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This paper gives a new approach to show the existence and regularity of linear equilibrium established by Lou ⓡ al. (2019) for a noisy rational expectations economy. Different from the existing method which essentially requires to find a fixed point of a system of nonlinear algebraic equations, the new approach is operated directly on an alternative form of market-clearing conditions. One main advantage of the new approach is that besides homogeneous-valuation economies, it can also handle the existence of equilibrium in economies with heterogeneous valuations where the existing method for dealing with homogeneous-valuation economies fails to work.

We study a financial market with asymmetric, multidimensional trader signals that have general correlation structure. Each of a continuum of traders belongs to one of finitely many “information groups.” There is a multidimensional aggregate signal for each group. Each trader observes an idiosyncratic signal about the fundamental, built from this group signal. Correlations across group signals are arbitrary. Several existing models serve as special cases, and new applications become possible. We establish existence and regularity of linear equilibrium, and demonstrate that the equilibrium price aggregates information perfectly as noise trade vanishes.

The aim of this manuscript is to analyze the monotonicity and limit properties of the optimal investment strategy in a behavioral portfolio choice model under cumulative prospect theory over risk aversion coefficient, loss aversion coefficient, and the market opportunity. We show that the optimal investment strategy is nonincreasing of the loss aversion coefficient, and strictly increasing of the Sharpe ratio for normal distributions. The monotonicity properties over risk aversion coefficient depend on the position of the investor and the goodness of the actual and perceived market. The piecewise-linear utility is also discussed. An interesting finding is that when the excess return follows an elliptical distribution, the optimal investment strategy over small mean for piecewise-power and piecewise-linear utility exhibits different limit behavior.

We study the impact of a reference point determined by social comparison on wealth growth and inequality. The reference point of each individual investor contains both personal and social components. Whereas the personal component depends on the investor’s own history of wealth, the social component is determined by the wealth level of other investors in her network. In the benchmark case without social interactions and under the assumption of homogeneous preferences, each investor’s expected wealth grows at a common rate, the wealth gaps widens and the Gini coefficient remains constant. On the other hand, if the reference point is determined solely by social interactions, then it is possible that the network simultaneously experiences high wealth growth and a reduction in inequality. Finally, for the general case where the reference point incorporates both personal and social components and some extensions of the model, we numerically show that increasing the degree of social interactions is beneficial for both increasing wealth growth and reducing inequality.

Multiparty negotiations have drawn much research attention in recent years and an important problem is how to find a Pareto optimal solution or the entire Pareto frontier in a decentralized way. Privacy preservation is also important in negotiation analysis. The main aim of this paper is to find an approximate representation of the Pareto frontier in a decentralized manner and meanwhile, all parties’ privacy can be effectively protected. In this paper, we propose a decentralized discrete-time algorithm based on a weight sum method and the well-known subgradient optimization algorithm, where a mediator works as a coordinator to help negotiators. The proposed algorithm is easily executable, and it only requires the mediator to compute a weighted average of the noisy estimates received from negotiators and negotiators to follow a subgradient optimization iteration at this weighted average. The proposed algorithm can generate an approximate Pareto optimal solution for one particular weight vector and an approximate representation of the Pareto frontier by varying appropriately weight vectors. The approximation error between the obtained approximate representation and the Pareto frontier can be controlled by the number of iterations and the step-size. Moreover, it also reveals that the proposed algorithm is privacy preserving as a result of the random disturbance technique and the weighted average scheme used in this algorithm.

This paper presents a first-order distributed continuous-time algorithm for computing the least-squares solution to a linear equation over networks. Given the uniqueness of the solution, with nonintegrable and diminishing step size, convergence results are provided for fixed graphs. The exact rate of convergence is also established for various types of step size choices falling into that category. For the case where non-unique solutions exist, convergence to one such solution is proved for constantly connected switching graphs with square integrable step size. Validation of the results and illustration of the impact of step size on the convergence speed are made using a few numerical examples.

Recently cumulative prospect theory (CPT) serves as a dominant tool for descriptive decisions under risk and uncertainty which can capture investors' psychology and emotions. In this paper, we consider a two-period consumption-investment decision model based on CPT. The CPT investor needs to balance the allocation of her initial wealth among the current consumption in the first period, the investment on the risky asset and the investment on the risk-free asset in the second period to maximize her total CPT utility. We derive the optimal consumption-investment strategies for the two cases that the difference of the initial wealth and the sum of the two reference points in the two periods is larger than zero and less than zero, respectively. Furthermore, we also present the optimal consumption-investment strategies when the utility is linear.

In this paper, we investigate source localization for wireless sensor networks based on received signal strength. We first formulate the localization problem as the intersection computation of a group of sensing rings, and then convert this non-convex problem into two weighted convex optimization problems. We next propose a unified distributed alternating projection algorithm to solve the resulting weighted optimization problems, where sensor nodes can communicate only locally with their neighbors over a time-varying jointly-connected topology. We also show that sensor nodes' estimates can achieve consensus on a possible minimizer. Both theoretical analysis and some comparative simulations reveal that the proposed approach has good estimation performance in both the consistent and inconsistent cases.

