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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