Multi-player Multi-issue Negotiation with Mediator using CP-nets

Thiri Haymar Kyaw

, Sujata Ghosh

and Rineke Verbrugge

University of Technology (Yatanarpon Cyber City), Pyin Oo Lwin, Myanmar

Indian Statistical Institute, SETS Campus, MGR Knowledge City, CIT Campus, Chennai 600 113, India

Institute of Artiﬁcial Intelligence, University of Groningen, PO Box 407, 9700 AK Groningen, The Netherlands

Keywords:

Multi-agent Negotiation, Multi-issue Negotiation, Conditional Preference Networks, Incomplete Information.

Abstract:

This paper presents a simple interactive negotiation approach for conﬂicts in everyday life with incomplete

information. We focus on mediation to obtain an agreement while going through alternating offers over a ﬁnite

time bargaining game. The mediator searches and proposes a jointly optimal negotiation text for all players

participating in the negotiation process based on their conditional preference networks (CP-nets). The players

make a decision whether to accept or reject by examining their utility CP-nets. We develop two algorithms for

the mediator and the players. If the ﬁrst negotiation text cannot be accepted by all players, the mediator offers

the next negotiation texts by searching for jointly optimal solutions. This negotiation process continues until

an agreement is achieved or a deadline is reached. This proposed approach can support multi-issue, multi-party

negotiation to achieve an agreement during a ﬁnite number of rounds with near optimal outcomes.

1 INTRODUCTION

Negotiation occurs in several areas of real-world

problems: personal cases such as marriage, divorce,

and parenting; business cases such as pricing between

seller and buyer and sharing a market between orga-

nizations; international crisis cases like the Cuba mis-

sile crisis, the North Korean crisis, and Copenhagen

climate change control. Negotiation is a process for

agents to communicate and compromise in order to

reach beneﬁcial agreements. In such situations, the

agents have a common interest in cooperating, but

have conﬂicting interests over exactly how to coop-

erate (Fatima et al., 2005).

Bargaining is a simple form of a negotiation pro-

cess. It is used to establish a price to trade a ﬁxed and

deﬁned commodity between seller and buyer. One

party usually attempts to gain advantage over another

to obtain the best possible agreement. Splitting a pie

between two players is a simple bargaining exam-

ple. In such games with many periods of offers and

counteroffers, strategies are not just actions, but rather

ways for choosing actions based on the actions chosen

by both agents in earlier periods (Rasmusen, 2007).

In competitive bargaining, the process is viewed

as a competition that is to be won or lost. Positional

bargaining is a negotiation strategy that involves hold-

ing on to a ﬁxed idea, or position, of what you want

and arguing for it and it alone, regardless of any un-

derlying interests. The classic example of positional

bargaining is the haggling that takes place between

proprietor and customer over the price of an item. The

customer has a maximum amount she will pay and the

proprietor will only sell something for a price above a

certain minimum amount. Each side starts with an ex-

treme position, which in this case is a monetary value,

and proceeds from there to negotiate and make con-

cessions. Eventually a compromise may be reached

A position is usually determined by the interests of

a negotiating party, and reﬂected in a contract that it

puts forward to its counterpart.

Integrative bargaining (also called “interest-based

bargaining” or “win-win bargaining”) is a negotiation

strategy in which parties collaborate to ﬁnd a “win-

win” solution to accommodate their different inter-

ests. This strategy focuses on developing mutually

beneﬁcial agreements based on the interests of the

disputants. Interests include the needs, desires, con-

cerns, and fears important to each side (Raiffa et al.,

2002). Integrative bargaining usually produces more

satisfactory outcomes for the players involved than

does positional bargaining. Positional bargaining is

based on ﬁxed, opposing positions and tends to result

in a compromise or no agreement at all. Our negoti-

ation approach focuses on integrative bargaining for

achieving a satisfactory agreement for all players.

Kyaw T., Ghosh S. and Verbrugge R..

Multi-player Multi-issue Negotiation with Mediator using CP-nets.

DOI: 10.5220/0004256900990108

In Proceedings of the 5th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2013), pages 99-108

ISBN: 978-989-8565-38-9

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

Interest-based negotiation either can get the par-

ties to an agreement point where they can bargain or

even better, to a point where they do not need to bar-

gain at all. Interest-based negotiation typically entails

two or more issues to be negotiated. It involves an

agreement process that better integrates the aims and

goals of all the negotiating parties through creative

and collaborative problem solving.

