Bilateral Multi-Issue Negotiation

Bilateral Multi-Issue Negotiation

海盜問題

 5 個海盜搶到了 100 顆寶石，每一顆都一樣的大小和價 值，他們決定這麼分︰ 　　 1. 抽簽決定自己的號碼（ 1 ， 2 ， 3 ， 4 ， 5 ） 　　 2. 首先，由 1 號提出分配方案，然後大家 5 人進行 表決，且僅當半數或超過半數的人同意時，按照他的提 案進行分配，否則將被扔入大海喂鯊魚。 　　 3. 如果 1 號死後，再由 2 號提出分配方案，然後大 家 4 人進行表決，且僅當半數或超過半數人同意時，按 照他的提案進行分配，否則將被扔入大海餵鯊魚。 　　 4. 以次類推 ...

海盜問題–條件



條件︰每個海盜都是很聰明的人，都能很理

智的判斷得失，從而做出選擇。



問題︰第一個海盜提出怎樣的分配方案才能

夠使自己的收益最大化？



據統計，在美國，在 20 分鐘內能回答出這

道題的人，平均年薪在 8 萬美金以上。

海盜問題–分析

 當只有 4,5 二人時， 4 必定提出「 4-100 ； 5-0 」的方 案並順利通過，因只要 4 同意就行（不用解釋吧） 當只有 3,4,5 三人時， 3 必定提出「 3-99 ； 4-0 ； 5-1 」的方案並順利通過 5 答應的原因：若 5 不答案，則 3 要死，到 4 提出方案時 則會變成「 4-100 ； 5-0 」的局面，到時 5 就會啥都沒 有，故此 5 一定要答應 不給 4 的原因：只要 3 一死 4 就可提出「 4-100 ； 5-0 」的方案，所以不能給 4

海盜問題–解答



所以，正確的答案是：

當有 1,2,3,4,5 五人時， 1 必定提出

「 1-98 ； 2-0 ； 3-1 ； 4-0 ； 5-1 」

的方案並順利通過。

Negotiation



When people are trying to resolve conf

licts between several parties, they ne

gotiate.



single-issue two-party problems



multi-issue multi-party

Bilateral Multi-Issue Negotiatio

n

 Multi-issue, two party

 Multi-issue negotiations are considered inte

grative \cite{raiffa:negotiationart1982}, wh ere all parties may find mutually beneficial outcomes, i.e., win-win solutions. However, the complexity of a multi-issue negotiation increases rapidly as the number of issues in creases, which means that people need more t ime and rationale in handling the negotiatio n problem.

Negotiation Support Systems

 The development of Negotiation Support Syste

ms (NSSs) and negotiating software agents (N SAs) have been proved to be able to reduce s ignificantly the negotiation time and allevi ate the negative effects of human cognitive biases and limitations \cite{lomuscio:negotiat ionclassification2001} \cite{sandholm:negotiatio ncomponent2000} \cite{maes:agent1999} \cite{foro ughi:negotiationprocess1998}.

Negotiation Analysis



decision analysis



game theory



Negotiation analysis

\cite{sebenius:NA92} is used to gene

rate

prescriptive advice

to the suppor

ted party given a

descriptive assessme

nt

of the opposing parties.

Uncertainty

 Problems can arise if assessment of the oppo

sing parties are not available or vague.

 For one-sided uncertainty, there is a protoc

ol where the uninformed agent makes all the offers and the informed agent either accepts or rejects offers \cite{vincent:bargaining19 89}.

 Alternatively the uninformed agent can try t

o model the opponent using a Bayesian networ k or an influence diagram \cite{vassileva:bi lateralnegotiation2002}.

Types of Games

 Perfect

 Each information set is a singleton.  Certain

 Nature does not move after any player moves.  Symmetric

 No player has information different from othe

r players when he moves, or at the end nodes.

 Complete

 Nature does not move first, or her initial mo

A Game of Incomplete Information



The use of the word “uncertainty” in

this paper should not be confused with

a game of uncertainty. We are handling

an imperfect, asymmetric, incomplete b

ut certain game, where there is a

two-sided uncertainty about the preference

s of the opponents who are bargaining.

