Bilateral Multi-Issue Negotiation

1. Bilateral Multi-Issue Negotiation Speaker: Raymund J. Lin, D85725004

2. 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.......

3. 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。

4. 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

5. 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。

6. Negotiation • When people are trying to resolve conflicts between several parties, they negotiate. • single-issue two-party problems • multi-issue multi-party

7. Bilateral Multi-Issue Negotiation • Multi-issue, two party • Multi-issue negotiations are considered integrative \cite{raiffa:negotiationart1982}, where 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 increases, which means that people need more time and rationale in handling the negotiation problem.

8. Negotiation Support Systems • The development of Negotiation Support Systems (NSSs) and negotiating software agents (NSAs) have been proved to be able to reduce significantly the negotiation time and alleviate the negative effects of human cognitive biases and limitations\cite{lomuscio:negotiationclassification2001} \cite{sandholm:negotiationcomponent2000} \cite{maes:agent1999} \cite{foroughi:negotiationprocess1998}.

9. Negotiation Analysis • decision analysis • game theory • Negotiation analysis \cite{sebenius:NA92} is used to generate prescriptive advice to the supported party given a descriptive assessment of the opposing parties. \cite{Kersten:modeling2001}

10. Uncertainty • Problems can arise if assessment of the opposing parties are not available or vague. • For one-sided uncertainty, there is a protocol where the uninformed agent makes all the offers and the informed agent either accepts or rejects offers \cite{vincent:bargaining1989}. • Alternatively the uninformed agent can try to model the opponent using a Bayesian network or an influence diagram \cite{vassileva:bilateralnegotiation2002}.

11. 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 other players when he moves, or at the end nodes. • Complete • Nature does not move first, or her initial move is observed by every player.

12. 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 but certain game, where there is a two-sided uncertainty about the preferences of the opponents who are bargaining.

13. Two-sided Uncertainty • The problem of two-sided uncertainty can be addressed using recursive modeling \cite{gmy:recursivemodel1995}. Nevertheless, for complex multi-issue negotiations, it could be computationally intractable.

14. Incomplete Information • The solution to the bilateral negotiation problem of incomplete information is addressed in the literature by giving a continuous distribution or discrete probabilities over the other agent's \emph{type} and using Bayesian rule to learn the type during the negotiation. The negotiation protocol used is sequential alternating protocol (SAP) \cite{rubinstein:perfectequilibrium1982}.

15. Sequential Equilibra • For single issue negotiation that negotiates on price, the type of the other agent is represented by its reservation price. These distributions are common knowledge. As a result, there is no subgame-perfect equilibrium, which requires that the predicted solution to a game be a Nash equilibrium in every subgame. Rubinstein analyzed it using a stronger equilibrium concept of sequential equilibra \cite{rubinstein:perfectequilibrium1982}.

16. Sequential Equilibra cont. • It requires that each uncertain player's belief be specified given every possible history, and Bayes rules are used to make beliefs consistent. However, if the other agent's behavior deviates from the equilibrium path, an update problem may occur since non-equilibrium paths are assigned zero probability. • This may result in incentives for agents to deviate from the equilibrium, so as to increase the number of possible outcomes \cite{faratin:automatednegotiation2000}.

17. Analysis Difficulties • Derivation from equilibrium behavior cannot be ruled out in games via SAP with two-sided uncertainty. The situation could be worse for multi-issue negotiation, since types of agents 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.

18. Random Selection Process ? • In the decision making behaviors of human, people rely on random selection processes, such as flipping a coin, to handle a decision that involves too much uncertainty and subsequently it becomes difficult for them to rationally judge a decision. However, considering the problem of computation complexity, the question resides in the possibility for agents participating in a multi-issue negotiation with two-sided uncertainty to simply flip a coin to decide on every issue.

19. A Mediation Game • As commented in \cite{raiffa:negotiationart1982}, it is neither feasible nor logical to do so. Flipping a coin on every issue will not generate mutually beneficial outcomes. We propose a mediation protocol that is based the Single Negotiation Text (SNT) device suggested by Roger Fisher \cite{fisher:mediation1978}. This protocol presents a deal construction game to both protagonists, where they actively participate in this construction process to find a mutually beneficial agreement.

20. A Scenario • let us say that there are two companies A and B, which have control over different resources, such that they do same jobs with different costs. There is a case that some customer 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 negotiate a plan of task sharing, the profit can increase. However, how they can divide the set of tasks and the money at the same time fairly without disclosing too much confidential information, given that they may have to compete with each other in another case, becomes the main challenge.

21. Assumptions • Expected Utility Maximizer • One-off Negotiation • Asymmetric Abilities • Inter-independent Issues • Two-sided Uncertainty • Inter-agent Comparison of Utility

22. 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 an initial, single negotiating text---a text to be criticized by both sides and then modified in an iterative manner. Modifications to the SNT-1 will be made by the mediator based on the criticisms of the two sides.

23. Fairness • The SNT technique is to be used as a means of focusing the attention of both sides on the same composite text. The important thing is that this process appear to be fair to both sides, not divisive.

24. SNT-1 • SNT-1 can be generated by locating a converging point from a “dance”of packages (see Figure~\ref{fig:ju}) or a focal point (for example, the mid-value on each continuous factor). When trying to generate the SNT-1, both agents must know that they are not haggling about a final contract, but a starting point for the pursuit of joint gains.

