Belief Based Distributed Buyer Coalition Formation with Non-transferableUtility

Chi-Kong Chan and Ho-fung LeungThe Chinese University of Hong Kong

{chanck, lhf}@cse.cuhk.edu.hk

Abstract

Online buyer coalition formation problem is an applica-tion of e-commerce and distributed agent technology. So far,most works in this topic are social utility based approaches,which assume the agents’ utilities to be transferable andpublicly known. However, such assumptions are not validin many software agents applications where agents beliefsare private, and stability is often a more important con-cept than social utility. In this paper, we study the prob-lem from a private-belief based non-transferable utility ap-proach: instead of social utility maximization, our focusis on achieving stable solutions in term of the core, b-coreand Pareto efficiency; instead of assuming common knowl-edge, we assume the agents’ actions to be based on fallibleprivate beliefs. We then propose a distributed mechanismwhere agents are allowed to propose incremental improve-ments toward a stable solution. We show by experiment thatour mechanism is able to reach core stable solutions in over97% of the cases, and b-core stable solution in almost of allcases that we tested.

1. Introduction

Many sellers offer volume discounts to buyers buyinglarge quantities of their products. Typically there are morethan one products on sell, together with non-increasingprice schedules such that the more units of the same typethat are bought together, the bigger is the discount offered.In order to take advantage of such price schedules, newapplications, known as online buyer coalitions [1, 8, 14]are emerging in both B2C (e.g, [6, 7, 10]) and B2B plat-forms (e.g [13]), which allow online buyers agents to teamup so that they can collectively place a larger order to en-joy the volume discount. In many cases, the products aresubstitutable, meaning that each buyer needs exactly oneunit of any of the products, although he may have differ-ent preferences regarding the various items. (For example,a buyer may only want to buy exactly one camera from a

wholesaler, even if there are multiple models on sale, buthe may have different preferences for the various models).The challenge here is to find a good partition of the buyersinto non-overlapping buying coalitions, with each coalitionplacing an aggregate order for one product. Such a partitionis called a buyer coalition structure (C.S.).

Several multi-agent system based approaches are pro-posed for this problem, with the majority focusing on reach-ing optimal or near optimal C.S. in terms of social util-ity maximization. One of such early work is [15], whichis a centralized mechanism where the next most profitablecoalition is greedily formed in each step. Later, a geneticalgorithm based approach is proposed in [9] and a multi-attribute approach is proposed in [11] where each buyer’sdecision is based on an Analytic Hierarchy Process. In theseapproaches, each buyer agent is assumed to have his reser-vation prices for the products, which represent the highestamounts that the agent is willing to pay for the items. Abuyer’s utility is defined as the difference between his reser-vation price and the actual price it end up paying for the itemit acquires. The goal of such mechanisms is thus to finda C.S. that maximizes the sum of the individual memberagent’s utility (called the social utility). These social utilitybased approaches thus imply a transferable utility model,without fully considering the stability of the coalitions.

As partially pointed out in our previous work in [3], thereare several concerns with these social utility maximizationapproaches. The first concern is non-transferable utilities.A buyer’s utility, defined as above, may not be transferablefor the following reasons. First, a buyer’s saving is not acompletely accurate indication of an agent’s preference be-cause it ignores the difference in reservation prices amongthe different products when computing the aggregate sav-ing: a saving of $100 out of a $1000 product is probablymore satisfactory than saving $100 out of a $10000 prod-uct, as the former represents a 10% discount whereas thelatter represents a 1% discount only. This implies the util-ities are not transferable. Second, a shopper’s preferencemay also be influenced by other factors in addition to thesaving amount. For example, a buyer’s satisfaction may de-

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pend on the brand name, seller’s reputation as well as theamount saved. This means that the preference order of anagent should be a multi-objective problem. (e.g., a savingof $100 for a Brand-A camera to be delivered the follow-ing day may be a better outcome than a saving of $200 fora Brand-B camera to be delivered one month later, even ifthe reservation prices are the same). This again suggestedthat the more general non-transferable utility model is moreaccurate in describing the problem.