Two-sided Uncertainty



The problem of two-sided uncertainty c

an be addressed using

recursive modeli

ng

\cite{gmy:recursivemodel1995}. Neve

rtheless, for complex multi-issue nego

tiations, it could be

computationally

intractable

.

Incomplete Information

 The solution to the bilateral negotiation pr

oblem of incomplete information is addressed in the literature by giving a continuous dis tribution or discrete probabilities over the other agent's \emph{type} and using Bayesian rule to learn the type during the negotiatio n. The negotiation protocol used is sequenti al alternating protocol (SAP) \cite{rubinste in:perfectequilibrium1982}.

Sequential Equilibra

 For single issue negotiation that negotiates

on price, the type of the other agent is rep resented by its reservation price. These dis tributions are common knowledge. As a result , there is no subgame-perfect equilibrium, w hich requires that the predicted solution to a game be a Nash equilibrium in every subgam e. Rubinstein analyzed it using a stronger e quilibrium concept of sequential equilibra \ cite{rubinstein:perfectequilibrium1982}.

Sequential Equilibra cont.

 It requires that each uncertain player's bel

ief be specified given every possible histor y, and Bayes rules are used to make beliefs consistent. However, if the other agent's be havior deviates from the equilibrium path, a n update problem may occur since non-equilib rium paths are assigned zero probability.

 This may result in incentives for agents to

deviate from the equilibrium, so as to incre ase the number of possible outcomes \cite{fa ratin:automatednegotiation2000}.

Analysis Difficulties

 Derivation from equilibrium behavior cannot

be ruled out in games via SAP with two-sided uncertainty. The situation could be worse fo r multi-issue negotiation, since types of ag ents increase dramatically as the number of issues increases. Besides, in a multi-issue negotiation, it is not necessarily true that the agreement that is worst for one agent is best for another, or vise versa. This makes the analysis of the intentions of each offer proposed by the opponent significantly more difficult.

Random Selection Process ?

 In the decision making behaviors of human, p

eople rely on random selection processes, su ch as flipping a coin, to handle a decision that involves too much uncertainty and subse quently it becomes difficult for them to rat ionally judge a decision. However, consideri ng the problem of computation complexity, th e question resides in the possibility for ag ents participating in a multi-issue negotiat ion with two-sided uncertainty to simply fli p a coin to decide on every issue.

A Mediation Game

 As commented in \cite{raiffa:negotiationart1

982}, it is neither feasible nor logical to do so. Flipping a coin on every issue will n ot generate mutually beneficial outcomes. We propose a mediation protocol that is based t he Single Negotiation Text (SNT) device sugg ested by Roger Fisher \cite{fisher:mediation 1978}. This protocol presents a deal constru ction game to both protagonists, where they actively participate in this construction pr ocess to find a mutually beneficial agreemen t.

A Scenario

 let us say that there are two companies A and B, which ha

ve control over different resources, such that they do sa me jobs with different costs. There is a case that some c ustomer would like to have his tasks being done with the price of m dollars. A and B discover that if one of them is going to do the tasks alone, the profit is almost zero (equal bargaining power). On the contrary, if they can ne gotiate a plan of task sharing, the profit can increase. However, how they can divide the set of tasks and the mon ey at the same time fairly without disclosing too much co nfidential information, given that they may have to compe te with each other in another case, becomes the main chal lenge.

Assumptions



Expected Utility Maximizer



One-off Negotiation



Asymmetric Abilities



Inter-independent Issues



Two-sided Uncertainty

Single Negotiation Text

 A mediation device suggested by Roger Fisher

\cite{fisher:mediation1978}.

 During the negotiation, the mediator firstly

devised and proposed a deal (SNT-1) for the consideration of the two protagonists.

 The mediator is not trying to push the first

proposal, but that it is meant to serve as a n initial, single negotiating text---a text to be criticized by both sides and then modi fied in an iterative manner. Modifications t o the SNT-1 will be made by the mediator bas ed on the criticisms of the two sides.

Fairness



The SNT technique is to be used as a m

eans of focusing the attention of both

sides on the same composite text. The

important thing is that this process a

ppear to be

fair

to both sides, not di

visive.

SNT-1



SNT-1 can be generated by locating a c

onverging point from

a “dance”

of pa

ckages (see Figure~\ref{fig:ju}) or a

focal point

(for example, the mid-valu

e on each continuous factor). When try

ing to generate the SNT-1, both agents

must know that they are not haggling a

bout a final contract, but a starting

point for the pursuit of joint gains.