Mediation usually consists of a negotiation pro-

cess that employs a mutually agreed upon third party

to settle a dispute between negotiating parties in or-

der to ﬁnd a compatible agreement to resolve disputes

(Fisher, 1978). In negotiation, the parties agree to

work with each other to resolve a dispute. In me-

diation, the parties agree to work with a facilitator

or mediator to resolve a conﬂict. In many cases, in-

ternational negotiations aim to achieve an agreement

on various issues between multiple parties. “Camp

David” is an interesting example of a negotiation that

happened between Egypt and Israel in 1978, resulting

in a more or less successful agreement with the help

of a mediator, the United States.

In this paper, we propose a simple mediated ap-

proach for multi-issue negotiation with incomplete in-

formation based on adjusting the players’ preferences.

In this approach, the mediator searches for a jointly

optimal negotiation text for all players in their condi-

tional preference networks (CP-nets), using a depth-

ﬁrst search-based algorithm. The players use utility-

based CP-nets to make a decision for agreement. The

purpose of the paper is to achieve near optimal joint

preference for all players while each player has im-

perfect information about his opponents.

The rest of the paper is organized as follows. In

Section 2, we brieﬂy explain preliminaries about CP-

nets, which are conditional preference networks for

representing and reasoning with qualitative prefer-

ences. We also discuss utility CP-nets and mediation

approaches using a single negotiation text in Section

2. In Section 3, we describe the proposed negotiation

approach with algorithms and illustrations. We dis-

cuss other closely related approaches to negotiation

based on CP-nets and compare our approach to them

in Section 4. Finally, in Section 5, we conclude the

paper and mention some future directions.

2 PRELIMINARIES

We begin with background concepts of conditional

preference networks (CP-nets), their induced prefer-

ence graphs and the utility-based CP-nets in this sec-

tion. We will also discuss a mediation approach using

single negotiation text (SNT).

2.1 CP-nets and UCP-nets

Boutilier and colleagues introduced CP-nets as a

graphical representation of conditional preference

networks that can be used for specifying prefer-

ence relations in a relatively compact, intuitive, and

structured manner using conditional ceteris paribus

(all other things being equal) preference state-

ments (Boutilier et al., 2004a; Boutilier et al., 2004b).

CP-nets can be used to specify different types of pref-

erence relations, such as a preference ordering over

potential decision outcomes or a likelihood ordering

over possible states of the world.

CP-nets are similar to Bayesian networks (Pearl,

1988). Both utilize directed graphs; however, the aim

of CP-nets in using the graph is to capture statements

of qualitative conditional preferential independence.

A CP-net over variables V = X

, ..., X

is a directed

graph G over X

, ..., X

whose nodes are annotated

with conditional preference tables CPT (X

) for each

∈ V . Each conditional preference table CPT (X

)

associates a total order 

with each instantiation u

of X

’s parents Pa(X

) = U (Boutilier et al., 2004a).

Let V = X

, ..., X

be a demand set of m attributes;

∈ V (i = 1 to m). D(X

) is the domain of X

and is

represented as D(X

) = x

, .., x

. There are D(X

)×

D(X

)×...×D(X

) possible alternatives (outcomes),

denoted by O. Elements of O are denoted by o, o0, o00

etc. and represented by concatenating the values of

the variables (Li et al., 2011b). For example, if

V = {A, B,C}, D(A) = {a

, a

}, D(B) = {b

, b

}

and D(C) = {c

, c

}, then the assignment a

assigns a

to variable A, b

to B and c

to C.

The preference information captured by an acyclic

CP-net N can be viewed as a set of logical assertions

about a user’s preference ordering over complete as-

signments to variables in the network. These state-

ments are generally not complete, that is, they do not

determine a unique preference ordering. Those order-

ings consistent with N can be viewed as possible mod-

els of the user’s preferences, and any preference asser-

tion that holds in all such models can be viewed as a

consequence of the CP-net (Boutilier et al., 2004b).

The set of consequences o  o0 of an acyclic CP-

net constitutes a partial order over outcomes: o is pre-

ferred to o0 in this ordering iff N |= o  o0. This partial

order can be represented by an acyclic directed graph,

referred to as the induced preference graph:

• The nodes of the induced preference graph cor-

respond to the complete assignments to the vari-

ables of the network; and

• There is an edge from node o0 to node o if and only

if the assignments at o0 and o differ only in the

value of a single variable X , and given the values

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Figure 1: CP-net, for Player 1.

assigned by o0 and o to Pa(X), the value assigned

by o to X is preferred to the value assigned by o0

to X.