The second concern is stability. In buyer coalition prob-lems, as in many other semi-competitive games, the stabilityof coalitions is more important than social utility. In short,a stable solution is one that no agents would have the desireto deviate from the agreed C.S. Various stability conceptsare proposed in game theory, including the core, Pareto effi-ciency, the kernel [12], and also the b-core [4]). The reasonsthat coalition stability is more suitable as solution conceptsfor group buying problem are that, 1) software agents areassumed to be selfish so that they are interested in their ownutility only. That is, their only concern is that whether thereexists some other solution where they (either alone or as asub-group) are better off, regardless the social utility. 2) Wecannot force any agents to sacrifice for the sake of globaloptimality, a point that is either not considered in severalworks, or only given limited handling. An unstable coali-tion is still meaningless even if it can achieve a good socialutility.

The third concern is private information. Softwareagents are typically modeled as individuals with distinct andprivate beliefs. For example, a buyer agent’s preference or-der for some items is private and known to that agent only,and he may not be willing to disclose his preference forprivacy concerns. However, most of the approaches so farhave a public knowledge assumption in that each individ-ual agent’s preference are either known to all agents (or atleast known by a central agent that computes the solution),or that the agents are willing to disclose such informationfor the purpose of computing a solution. These are not re-alistic assumptions. Thus, instead of assuming each agentto know each other’s preferences, it is more reasonable tomodel such information as each agent’s beliefs regardingthe others’ preferences,as done in many traditional A.I. andepistemological approaches. The difference here is that be-liefs, unlike knowledge, are fallible, and may differ fromagent to agent.

The first two concerns are partly handled in [2] and [3].In [2], a non-transferable payoff model is assumed, and thebuyer agents send their preferences to a central agent, wherePareto optimal solutions are found using an exact set coveralgorithm. This work targets Pareto optimal solutions andnot the stricter and more stable criterion of the core. Afterthat, in [3], we proposed a distributed mechanism and cor-responding strategies that aim at reaching core-stable solu-

Table 1. Example of Non-transferable UtilityPreference of buyer agents

a1 a2 a3 a4

Item Size Item Size Item Size Item Sizeg1 4 g1 4 g3 4 g3 4g1 3 g2 4 g3 3 g2 4g1 2 g1 3 g2 4 g3 3g1 1 g2 3 g3 2 g2 3g2 4 g1 2 g2 3 g3 4g2 3 g2 2 g3 1 g3 1g2 2 g2 1 g2 2 g2 2g2 1 g1 1 g2 1 g2 1g3 4 g3 4 g1 4 g1 4g3 3 g3 3 g1 3 g1 3g3 2 g3 2 g1 2 g1 2g3 1 g3 1 g1 1 g1 1

tions. Our current work is an extension of these two worksby also taking the third concern (private information)intoaccount. Instead of assuming the buyer agents to have reser-vation prices for each item, we only require them to haveexplicit preference order regarding possible types of coali-tion. Instead of using social utility, we use the game theo-retical concepts of the core, and Pareto efficiency, as wellas a belief-based stability concept known as the b-core [4],as measurements of the coalitions’ quality. We follow amechanism plus strategy approach: we first introduce a dis-tributed non-transferable utility mechanism and discuss twostrategies, one of them is belief-based and the other one not.We first show that both of these two strategies, if followedby the agents, can reach core and b-core stable solutionsin over 97% of the time according to a series of test cases,which make our mechanism a stable one. We then comparethe two strategies from the agents’ point of view, and showthat they achieve similar results in term of agents prefer-ences, but with the belief-based strategy being much moreefficient in term of message costs, and that it is to the bene-fits of the agent to adopt that strategy.

2. Background

2.1 Unit Price Schedule and Agent Preference

We follow the problem setting of [2] and [3]. There arem type of substitutable products g = {g1,g2, . . . ,gm} beingsold, and n buyers agents A = {a1,a2, . . ., an}, each inter-ested in buying one unit of any of the products. A non-increasing unit-price schedule is associated with each prod-uct so that the unit price decreases with the number of unitsbought together in the same order.