Iteration



After the SNT-1 has been located, both

protagonists then try to improve it si

multaneously or by taking turns.



The mediator will take both protagonis

ts' criticisms or suggestions into con

sideration, and generate a new version

of SNT for further revisions. This pro

cess continues until all the issues ar

e settled.

The Challenge

 The SNT technique had been applied by U.S. o

n the mediation of Egyptian-Israeli conflict in early 1977, known as Camp David Negotiati ons. Part of the story can be found in \cite {raiffa:negotiationart1982}.

 The challenge of SNT, as noted by Raffia, is

: How can we devise negotiating processes th at will encourage more honest revelations an d less strategic behavior?

Def – deal

 A deal D is represented by a binary string i

n A's view:

 h b₁,b₂,…,b_ni = h c₁,c₂,…,c_p,r₁,r₂,… ,r_qi

 c_i represents a bit that will result in nega

tive utility (cost) for A when false (c_i = 0), while r_i is a bit that will generate pos itive utility (revenue) for A when true (c_i = 1).

Def – deal cont.

 The money earned is mapped to the r_i bits. Each r_i represe

nts a bag of money used for exchange. As suggested in cit e[p. 216]{raiffa:negotiationart1982}, the issues to be ne gotiated should be in comparable magnitude of importance, so that protagonists might then agree to resolve each iss ue separately by the toss of a coin. Therefore we let the amount of a bag of money be:

Def – profits

 The total cost (TC) and total revenue (TR) o

f a deal D is:

 TC_A(D)=_{1· j· p} Cost_A(c_j)£ (1-c_j).  TR_A(D)=_{1· j· q} Revenue_A(r_j)£ r_j.  TC_B(D)=_{1· j· p} Cost_B(c_j)£ (c_j).

 TR_B(D)=_{1· j· q} Revenue_B(r_j)£ (1-r_j).

 The profit gained is:

 Profit_i(D) = TR_i(D) - TC_i(D).

 For each agent, a rational deal D must satisfy Pr

Def – utility



The utility of a deal D is:

 Utility_i(D) = Profit_i(D) + _{1· j· p} Cost_i(c_j)

.



The utility generated by a set of bits

O=C[R, where C is a set of cost bits a

nd R is a set of revenue bits, is:

Game



Players – two players in our scenario



Actions



Information



Strategies



Payoff



Outcome



Equilibrium

Fairness & Constructiveness

 This mediation process can produce a fair an

d constructive outcome if both agents do cho ose their preferred bits sequentially.

 For a fair outcome, we mean that each agent

will have a probability 0.5 of getting the u tility from a bit if they are faced with a d irect conflict. Otherwise, they can trade th eir less preferred bits for more preferred b its to get a constructive outcome, which mea ns that both agents get higher final utility than flipping a coin on every bit.

Strategic Move

 However, an agent can strategically choose a less pr

The DOT Strategy

 We named it the Delay of Trades (DOT) strate

3-DOT

 4-DOT

 O₂(Conflict  Win) xg. O₃(Conflict  Lose)

 3-DOT

Note



In the DOT abstraction, when we say U

(O

₁

) ¸ U(O

₂

), we mean that:

 For Every bit b_j 2 O₁ and every bit b_k 2 O₂  U(b_j) ¸ U(b_k)



Also:

Utility Gain in the DOT Strategy



4-DOT

 From:

 To:

Utility Gain in the DOT Strategy

 3-DOT

 From:

 To:

Success Condition of DOT



For A to success:

 A must know B’s utility function

 B must report his preference honestly

The Binary Match Game



Since A and B may partially agree on t

he some of the issues, some part (bit

s) in SNT-2 will be

fixed

, which means

that these bits can not be further mod

ified in the next stage. A and B then

concentrate on the resolution of the r

emaining issues until all issues are s

ettled.

Trades

 The bits b_i and b_j (i<> j) is said to be traded i

f one of them is assigned a value 1 (in A's favo r) and the other is assigned a value 0 (in B's f avor).

 A good trade occurs when a less preferred bit is

traded for a more preferred bit from both agent s' view. On the contrary, a bad trade may occur when a more preferred bit is traded for a less p referred.

