Designing Flexible Negotiation Agent with Relaxed Decision Rules Kwang Mong Sim and Shi Yu Wang Department of Information Engineering Chinese University of Hong Kong, Shatin, NT, Hong Kong E-mail:
[email protected],
[email protected] Abstract This article presents a model for designing negotiation agent with two distinguishing features: 1) a marketdriven strategy and 2) a set of relaxed decision rules. Market-driven agents make adjustable rates of concession by reacting to changing market situations. In determining the amount of concession for each negotiation round, a market-driven agent is guided by three mathematical functions of remaining trading time, trading opportunity, and competition. Furthermore, agents in this research are also guided by a set of fuzzy rules for determining whether to relax their trading aspirations in the face of (very) high negotiation pressures such as fast approaching deadlines, and high competitions. Empirical results suggest that on average, when compared to Sim’s market-driven, agents in this research achieved 1) higher success rates in reaching a deal and 2) higher expected utilities.
1. Introduction The problem for negotiating parties to resolve conflicting objectives is both difficult and time consuming. Although there are many extant agent-based systems for negotiation in e-commerce [1-3], the strategies of some of these agents are mostly static. The strategies adopted by these agents may not necessarily be the most appropriate for changing market situation. As products and services become available, and traders enter and leave a market, the condition for deliberation changes as new opportunities and threats are constantly being introduced. In addition, since deadline puts negotiators under pressure [4], deliberation is also bounded by time. In pre-specified negotiation strategies, these issues are not addressed and agents increase/relax their bids at a constant rate. Previous empirical results by Sim and Wong [5], show that in general, more flexible negotiation agents [5-8], outperform fixed strategy negotiation agents in many situations. To built agents that are capable of flexible and more sophisticated negotiation, Faratin et al. [6] devised a negotiation model that defines a range of strategies and tactics for generating (counter-)proposals. Although Faratin et al.’s strategies are based on time, resource and behaviors of negotiators, other essential
factors such as competition (for multi-lateral negotiation), trading alternatives and differences of negotiators were not considered. In a multilateral negotiation, it is intuitive to think that a negotiator’s bargaining “power” is affected by the number of competitors, and trading alternatives (partners). Good options give a negotiator more “power” because the negotiating party need not pursue the negotiation with any sense of desperation [9, p157]. In addition, negotiations may break down because the parties cannot resolve their differences [10, p94]. Factors such as competition, trading alternatives, differences of negotiators and deadline are considered in [5,7-8] and this research. However, even the market-driven agents in [5,78] are not sufficiently flexible. For instance, in [5,7-8], the decision to deal is made only when the current (counter-)proposal(s) of two agents co-inside. In [1-3,58], agents were not programmed with the flexibility to relax their decision to (quickly) complete a deal in the face of intensive negotiation pressure. In real-life trading, relaxing decisions in the face of trading pressure is common. For example, a grocery store is more likely to offer discounts when there are many nearby stores selling the same goods. This motivating consideration provides the impetus for designing enhanced market-driven agents that augment the market-driven strategy (section 2.1) with a set of fuzzy rules (section 2.2). While theoretical analyses in section 2.1 have proven some properties of the enhance market-driven agent, a series of experiments was carried out to evaluate the performance of these agents. Empirical results reported in section 3 show that, on average, agents in this research achieved both 1) higher success rates in reaching a deal, and 2) higher expected utilities, when compared to Sim’s market-driven agents [7,8]. Section 4 compares this work with related systems.
2. Flexible Negotiation Agent Model The negotiation agent model in this work has two distinguishing features: i) an enhanced market-driven strategy (section 2.1), and ii) a set of fuzzy decision rules (section 2.2). Market-driven agents [5,7-8] are agents that make adjustable amounts of concession by reacting to different market situations (eg, amount of competition) and trading constraints (eg, deadline). To enhance the
Proceedings of the IEEE/WIC International Conference on Intelligent Agent Technology (IAT’03) 0-7695-1931-8/03 $ 17.00 © 2003 IEEE
flexibility of negotiation agents, the set of fuzzy rules are used to guide agents in relaxing their trade conditions (e.g., aspiration price) in the face of (intense) negotiation pressure (e.g., very stiff competition).
2.1. Market-driven Strategy Like Sim’s market-driven agents [7,8], agents in this research also make concession by narrowing the spread (difference in the proposals) between itself and others. The differences between agents in [7,8] and this work are: 1) An agent in [7,8] concede by attempting to reduce the expected spread in the next round i+1 to a fraction of the actual spread in the current round i by considering 4 functions: opportunity, competition, time and eagerness. 2) An agent in this research concede by attempting to reduce the expected spread in the next round i+1 to a fraction of either the actual spread in the current round i or the actual difference between it’s trading partner’s proposal and its own reservation value. Additionally, only 3 functions are used: opportunity, competition, and time. Difference between current and reserve values: Let [lamin , lamax] be the negotiation interval of issue l for agent a. Let a’s proposal for issue l in round i be lai∈[lamin , lamax]. The function to evaluate a’s payoff in round i is defined as: ° (l a max − l a i ) (l a max − l a min ) a is buyer va (l a i ) = ® a a a a °¯ (l i − l min ) (l max − l min ) a is seller If a is a buyer, the ratio of the change of a’s payoff for two consecutive negotiation rounds is: v (l a ) − v (l a ) l a − l a R (i, i + 1) = a i aa i +1 = a i +1 ai va (l i ) l max − l i R(i,i+1) measures the amount of concession with respect to the difference between an agent’s current bid and its reserve value lamax-lai. For example, let issue l be the price in round i. For 2 buyer agents a and b, with lamax-lai=$100 and lbmax-lbi=$200, a concedes more if both a and b increase their bids by the same amount (e.g., $8). Spread: Let the proposals of buyer a and seller b for issue l in round i be lai and lbi respectively. The spread kaļbi between a and b for issue l in round i is lbi-lai. a concedes by determining the value lai+1 for round i+1 using lbi and kai+1, where kai+1 is the expected spread that a hopes to achieve in the next round. The actual spread kaļbi+1 between a and b can only be determined in round i+1 since it depend on the proposal lbi+1 of b. Hence, a’s proposal for issue l in the next round is lai+1= lbi-kai+1. kai+1 can be determined by assessing the current market situation. If l b i ∈ [l a min , l a max ] , kai+1 is determined as a aļb
fraction of k
i.