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In order to take advantage of the price schdeule, the buy-ers form coalitions, such that for each product gi, there isone (possibly empty) coalition Ci ⊆ A where the agents inCi place a joint order for the product gi. The mutually dis-joint coalitions for the products form a coalition structureCS = {C1,C2, . . . ,Cm}, where each Ci is the coalition forproduct gi. For example, if there are three products, g1 tog3, being sold, and four buyer agents a1 to a4. If each of theagents separately buys one unit of product g1, the C.S. canbe represented as CS1 ={{a1,a2,a3,a4}, /0, /0}. Each C.S.represents a candidate solution to the buyer coalition prob-lem.

The stability of the coalitions depends on both the priceschedules and the preference of each agent. For simplic-ity, we have the following assumptions regarding the agentspreferences:

• Assumption 1. For any two coalitions buying the sameproduct, the preference of an agent is the coalitionthat can achieve the lower price according to the priceschedule.

• Assumption 2. The preference of an agent depends ononly the product it ends up purchasing and the price itpays.

• Assumption 3. Each agent in the same buyer coalitionpays the same amount.

Assumption 1 is natural enough and self-explaining. As-sumption 2 says that all buyer coalitions that produce thesame result are treated as equals by any agent, in particu-lar, each agent has no preference over who its partners are,and treats all coalitions of equal size as equally preferred.Assumption 3 concerns the fairness of the coalitions, as weconsider that any coalition that requires some members topay more than the others for the same item as unfair andwill discourage people from joining.

An implication of these assumptions, as first pointed outin [2], is that we can define the preferences of the agentssolely in terms of the obtained unit price and what prod-uct is being bought. We follow the notation of [3]: sincethe resulting price of an item depends only on the coalitionsize and the item, we can describe each buyer coalition us-ing a (item,size) couple, which we called a bid. Any coali-tion with the same (item, size) bid is thus indistinguishablefrom an agent’s point of view. Obviously, there are onlym ∗ n possible bids where m is the number of products andn is the number of participating agents, so the number ofpreferences to be specified is manageable. Thus we can fur-ther simplify the agents’ preference by defining each agent’spreference to be a total ordered relation on all possible bids(ties are broken arbitrarily). One of such example is givenin table 1, which says that agent a2’s most preferred bid is(g1,4), followed by (g2,4), and so on.

But before we define agents preferences, we first definethe idea of valid bids.

Definition 1 (valid bids). A bid is a couple (item,size),where item is a product being sold, and size is the num-ber of units requested by a buyer coalition, which is alsoequal to the size of the coalition. Furthermore, we say a bidb = (gi,s) is a valid bid if size is less than or equal to boththe number of participating buyer agents, and the number ofitems available for sale from the seller.

For a given coalition C, we write bC = (gC, |C|) to rep-resent the bid of that coalition. Similarly, given an agenta and a coalition structure CS, we write ba|CS to repre-sent the bid of the coalition C in CS where a is a member,i.e.,ba|CS = (gC, |C|) where a ∈C and C ∈CS.

We then define an agents’ preference as:

Definition 2 (Agent preference �i ). The preference of anagent i is defined by a total ordered relation �i on the setof all valid bids, such that, for any two valid bids b1 and b2,we have b1 �i b2 if agent i prefers b1 to b2.

2.2. Agent Beliefs

Most existing works assume each individual agents pref-erences is known to each other, an assumption that we donot follow in this work. Instead, we employ a more realis-tic belief-based models. In many situation, the agents canhave some beliefs regarding the other agents’ preferenceseven though they do not have access to the others’ prefer-ence directly. For example, in a repeated coalition forma-tion game, an agent can have models of his peers’ prefer-ences by observing their past actions in previous games orduring the negotiation process, or an agent may simply as-sume his peers to have the same preference as himself. Suchinformation form an agent’s belief, defined as follows.

Definition 3 (Agent belief bel). For two agents i and j, forany two valid bids b1 and b2, we say beli(b1 � j b2) if agenti believes agent j prefers b1 to b2.

Of course, the beliefs, unlike public knowledge, are fal-lible: beli(b1 � j b2) does not imply b1 � j b2.