26. Iteration • After the SNT-1 has been located, both protagonists then try to improve it simultaneously or by taking turns. • The mediator will take both protagonists' criticisms or suggestions into consideration, and generate a new version of SNT for further revisions. This process continues until all the issues are settled.

27. The Challenge • The SNT technique had been applied by U.S. on the mediation of Egyptian-Israeli conflict in early 1977, known as Camp David Negotiations. 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 that will encourage more honest revelations and less strategic behavior?

28. Def – deal • A deal D is represented by a binary string in A's view: • h b1,b2,…,bni = h c1,c2,…,cp,r1,r2,… ,rqi • ci represents a bit that will result in negative utility (cost) for A when false (ci = 0), while ri is a bit that will generate positive utility (revenue) for A when true (ci = 1).

29. Def – deal cont. • The money earned is mapped to the ri bits. Each ri represents a bag of money used for exchange. As suggested in cite[p. 216]{raiffa:negotiationart1982}, the issues to be negotiated should be in comparable magnitude of importance, so that protagonists might then agree to resolve each issue separately by the toss of a coin. Therefore we let the amount of a bag of money be:

30. Def – profits • The total cost (TC) and total revenue (TR) of a deal D is: • TCA(D)=1· j· p CostA(cj)£ (1-cj). • TRA(D)=1· j· q RevenueA(rj)£ rj. • TCB(D)=1· j· p CostB(cj)£ (cj). • TRB(D)=1· j· q RevenueB(rj)£ (1-rj). • The profit gained is: • Profiti(D) = TRi(D) - TCi(D). • For each agent, a rational deal D must satisfy Profit(D) > 0.

31. Def – utility • The utility of a deal D is: • Utilityi(D) = Profiti(D) + 1· j· p Costi(cj). • The utility generated by a set of bits O=C[R, where C is a set of cost bits and R is a set of revenue bits, is:

32. Game • Players – two players in our scenario • Actions • Information • Strategies • Payoff • Outcome • Equilibrium

33. Sequtial Match

34. Fairness & Constructiveness • This mediation process can produce a fair and constructive outcome if both agents do choose their preferred bits sequentially. • For a fair outcome, we mean that each agent will have a probability 0.5 of getting the utility from a bit if they are faced with a direct conflict. Otherwise, they can trade their less preferred bits for more preferred bits to get a constructive outcome, which means that both agents get higher final utility than flipping a coin on every bit.

35. Strategic Move • However, an agent can strategically choose a less preferred bit firstly to increase its final utility.

36. The DOT Strategy • We named it the Delay of Trades (DOT) strategy. (4-DOT, 3-DOT)

37. 3-DOT • 4-DOT • O2(Conflict  Win) xg. O3(Conflict  Lose) • 3-DOT • O2(Lose  Conflict) xg. O3(Conflict  Lose)

38. Note • In the DOT abstraction, when we say U(O1) ¸ U(O2), we mean that: • For Every bit bj2 O1 and every bit bk2 O2 • U(bj) ¸ U(bk) • Also: • #(O1) = #(O2)

39. Utility Gain in the DOT Strategy • 4-DOT • From: • To: • Gain:

40. Utility Gain in the DOT Strategy • 3-DOT • From: • To: • Gain:

41. Success Condition of DOT • For A to success: • A must know B’s utility function • B must report his preference honestly (B does not know A’s utility function)

42. The Mediation Protocol

43. The Binary Match Game • Since A and B may partially agree on the some of the issues, some part (bits) in SNT-2 will be fixed, which means that these bits can not be further modified in the next stage. A and B then concentrate on the resolution of the remaining issues until all issues are settled.

45. Trades • The bits bi and bj (i<> j) is said to be traded if one of them is assigned a value 1 (in A's favor) and the other is assigned a value 0 (in B's favor). • A good trade occurs when a less preferred bit is traded for a more preferred bit from both agents' view. On the contrary, a bad trade may occur when a more preferred bit is traded for a less preferred. • If a trade is beneficial to only one of them, but indifferent to the other, we call it a single-interest trade.

46. Negotiation Strategy • A negotiation strategy is a function from the history of the negotiation to the current action (bits relocation) that is consistent with the mediation protocol.

47. The BH Strategy • Since the SNT-1 is randomly traded, it is fair but not constructive. A and B may have different ideas on how the 1-bits and 0-bits should be relocated. • According to A's utility function, there exists a better half ½, such that #()=d#()/2e and 8 bi2, 8 bj2bar{}, UA(bi)>=UA(bj). If 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.

48. Further Trades • If both agents use the BH strategy to relocate 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 into two divisions  (better half) and  (lesser half), marked as B and L in Figure. • Now both  and  are treated as sets of conflicting bits to be resolved separately (enforcing trades within each division).

49. Final Match • When there are two bits left in SNT-t, both agents must submit their preferences over the combinations of h 1,0i and h 0,1i. If they agree on the same combination, the mediation is done. Otherwise, the mediator must flip a coin to decide that which combination is selected. We named this mediation protocol the Match-Game Mediation (MGM) Protocol.

50. Pareto Optimal • We say that a deal D dominates D0, and write DÂ D0, if and only if (UtilityA(D), UtilityB(D))À(UtilityA(D0), UtilityB(D0)) (it is better for at least one agent and not worse for the other). • A deal D is called pareto optimal if there does not exist another deal D0 such that D0Â D.