For example, consider the CP-net given in Fig-

ure 1, whose variables are A, B and C. The prefer-

ence statements mean that a

is strictly preferred to

while the preferences of variable B are conditioned

on the variable A. If a

is chosen, then b

is preferred

to b

and if a

is chosen, b

is preferred to b

. The

preferences of variable C are also conditioned on the

variable B. The preference graph induced by the CP-

net of Figure 1 is shown in Figure 2.

The concept of Utility CP-net (UCP-net) was also

introduced by Boutilier and colleagues. It extends

the concept of CP-net by allowing quantiﬁcation over

nodes with conditional utility information. Semanti-

cally, Boutilier et al. treat the different factors V =

, ..., X

as generalized additive independent of one

another for an underlying utility function u (Boutilier

et al., 2001); this means intuitively that the expected

value of u is not affected by correlations between the

variables, and implies that u can be decomposed as

a sum of factors over each set of variables X

(Bac-

chus and Grove, 1995). For example, the CP-net

in Figure 1 can be extended with utility information

by including a factor ( f ) for each variable in the

network, speciﬁcally f

(A), f

(A, B) and f

(B,C) as

shown in Figure 3. We calculate the total utility of

preference strings as follows: u(A, B,C) = f

(A) +

(A, B)+ f

(B,C). Each of these factors is quantiﬁed

by the CPT (conditional preference tables) in the net-

work. For example, the utility of a

is as follows:

u(a

, b

, c

) = f

) + f

, b

) + f

, c

) = 5 +

0.6 +0.6 = 6.2 according to Figure 3.

F. Rossi and colleagues presented an extension of

the CP-net, called mCP-nets (Rossi et al., 2004), to

model the qualitative and conditional preferences of

multiple agents. They allowed the individual agents

to vote to obtain mCP-nets by combining several par-

tial CP-nets. K.R. Apt and colleagues proposed an

approach for analyzing strategic games that can be

used to study CP- nets (Apt et al., 2005). They in-

Figure 2: Induced Preference Graph of the CP-net of Fig-

ure 1, for Player 1.

Figure 3: UCP-net corresponding to the CP-net of Figure 1,

for Player 1.

troduced a generalization of strategic games in which

each player has to his disposal a strict preference re-

lation on his set of strategies, parameterized by a joint

strategy of his opponents. They showed that optimal

outcomes in CP-nets are Nash equilibria of strategic

games with parameterized preferences. Z. Liu and

colleagues also focused on the relationship between

CP-nets and strategic games (Liu et al., 2012). They

proposed a solution to resolve the optimal outcomes

of CP-nets by transforming a CP-net to a game tree

and using a tree algorithm to ﬁnd Nash equilibria.

Multi-playerMulti-issueNegotiationwithMediatorusingCP-nets

101

2.2 Mediation using a Single

Negotiation Text (SNT)

The concept of a single negotiation text (SNT) was

suggested as a mediation device by Roger Fisher

(Fisher, 1978). SNT is often employed in interna-

tional negotiations, especially with multi-party nego-

tiations (Raiffa et al., 2002), (Korhonen et al., 1995)

and (Ehtamo et al., 2001). For example, the SNT

approach was applied by the United States in mediat-

ing the Egyptian–Israeli conﬂict, which is known as

the Camp David Negotiations (Raiffa 1982). During

an SNT negotiation, a mediator ﬁrst devises and pro-

poses a deal (SNT-1) for the two protagonists to con-

sider. The ﬁrst proposal is not intended as the ﬁnal

agreement. It is meant to serve as an initial, single ne-

gotiating text : a version to be privately criticized by

both sides and then modiﬁed in an iterative manner.

The SNT is utilized as a method of focusing the

parties’ attention on the same composite text (Raiffa

et al., 2002). The important aspect of the process

is that it appears to be fair to both sides, and not

divisive. Based upon the criticisms by the parties,

the mediator prepares another proposal, which is not

perfect, but which improves both parties’ positions.

Again, both parties provide suggestions on improving

the proposal, and this new proposal is again criticized

by the parties. This process continues until all the is-

sues are settled and the ﬁnal agreement is achieved or

it is clear that no agreement is achievable. P. Korho-

nen and colleagues discussed the importance of the

starting point of the single negotiation text (Korho-

nen et al., 1995). They argue that, if the path taken

in subsequent steps does not compensate for a biased

starting point, the bias will have considerable impact

on the ﬁnal outcome of the negotiations.