If l i ∉ [l b
a
min
,l
a
a
max
] , then k
i+1
is
determined as a fraction of lamax-lai (respectively, lai-lamin for seller) as follows: k a i +1 = [O (n a i , < w j → a i >, < v a → j i >)C (m a i , n a i )T (t ,τ , ε )] (2.1) ×[ β (l a max − l a i ) + (1 − β )k a ↔b i ] a is buyer, ȕ=1 when kaļbi>(lamax-lai) otherwise ȕ=0
For a buyer agent, kai+1 is determined as follows: k i +1 = [O ( n a i , < w j → a i > , < v a → j i > )C ( m a i , n a i )T (t , τ , ε )] (2.2) a
×[ β (l a i − l a min ) + (1 − β ) k a ↔ b i ]
a is seller, ȕ=1 when kaļbi>(lai-lamin) otherwise ȕ=0 O(nai,,vaĺji) determines the amount of concession based on trading alternatives (number of trading partners) and differences in offers/bids [7,8]. Suppose agent a engages j in round i. a’s last proposal generates a payoff of vaĺji for itself, and j’s last counterproposal generates a payoff of wjĺai for a. If a accepts j’s counter-proposal, it will obtain wjĺai with certainty. If a insists on its last proposal and i) if j accept it, a will obtain vaĺji and ii) if j do not accept it, a may be subject to a conflict utility ca. ca is the worst possible utility for a (i.e. an agent’s payoff in the absent of agreement [11,12]), hence, wjĺai ca . If j does not accept a’s last proposal, a may ultimately have to settle with lower utilities (the lowest possible being the conflict utility), if there are changes in the market situation in subsequent cycles. For instance, a may face more competitions in subsequent cycles, and may have to ultimately accept a utility that is lower than wjĺai (possible as low as ca). If the subjective probability of a obtaining ca is paļjc,i (conflict probability) and the probability that a achieving vaĺji is1- paļjc,i, then, according to Zeuthen’s analysis [11,12], if a insists on holding its last proposal, a will obtain a payoff of (1paļjc,i)vaĺji + paļjc,ica. Hence, a will find that it is advantageous to insist on its last proposal only if (1paļjc,i)vaĺji + paļjc,icawjĺai. Consequently, the maximum value of paļjc,i is the highest probability of a conflict which a may encounter in round i, given as follows [7,8]: (2.3) pa↔ j c,i = (va→ j i − w j→a i ) (va→ j i − ca ) aļj p c,i is a ratio of two utility differences. While vaĺjiwjĺai measures the cost of accepting the trading agent’s last offer (the spread k or difference between the (counter)proposals of a and j), vaĺji- ca measures the cost of provoking a conflict. Hence, the probability that a will obtain a utility vaĺji, with at least one of its partners is
a ni trading
nia
vi a→ j − wi j→a a→ j − ca ) j =1 (vi
O(ni a , < wi j→a >, vi a→ j ) = 1− ∏
(2.4)
With ample trading options, O(nai,,vaĺji) approaches 1, an agent has more bargain power, and hence, makes smaller compromises. Since there are a very
Proceedings of the IEEE/WIC International Conference on Intelligent Agent Technology (IAT’03) 0-7695-1931-8/03 $ 17.00 © 2003 IEEE
large number of trading partners, the likelihood that some other agent’s proposals is potentially close to the agent’s proposal is higher (proposition 1). As O(nai,,vaĺji) reduces to 0, an agent’s bargaining power decreases. It makes larger compromises. Even with a very large number of trading options, it would be difficult for an agent to reach a consensus if there are large differences between itself and other negotiating parties (proposition 2). Proposition 1. In round i, the probability that an agent will complete a deal approaches 1 as the number of partners nai approaches a very large number d1. Proposition 1 was proven in [8], and the proof is omitted due to space limitation. Proposition 2. The probability that an agent a will complete a deal approaches 0, if for all nai1, the best spread between a and its trading partners approaches a’s negotiation interval [lamin , lamax]. Proof: Let k*i be the best spread in round i. k*i= lai-l*i such that l*i is the best proposal from a’s partners and if a is a buyer, l*i lj, ∀jęnai . Hence, k*iĺ[lamin , lamax] when l*i ĺ lamax and lai ĺ lamin . Let the payoffs generated by lai and l*i be vaĺji and w*i respectively. For a buyer, if l*i ĺ lamax then w*iĺ ca. To show that the probability of agent a to complete a deal approaches 0 is to show that O(nai,,vaĺji)ĺ0, hence, it suffices to prove that the following holds: v a → j i − w j →a i →1 . va → j i − ca j =1
nai
If w*i ĺca,
∏
(2.5)
The proof proceeds by induction on nai as follows Base case: nai=1. a has one partner, hence, w*i=w1ĺai. § v a →1i − w1→ a i · lim ¨ a →1 ¸ =1 a w1→ a i → c a © v i −c ¹
Induction case: When nai>1, assume as an induction hypothesis that (2.5) holds for nai
