Backward induction

Abstract: Demand for electricity varies throughout the day, increasing the average cost of power supply. Time-of-use (TOU) pricing has been proposed as a demand-side management (DSM) method to influence user demands. In this paper, we describe a game-theoretic approach to optimize TOU pricing strategies (GT-TOU). We propose models of costs to utility companies arising from user demand fluctuations, and models of user satisfaction with the difference between the nominal demand and the actual consumption. We design utility functions for the company and the users, and obtain a Nash equilibrium using backward induction. In addition to a single-user-type scenario, we also consider a scenario with multiple types of users, each of whom responds differently to time-dependent prices. Numerical examples show that our method is effective in leveling the user demand by setting optimal TOU prices, potentially decreasing costs for the utility companies, and increasing user benefits. An increase in social welfare measure indicates improved market efficiency through TOU pricing.

Abstract: Public blockchain networks using proof of work (PoW)-based consensus protocols are considered as a promising platform for decentralized resource management with financial incentive mechanisms. In order to maintain a secured, universal state of the blockchain, PoW-based consensus protocols financially incentivize the nodes in the network to compete for the privilege of block generation through cryptographic puzzle solving. For rational consensus nodes, i.e., miners with limited local computational resources, offloading the computation load for PoW to the cloud/fog providers (CFPs) becomes a viable option. In this paper, we study the interaction between the CFPs and the miners in a PoW-based blockchain network using a game theoretic approach. In particular, we propose a lightweight infrastructure of the PoW-based blockchains, where the computation-intensive part of the consensus process is offloaded to the cloud/fog. We formulate the computation resource management in the blockchain consensus process as a two-stage Stackelberg game, where the profit of the CFP and the utilities of the individual miners are jointly optimized. In the first stage of the game, the CFP sets the price of offered computing resource. In the second stage, the miners decide on the amount of service to purchase accordingly. We apply backward induction to analyze the subgame perfect equilibria in each stage for both uniform and discriminatory pricing schemes. For uniform pricing where the same price applies to all miners, the uniqueness of the Stackelberg equilibrium is validated by identifying the best response strategies of the miners. For discriminatory pricing where the different prices are applied, the uniqueness of the Stackelberg equilibrium is proved by capitalizing on the variational inequality theory. Further, the real experimental results are employed to justify our proposed model.