3 THE PROPOSED APPROACH

TO NEGOTIATION

In real-world negotiations, negotiators need to

achieve an agreement on multiple issues with mul-

tiple players. Sometimes, a mediator is included to

facilitate the negotiation process. Some negotiations

fail because the parties have too many conﬂicts and

they cannot work with each other. Therefore, a medi-

ator may be used if the parties prefer a third party who

is neutral and does not represent any party’s interests.

Also in situations where the parties cannot meet to

negotiate directly, a mediator may be needed.

Our approach is based on a natural way to nego-

tiate in the real world. The proposed framework con-

sists of two types of individuals: the mediator and

the players. All players and the mediator specify the

issues that they need to negotiate before the negotia-

tion process starts. Each player keeps his own private

information and he does not know his opponents’ pri-

vate information. Each player reports his partial CP-

nets to the mediator. They do not directly come to

know their opponents’ preferences at any stage. In

addition, each player deﬁnes his own utility values

for each attribute and calculates his total utility for all

combinations of variables, as we mentioned in Sec-

tion 2.1. Each player creates his own UCP-net that

is used for proposing a maximum preference and for

deciding to accept or reject the proposal by the me-

diator. The mediator seeks to propose a single ne-

gotiation text that gains optimal joint outcomes for

all players by comparing all players’ proposed pref-

erences based on depth-ﬁrst search (Li et al., 2011b;

Li et al., 2011a; Aydogan et al., 2011).

3.1 Case Study: Negotiation with

Mediation

As an example case, let us consider three players and

one mediator for three issues. In this framework, the

players and mediator can be run on different comput-

ers. When starting the negotiation process, all play-

ers report their overall preference information about

negotiation issues to the mediator. Then, the media-

tor creates induced preference graphs for all players

based on their CP-nets. Let N be the set of play-

ers: N = {1, 2, 3}. We consider the three variables

A, B,C as three negotiation issues. The domains of

the variables are D(A) = a

, a

; D(B) = b

, b

; and

D(C) = c

, c

This can be seen as a simple real-world negoti-

ation between different preferences of family mem-

bers. Suppose that a family including father, mother

and 20 years old son decided to buy a new house.

They have different preferences for the three issues:

Type of House (A): house with small garden (a

);

condo apartment (a

);

Place Near (B): market (b

); park (b

);

Price Range (C): high (c

); medium (c

); low (c

For example, mother (Player 1) prefers to buy a

house with a small garden, situated near a market. If

the house is situated near a market, she prefers a high

price to a medium or a low price. If it is situated near

a park, she prefers a low price to a high price. Fa-

ther (Player 2) prefers a place near a park to a place

near a market. If the place is near a park, he prefers a

condominium apartment and if it is near a market, he

prefers a house with a small garden. He prefers a high

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Table 1: Utility Table for Players 1, 2 and 3.

Strings Player 1 Player 2 Player 3

6.2 3 7.2

6 2.7 7.4

5.8 2.9 7.6

5.5 5.8 8.6

5.7 5.5 8.4

5.9 5.7 8.2

2.9 2.7 4.6

2.7 2.4 4.4

2.5 2.6 4.2

2.8 6.1 4.2

3 5.8 4.4

3.2 6 4.6

Average 4.35 4.27 6.15

price or a low price rather than a medium price. Their

son (Player 3) prefers a house with a small garden to

a condo apartment. He also prefers a place situated

near a park to a place near a market. If it is a house

near a market or an apartment near a park, he prefers

a high price to a medium or a low price. Otherwise,

he prefers a low price to a medium or a high price.

Let us suppose that the real estate agent acts as a

mediator and that the family members do not want

to share their preferences with one another. After

proposing three negotiation texts by the mediator, all

family members agree to buy a new house near a park

with low price. We will illustrate the details of the

negotiation process in Section 3.2.

3.2 Negotiation Process for Case Study

Assume that the CP-net and the induced preference

graph given in Figure 1 and Figure 2 have been pro-

posed by Player 1. The CP-nets and their induced

graphs of Player 2 are shown in Figures 4, 5, and

those for Player 3 in Figure 7, 8. All players prepare

UCP-nets with their private utility values as illustrated

in Figure 3, 6 and 9. Each player also calculates the

total utility of strings in their UCP-nets, as shown in

Table 1. They pick up one string with maximum util-

ity outcomes in their UCP-nets and propose it to the

mediator. In this example, Players 1, 2 and 3 propose

, b

, c

, a

, b

, c

and a

, b

, c

respectively. These

are the bottommost strings of the induced graphs (see

Figure 2, 5 and 7).

