View
50
Download
0
Category
Preview:
DESCRIPTION
Modeling Complex Multi-Issue Negotiations Using Utility Graphs. Valentin Robu , Koye Somefun, Han La Poutr é CWI, Center for Mathematics and Computer Science, Amsterdam, The Netherlands. Multi-issue (multi-item) negotiation. - PowerPoint PPT Presentation
Citation preview
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
TFG - MARA, Budapest, September 2005 2
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
TFG - MARA, Budapest, September 2005 3
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 *
TFG - MARA, Budapest, September 2005 4
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
TFG - MARA, Budapest, September 2005 5
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
TFG - MARA, Budapest, September 2005 6
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)
TFG - MARA, Budapest, September 2005 7
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
TFG - MARA, Budapest, September 2005 8
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)
TFG - MARA, Budapest, September 2005 9
Learning in utility graphs (1)
• Seller has a super-graph for possible inter- dependencies 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 ,
TFG - MARA, Budapest, September 2005 10
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
TFG - MARA, Budapest, September 2005 11
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
TFG - MARA, Budapest, September 2005 12
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
TFG - MARA, Budapest, September 2005 13
Experimental results
TFG - MARA, Budapest, September 2005 14
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
TFG - MARA, Budapest, September 2005 15
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
TFG - MARA, Budapest, September 2005 16
Architecture & simulation model view
TFG - MARA, Budapest, September 2005 17
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
TFG - MARA, Budapest, September 2005 18
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
TFG - MARA, Budapest, September 2005 20
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
TFG - MARA, Budapest, September 2005 21
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
TFG - MARA, Budapest, September 2005 22
Results: Correlation-based similarity
TFG - MARA, Budapest, September 2005 23
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
TFG - MARA, Budapest, September 2005 24
Questions
• Thank you very much for your attention!
• Full paper(s) available from:
• homepages.cwi.nl/~robu
Recommended