Agents with Nonlinear Utility Function

1. Many to Many Matching of Agents
with Nonlinear Utility Function
Nagoya institute of technology
Takayuki Ito, Moustafa laboratory
QIAO SENSEN, Takayuki Ito, Ahmed Moustafa, Shun Okuhara
(Many to Many Matching of Agents with Nonlinear Utility Function)

2. Introduction
Many to many matching problem
The result of negotiation between two agent sets is the problem of finding the best combination.
Many to many negotiation problem has many applications in the real world.
・Matching application finds the best pair of men and women.
・Transportation problems optimize how much product is shipped from
a factory to a store.
・Negotiations conducted by people are also matching of offers proposed by buyers and sellers from their respective utility
spaces.
Fig.1： many to many matching of Sellers and Buyers
Existing studies do not consider the interdependence of issues.
Seller １ Buyer ２
Buyer １
Seller ２
Buyer ４
Buyer ３
1

3. Multi-issues negotiation problem
Negotiations are an important social activity for conflict resolution and consensus building for mutual benefit
multi-issue negotiation problem :It is a negotiation problem with multiple issues.
– Independent issue: Utility can be expressed as a linear function.
– Interdependence issues: Utility can be expressed by a non-linear function
that requires multi-objective optimization
Utility function using constraint expression
– Constraints express the range of utility for each issue [1]
Non-linear utility space
– The utility space using the constraint expression
becomes an uneven nonlinear space.
[1] Hiromitsu HATTORI, Takayuki ITO, and Mark KLEIN ，and M. Klein, “An Auction-Based Negotiation Protocol for Agents with Nonlinear Utility Functions”,
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, J89-D, No. 12, pp.2648-2660, 2006(in Japanese).
0
20
40
60
80
100
Isuue１
Isuue2
Practical negotiation matching needs to consider complex dependencies between issues.
Utility
2
Fig.2： Non-linear utility space

4. research object
Task (Issue)
• Given the complex utility space, it is very difficult to find the best combination in
many-to-many negotiation matching.
Proposed method
• The agent presents preferences (bid) to a mediator and the mediator finds an
agreement combination that maximizes social utility.
Proposal of negotiation matching protocol based on complex utility space
3

5. Related research ｜ Particle Swarm Optimization and Kernel Density Estimator in Concurrent Negotiations
Kostas et al. [2] study a simultaneous negotiation of one buyer and multiple sellers.
The buyer and the seller aim to reach an agreement by repeating the proposal and response based on the proposal response
protocol. In this paper, particle swarm optimization is used to search for the optimal solution. Buyer’s method is to change the
negotiation strategy for each seller based on the optimal solution.
The above research assumes independence between issues but does not consider complex utility spaces.
[2]Kostas Kolomvatsos, Stathes Hadjiefthymiades. On the Use of Particle Swarm Optimization and Kernel Density Estimator in Concurrent Negotiations[J].
Information Sciences 262.November 2013.
Offer１
Offer２
Offer３
seller 1
seller 2
seller 3
buyer
N
N
N
Fig.３：Negotiation using particle swarm optimization.
Negotiation matching method that assumes realistic and complicated utility space is required
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6. Proposed method |Proposal of negotiation matching protocol based on mediator
• Mediator accepts bids and matches efficiently.
Agent
d
Agent
c
Agent
a
Agent
b
Group A Group B
mediator
a’s bid c’s bid
b’s bid d’s bid
Fig4：Many to Many matching
5

7. Approach ｜ Matching process
agent A
agent B
agent C
agent D
agent a
agent b
agent c
agent d
The mediator calculates the negotiation result of all pairs for each agent bid.
A B C D
a 50 40 X 60
b 60 X X 20
c 80 110 20 50
d 90 X 60 70
For example
・Agent A and Agent a can negotiate and agree to get 50 social benefits
・Agent B and Agent b cannot agree ….
6
Fig5：Matching
Fig6： Utility value in the matrix

8. Approach｜step1 Searching local optimal solution by SA
Searching by Simulated Annealing
(SA)
[step1] Agents explore local
optimal solutions in their
nonlinear utility spaces by SA.
ContractsUtility
local optimum solution
[1] Hiromitsu HATTORI, Takayuki ITO, and Mark KLEIN ，and M. Klein, “An Auction-Based Negotiation Protocol for Agents with Nonlinear Utility Functions”,
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, J89-D, No. 12, pp.2648-2660, 2006(in Japanese). 7
Fig7： Search high utility point

