Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets

Outline Consensus Networks Water distribution Conclusions

Consensus Networks as Agreement Mechanism
for Autonomous Agents in Water Markets

M. Rebollo, A. Palomares and C. Carrascosa

Dept. Sistemas Informáticos y Computación
Univ. Politécnica de Valencia (Spain)

Math. Models of Addictive Behav., Medicine & Engineering
Valencia, 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 complex
negotiations
Social-ecological systems (SES) suggest self-organized solutions as
the most sustainable in the long term (Ostrom, 2009)



Our Proposal

The challenge
Design a procedure that allows a set of self-organised agents to
achieve 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 water
distribution problems



Outline

1 Outline

2 Consensus Networks as Agreement Mechanism

3 Water Distribution as a Consensus Problem

4 Conclusions and Future Work




Consensus networks
Let (G, X ) be the state of a network with value X and topology G,
where X = (x1 , . . . , xn ) ∈ Rn , where xi is a real value associated
with 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 (aij (xj (k) − xi (k))),
j∈Ni

The collective dynamics of the network for this algorithm can be
written as
x(k + 1) = Px(k)
where P = I − εL is the Perron matrix of a graph with parameter ε

1
Agents with discrete-time model



Simple Consensus Protocol with Initiator

Initiator: Facilitator Participant-i Participant-j

request n
inform-value k
consensus
not-understood value
n inform-value calculation

k'
refuse

inform-agree
n

inform-disagree




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 can
represent concepts as reputation or trust, so the most relevant
agents can have higher importance in the consensus process
and they can influence the final consensus value.
stubborn agents: if an agent does not change the value for
the dimension the final value of the consensus clearly converge
to this value, distorting the result.
the behavior of stubborn agents can be used to create some
kind of decentralized control (for example, fulfillment of
norms)



Water Distribution as a Consensus Problem

Problem Description

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

Ei = {Ri , Ri , Pi , Pi }

where
Ri rights that Ei owns
Ri rights that entity Ei desires
Pi initial price proposed byEi
Pi upper/lower price bound for Ei .2 This parameter is private

2
It depends on been a buyer or a seller



Problem Dynamics

PiS (k + 1) = PiS (k) + ε (Bj (PjB (k) − PiS (k))),
j∈Ni

PiB (k + 1) = PiB (k) +ε (Sj (PjS (k) − PiB (k)))
j∈Ni

where the index S and B denotes seller and buyers respectively.
Agents will disconnect from the network if
PiS (k) < PiS for seller agents.
PiB (k) > PiB for buyer agents.




Model Reformulation
Detected problem: convergence of water rights

PiS (k + 1) = PiS (k) + ε (Bj (PjB (k) − PiS (k) + Ci (k))),
j∈Ni

PiB (k + 1) = PiB (k) +ε (Sj (PjS (k) − PiB (k) + Ci (k)))
j∈Ni

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

j∈Ni Bj
Ci (k) = δ ·
j∈Ni Sj

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



Experiments Desgin
Parameter Exp. 1 Exp. 2 Exp. 3
n 2000 2000 2000
Rmax 4 4 4
RT 4000 4000 4000
Rmax 4 4 4
All RT 4000 4000 4000
ε 0.01 0.01 0.01
δ 0 1 1
S
P 10 10 10
Sellers σS 0.2 0.2 0.2
FS 1.25 1.25 1.25
λB 0.5 0.5 0.5
Buyers B
P 12 12 6
M. Rebollo et al. FB 0.9 0.9 0.9 DSIC-UPV



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 achieve
agreements 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

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Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets