction from the such that |G1(σ) − G2(σ)| = kmax − kmin < maxi=1,2 Gi(o) − vendor not to sell the house for less than the asking price and this . Thus, according mini=1,2 Gi(o) = |G1(o) − G2(o)| o∈ / f(BP ) is common knowledge), and B1 = {E} and B1 = {¬E}. Also, to axiom FR. Contradiction. suppose that ¯ and ¯. Clearly, E P E ≻1 P E P E ≻2 P E C∪ B1 ∪ B2 That f (BP ) \ f(BP ) 6= ∅ follows trivially from axiom UB is inconsistent, but D = P E and D = P E¯, and AFBP = 1 2 and Lemma 2. {PE,P E¯}. It’s also straightforward to see that f E satisfies axiom IIA.

COROLLARY 1. The bargaining solution f E defined in Theo- Given Lemma 3, one fairly naive idea is to perform belief merg- rem 1 satisfies IIA. ing (see e.g., [7]) with integrity constraint (i.e., merging B1 and B2 with the integrity constraint C) to obtain a consistent belief base X So far in this section, the agent’s beliefs Bi are only used to which will be treated as the shared bottom line for both agents 1 and define the agent’s disagreement point and don’t play much role 2. Subsequently, the bargaining model described in Section 3.1 can in characterising the negotiation outcome. The problem becomes be applied to characterise the negotiation outcome. Unfortunately, more challenging when we allow the agents’ beliefs to change ac- this straightforward idea will not work, for the simple reason that cording to the bargaining situation. belief merging takes into account the two belief bases B1 and B2 3.2 Bargaining with revisable agents’ beliefs when merging them with respect to the integrity constraints C. But it fails to take into consideration the agents’ preferences 1 and 2 According to the bargaining model introduced in the preceding regarding the preferred outcomes. section, when the agents’ beliefs Bi define the bottom-lines Di that result in an empty set of agreement-feasible outcomes (i.e., AF EXAMPLE 4. In the running example, consider a bargaining {o ∈ O : o 1 D1}∩{o ∈ O : o 2 D2} = ∅), agreement is situation in which C = {⊤} (i.e., no hard constraints), B1 = not possible. Nevertheless, in most negotiations, the agents’ beliefs {P, ¬E}, and B2 = {E}. Furthermore, the agents preferences represent their inclination toward a particular position rather than are: P E ≻1 P E¯ ≻1 P¯ E ≻1 P¯E¯ and P¯ E ≻2 P¯E¯ ≻2 P E ≻2 unmovable. For instance, a buyer of a house may know for certain P E¯ with D1 = P E¯ and D2 = P E. Clearly, C ∪ B1 ∪ B2 is in- that an identical house was sold a month ago for $y, and thus is consistent and most standard belief merging mechanisms (see [7]) not too willing to pay much more than $y for this house. However, would result in a merge belief base X =∆C(B1 ⊔B2) = {P }. By knowing that there is no other house left in the area that he can buy, taking X to define the common bottom line for both agents, the new he is perhaps willing to pay more than $y, if there are compelling ′ ′ disagreement points for them become D = D1 and D = P E¯. reasons for him to do so (e.g., there are other buyers who would 1 2 like to buy a house in the area). In this situation, if P denotes the 6When clear from the context, we will omit the subscript and write proposition that the vendor agent’s asking price is higher than $y, AF instead. j j+1 Clearly, this has disadvantaged agent 2. Moreover, it has also im- Θ~ ⊆ Θ~ ), where S~ = (S1,...,Sn) ⊆ (T1,...,Tn) = T~ posed agent 1’s bottom line regarding attribute P on agent 2 with- if and only if Si ⊆ Ti for all i ∈ {1,...,n}. A more precise out any reasonable justification and compensation. bargaining protocol will be introduced later in this section. From the discussion in the preceding section, a bargaining out- On the other hand, any mechanism that searches for a negoti- come should be based on the agents’ beliefs about the bargaining ation outcome based only on the agents’ preferences 1 and 2 situation at hand while taking into account their preferences. To without taking into account the agents’ beliefs is likely to produce formulate the idea that an agent’s beliefs that define her position on impractical outcomes as well. For instance, in the bargaining situa- certain bargaining issues can be undermined or dominated by her tion discussed at the beginning of this section, assume that P E ≻1 opponent’s beliefs, we appeal to the argumentation-based frame- ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ P E ≻1 P E ≻1 P E and P E ≻2 P E ≻2 P E ≻2 P E. Assume work [4, 1]. In a strategic model of bargaining, a bargaining prob- also that the vendor’s asking price is $x > $y, and B1 = {P, ¬E} lem BP = (C, hB1, 1i, hB2, 2i) is based on a common lan- (i.e., the vendor agent knows that the house he is selling is currently guage L and a common outcome alphabet PO, with the set of hard the only house for sale in the area and there are several buyers who constraints C being common knowledge while the agents beliefs are looking for a house in the area, while a recent government regu- and preferences hBi, ii for i =1, 2 are their private information. lation requires mortgage lender to carry out a number of checks be- We will first reproduce some notions of argumentation theory. fore releasing the fund for settlement), and B2 = {¬P, ¬E} (i.e., the buyer knows that an identical house was sold for $y last month). DEFINITION 4. ([1]) An argument of a set of sentences X ⊆L As the set of agreement-feasible outcomes AF is empty, the agents (aka. X-argument) under the constraints C is a pair (H,h), where need to make concessions to possibly reach an agreement. With- h ∈L and H ⊆ X, satisfying: out taking into account the agents’ beliefs, the bargaining strategy (i) C∪ H is consistent, of minimal concession (see [5]) suggests the following negotiation process: First, agent 2, the buyer, will make a minimal concession (ii) C∪ H |= h, and ¯ ¯ ¯ from its current offer of P E to P E; then, agent 1 makes a minimal (iii) H is minimal (i.e., no strict subset of H satisfies (i) and (ii)). concession, and accept the offer P¯ E. This outcome is certainly not justified since the right price in this case should be $x (thus, H is called the support and h the conclusion of the argument (H,h). Moreover, given two arguments (H,h) and (H′,h′), if agreeing on P ) while the agents can also agree on ¬E. ′ ′ Therefore we argue that a reasonable mechanism should enable H ⇔ H and h ⇔ h then we treat them as the same argument. the agents to use their beliefs to make the decision on what conces- That is, a set of arguments can not contain both arguments. An argument (H′,h′) is a subargument of the argument (H,h) sion to make, taking into account their preferences. Consequently, ′ we’ll investigate a strategic model for bargaining in the following iff H ⊆ H. section. Given a set of arguments Γ, the base of Γ is the set: BΓ = (H,h)∈Γ H. S 4. AN ARGUMENTATION-BASED MODEL We denote by AC(X) the set of all X-arguments under the con- OF BARGAINING straints C. ′ ′ The axiomatic bargaining model introduced above is inherently DEFINITION 5. ([1]) Let (H,h) and (H ,h ) be two arguments static in the sense that only the outcome, and not the bargaining of AC(X): process, is analysed. This ensures a number of advantages such as ′ ′ ′ tractability. Nevertheless, in most circumstances, it’s important to • (H,h) rebuts (H ,h ) if and only if h ⇔¬h . study the bargaining process as well as the bargainers’ strategies. • (H,h) undercuts (H′,h′) if and only if h ⇔¬h′′ for some For instance, we may be interested in knowing how the bargaining h′′ ∈ H′. outcome is affected by changes in the bargaining procedure, and ′ ′ what would be the best strategy or decision a bargainer should take When (H,h) rebuts or undercuts (H ,h ), we also say that (H,h) attacks (H′,h′). When (H,h) attacks (H′,h′) and (H′′,h′′) at- in a given situation. ′′ ′′ ′ ′ The bargaining protocol to be used by the agents to reach an tacks (H,h), we say that (H ,h ) defends (H ,h ). agreement is based on the belief negotiation models proposed by We are now in a position to formally define our bargaining pro- Booth [3]. In this model, the negotiation proceeds in rounds. The tocol. First, we define the notion of bargaining proposal. ~ 0 0 0 negotiation starts off with the initial offer profile Θ = (Θ1, Θ2), j DEFINITION 6. Let BP = (C, hB1, 1i, hB2, 2i) be a bar- where an offer Θi is a subset of the set of feasible outcomes O, in- dicating the outcomes agent i is willing to accept after j rounds of gaining problem and OC denote the set of feasible outcomes. A ~ j bargaining proposal (or, proposal) by agent i at stage j, denoted negotiation. If Θ (j ≥ 0) is consistent then the set of agreement- j j j j j j feasible outcomes AF = Θ ∩ Θ is non-empty and a physical by ρi is a pair (Θi , Γi ), where Θi ⊆ OC is the set of outcomes 1 2 j A ~ j agent i is willing to agree on and Γi ⊆ C(L) is the set of argu- agreement can be selected from AF . If Θ (j ≥ 0) is inconsistent j then a “contest” between the agents will be carried out to select a ments agent i has used to defend her offers Θi . 