We present the first combinatorial polynomial time algorithm for computing the equilibrium of the Arrow-Debreu market model with linear utilities.
All algorithms mentioned so far are centralized in the sense that the algorithms need to know all the utilities at the start of the algorithm and, in the case of iterative algorithms, the global state of the market, i.e., all prices and the demand for each good. Cole and Fleischer @cite_4 and Cheung, Cole, and Rastogi @cite_13 explore local algorithms, where each agent only knows the price and demand of his good and, moreover, the market is run for many periods.
@cite_13 @cite_4
1970021516 136334786 2192253623 1760997490
Why might markets tend toward and remain near equilibrium prices? In an effort to shed light on this question from an algorithmic perspective, this paper formalizes the setting of Ongoing Markets, by contrast with the classic market scenario, which we term One-Time Markets. The Ongoing Market allows trade at non-equilibrium prices, and, as its name suggests, continues over time. As such, it appears to be a more plausible model of actual markets. For both market settings, this paper defines and analyzes variants of a simple tatonnement algorithm that differs from previous algorithms that have been subject to asymptotic analysis in three significant respects: the price update for a good depends only on the price, demand, and supply for that good, and on no other information; the price update for each good occurs distributively and asynchronously; the algorithms work (and the analyses hold) from an arbitrary starting point. Our algorithm introduces a new and natural update rule. We show that this update rule leads to fast convergence toward equilibrium prices in a broad class of markets that satisfy the weak gross substitutes property. These are the first analyses for computationally and informationally distributed algorithms that demonstrate polynomial convergence. Our analysis identifies three parameters characterizing the markets, which govern the rate of convergence of our protocols. These parameters are, broadly speaking: 1. A bound on the fractional rate of change of demand for each good with respect to fractional changes in its price. 2. A bound on the fractional rate of change of demand for each good with respect to fractional changes in wealth. 3. The closeness of the market to a Fisher market (a market with buyers starting with money alone). We give two types of protocols. The first type assumes global knowledge of only (an upper bound on) the first parameter. For this protocol, we also provide a matching lower bound in terms of these parameters for the One-Time Market. Our second protocol, which is analyzed for the One-Time Market alone, assumes no global knowledge whatsoever. We design new algorithms for the problem of allocating uncertain flexible, and multi-unit demand online given uncertain supply, in order to maximise social welfare. The algorithms can be seen as extensions of the expectation and consensus algorithms from the domain of online scheduling. The problem is especially relevant to the future smart grid, where uncertain output from renewable generators and conventional supply need to be integrated and matched to flexible, non-preemptive demand. To deal with uncertain supply and demand, the algorithms generate multiple scenarios which can then be solved offline. Furthermore, we use a novel method of reweighting the scenarios based on their likelihood whenever new information about supply becomes available. An additional improvement allows the selection of multiple non-preemptive jobs at the same time. Finally, our main contribution is a novel online mechanism based on these extensions, where it is in the agents' best interest to truthfully reveal their preferences. The experimental evaluation of the extended algorithms and different variants of the mechanism show that both achieve more than 85 of the offline optimal economic efficiency. Importantly, the mechanism yields comparable efficiency, while, in contrast to the algorithms, it allows for strategic agents. A key challenge to create a sustainable and energy-efficient society is in making consumer demand adaptive to energy supply, especially renewable supply. In this paper, we propose a partially-centralized organization of consumers, namely, a consumer cooperative for purchasing electricity from the market. We propose a novel multiagent coordination algorithm to shape the energy consumption of the cooperative. In the cooperative, a central coordinator buys the electricity for the whole group and consumers make their own consumption decisions based on their private consumption constraints and preferences. To coordinate individual consumers under incomplete information, we propose an iterative algorithm in which a virtual price signal is sent by the coordinator to induce consumers to shift demand. We prove that our algorithm converges to the central optimal solution. Additionally we analyze the convergence rate of the algorithm via simulations on randomly generated instances. The results indicate scalability with respect to the number of agents and consumption slots. A market-based algorithm is presented which autonomously apportions complex tasks to multiple cooperating agents giving each agent the motivation of improving performance of the whole system. A specific model, called “The Hayek Machine” is proposed and tested on a simulated Blocks World (BW) planning problem. Hayek learns to solve more complex BW problems than any previous learning algorithm. Given intermediate reward and simple features, it has learned to efficiently solve arbitrary BW problems. The Hayek Machine can also be seen as a model of evolutionary economics.