In this paper, we assume each agent’s belief is com-plete in the sense that for any two agents i and j, for anytwo valid bids b1 and b2, we have either beli(b1 � j b2) orbeli(b2 � j b1). In cases where no such prior knowledgeexists, an agent can still maintain a rough estimation bymodeling his fellow’s preference based on his own, or ac-cording to some commonly known principle (e.g. for thesame item, a larger coalition is always more preferable to asmaller one).

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2.3. Buyer Coalition Stability

We can now define the stability concepts which will beused as measurement of solution quality of a buyer coalitionformation mechanism. We first briefly describe two of themore popular non-belief based stability concepts in gametheory, namely Pareto optimality, and the core. Then wewill describe a belief based concept known as the b-core.

The more basic (and less strict) concept of these conceptsis Pareto optimality, which defines a C.S. to be efficient iffno agent can be better off without sacrificing at least oneother member. So for the buyer coalition problem, we have:

Definition 4 (Pareto optimality in buyer coalition problem).Given n agents A = {a1,a2, . . . ,an} and a coalitionstructure CS = {C1,C2, . . . ,Cm}, we say CS is Paretooptimal if there does not exist another coalition structureCS = {C

′1,C

′2, . . . ,C

′m} such that for each agent a ∈ A , we

have ba|CS′ �a ba|CS, and ba|CS′ �a ba|CS for at least oneagent.

The second solution concept that we will use is the core[12], which is defined as the set of all coalition structureswhere no subset of agents can defect from their originalcoalition and join another coalition such that every defect-ing agent is better off after defecting. So, for the buyercoalition problem:

Definition 5 (The core in buyer coalition problem). Givenn agents A = {a1,a2, . . . ,an} and a coalition structureCS = {C1,C2, . . . ,Cm}, we say CS is in the core if theredoes not exist another coalition structure CS′ and a subsetof agents C ⊆ A such that for each agent a ∈ C, we haveba|CS′ �a ba|CS.

These two solution concepts are the traditional crite-ria, and are used by existing stability based works such as[15, 2]), so we will also include them for comparison pur-poses. However, as pointed out in [4], there is a drawbackwhen applying them to games where the agents’ actions isdetermined by private beliefs instead of public knowledge:a coalition should also be viewed as stable as long as everyagent in that coalition believes that there are no better alter-native, no matters whether those beliefs are accurate or not.This idea has led to the b-core criteria:

Definition 6 (The b-core in buyer coalition problem).Given n agents A = {a1,a2, . . . ,an} and a coalition struc-ture CS = {C1,C2, . . . ,Cm}, we say CS is in the b-core ifthere does not exist another coalition structure CS′ and asubset of agents C ⊆ A such that i) for each agent a ∈C, wehave ba|CS′ �a ba|CS, and ii), there exists at least one agent

j ∈ C such that for each agent k ∈ C, we have bel j( bk|CS′�k bk|CS ).

Intuitively, we say a coalition structure is in the b-core ofa game if there does not exist any alternative coalition struc-ture that satisfies the following two conditions: 1) everymember of at least one coalition in the alternative prefersthe alternative to the original profile and 2) at least one agentin that coalition correctly believes that point 1 is the case. Itcan be easily shown the the core is a subset of the b-core.

2.4. On mechanism designs

We conclude this section with a brief note on the prin-ciple of mechanism design [5]. In game theory, the idea ofmechanism design is to decide the rules of a game, so thatthe achieved outcomes have some desired properties, whichin our case is stability. In mechanism design the agentsare typically assumed to be self-interested, so the goal isto design a mechanism such that the desired properties canstill be achieved even if each agent act selfishly to achievetheir own goal only. Ideally, this is done by having incen-tive compatible strategies that can achieve the desired goal( a strategy is said to be incentive compatible if it is to theagents’ own benefit to adapt that strategy).

In the following sections, we will follow the mechanismdesign principle by first proposing a mechanism that canachieve our goal (stability) provided the agents follow cer-tain strategies, then we will provide preliminary evidencethat it is incentive compatible to follow the proposed strat-egy by means of experiments, where we show that our pro-posed strategy is the best one to follow amongst a numberof reasonable approaches.

