Optimal Budget Allocation in Social Networks: Quality or Seeding

In this paper, we study a strategic model of marketing and product consumption in social networks. We consider two competing firms in a market providing two substitutable products with preset qualities. Agents choose their consumptions following a myopic best response dynamics which results in a local, linear update for the consumptions. At some point in time, firms receive a limited budget which they can use to trigger a larger consumption of their products in the network. Firms have to decide between marginally improving the quality of their products and giving free offers to a chosen set of agents in the network in order to better facilitate spreading their products. We derive a simple threshold rule for the optimal allocation of the budget and describe the resulting Nash equilibrium. It is shown that the optimal allocation of the budget depends on the entire distribution of centralities in the network, quality of products and the model parameters. In particular, we show that in a graph with a higher number of agents with centralities above a certain threshold, firms spend more budget on seeding in the optimal allocation. Furthermore, if seeding budget is nonzero for a balanced graph, it will also be nonzero for any other graph, and if seeding budget is zero for a star graph, it will be zero for any other graph too. We also show that firms allocate more budget to quality improvement when their qualities are close, in order to distance themselves from the rival firm. However, as the gap between qualities widens, competition in qualities becomes less effective and firms spend more budget on seeding.Comment: 7 page

Connectivity Preservation in Distributed Control of Multi-Agent Systems

The problem of designing bounded distributed connectivity preserving control strategies for multi-agent systems is studied in this work. In distributed control of multi-agent systems, each agent is required to measure some variables of other agents, or a subset of them. Such variables include, for example, relative positions, relative velocities, and headings of the neighboring agents. One of the main assumptions in this type of systems is the connectivity of the corresponding network. Therefore, regardless of the overall objective, the designed control laws should preserve the network connectivity, which is usually a distance-dependent condition. The designed controllers should also be bounded because in practice the actuators of the agents can only handle finite forces or torques. This problem is investigated for two cases of single-integrator agents and unicycles, using a novel class of distributed potential functions. The proposed controllers maintain the connectivity of the agents that are initially in the connectivity range. Therefore, if the network is initially connected, it will remain connected at all times. The results are first developed for a static information flow graph, and then extended to the case of dynamic edge addition. Connectivity preservation for problems involving static leaders is covered as well. The potential functions are chosen to be smooth, resulting in bounded control inputs. These functions are subsequently used to develop connectivity preserving controllers for the consensus and containment problems. Collision avoidance is investigated as another relevant problem, where a bounded distributed swarm aggregation strategy with both connectivity preservation and collision avoidance properties is presented. Simulations are provided throughout the work to support the theoretical findings

Slopey quantizers are locally optimal for Witsenhausen's counterexample

We study the perfect Bayesian equilibria of a leader-follower game of incomplete information. The follower makes a noisy observation of the leader's action (who moves first) and chooses an action minimizing her expected deviation from the leader's action. Knowing this, leader who observes the realization of the state, chooses an action that minimizes her distance to the state of the world and the ex-ante expected deviation from the follower's action. We show the existence of what we call “near piecewise-linear equilibria” when there is strong complementarity between the leader and the follower and the precision of the prior is poor. As a major consequence of this result, we prove local optimality of a class of slopey quantization strategies which had been suspected of being the optimal solution in the past, based on numerical evidence for Witsenhausen's counterexample

Connectivity Preservation in Nonholonomic Multi-Agent Systems: A Bounded Distributed Control Strategy

This technical note is concerned with the connectivity preservation of a group of unicycles using a novel distributed control scheme. The proposed local controllers are bounded, and are capable of maintaining the connectivity of those pairs of agents which are initially within the connectivity range. This means that if the network of agents is initially connected, it will remain connected at all times under this control law. Each local controller is designed in such a way that when an agent is about to lose connectivity with a neighbor, the lowest-order derivative of the agent's position that is neither zero nor perpendicular to the edge connecting the agent to the corresponding neighbor, makes an acute angle with this edge, which is shown to result in shrinking the edge. The proposed methodology is then used to develop bounded connectivity preserving control strategies for the consensus problem as one of the unprecedented contributions of this work. The theoretical results are validated by simulation

Stochastic Opinion Dynamics under Social Pressure in Arbitrary Networks

Social pressure is a key factor affecting the evolution of opinions on networks in many types of settings, pushing people to conform to their neighbors' opinions. To study this, the interacting Polya urn model was introduced by Jadbabaie et al., in which each agent has two kinds of opinion: inherent beliefs, which are hidden from the other agents and fixed; and declared opinions, which are randomly sampled at each step from a distribution which depends on the agent's inherent belief and her neighbors' past declared opinions (the social pressure component), and which is then communicated to their neighbors. Each agent also has a bias parameter denoting her level of resistance to social pressure. At every step, the agents simultaneously update their declared opinions according to their neighbors' aggregate past declared opinions, their inherent beliefs, and their bias parameters. We study the asymptotic behavior of this opinion dynamics model and show that agents' declaration probabilities converge almost surely in the limit using Lyapunov theory and stochastic approximation techniques. We also derive necessary and sufficient conditions for the agents to approach consensus on their declared opinions. Our work provides further insight into the difficulty of inferring the inherent beliefs of agents when they are under social pressure

Estimating True Beliefs from Declared Opinions

Social networks often exert social pressure, causing individuals to adapt their expressed opinions to conform to their peers. An agent in such systems can be modeled as having a (true and unchanging) inherent belief but broadcasts a declared opinion at each time step based on her inherent belief and the past declared opinions of her neighbors. An important question in this setting is parameter estimation: how to disentangle the effects of social pressure to estimate inherent beliefs from declared opinions. To address this, Jadbabaie et al. formulated the interacting P\'olya urn model of opinion dynamics under social pressure and studied it on complete-graph social networks using an aggregate estimator, and found that their estimator converges to the inherent beliefs unless majority pressure pushes the network to consensus. In this work, we study this model on arbitrary networks, providing an estimator which converges to the inherent beliefs even in consensus situations. Finally, we bound the convergence rate of our estimator in both consensus and non-consensus scenarios; to get the bound for consensus scenarios (which converge slower than non-consensus) we additionally found how quickly the system converges to consensus