We have developed two algorithms: Algorithm 2

(see page 10) that helps the mediator decide which

single negotiation texts to propose, given the play-

ers’ CP-nets and their answers to previous proposals;

and Algorithm 1 (see 10), that helps each of the other

players to decide whether to accept a proposal by the

mediator or not. We now proceed to show how the

Figure 4: CP-net for Player 2.

Figure 5: Induced Preference Graph of the CP-Net of Fig-

ure 4, for Player 2.

algorithms work for the case study.

The mediator generates a single negotiation text

that we call “the proposal”, by searching jointly op-

timal gains of all players according to Algorithm 2

(see page 10). After receiving the maximum preferred

strings of all players, the mediator searches accept-

able probability to the other players’ strings (line 11,

Algorithm 2, see page 10). Starting point is the bot-

tommost string of the induced preference graph and

we assume that the string whose edge directly points

to the bottommost string gets the acceptable proba-

bility 0.9. We deﬁne probability of a string on the

graph by going backward from the bottommost or

maximum preferred string. We count the intermedi-

ate edges from the bottommost string to the particular

string by reducing the probability by 0.1 for one edge.

This searching process continues until the acceptable

probability 0.5 is reached. Otherwise, the probability

is assigned to zero.

In this case study, the acceptable probability from

Player 1’s preferred string a

, b

, c

to Player 2’s pre-

ferred string a

, b

, c

is 0.8 and to Player 3’s pre-

ferred string a

, b

, c

it is 0.9 on Player 1’s in-

duced graph (see Figure 2). For Player 2, probability

from his string a

, b

, c

to Player 1’s proposed string

, b

, c

is 0.8 and probability to Player 3’s proposed

string a

, b

, c

is 0.9 (see Figure 5). For Player 3,

probability from his string a

, b

, c

to Player 1’s pro-

posed string a

, b

, c

is 0.9 and probability to Player

2’s proposed string a

, b

, c

is 0.9 (see Figure 7). Ac-

Multi-playerMulti-issueNegotiationwithMediatorusingCP-nets

103

Figure 6: UCP-net for Player 2.

Figure 7: CP-net for Player 3.

cording to this example, a

, b

, c

is jointly optimal

for all players because it obtains reachable probability

0.9 from their maximum preference strings. There-

fore, the mediator proposed a

, b

, c

as a ﬁrst jointly

optimal negotiation text.

Moreover, the mediator marks all acceptable prob-

abilities less than threshold (line 14, Algorithm 2,

see page 10). This threshold can be changed when

the mediator cannot ﬁnd any jointly optimal proposal

within the threshold. If the players reject the proposal

and the mediator has an alternative jointly optimal

proposal in his previous marked list, the mediator can

use the alternative as the next proposal.

If there is no jointly optimal proposal among the

players’ proposals, the mediator tries to search for

an alternative jointly optimal proposal (line 21, Al-

gorithm 2, see page 10) that has the same acceptable

probability of the players’ previous proposals. The

mediator searches all strings that have one backward

edge from the maximum preferred string (probabil-

ity 0.9) for all players. He then searches a common

string of all players. If there is no common string, the

mediator continues to search all possible strings for

probability 0.8. This process continues until an aver-

age maximum probability is found.

After receiving a new negotiation text from the

mediator, all players checks their utility outcomes

(see Table 1) of the text to make a decision “accept”

Figure 8: Induced Preference Graph of the CP-Net of Fig-

ure 7, for Player 3.

Figure 9: UCP-net for Player 3.

or “reject”. Player 1 rejects the proposal a

, b

, c

be-

cause the difference between her maximum utility and

the utility of the current text is greater than the thresh-

old (maxU −a

, b

, c

= 6.2 −5.5 = 0.7) according to

Algorithm 1 (page 10). We assume that the starting

threshold is 0.4 in Algorithm 1 (page 10). The thresh-

old can be changed according to the type of utility

values. Player 3 strongly accepts the current proposal

because he gets his maximum utility. Player 2 also

accepts because the difference between his maximum

utility and the current text is less than the threshold

(maxU − a

, b

, c

= 6.1 − 5.8 = 0.3).

