Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets

Es el título de nuestro paper en las Jornadas que organiza el \(im^2\) (Instituto de Matemática Multidisciplinar) de la UPV: Mathematical Models for Addictive Behaviour, Medicine & Engineering. El tema es el uso de redes de consenso para alcanzar acuerdos de forma descentralizada, aplicado en concreto a problemas de gestión de recursos hídricos. A continuación te dejo el resumen (en inglés) y las trasparencias de la presentación. En cuanto esté publicado dejaré también la referencia completa al artículo y, si puedo por temas de licencia, el enlace.


The aim of this paper is to present a way of share opinions in a decentralized way by a set of agents that try to achieve an agreement by means of a Consensus Network, allowing them to know beforehand if there is possibilities to achieve such an agreement or not.
The theoretical framework for solving consensus problems in dynamic networks of agents was formally introduced by Olfati-Saber and Murray (2004). The interaction topology of the agents is represented using directed graphs and a consensus means to reach an agreement regarding a certain quantity of interest that depends on the state of all agents in the network. This value represents the variable of interest in our problem.

A consensus network is a dynamic system that evolves in time. Consensus of complete network is reached if and only if \(x_i = x_j \forall i, j\). Has  been de demonstrated that a convergent and distributed consensus algorithm in discrete-time can be written as follows:

\(x_i(k+1)=x_i(k) + \varepsilon \sum_{j \in N_i} a_{ij}(x_j(k)-x_i(k))\)

where \(N_i\) denotes the set formed by all nodes connected to the node i (neighbors of i). The collective dynamics of the network for this algorithm can be written as \(x(k+1)=Px(k)\), where \(P=I-\varepsilon L\) is the Perron matrix of a graph with parameter \(\varepsilon\). The algorithm converges to the average (or other functions) of the initial values of the state of each agent and allows computing the average for very large networks via local communication with their neighbors on a graph.
The convergence of this method depends on the topology of the network and its convergence is usually exponential. But sometimes it not needed to reach a final agreement on a concrete value. This proposal uses consensus networks to determine if an agreement is possible among a set of entities. Agents can leave the agreement if its parameters are out of the expected bounds, so the consensus network can be used to detect the candidate agents to be members of the final agreement. All this process is solved in a self-organized way and each individual agent decides to belong or not to the final solution.
To show the validity of the present approach, a water market is presented as case of study. The water market is a case of complex social-ecological system (SES), where centralized and hierarchical approaches trend to fail and self-organized solutions seems to be more sustainable in the long term (Ostrom, 2009). In general, agreements related to natural resource management involve very complex negotiations among agents. Water demands and regulation is a very complex distributed domain appropriated for MAS.
An important question is if this kind of markets requires some regulation or not. From an exclusively economic point of view the dominant strategy for agents in deregulated markets is not cooperative because each agent wants to maximize exclusively his payoff, and therefore they are not interested in the global and socially efficiency of the natural resources.