3 A Distributed Mechanism

In this section, we present a distributed mechanism forthe buyer coalition problem. The mechanism, which is anenhancement of the mechanism describe in [3], is describedbelow in the following subsection. After that, some agentsstrategies is proposed.

3.1. Distributed Non-transferable utility CoalitionFormation Mechanism (DNCF)

The DNCF Mechanism. For ease of description, we sup-pose there are m + 1 rooms labeled r0,r1, . . . ,rm respec-tively. The rooms r1, . . . ,rm function as virtual gatheringplaces for agents currently interested to form coalition tobuy the ith product, whereas the remaining room r0 func-tions as a non-cooperative room, meaning the agents in thisroom will buy the items individually instead of buying in

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a group. Initially, all agents are in r0. For simplicity, inthe following discussion, we will use the phrase “agent a isin room ri” as synonym for “agent a is in the buyer coali-tion for product gi”, and “agent a switches to room ri” tomean “agent a, which is currently not in the coalition for gi,leaves its current coalition and join the coalition for productgi”, and “size of room ri” as synonym for “size of coalitionfor the product gi”, except for the room r0 which does notrepresent any coalition.

The agents are randomly ordered to make proposals inturn. The mechanism is divided into rounds, and the currentcoalition structure at the beginning of a round is labeled CS.In turns, each agent a performs either one of the followingthree options, according to the proposal order:

Opt 1 Do nothing and maintain the status quo.

Opt 2 Leave its current room and switch to room r0.

Opt 3 Invite other agents to switch together to a certain roomri, which is achieved by the following four steps:

3.1 Select a room ri that is not already proposed be-fore in the current round by the proposing agent.

3.2 Select a set of agents C that are not currently inri, and send a proposal message to each, askingeach recipient to indicate:

i Whether he is interested in switching to ri.

ii The minimum coalition size that must beachieved before he can agree to switch to ri.

3.3 Upon receiving the response, the proposing agenttry to construct a new coalition structure CS2 withthe following four properties:

i The proposing agent is switched to ri (if itwas not already in ri at the beginning of theround) and that it prefers the resulting bidba|CS2

to the current bid ba|CS.

ii Amongst those agents that indicated theywould consider a switch in 3.2, select zeroor more of them to switch to ri, such that theresulting size of ri after the switches is noless than the requested coalition size of eachselected agent.

iii CS2 has not been proposed before since thebeginning of the mechanism by any agent.

iv All other non-selected agents remain in thesame coalition as in CS.

3.4 Send confirmation messages to the selectedagents in sub-step ii of step 3.3 above, and broad-cast the resulting new coalition structure CS2 toall agents. By doing this, the proposal is said tobe successful and this marks the end of a round

(explained below). If the agent cannot constructsuch a new coalition structure, however, the agentcan either choose to maintain status quo (option1) and the current round continues, or switch tor0 (option 2).

After the last agent in the proposal order has proposed,the first agent then proposes again and the process repeats.A round ends and the next round begins only when one ofthe proposing agents is able to make a successful proposal(option 3) or chooses to switch to r0 (option 2) so that thecoalition structure is changed as a result. Note that the be-ginning of a new round does not mean we jump back tothe first agent in the proposal order. Rather, the next agentto propose is still the next one in the proposal order, andthe only difference is that the restriction in step 3.1, whichsays that an agent is not allowed to repeatedly make propos-als for the same product in the same round, is now lifted.However, the restriction in point iii in step 3.3, which re-quires that the resulting coalition structure CS2 has not beenreached before since the beginning of the mechanism, stillapplies. (The reason to have such restriction is to make surethe mechanism does not run into permanent loops). Themechanism ends after there is no successful proposal in overn∗m moves in a row, where m is the number of products andn is the number of agents.

Note that, unlike the previous version, this mechanismdoes not assume the proposing agent would send the pro-posal messages to all agents that is not currently in the tar-get room (step 3.2 of option 2). Rather, the proposing agentshould select the recipients of the messages according totheir preferred strategy. There are two strategies worth con-sidering in here: a non-belief based (labeled nb-strategy)and a belief Based one (labeled b-strategy).