The negotiation process continues until the agree-

ment is achieved by the mediator acting according to

Algorithm 2 and the Players according to Algorithm

1. If there is no agreement until the ﬁnal round be-

fore the deadline, the mediator can announce this to

all players and the players can evaluate all negotia-

tion texts and consider if they are willing to reduce

their maximum utility.

Finally, in this case, the mediator proposes a

as a jointly optimal negotiation text. Player 1 ac-

cepts the proposal because the difference between

her maximum utility and the current text is less than

the threshold; (maxU ) − a

, b

, c

= 6.2 − 5.9 = 0.3).

Player 2 and 3 also accept the proposal because the

difference between their maximum utilities and the

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104

current text is equal to threshold; (maxU −a

, b

, c

6.1 − 5.7 = 0.4 and maxU − a

, b

, c

= 8.6 − 8.2 =

0.4, respectively). All players achieve a jointly op-

timal outcome, which is greater than their average

outcomes (see Table 1), although they do not achieve

their maximum outcomes.

3.3 Cyclic CP-nets

In addition, our approach can achieve a negotiation

outcome even when players’ CP-nets are cyclic as

shown in Figure 10 (a). Let us show a simple ex-

ample of negotiation between two players. Player 1’s

preferences and induced graph are shown in Figure 10

(b) and 10 (c). Player 1’s private utilities are: a

(4),

(4), a

(2) and a

(2). Player 2’s preferences

and induced graph are shown in Figure 10 (d) and 10

(e). Player 2’s private utilities are: a

(3), a

(3),

(3) and a

(3). Actually, player 2’s preference

utilities are all the same and its induced graph is not

satisﬁable (Boutilier et al., 2004a).

Figure 10: Examples of cyclic CP-nets and their induced

preference graphs.

In our negotiation process, player 1 proposes a

as her maximum preferred string and player 2 pro-

poses a

. When the mediator computes the accept-

able probability, a

has the same probability as a

for Player 1 and a

has 0.8, an acceptable probabil-

ity for Player 2. If there is no backward edge from

the maximum preferred string to a particular string,

the two strings may have the same probability. For

instance, if two strings, a

and a

(see Figure

10 (c)) have a backward edge from the same string

), then, a

and a

have the same probabil-

ity. This reasoning can easily be applied to search on

the preference graphs given from the cyclic CP-nets.

In our example, the mediator proposes a

which

is Player 2’s proposed string. It also has the same

acceptable probability as Player 1’s proposed string

. Both players accept the mediator’s proposal be-

cause it meets their maximum preference utility.

3.4 Negotiating International Conﬂict

Our approach can be applied to international conﬂict

resolution as well. Camp David is a well-known ne-

gotiation process that happened between Egypt and

Israel in 1978. The negotiation process lasted for 13

days and the United States acted as a mediator. U.S

mediators had already known deeply about the pre-

ferred solutions of Egypt and Israeli and they decided

to use a single negotiation text (SNT). The U.S started

by offering its ﬁrst SNT-1 but was not trying to push

this ﬁrst proposal. It was meant to serve as an initial

SNT; a text to be criticized by both sides, then mod-

iﬁed, and remodiﬁed in an iterative manner. These

modiﬁcations would be made by the U.S based on

the recommended changes by both sides. After play-

ing six rounds, a satisfactory agreement, the Camp

David accord, was reached (Raiffa et al., 2002). We

can simulate this negotiation using the proposed ap-

proach. In Camp David, there are two players, Egypt

and Israel, and four basic negotiation issues (Telhami,

1993; Oakman, 2002):

1. A peace treaty and normalization of relations be-

tween Egypt and Israel (A);

2. Demilitarization and removal of Israeli settle-

ments from Sinai (B);

3. Linkage between these issues and the future of the

West Bank and Gaza (C);

4. A statement on principles, including Israeli with-

drawal from all occupied territories and the right

of Palestinians to self-determination (D).

The Camp David negotiation process concerns in-

ternational issues and foreign affairs and we omit

the details of Egyptian and Isreali preferences. Let

us consider that D(A) =(a

), D(B) =(b

D(C) =(c

), and D(D) =(d

). There is a

total combination of 36 possible agreements. The

mediator, representing U.S., prepares CP-nets and

induced preference graphs for both players. We

can deﬁne the utility values of all variables for

both players. Then, negotiation process contin-

ues according to Algorithms 1 and 2. We found

that indeed, a ﬁnal agreement is achieved within

a ﬁnite number of rounds. For details, see

http://www.ai.rug.nl/SocialCognition/experiments/.

