Modeling Complex Multi-Issue Negotiations Using Utility Graphs

TFG - MARA, Budapest, September 2005 1

Modeling Complex Multi-Issue Negotiations Using Utility Graphs

Valentin Robu, Koye Somefun, Han La Poutré

CWI, Center for Mathematics and Computer Science, Amsterdam, The Netherlands


Multi-issue (multi-item) negotiation

• Negotiation = method of competitive (or partially cooperative) allocation of goods, resources, tasks between agents

• Applications:• E-commerce: Bundling can be an effective method to increase

sales (use in recommender systems)

• High degree of customization – possible through negotiations

• Logistics: mechanism for task allocation

• Many deals are negotiated bilaterally or in closed groups of companies (e.g. transportation contracts)

• Utility functions are not (or partially) revealed => indirect revelation mechanism

• Search with incomplete information


Utility functions for multi-issue negotiations

• Linearly additive: • Linear combination of issue utilities:• Search space is structured -> more accesible to heuristics

[Faratin Sierra & Jennings. 2002], [Jonker & Robu 2004], [Coehoorn & Jennings 2004] [Gerding & La Poutre, 2004]

• “Auction-type”: XOR of ANDs • K-additive:

• Captures local substitutability/complementarity effects between k issues

• Finding optimal allocation can become hard even for the 2-additive case

• Exiting solutions: assume a trusted mediator, computationally expensive (3000-5000 bids for 50 issues)

• [Klein, Faratin, Sayama & Bar-Yam, 2003] [Lin 2004]

iiB UwU *


Utility graphs: basic ideas

• Inspiration: probabilistic graphical models• Each node = one issue under negotiation (or item in

a bundle)• Nodes grouped into clusters of connected nodes• Cost of representation

• Exponential in size of the cluster• Linear in the number of clusters

• Use in negotiation• Opponent modelling: seller maintains & updates a model

of buyer’s preferences


Utility graphs: an example

• Global utility is a sum of utility over clusters, rather than individual issues

• Buyer - cluster potentials:

u(I1) = $7, u(I2) = $5, u(I3) = $0

u(I4) = $0, u(I1, I2)= - $5,

u(I2, I3)=$4, u(I2, I4)=$4• Seller - all items have cost $2.

uBUYER(I1=1, I2=0, I3=1, I4=0) = $7

Gains from Trade = Buyer_utility – Seller_Cost

Optimal combination?

GT(I1=0, I2=1, I3=1, I4=1)=$13 - 3*$2 = $7


Utility graphs: Use in negotiation

• Bundles with maximal G.T. Pareto-optimal bundles [Somefun, Klos & La Poutré 2004]

• Seller keeps a model of the utility graph of the buyer and aims for a bundle with maximal GT

• After each counter-offer, he updates this model (true graph of the buyer remains hidden)

• Seller knows a super-graph of possible buyer utility graphs (qualitative assumption)


Partitioning a utility graph

• Q: How to select the bundle with a maximal GT, with respect to a utility graph learned so far?

• A1 (Brute force answer): generate all possible bundles and select the best one.

• Complexity for 50 issues: 250 > 1015 bundles• A2: Partition the graph into sub-graphs• Nodes belonging to more than 1 subgraph = cutset nodes• For all possible instantiations of cutset nodes, compute local

sub-bundle combination• Merge them, such that a local optimum is achieved


Partitioning a utility graph (2)

• Complexity of exploring all bundles: 2c * (2p + 2q)• Partitions can be found in polynomial time (always for

graphs of tree-width 2)


Learning in utility graphs (1)

• Seller has a super-graph for possible interdependencies in the buyer population

• This graph contains tables for each cluster, with size 2 at the power of size of the cluster

• Initial values = proportional to the Hamming distance

Values are adjusted as follows:

))(1(*)()( ,, icucu biibii

, for the combination induced from buyer’s bid

, for all other combinations))(1(*)()( icucu ii

bic ,


Learning: a simple example

• Two complementary issues: I1 and I2

I1 I2 time t t+1 t+20 0 0 0 0

0 1 $7 $8.4 $10

1 0 $5 $4 $3.2

1 1 $17 $13.6 $10.9

Buyer asks, for several rounds: I1=0, I2=1

This combination gets updated with (1+α), the

others with (1-α)