3.2. Non-belief based proposal strategy (nb-strategy)

The nb-strategy is described as follows. When it is anagent ai’s turn to make proposal, he will: i), he should checkwhether there is a bid b′ = (gi,s) where g ∈ G, and s > 1is larger than the current size of ri , such that b′ �i bai|CS,where CS is the current coalition structure as mentionedabove; ii a), if such a bid b′ exists and the agent has notalready proposed product gi in the current round, then heshould select option 3 and send proposal messages to allagents that is not already in room ri; ii b). Otherwise, if nosuch bids exists, it should check if there exists b′′ = (gi,1),such that b′′ �i bai|CS. If such a bid b′′ exists, selects option2, otherwise, selects option 1.

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3.3. Belief based proposal strategy (b-strategy)

The b-strategy is similar to the nb-strategy, except thatstep ii a) is modified as follows:

ii a), if such a bid b′ exists and the agent has not alreadyproposed product gi, then current round, then he should se-lect option 3 and invite other agents to switch to ri by send-ing proposal messages to all agents j such that beli( b j|CS′� j b j|CS ).

The two proposal strategies differ only in the agents theproposer should propose to. The nb-strategy based strat-egy propose to every agent that is not currently in the targetroom, whereas in the b-strategy, the agent selectively pro-pose only to those agents which, according to the proposer’sprivate belief, will agree to switch to the desired room. Welabel agents employing the nb-strategy and b-strategy as nb-agents and b-agents respectively.

For the recipient agents, on the other hand, there isonly one rational strategy, that is, accept the proposal if heprefers it to his current coalition, and rejects it otherwise.This responding strategy is employed by the nb-agents andb-agents alike.

3.4. Responding agent strategy

An agent a j that receives a proposal to switch to roomri (step 3.2 above) will response as follows. First, it shouldcheck whether there exists a bid b = (gi,s), where gi is thecorresponding product for room ri, such that b � j ba j |CS .If such a bid is found, response to the proposing agent thatit is interested, provided that it the new coalition containsat least s members. (If there are more than one such bids,choose one with the smallest coalition size.) If no such bidis found, then indicate to the proposing agent that is notinterested.

3.5. Properties of the Proposed Mechanism andStrategies

The idea of the proposed mechanism and strategies isthat they allow the agents to make incremental improve-ments of the C.S. toward a core-stable solution. In fact,from the proposing agent’s point of view, each successfullyproposal means that a more preferred C.S. is reached. Fromthe game stability point of view, every successful proposalrepresents an objection to the previous C.S. according to theconcept of the core (in case of nb-strategy) or the b-core (incase of the b-strategy), and the mechanism continues un-til no agents can make any further objection to the existingC.S. In fact, the only case that the mechanism fails to reacha core-stable solution is when the core is empty, and in therare cases where the only paths that can lead to a core-stablesolution are all blocked by the restriction in point iii of op-tion 3.3 of the mechanism.

Figure 1. Percentage of results in core

Figure 2. Percentage of results that are Paretooptimal

4. Experiment

We conduct a two parts experiments to test the perfor-mance of the distributed buyer coalition mechanism. In partone, we first demonstrate that the mechanism is a stable one,according to the criteria of the core, b-core and Pareto op-timality, no matters which strategies are employed by theagents, and we also check the scalability of the mechanismby studying the number of messages sent in larger games,particularly, for the belief-based mechanism. This way, wesee that the mechanism can achieve its goal, provided thatthe assumed strategy is incentive compatible. Then in parttwo of the experiment, we will provide preliminary evi-dence that it is incentive compatible by showing our pro-posed strategy is the best one to follow amongst a numberof reasonable approaches.