4 DISCUSSION ON RELATED

WORK

This paper presents an approach for negotiation over

Multi-playerMulti-issueNegotiationwithMediatorusingCP-nets

105

multiple players on multiple issues with the support

of a mediator. To achieve a jointly optimal agreement,

the mediator offers a single negotiation text based on

all players’ preference graphs given from their CP-

nets. Every player can decide to accept or reject the

offer by checking the negotiation text’s utility on his

or her private UCP-nets. We proposed two algorithms

for the mediator and the players. For successful ne-

gotiation between players, they often need to give up

their maximum expected preferences because other-

wise the negotiation process may not achieve a sat-

isfactory agreement within a ﬁnite number of rounds.

The proposed approach is appropriate for players who

are willing to accept a jointly optimal choice.

M. Li and colleagues presented a protocol for

negotiation in combinatorial domains (Li et al.,

2011b), which can lead rational agents to reach opti-

mal agreements under an incomplete information set-

ting. They proposed POANCD (Protocol to reach Op-

timal Agreement in Negotiation over Combinatorial

Domains), which has two phases. The ﬁrst phase of

POANCD consists of distributed formation of a ne-

gotiation tree by the participating agents, based on

CP-nets of agents. After the ﬁrst phase, the agents

make a few initial agreements. In the second phase,

the agents act cooperatively to achieve best possi-

ble agreement by exploring possible mutually ben-

eﬁcial alternatives. Li and colleagues also proved

their approach to dominance testing in CP-nets (Li

et al., 2011a). Their approach did not have a solu-

tion on cyclic CP-nets yet in (Li et al., 2011b). Re-

cently, however, Li and colleagues proposed an ap-

proach called MajCP (Majority-rule-based collective

decision-making with CP-nets) that can work with

cyclic CP-nets as well (Li et al., 2011c).

The purpose of the proposed approach in this pa-

per is to deal with similar situations, players do not

only have preference orderings in their CP-nets, but

they also have their private utility values (possibly dif-

ferent utilities for the same preference order). Our

approach provides negotiation in an interactive way,

with mediator and players as a game where the medi-

ator proposes a single negotiation text and players can

decide about agreement themselves.

K. Purrington and E. H. Durfee also proposed an

algorithm to ﬁnd social choices of two agents by ex-

ploiting the CP-net structure (Purrington and Dur-

fee, 2009). Their algorithm searches agents’ outcome

graphs from the top down, using the satisfaction inter-

val associated with each tier, and it proceeds for each

agent. A set of candidate outcomes is maintained for

each agent. It contains all the outcomes that an agent

is willing to accept. For each agent, the algorithm ex-

amines the highest tier of outcomes that are not cur-

rently in its candidate set, and for the agent(s) with the

highest minimum for this next tier, adds those out-

comes to the candidate set(s). If the intersection of

the agents’ candidate sets is non-empty, one of the

outcomes in the intersection has maximin optimality.

Their algorithm considers only the level of the pref-

erence graphs. Our framework provides not only the

mediator’s joint choice, but also the players’ decisions

based on their private UCP-nets.

R. Aydogan and P. Yolum developed a negotia-

tion approach using heuristics for CP-nets with partial

preferences (Aydogan et al., 2011). They observed

three negotiation strategies: depth-based, ranking-

based and utility-based. They showed an example

of negotiation between a producer and a consumer

agent over a service. Negotiation takes place in a

turn-taking fashion, where the consumer agent starts

the negotiation with a particular service request. A

service request can be considered as a vector of is-

sues (discrete or continuous), which represents the

service. If the producer agent does not prefer to sup-

ply this service, then the producer generates an alter-

native service. The consumer agent can accept this

alternative service or continue negotiation to pursue

a better one. This process continues until reaching a

consensus or a deadline. Aydogan and Yolum focus

only on one player’s preferences and do not mention

the other player’s preferences. They do not deal with

negotiation for multiple (more than two) players. The

purpose of our approach is to deal with multi-player,

multi-issue negotiation via a mediator.

M. Chalamish and S. Kraus presented AutoMed,

an automated mediator for bilateral negotiation un-

der time constraints, which uses a qualitative model.

