Multi-Agent Systems

Negotiation

Shari Naik

Negotiation

Inter-agent cooperationConflict resolutionAgents communicate respective desiresCompromise to mutually beneficial agreement

Negotiation in Cooperative domains

Jefferey Rosenschein Gilad Zlatkin

Domains

Distributed problem solving Distributed but centrally designed AI systems Global problem to solve

Multiagent systems Distributed, with different designers Agents working for different goals Task Oriented State Oriented Worth Oriented

Task Oriented Domain

Non-conflicting jobsNegotiation : Redistribute tasks to everyone’s mutual benefitExample - Postmen domain

State Oriented DomainGoals are acceptable final statesHave side effects - agent doing one action might hinder or help another agentNegotiation : develop joint plans and schedules for the agents, to help and not hinder other agentsExample – Slotted blocks world

Worth Oriented DomainRates the acceptability of final statesNegotiation : a joint plan, schedules, and goal relaxation. May reach a state that might be a little worse that the ultimate objectiveExample – Multi-agent Tile world

Task Oriented Domain

Tuple <T, A, c>T - set of tasks,A – List of agentsC - cost function from any set of tasks to a real numberEncounter(goal) - a list, T1, … Tn, of finite sets of tasks from the task set T, such that each agent needs to achieve all the tasks in its set.

Building blocks

Precise specification of the domainNegotiation protocolNegotiation strategy

Assumptions Expected Utility Maximizer Complete Knowledge No History Commitments are Verifiable

Domain Definitions Graph (City Map) G = G(V,E)

v V => nodes (address / Post office) e => edges (roads)

Weight function (Distance of road) W : EIN

Letters for agent A : LA Agent Li : I Letters (LA LB) =

Cost(L) IN => weight of minimum weight cycle that starts at PO and visits all vertices of L and ends at PO

Definitions

Deal – Division of LAULB to two disjoint subsets, (DA,DB) such that

DAUDB= LAULB DADB=

Utility – Difference between the cost of achieving his goal alone and the cost of his part of the deal

Utilityi(DA,DB) = Cost(Li) – Cost(Di)

Properties of a Deal ()

Individual rational {A,B}, Utilityi() >= 0

Pareto optimal – there does not exist another deal such that

Negotiation set – set of deals that are individual rational and pareto optimal

() – Product of the two agent utilities from

Negotiation ProtocolA product maximizing ngotiation protocol

One step protocol Concession protocol

At t >= 0, A offers (A,t) and B offers (B,t), such that Both deals are from the negotiation set i andt >0, Utilityi((i,t)) <= Utilityi((i,t-1))

Negotiation ending Conflict - Utilityi((i,t)) = Utilityi((i,t-1)) Agreement, j !=i Utilityj((i,t)) >= Utilityj((j,t))

Only A => agree (B,t) Only B => agree (A,t) Both A,B => agree (k,t) such that ((k))=max{((A)),((B))} Both A,B and ((A))=((B)) => flip a coin

Pure deals

Mixeddeal

Negotiation StrategiesHow an agent should act given a set of rules.

Definition – Function from the history of the negotiation to the current message

Risk - an indication of how much an agent is willing to risk a conflict by sticking to its last offer

Risk(A,t) = Utility, A loses accepting B’s offer Utility, A loses by causing a conflict

Risk Loss

Rational Negotiation Stratergy – At any step t+1, A sticks to his last offer if, Risk(A,t) > Risk(B,t)

Negotiation Strategies Cont

Zeuthen Strategy – Start – A offers B the minimal offer

UtilityB((A,1)) = minNS{UtilityB() }

Next - A will make a minimal sufficient concession at step t+1 iff Risk(A,t)<=Risk(B,t)

If both agents follow the above stratergy, they will agree on a deal NS, such that (*)=maxNS {()}

Equilibrium

A negotiation strategy s will be in equilibrium if under the assumption that A uses s, B prefers s to any other strategy

Zeuthen strategy is not in equilibrium

Mixed deal

Element of probability – Agents will perform (DA,DB) with probability p or (DA,DB) with probability 1-pCosti([(DA,DB):p]) = pCost(Di) + (1-p)Cost(Dj)

Utilityi([:p]) = Cost(Li) – Costi([:p])

All or nothing deal – 0<=p<=1 such that mixed deal m = [({LA,LB}, ):p] NS (m) = maxNS(d)

Incomplete InformationG and w – common knowledgei knows Li, not Lj : j!=I

Solution Exchange missing information Penalty for lie

Possible lies False information

Hiding letters Phantom letters

Not carry out a commitment

Hidden lettersUtility of A

Expected(on telling the truth) = 4 Pure deal – [(,] = 6 Mixed deal - [(,] = 33/4

Phantom lettersUtility of A

Expected(on telling the truth) = 3 Pure deal – [(,] = 4 Mixed deal – possibility of being caught (all or

nothing deal)

Subadditive Task Oriented Domain

the cost of the union of tasks is less than or equal to the sum of the costs of the separate setsfor finite X,Y in T, c(X U Y) <= c(X) + c(Y)).Example of non additive TOD

Incentive compatible Mechanism

L lying is beneficialT Honesty is betterT/P Lying can be beneficial, but chances of being caught

Concave Task Oriented Domain

We have 2 tasks X and Y, where X is a subset of YAnother set of task Z is introduced

c(X U Z) - c(X) >= c(Y U Z) - c(Y).

Modular TOD

c(X U Y) = c(X) + c(Y) 2 c(X Y).

Multi Agent Compromise via Negotiation

Katia Sycara

Negotiation process for conflicting goals

Identify potential interactionsModify intentions to avoid harmful interactions or create cooperative situations

Techniques required Representing and maintaining belief models Reasoning about other agents beliefs Influencing other agents intentions and beliefs

PERSUADER

Program to resolve problems in labor relations domainAgents

Company Union Mediator

Tasks Generation of proposal Generation of counter proposal based on feedback

from dissenting party Persuasive argumentation

Negotiation Methods

Case based ReasoningPreference analysis

Case Based Reasoning

Uses past negotiation experiences as guides to present negotiationProcess

Retrieve appropriate precedent cases from memory Select the most appropriate case Construct and appropriate solution Evaluate solution for applicability to current case Modify the solution appropriately

Case Based Reasoning

Cases organized and retrieved according to conceptual similarities.Advantages

Minimizes need for information exchange Avoids problems by reasoning from past failures.

Intentional reminding. Repair for past failure is used. Reduces computation.

Preference Analysis

From scratch planning methodBased on multi attribute utility theoryGets a overall utility curve out of individual ones.Expresses the tradeoffs an agent is willing to make.Property of the proposed compromise

Maximizes joint payoff Minimizes payoff difference

Persuasive argumentation

Argumentation goals Ways that an agents beliefs and behaviors can be

affected by an argument

Increasing payoff Change importance attached to an issue Changing utility value of an issue

Narrowing differences

Gets feed back from rejecting party Objectionable issues Reason for rejection Importance attached to issues

Increases payoff of rejecting party by greater amount than reducing payoff for agreed parties.

Experiments

Without Memory – 30% more proposalsWithout argumentation – lesser proposals and better solutionsNo failure avoidance – more proposals with objectionsNo preference analysis – Oscillatory conditionNo feedback – communication overhead by 23%

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