7 j j subset of agents who are required to “make some concessions”. The pair (ρ1, ρ2) of the agents’ proposals in stage j is called the The new offer profile Θ~ j+1 after the selected agents making the bargaining context at stage j. concession allows the negotiation to proceed to the next round. It’s important to note that, in a strategic model of bargaining the Under “certain predefined conditions”, a disagreement is reached. agents beliefs and preferences hBi, ii are their private informa- Furthermore, under a monotonic concession protocol (see [13]), the tion. Therefore, agent i can introduce arguments that are not based j+1 new offer profile Θ~ is required to include the previous one; i.e., on her beliefs if she thinks that they would give her an advantage. In 7The generality of this protocol allows it to encompass many com- the following, we’ll write Θ instead of o. We will now define mon negotiation protocols including the alternating-offer protocol W o_∈Θ and the simultaneous-concession protocol. the notion of an argument being relevant to a bargaining context. DEFINITION 7. Let (ρ1, ρ2) be a bargaining context, where ρi = The negotiation starts off with the initial proposal profile (({P E}, (Θi, Γi) is agent i’s proposal, for i =1, 2. An argument (H,h) is ∅), ({P¯ E}, ∅)). In the next round, the buyer introduces the argu- relevant to this bargaining context (for agent i) if (H,h) ∈/ Γi and ment ({Y,M,Y ∧ M ⇒ ¬P }, ¬P ) and the seller introduces the argument ({Y,S,D,S ∧D ⇒ C,Y ∧C ⇒ P },P ) to defend their • Θ−i ∧ h |= ⊥; or respective positions. In the consecutive round, the seller then defeat W ′ ′ • (H,h) attacks an argument (H ,h ) ∈ Γ−i. the buyer’s argument with the argument ({C,C ⇒ ¬M}, ¬M). This settles the issue on the price of the house with the buyer mak- Of the two non-trivial conditions above, the former says that ing a concession and willing to accept any outcome from the set agent i rejects agent −i’s current offered outcomes Θ−i by ad- {PE,¯ P¯E,P¯ E¯}. However, since the set of agreement-feasible out- vancing an argument (H,h) that contradicts Θ−i and thus requires come AF = {P E}∩{PE,¯ P¯E,P¯ E¯} remains empty, the buyer agent −i to make a concession. The latter allows agent i to ad- then advances the argument ({P,R,P ∧ R ⇒ ¬E}, ¬E). Since vance an argument (H,h) to defeat a relevant argument (H′,h′) the seller has no counter-argument, he makes a concession and is advanced by agent −i in previous rounds of bargaining. willing to accept any outcome from the set {PE,P E¯}. They settle In the bargaining protocol informally described at the beginning with the outcome P E¯. of this section, for the “contest” to select who need to make a revised proposal during a negotiation round, we assume that all To formalise the notion of winning argument in an exchange be- agents will have to submit the updated proposal in each round. tween bargainers, we’ll appeal to the argument-based semantics of Furthermore, an agent i’s proposal in round 0 has to contain a non- admissibility, introduced by Dung [4]. 0 empty offer Θi 6= ∅ and, to simplify the protocol, it also contains 0 DEFINITION 8. A set Γ of arguments is conflict-free if there an empty set of arguments Γi = ∅. Agent i’s proposal in round ′ ′ j > 0, ρj = (Θj , Γj ) is required to meet the following conditions: are no two arguments (H,h) and (H ,h ) such that (H,h) attacks i i i (H′,h′) or (H′,h′) attacks (H,h). j j−1 A set Γ of arguments is admissible if it is conflict-free and it • Θi ⊇ Θi ; defends all of its members against all attackers. j j−1 j j−1 • Γi ⊇ Γi such that B j ∪C is consistent and, if Γi 6=Γi A set Γ of arguments is strongly admissible if it is admissible Γi then the new arguments have to be relevant to the previous and none of the arguments that attack its members belong to an j−1 j−1 admissible set of arguments. bargaining context (ρ1 , ρ2 ).

