Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets

Outline Consensus Networks Water distribution Conclusions

Consensus Networks as Agreement Mechanismfor Autonomous Agents in Water Markets

M. Rebollo, A. Palomares and C. Carrascosa

Dept. Sistemas Informáticos y ComputaciónUniv. Politécnica de Valencia (Spain)

Math. Models of Addictive Behav., Medicine & EngineeringValencia, September 2010

M. Rebollo et al. DSIC-UPV

Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets


Water Distribution Problem

Motivation

Water management is a complex task

centralised solutions trend to fail: low implication of users

WUA valid for small and medium domains

pure market solutions result in unfair distribution

agreements related with natural resources involve complexnegotiations

Social-ecological systems (SES) suggest self-organized solutions asthe most sustainable in the long term (Ostrom, 2009)




Our Proposal

The challenge

Design a procedure that allows a set of self-organised agents toachieve agreements

What is needed. . .

to obtain a theoretical model of agreement

to define protocol to achieve agreements by consensus

to design a self-regulated system to deal with waterdistribution problems




Outline

1 Outline

2 Consensus Networks as Agreement Mechanism

3 Water Distribution as a Consensus Problem

4 Conclusions and Future Work




Consensus Networks as Agreement Mechanism

Consensus networks

Let (G ,X ) be the state of a network with value X and topology G ,where X = (x1, . . . , xn) ∈ R

n, where xi is a real value associatedwith the node Ei .

a b c

d e f g

h i





Theoretical Model (Olfaty, 2004)

The consensus problem can be formulated as:1

xi(k + 1) = xi(k) + ε∑

j∈Ni

(aij(xj(k) − xi(k))),

The collective dynamics of the network for this algorithm can bewritten as

x(k + 1) = Px(k)

where P = I − εL is the Perron matrix of a graph with parameter ε

1Agents with discrete-time model





Simple Consensus Protocol with Initiator

Initiator: Facilitator Participant-i

request

not-understood

refuse

inform-disagree

inform-agree

consensus value

calculation

n

n

n

Participant-j

inform-value

inform-value

k

k'





Consensus Protocol for Agreement Spaces

But sometimes we do not need to know a common value

just the existence of a possible consensus is needed

definition of an agreement space

So the process can be interrupted when some conditions are met

deliberation time is over

one agent leaves the process

a percentage of agents leave the network

a threshold has been reached





Additional considerations

weighted agents: weights in consensus networks canrepresent concepts as reputation or trust, so the most relevantagents can have higher importance in the consensus processand they can influence the final consensus value.

stubborn agents: if an agent does not change the value forthe dimension the final value of the consensus clearly convergeto this value, distorting the result.

the behavior of stubborn agents can be used to create somekind of decentralized control (for example, fulfillment ofnorms)




Water Distribution as a Consensus Problem

Problem Description

The market consist of n entities (agents) Ei , i = 1, . . . , n

Ei = {Ri , R̃i ,Pi , P̃i}

where

Ri rights that Ei owns

R̃i rights that entity Ei desires

Pi initial price proposed byEi

P̃i upper/lower price bound for Ei .2 This parameter is private

2It depends on been a buyer or a seller





Problem Dynamics

PSi (k + 1) = PS

i (k) + ε∑

j∈Ni

(Bj(PBj (k)− PS

i (k))),

PBi (k + 1) = PB

i (k) + ε∑

j∈Ni

(Sj(PSj (k)− PB

i (k)))

where the index S and B denotes seller and buyers respectively.Agents will disconnect from the network if

PSi (k) < P̃S

i for seller agents.

PBi (k) > P̃B

i for buyer agents.





Model Reformulation

Detected problem: convergence of water rights

PSi (k + 1) = PS

i (k) + ε∑

j∈Ni

(Bj(PBj (k) − PS

i (k) + Ci(k))),

PBi (k + 1) = PB

i (k) + ε∑

j∈Ni

(Sj(PSj (k) − PB

i (k) + Ci (k)))

where the added term Ci(k) is proportional to rights bought andsold by agents in each iteration, and is calculated as follows:

Ci (k) = δ ·

∑j∈Ni

Bj∑j∈Ni

Sj

where δ > 0. In this experiment the algorithm converges and stopswhen the mean prices of buyers and sellers are approximately equal.





Experiments Desgin

Parameter Exp. 1 Exp. 2 Exp. 3

All

n 2000 2000 2000Rmax 4 4 4RT 4000 4000 4000

R̃max 4 4 4

R̃T 4000 4000 4000ε 0.01 0.01 0.01δ 0 1 1

SellersP̃

S

10 10 10σS 0.2 0.2 0.2F S 1.25 1.25 1.25

BuyersλB 0.5 0.5 0.5

P̃B

12 12 6

F B 0.9 0.9 0.9

Table: Parameters used in the experiments.





Experiment 1: full connected, fixed topology





Experiment 2: scale free α = 2.5, fixed topology





Experiment 3: full connected, switching topology, unbiased

rights





Experiment 4:full connected, switching topology, biased

rights





Experiment 5: scale free α = 2.5, switching topology,

unbiased rights






biased rights




Conclusions and Future Work

What we have done

test theoretical consensus model

design a protocol that allow intelligent agents to achieveagreements based on consensus

model a self-regulated, water rights ’market’




Conclusions and Future Work

Future Work

multidimensional

time delay

re-entry of agents

group identification

study the impact of other network models



Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets

Technology

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