• Supposing costs are c(I1)=c(I2)=$3, α=0.2 the bundle with maximal GT changes from (1,1) to (0,1) after 2 steps


Learning in utility graphs (2)

• The cluster update factor is clique-specific:

• |C| = total number of cliques; α, β = learning parameters

• Where the clique Gains from Trade Ratio is defined as ratio of “local” (per clique) vs. total (bundle-wide) GT:

• We adjust the model more towards the other’s value for clusters which are less important, and less for the others

|)|/1)((var 1

1*)(

CiGTRfixed ei

)(

)()( ,

b

bii

bGT

cGTiGTR


Experimental validation: set-up

• Graph with 50 issues, 28 clusters: 3 of size 4, 16 of size 3, 6 of size 2, 3 of size 1

• Costs and strength of interdependencies: drawn from a independent, normal distributions (i.i.d-s): • Means around 1*(Hamming Distance)• Spreads between 0 and 5 • => highly non-linear search space

• Results averaged for 100 tests/configuration


Experimental results


Negotiation part: Conclusions

• It is possible to reach Pareto-efficient outcomes reasonably fast, by exploiting the decomposable structure of utility functions

• Consequence:• We can handle complex negotiations even in time

constrained domains / with buyer impatience • Assumption: A structure of the super-graph for the

population of likely buyers• Solution: collaborative filtering past negotiation data


Structure of the initial utility graph

• Preferences of buyers are in some way clustered • Class (population) of buyers with similar preference

structures => largely overlapping utility graphs• Can we estimate which items can be potentially

complementary/substitutable by looking at previous buying patterns?

• Collaborative filtering asks the same questions !• Not all relationships hold for all users – only a

super-graph of these relationships is required


Architecture & simulation model view


Collaborative filtering: Overview

• Output recommendations to buyers, based on previous buy instances

• User-based: for each user, select a neighbourhood of users with a similar preferences

• Item-based: identify relationships between items, based on previous buying patterns

• In our case, recommendation step is completely replaced by negotiation => more customization possible


Step 1: Data preparation

Items

Previous negotiations

I1 I2 IK... I50

Neg. 1 0 1 1 0

Neg. 2

…

1

…

1

…

0

…

1

…

Neg. N(eg. N=2000)

1 1 0 0

Negotiation outcomes matrix

Item

pairs

I1 I2 IK... I50

I1 N 134 … 220

I2 134 N … …

IK … … … …

I50 220 … … N

•1-1 pairs: Ni,j(1,1)

•1-0 pairs: Ni,j(0,1)

•0-1 pairs: Ni,j(1,0)

•0-0 pairs: Ni,j(0,0)

Total no. buys

(out of N)

N1(1) N2(1) NK(1).. N50(1)

260 130 … 50

4 Item-item matrixes


Criteria 1: Cosine-based similarity

• Measure of distance between the buying vectors for two items i, j

• Intuitive, but not so precise• Complementarity effect:

• Substitutability effect:

)1()1(

)1,1(),( ,

ji

jicompl

NN

NjiSim

)1()1(

)0,1()1,0(),( ,,

ji

jijicompl

NN

NNjiSim


Criteria 2: Correlation-based similarity

• Average buys per item:

• Similarity between items i and j:

N

NiAv i )1()(

)1)(1)(1,1()1)(0,1(

)1()1,0()0,0(

,,

,,1

jijijiji

jijijiji

AvAvNAvAvN

AvAvNAvAvN

N

NN

N

NN jjii)1()0()1()0(

2

2

1),(

jiSim


Results: Correlation-based similarity


Conclusions & discussion

• Utility graphs efficient way to guide online learning of buyer preferences in electronic negotiations

• Learning a starting structure of these graphs – possible through collaborative filtering

• By combining the two techniques => relatively short negotiations (around 20 steps/50 issues)

• Intuition: we explicitly utilize the clustering effect between utility functions of typical buyers

• Personalization techniques used in collaborative filtering can be successfully combined with personalization through agent-mediated negotiation


Questions

• Thank you very much for your attention!

• Full paper(s) available from:

• homepages.cwi.nl/~robu

http://www.cwi.nl/~robu

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Modeling Complex Multi-Issue Negotiations Using Utility Graphs

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