AutoMed produces the negotiators’ preferences using

Weighted CP-networks (WCP-nets). Each disputant

speciﬁes her preferences by creating her WCP-net us-

ing a graphical interface. Next, AutoMed sorts all

possible agreements according to the WCP-nets and

removes all non-optimal sets. During the negotia-

tion process, AutoMed searches for an optimal offer

by ﬁnding all agreements preferred to the offer made

by the opponent in each list (Chalamish and Kraus,

2012). In our approach, the mediator does not use

weighted or utility CP-nets but only CP-nets based on

partial preferences of the players, because the players

do not want to show their private utility values.

5 CONCLUSIONS

In this paper, we present a simple negotiation ap-

proach that is useful in a practical way for negotia-

tions. The mediator offers jointly optimal negotiation

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texts based on CP-nets over a ﬁnite number of rounds

and the rest of the players are willing to adjust their

private interests at each round using UCP-nets. In this

interactive framework, the mediator and the players

can play on different machines by sending messages.

The framework can deal with negotiation for multi-

ple issues, and with multiple players who have dif-

ferent preferences even when their preference graphs

are cyclic. The proposed approach provides a satis-

factory agreement for all players with their optimal

outcomes, which are not less than average utility. As

a future direction, we plan to test the performance of

this approach and hope to construct a more efﬁcient

search algorithm on preference graphs. For the result-

ing negotiation algorithms, we intend to prove formal

properties related to correctness and complexity.

ACKNOWLEDGEMENTS

We gratefully acknowledge NWO research grant

600.065.120.08N- 201, Vici grant NWO 227-80-001,

and a Lotus II grant from the EU.

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APPENDIX

Algorithm 1: Negotiation Decision by Player.

1: Input :

2: UCP-nets with total utility and ordering

3: Agreement ← 0; maxU ← maximum utility

4: threshold ← 0.3; Proposals ←

5: while Agreement 6= 1 and t 6 f inalRound do

6: Search S (U(S) = maxU) in UCP-net //search

maximum preferred proposal

7: Send (S) to Mediator

8: currentProposal ← Receive(Proposal by Media-

tor)

9: Proposals ← Proposals ∪ currentProposal

10: if (maxU−U(currentProposal)) 6 threshold

then

11: accept Proposal

12: Agreement ← 1

13: else

14: reject Proposal

15: Update maxU //search and update the utility

less than current maxU

16: Send (S)

17: end if

18: end while

19: if Receive(ﬁnalRound) then

20: while maxU > avgU do

21: Update maxU //search and update the utility

less than current maxU

22: Evaluate proposals

23: if ∃proposal : (maxU−U(proposal) 6

threshold) then

24: accept proposal;

25: Agreement ← 1

26: end if

27: end while

28: end if

Algorithm 2: Negotiation by Mediator.

1: Input:

2: Player N: N = (1, 2, . . . , n)

3: CPN

,CPN

, . . . ,CPN

//Players’ CP-nets

4: S

, S

, ..., S

//Players’ proposals

5: maxP ← 0.9; avgP ← 0.5 //maximum and av-

erage acceptable probability

6: Agreement ← 0; JointOptimal ←

0;t ← 0

7: threshold ← 0.3

8: while Agreement 6= 1 and t 6 f inalRound do

9: for i = 1 to n do

10: for j = 1 to n do

11: Search acceptableProbability (S

, S

: i 6= j)

12: end for

13: end for

14: Mark all S

, S

: acceptableProbability <

threshold;

15: while maxP > avgP do

16: for i = 1 to n do

17: for j = 1 to n do

18: if ∃S

: (acceptableProbability(S

, S

) =

maxP ; i 6= j) then

19: JointOptimal ← S

20: else

21: Search alternativeOptimal(S

) //Other

proposals with same maxP

22: if ∃S

: (acceptableProbability(S

, S

) =

maxP; l 6= j) then

23: JointOptimal ← S

24: else

25: maxP ← maxP − 0.1

26: end if

27: end if

28: end for

29: end for

30: end while

31: Propose JointOptimal

32: if ∀ Player k ∈ N accept Proposal then

33: Agreement ← 1

34: else if Player k (k ∈ N) rejects S

then

35: Ask new proposal to Player k

36: Update S

(i = k)

37: maxP ← 0.9

38: end if

39: end while

40: if Agreement = 0 and t = ﬁnalRound then

41: Announce ﬁnalRound and Ask for evaluating all

proposals to Players

42: end if

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