9. Approach｜step2 Generating bids
The agent calculates the utility value, which is the total utility of the constraints satisfied by the agreement.
[step2] The mediator set the number of bids. The agent selects bids with a high utility value and submit it to the mediator.
𝑢𝑖 s =
𝑐 𝑘
∈𝐶, 𝑠∈𝑥(𝑐𝑘)
𝑤𝑖(𝑐𝑘, 𝑠)
s : agreement
𝑢𝑖： Utility
𝑐 𝑘： constraint
𝑥(𝑐𝑘)：A set of consensus proposals that can satisfy constraint 𝑐 𝑘
𝑤𝑖(𝑐𝑘, 𝑠)：Utility value when constraint 𝑐 𝑘 is satisfied by agreement s
[1] Hiromitsu HATTORI, Takayuki ITO, and Mark KLEIN ，and M. Klein, “An Auction-Based Negotiation Protocol for Agents with Nonlinear Utility Functions”,
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, J89-D, No. 12, pp.2648-2660, 2006(in Japanese).
Contracts
Utility
1
43 2
8
Fig8： Draft agreements

10. Approach｜step3 Deciding consensus
[step3]
• Each agent submits a set of bids.
• The mediator checks the overlap between the bid sets,
and if there is an overlap, the matching is successful.
• For every pair, the mediator check the overlap and store
its utility value in the matrix.
Fig9： Consensus decision
Contracts
UtilityUtility
Agent１
Agent２
A B C D
a 50 40 X 60
b 60 X X 20
c 80 110 20 50
d 90 X 60 70
9
Fig.10 : Sample of utility value in the matrix

11. Overview of evaluation experiment
Experimental settings
The constraint is generated as follows
(Issue 1, Issue 2, Issue 3) = ([3,6], [2,9], [3,8])
The constraints are single constraints, binary constraints, triple constraints ...
Agent_num 4, 6,8,10,12,14,16
Issue_num 3,4
Issue_value_num [ 0, 9]
Constraints_num 10
Bids_num [ 0, 100 ]
Table 1: Parameter values
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12. Result
Experimental results with 3 issues
The Social welfare is increasing with the increase in the number of bids, and the common part of the value of the
issue can be searched.
0
1000
2000
3000
4000
5000
6000
7000
8000
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Relationship between number of agent and
social welfare
agent_num_4
agent_num_6
agent_num_8
agent_num_10
agent_num_12
agent_num_14
agent_num_16
Number of bids: Number of bids submitted by one agent
Socialwelfare
11

13. Result
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
10 20 30 40 50 60 70 80 90
Relationship between the number of agents and social utility
agent_num_4
agent_num_6
agent_num_8
agent_num_10
agent_num_12
agent_num_14
agent_num_16
Experimental results with 4 issues
Social utility increases as the number of bids increases. It is difficult to agree with a small number of issues.
Number of bids
Socialwelfare
12

14. Experimental result
Results of comparison between the proposed method and full search
3 issues: Approximate solution of optimal solution can be obtained.
4 issues: Utility value has decreased significantly. It is difficult to agree with a small number of bids
0
500
1000
1500
2000
2500
3000
3500
4000
0 10 20 30 40 50 60 70 80 90
Average utility
ALL_average_utility SA_average_utility
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60 70 80 90
Average utility
ALL_average_utility SA_average_utility
Comparison result of issue 3 Comparison result of issue 4
Number of bids
Socialwelfare
Number of bids
Socialwelfare
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15. Conclusion
Proposed method
• Explored local optimal solution in nonlinear utility space by SA
• Set a mediator between each agent to search for a matching combination that maximizes social utility.
Evaluation experiment results
• It is possible to omit some of the information required for bidding to be disclosed to mediators.
• The effectiveness of the proposed method is presented experimentally.
Future work
• Experiment that the proposed method is effective for larger-scale matching problem is essential.
Proposed Matching Protocol with Non-linear Utility Function by Mediator
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