 If a trade is beneficial to only one of them, bu

t indifferent to the other, we call it a single-interest trade.

Negotiation Strategy



A negotiation strategy is a function f

rom the history of the negotiation to

the current action (bits relocation) t

hat is consistent with the mediation p

rotocol.

The BH Strategy

 Since the SNT-1 is randomly traded, it is fa

ir but not constructive. A and B may have di fferent ideas on how the 1-bits and 0-bits s hould be relocated.

 According to A's utility function, there exi

sts a better half ½, such that #()=d#() /2e and 8 b_i2, 8 b_j2bar{}, U_A(b_i)>=U_A(b_j). I f A generates SNT(A,t) by relocating 1-bits in SNT-t to the better half , we say that A is using the BH (better half) strategy.

Further Trades

 If both agents use the BH strategy to reloca

te the 1-bits and 0-bits in SNT-t, some good trades may be found.

 To encourage further trades, the conflicting

bits (those not yet grayed) are separated in to two divisions  (better half) and  (les ser half), marked as B and L in Figure.

 Now both  and  are treated as sets of con

flicting bits to be resolved separately (enf orcing trades within each division).

Final Match



When there are two bits left in SNT-t,

both agents must submit their preferen

ces over the combinations of h 1,0i an

d h 0,1i. If they agree on the same co

mbination, the mediation is done. Othe

rwise, the mediator must flip a coin t

o decide that which combination is sel

ected. We named this mediation protoco

l the Match-Game Mediation (MGM) Proto

col.

Pareto Optimal



We say that a deal D dominates D0, and

write DÂ D0, if and only if (Utility

_A

(D), Utility

_B

(D))À(Utility

_A

(D0), Utilit

y

_B

(D0)) (it is better for at least one

agent and not worse for the other).



A deal D is called pareto optimal if t

here does not exist another deal D0 su

ch that D0Â D.

Preference Indifference

 If both agents use the BH strategy, and the

information of preference indifference is di sclosed, then both good trades and single-in terested trades can be discovered by the pro posed mediation protocol, since the MGM prot ocol exhaustively searches all possible trad es utilizing the concept of divide and conqu er. Therefore, the mediation result is paret o optimal.

Strategic Behavior in MGM



The DOT strategy will generate higher

utility if O

₁

can be successfully trade

d for O

₂

and O

₃

can be successfully tra

ded for O

₄

at the same time. In case 1,

if B wants to use the DOT strategy, it

must strategically cause a conflict in

the first match by reporting that O

₄

is

not in his better half.

Causing a Conflict



There are two ways of causing a confli

ct:

 Causing a full conflict: B reports that i

ts better half and A's are the same. B ca n strategically do so to delay the trades of bits till the second round.

 Causing a partial conflict: In case 1, B

can report that its better half does not include O₄ (but include O₂), then O₁ is tr aded for O₂.

Beneficial Strategic Behaviors

 In case 1, if B strategically cause a full c

onflict, O₁ will be in the  division of the next SNT, while O₂ is in the  division. We know that trades are only allowed within eac h division, but not across divisions. This s trategic behavior does not benefit B. The sa me rationale forbids the strategic bahvior o f causing a full conflict in case 3. But B c an benefit from causing a full conflict in c ase 2.

 B can benefit from causing a partial conflic

Nash Equilibrium



A negotiation strategy s will be in eq

uilibrium if the following condition h

olds: under the assumption that A uses

s, B prefer s to any other strategy.

Case 1



We assume K(E) µ E µ P(E)



K

_A

(U

_A

), K

_B

(U

_B

), K

_A

(~K

_B

(U

_A

)) and K

_B

(~K

_A

(U

B

)) is common knowledge.

Case 1 Proof

 Assume that A uses the BH strategy. If B also uses the

BH strategy, the utility gained is the sum of the util ity gained from good trades. Let Og denotes the bits th at are traded, and O_c denotes the bits that cause confl icts. The deal is D=Og[ Oc. Let BH(Og) denotes the bits in Og that are also in the better half, and LH(Og) deno tes the bits in O_g that are also in the lesser half.

 If a trade occurs, the 0-bit is in BH(O_g) and the 1-bit

is in LH(Og), since B has relocated all the 0-bits to t he better half. Therefore, that trade is a good trade.