If Θ~ j (j ≥ 0) is consistent then the set of agreement-feasible The following lemma is trivial since at all stages j of the bar- j j gaining, B j ∪C is required to be consistent. outcomes AF = Θ1 ∩ Θ2 is non-empty and an agreement can Γi be selected from AF . If Θ~ j (j ≥ 0) is inconsistent then the bargaining proceeds to the next round. If in two consecutive rounds LEMMA 5. The sets of arguments contained in the bargaining of bargaining, the bargaining context is not updated, i.e., for some proposals introduced by the agents according to the bargaining j j+1 j+2 protocol defined above are conflict-free. j ≥ 0, ρi = ρi = ρi for i =1, 2, then the bargaining reaches a disagreement. The following axiom requires that bargainers do not ignore strongly LEMMA 4. The bargaining protocol defined above terminates. admissible sets of arguments that support an agent’s position.

Proof: Since the alphabet P of the language L (and the alphabet SA. If BP = (C, hB1, 1i, hB2, 2i) is a bargaining problem and AG ⊆ OC the agreement reached after j rounds of bar- PO ⊆ P of the bargaining outcomes) is finite, there can only a gaining. If the set of arguments j is strongly admis- finite number of logically different arguments; i.e., the set AC(L) Γ ⊆ Γi sible then for each outcome is is finite and the set OC is also finite. Thus, if the bargaining does o ∈ AG : (H,h)∈Γ h ∧ o not terminate with a disagreement then at some point, both agents consistent. V will have exhausted the set of arguments and thus will have to in- Intuitively, axiom SA requires that, if agent i can present an ar- creasingly add the members of OC in their respective offers and the gument that agent −i can not defeat, then every agreed outcome bargaining terminates with a non-empty set of agreement-feasible has to be consistent with the conclusions obtainable from this set outcomes AF . of arguments. Given the negotiation protocol, our example in the preceding We can now state a lemma trivially derived from the definition section about the vendor agent who argues to convince the buyer of strongly admissible sets of arguments. to change her position on the price of the house can be described as follows. LEMMA 6. Given a bargaining problem BP , if a sentence h is supported by a strongly admissible set of arguments Γ from the EXAMPLE 5. We will assume that the alphabet P also contains j j current bargaining context (ρ1, ρ2) then there does not exist any the following propositional symbols: Y for “a similar house was admissible set of arguments Γ′ from the current bargaining context sold for $y last month”, M for “house prices last month reflects that supports ¬h. today market”, S for “houses in the area have become scarce”, D for “there has been an increase in the demand for houses in Equipped with Lemma 6 and assuming a belief revision operator the area”, C for “the market has changed with the price on the ∗AGM that satisfies the AGM axioms (see [6]), we can now define up”, and R for “bargainers should exercise reciprocity”. We’ll also an argument-augmented bargaining problem of a given bargaining ¯ ¯ ¯ ¯ ¯ ¯ ¯ assume that P E ≻1 P E ≻1 P E ≻1 P E and P E ≻2 P E ≻2 problem. P E¯ ≻2 P E are the agents’ respective preferences. That is, the buyer’s preference on the value of E (whether it should be an early DEFINITION 9. Let BP = (C, hB1, 1i, hB2, 2i) be a bar- settlement) is conditional on the value of P (whether she has to pay gaining problem and given a a belief revision operator ∗AGM that the higher price). satisfies the AGM axioms. Consider the set of all admissible sets of arguments of the base B1 ∪B2: ASBP = {Γ ⊆ AC(B1 ∪B2): Γ Acknowledgements is strongly admissible }. Let α denote h, then This work was supported by the Australian Research Council (ARC) (H,h)∈S^Γ Γ ∈ASBP grants DP0987380 and DP110103671. The authors would like to the argument-augmented bargaining problem of BP , denoted thank Prof Ryszard Kowalczyk for his support. The authors would BP ∗AGM ∗AGM by A is defined to be (C, hB1 α, 1i, hB2 α, 2i). also like to thank the four anonymous reviewers of AAMAS-2012 for their very detailed comments and suggestions. 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