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Learning under Random Updates
Published in Hamidou Tembine, Distributed Strategic Learning for Wireless Engineers, 2018
There are different models of stochastic games. Stochastic games in the sense of Shapley, stochastic differential games, stochastic difference games etc. Here we present the first type. A stochastic game, introduced by Lloyd Shapley in the 1950s [144], is a dynamic game with probabilistic transitions played by one or several players. The game is played in a sequence of runs. At the beginning of each step (time slot), the game is in some state (which can be unknown to the players). The players select actions, and each player receives a payoff that depends on the current state and the chosen actions. The game then moves to a new random state whose distribution depends on the previous state and the actions chosen by the players (Markovian model). However, the strategies do not need to be Markovian. The procedure is repeated at the new state and play continues for a finite or infinite number of stages. The total payoff, discounted payoff or the limit inferior of the averages of the stage payoffs to a player are often considered.
Optimum risk/reward sharing framework to incentivize integrated project delivery adoption
Published in Construction Management and Economics, 2023
Qiuwen Ma, Sai On Cheung, Shan Li
Most of the allocation approaches assume that the cost or revenue to be shared is deterministic. However, not all eventualities and their risks can be foreseen so that respective allocations can be made ex ante. Considering the allocation solution for uncertain outcomes, stochastic cooperative game theory (Suijs 2012) was applied in this study. The most appealing value of stochastic cooperative game is that it can explicitly include the risk preferences of decision-makers. Any response to risks, from taking to avoiding, are manifestations of these preferences (Suijs 2003, 2012). In a stochastic game, the random benefits can be represented by a deterministic number, also known as the certainty equivalent. Using the concept of certainty equivalent, a stochastic game can be reduced to a game with deterministic outcomes. The “core” in the stochastic cooperative game therefore refers to the allocations that the certainty equivalent of the future random benefit that every coalitional member perceives is at least as much as the member can obtain on their own or by joining any other coalition. With the capability to capture the individual’s risk preferences and approach stochastic outcomes, the stochastic approach embedded in a stochastic game (Suijs 2012) is used in this study.
Enhancing cyber-physical security in manufacturing through game-theoretic analysis
Published in Cyber-Physical Systems, 2018
Zach DeSmit, Aditya U. Kulkarni, Christian Wernz
Stochastic games were originally developed by Shapley [66] as a generalisation of repeated games with perfect information. Shapley developed a framework where the players would play a unique game given the state of the world. A stochastic game permits the description of multiple states of the world, and for each state of the world, there is a unique game with potentially unique strategies for the players. Consider the following stochastic game extension of the attacker–defender game that has been discussed in this article. The state of the world is defined by the demand for the products of the cyber-physical manufacturing company. There are two possible states: (1) high, where the demand for the products is high; and (2) low, where the demand for the products is low. When the state of the world is high, the cyber-physical manufacturing system is used heavily, and the defender is allocated a large budget for securing the cyber-physical system. When the state of the world is low, the cyber-physical system is used sparingly, and the defender must do without any additional funds. The state of the world also affects the attacker. If the state is high, then the cyber-physical system is online more frequently, and the attacker has many chances to compromise the system. While when the state of the world is low, then attacker has to either monitor the system all the time to determine when the system goes online or choose not to attack. In this manner, stochastic games can combine multiple game scenarios into one compact analytical framework. Dynamic programming is used to determine equilibrium strategies in stochastic games [67].