 The utility gained in this stage is U(BF(O_g))-(U(O_g)/2)

Case 1 Proof



Let

 denotes the average utility if b

oth agents flip a coin to decide on ev

ery issue.



The expected utility in the end of the

mediation if equal to or greater than



 + U(BF(O

_g

))-(U(O

_g

)/2), denoted as



Case 1 Proof

 If B does not use the BH strategy, there is a 0-bit at

the position bj that is not relocated to the position b i in the better half. It makes:

 a good trade becomes a conflict or  creates a smaller good trade

 A good trade becomes a conflict when some lesser bit c

an be traded for the bi but now a conflict occurs at th e bi. A smaller good trade is resulted from the success fully trading of some lesser bit for b_j. But since U(b i) > U(bj), a better trade is missed. In Theorem thm:HM GM, it is proved that a delay of good trade is not pos sible, so that the benefits of trading some lesser bit for b_i is lost forever. Note that there will not be a b ad trade since A uses the BH strategy.

Case 2

 K_A(U_A), K_B(U_B), K_B(U_A) and K_A(U_B) is common kno

wledge.



K

_A

(U

_A

) and K

_B

(U

_B

) are common knowledge.



For A: (BH+DOT)

 K_A(U_B) and K_A(~K_B(U_A))  DOT

 K_A(U_B) and K_A(K_B(U_A))  DOT (Signaling)  K_A(U_B) and P_A(~K_B(U_A))  DOT

 ~K_A(U_B)  BH

Case 3

Proof



No matter what A’s or B’s private kn

owledge is, A’s best strategy is:

BH+DOT.

Information Disclosed



Minimun preference information.



For A,

Bargaining Power

 The design objective of the proposed mediati

on protocol is to provide a fair and constru ctive conflict-resolution mechanism between two agents assuming that they have same barg aining power. This assumption seems to limit the applications of our mediation mechanism at first, but we find that actually most neg otiation cases occur when both negotiating p layers are having same bargaining power. Thi s is because that if one of the players has greater bargaining power, he can force the o ther player to concede in the negotiation.

Bargaining Power (cont.)

 cite{rubinstein:perfectequilibrium1982,fatima:timecons

traints2002} showed that the stronger player will clai m all the pie while the weaker player gets almost noth ing.

 In our scenario, both agents are competing the same ca

se from the same customer. Therefore, they should have a same deadline. However, if one of them has lower cos ts of performing all the tasks in the case, it is cons idered stronger than the one with higher costs. The st ronger agent does not need to negotiate with the weake r agent since there is no doubt that it will win the c ase.

Forming Coalitions

 There has been a significant amount of work in game th

eory and multi-agent systems field in the area of coal ition formation. However, as commented by Banerje \cit e{banerje:partners2002}, almost all of them ignores th e issues of how the coalitions generate their revenues or the nature of the problem solving adopted by indivi dual agents after they form a coalition. This paper, o n the other hand, addresses the problem of how the tas ks and the money can be redistributed fairly and const ructively when (or after) forming a coalition.

Related Work



Although various approaches \cite{zhen

g:learning1997,carmel:learningmodels19

96,gmy:recursivemodel1995} of modeling

the opponent in the negotiation have b

een proposed in game theory and distri

buted artificial intelligence (DAI), t

hey cannot produce a mutually benefici

al negotiation outcome in a one-off mu

lti-issue negotiation.

Related Work (cont.)

 Some research therefore gives up the idea of

modeling the opponent, but try to make an of fer similar to the opponent's directly \cite {faratin:similarity2002,lin:fixedpie2002}, s o as to approximate the opponent's preferenc e structure and generate a mutually benefici al outcome. However, these mechanisms cannot satisfy the requirement of game theoretic ra tionality, since there are no rational strat egies in making concessions in a negotiation with two-sided uncertainty.

Conclusion

 We believe that the idea of creating a fair

and constructive mediation game to resolve t he conflicts in a multi-issue negotiation ma y point out another possible research direct ion.

 The mediator in this game does not necessari

ly exist, because the mediation process is f air and can be verified by both agents. Neve rtheless, some fair random selection service is required to simulate a fair “coin”.

Future Work