This paper presents a first-order continuous-time distributed step-size algorithm for computing the least squares solution to a linear equation over networks. Given the uniqueness of the solution and nonintegrable step size, the convergence results are provided for fixed graphs. For the nonunique solution and square integrable step size, the convergence is shown for constantly connected switching graphs. We also validate the results and illustrate possible impacts on the convergence speed using a few numerical examples.

In this paper, some privacy-preserving features for distributed subgradient optimization algorithms are considered. Most of the existing distributed algorithms focus mainly on the algorithm design and convergence analysis, but not the protection of agents’ privacy. Privacy is becoming an increasingly important issue in applications involving sensitive information. In this paper, we first show that the distributed subgradient synchronous homogeneous-stepsize algorithm is not privacy preserving in the sense that the malicious agent can asymptotically discover other agents’ subgradients by transmitting untrue estimates to its neighbors. Then a distributed subgradient asynchronous heterogeneous-stepsize projection algorithm is proposed and accordingly its convergence and optimality is established. In contrast to the synchronous homogeneous-stepsize algorithm, in the new algorithm agents make their optimization updates asynchronously with heterogeneous stepsizes. The introduced two mechanisms of projection operation and asynchronous heterogeneous-stepsize optimization can guarantee that agents’ privacy can be effectively protected.

This paper revisits the efficiency of a rational expectations equilibrium model of a competitive market from the perspective of the incentive to social communication. The classical result tells us that the equilibrium price perfectly reveals all dispersed information in the market when the supply of the risky asset is a constant, but does not in the environment of random supply. As opposite to this finding, in this paper we show that for some cases the equilibrium price still perfectly reveals all private information of traders when taking into account the incentive to social communication. Specifically, we show that when the communication cost and the difference between traders' signal precisions are small, traders have the incentive to take the average of all traders' demands, that is, the per-capita supply to improve social welfare and every individual trader's welfare. Knowing the supply information reduces the equilibrium price to be a perfect aggregator again even though the supply of the risky asset is stochastic.

We investigate reference point formation in a social network of multiple investors and study its impact on wealth growth and inequality under a framework of Prospect Theory. The reference point of each individual investor contains both personal and social components. Whereas the personal component depends on the investor’s own history of wealth, the social component is determined by the wealth level of other investors in her network. In the benchmark case without social interactions and under the assumption of homogeneous preferences, each investor’s expected wealth grows at a common rate, the wealth gaps widens and the Gini coefficient remains constant. On the other hand, if the reference point is determined solely by social interactions, then, while the specific behavior depends on the model parameters, it is possible that the network simultaneously experiences high wealth growth and a reduction in inequality. Finally, for the general case where the reference point incorporates both personal and social components, we numerically show that increasing the degree of social interaction is beneficial for both increasing wealth growth and reducing inequality.

In this paper, we study an approximate projected consensus algorithm for a network to cooperatively compute the intersection of convex sets, where each set corresponds to one network node. Instead of assuming exact convex projection that each node can compute, we allow each node to compute an approximate projection with respect to its own set. After receiving the approximate projection information, nodes update their states by weighted averaging with the neighbors over a directed and time-varying communication graph. The approximate projections are related to projection angle errors, which introduces state-dependent disturbance in the iterative algorithm. Projection accuracy conditions are presented for the considered algorithm to converge. The results indicate how much projection accuracy is required to ensure global consensus to a point in the intersection set when the communication graph is uniformly jointly strongly connected. In addition, we show that $\pi/4$ is a critical angle for the error of the projection approximation to ensure the boundedness. Finally, the results are illustrated by simulations.

In this paper, we propose an approximate gradient algorithm for the multi-agent convex optimization problem with constraints. The agents cooperatively compute the minimum of the sum of the local objective functions which are subject to a global inequality constraint and a global constraint set. Instead of each agent can get exact gradient, as discussed in the literature, we only use approximate gradient with some computation or measurement errors. The gradient accuracy conditions are presented to ensure the convergence of the approximate gradient algorithm. Finally, simulation results demonstrate good performance of the approximate algorithm.

In this paper, we propose a random sleep algorithm for a network to cooperatively find a point within the intersection of some convex sets, each of which is known only to a particular node. At each step, each node first chooses to project its own set or not at random by a Bernoulli decision independently. When a node has chosen to project its set, we assume that it can detect only the projection direction rather than the exact projection point, based on which the node obtains an estimate for the projection point. Then the agents update their states by averaging the estimates with their neighbors. Under directed and time-varying communication graph, sufficient and/or necessary stepsize conditions are presented for the considered algorithm converging to a consensus within the intersection set.

In this paper, the target aggregation is investigated for a multi-agent system consisting of second-order agents and multiple leaders. Sufficient conditions are proposed to make the agents approach the target set spanned by these moving leaders. With the condition on switching interconnection topologies, all mobile agents can asymptotically track the dynamical target set specified by multiple moving leaders with bounded error. Moreover, discussion on the case with static leaders is also given.

In this paper, the distributed containment control is considered for a second-order multi-agent system guided by multiple leaders with random switching topologies. The multi-leader control problem is investigated via a combination of convex analysis and stochastic process. The interaction topology between agents is described by a continuous-time irreducible Markov chain. A necessary and sufficient condition is obtained to make all the mobile agents almost surely asymptotically converge to the static convex leader set. Moreover, conditions on the tracking estimation are provided for the convex target set determined by moving multiple leaders.