In the first part of the experiment, We randomly gener-ated a series of scenarios with the number of agents rangingfrom 4 to 10. There are 3 products on sale, and in each testcase, the preferences of each agent are randomly ordered,

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Figure 3. Percentage of results in b-core

Figure 4. Number of messages sent

with the restriction that, among the possible bids for thesame products, a bid from a larger coalition is always pre-ferred over a bid from a smaller coalition, that is, we assume(gi,s1)�a (gi,s2) for any given agent a if s1 > s2, otherwise,the preferences are randomly ordered. In each case, threeagents compositions are tested: first with all agents beingb-agents, than with all agents being nb-agents, and finallywith a equal mix of b-agents and nb-agents. The belief ac-curacy of the b-agents are set at 80%, meaning that for anytwo agents i and j and any two bids b1 and b2, there is aprobability of 0.2 that agent i is wrong about agent j’s pref-erence order for the two bid. Each scenario is repeated 1000times and the number of stable results achieved (out of the1000 runs) is reported. The results are shown in figure 1 tofigure 3, where we also implemented a related approach [2]for comparison, which is a centralized algorithm for non-transferable payoff problems. It can be seen from figures1 and 3 that the result of our mechanism (labeled DNCF)is impressive, as it is able to reach a core-stable solutionin over 97% of all test cases, and reached b-core stable so-lutions in almost 100% of the cases in all composition ofagents strategies. Whereas, for comparison, the other ap-

Figure 5. b-strategy Vs. nb-strategy

Figure 6. b-strategy Vs. passive strategy

proach only achieved core-stable result in 93% of the testcases on average. The DNCF mechanism has passed thestability criterion. Our result is also good for the Pareto op-timality criterion as we also obtained Pareto optimal resultsin over 99% of the cases.

We then investigate on the scalability of DNCF by ex-tending the experiment to include up to 30 agents. Eachcase is repeated twice with all b-agents and all nb-agentsrespectively, and the total number of messages sent isrecorded. The result is shown in figure 4, which suggestthat the rate of growth of number of messages with respec-tive to the number of agents is close to linear for both theb-agents and the nb-agents cases. In particular, the resultfor the all nb-agents games are impressive.

Therefore the DNCF mechanism is both stable and scal-able, particularly if the agents are employing b-strategy. Sothere is one more task remaining: to show that it is to thebenefits of the agent to follow the b-strategy. Again, a se-ries of test cases with number of agents ranging from 4 to10 are generated. This time, in each case, half of the agentsare designated as experimenters. Each case is run twicewith identical settings, except that each experimenter em-

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ploys the b-strategy in the first run, and the nb-strategy inthe second run. Each experimenter then compares the out-come he obtains by using the two strategies according tohis preference order for the bids: each time that he obtainsa more preferable result by using the b-strategy, a ”win” isrewarded to the b-strategy, and vice versa. The experimentis repeated 1000 times and we count the number of ”wins”,”ties” and ”loses” obtained by each strategies. The resultis shown in figure 5, where we see that the results obtainedby using the two strategies are actually very close with thefar majorities of the results being in ties, despite the factthe number of proposals sent by nb-agents using nb-strategygreatly out-number those sent by b-agents (figure 4), whichindicates that the b-strategy is actually superior. Finally, forcompleteness, we also tested the b-strategy against a thirdstrategy which we called the passive strategy. An agent withthe passive strategy will simply propose nothing even whenit is his chance to propose, and he will therefore just pas-sively wait for other agent’s proposal for me. The resultis shown in figure 6, where we see that the b-strategy issuperior, as the number of wins obtained by the b-strategyexceeding those of the nb-strategy . Thus the experimentsshow that it is advantageous for the agent to follow the b-strategy

5 Conclusion

The online buyer coalition formation problem is a nat-ural e-commerce application for distributed agent technol-ogy, with both B2B and B2C examples emerging. Mostexisting works in this topic so far treated it as a transfer-able utility problem with the agent’s preferences treated ascommon knowledge, and used social utility as measurementof solution quality, whereas the more important criterionof coalition stability is either not dealt with, or only han-dled partially. In this work, developed a distributed non-transferable payoff approach. We follow the mechanismdesign principle by providing a stable and scalable mech-anism and also propose strategies. We show by experimentthat provided the proposed strategies are used by the agents,the mechanism can reach b-core results in over 99% of thetest cases, then we show by separate experiment that it isindeed advantageous for the agents to follow the proposedstrategy.

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