TU Delft Algorithmics. Intelligent autonomous agents, designed to automate and simplify many aspects of our society, will increasingly be required to also interact with other agents autonomously.
Where agents interact, they are likely to encounter resource constraints. For example, agents managing household appliances to optimize electricity usage might need to share the limited capacity of the distribution grid.
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This thesis describes research into new algorithms for optimizing the behavior of agents operating in constrained environments, when these agents have significant uncertainty about the effects of their actions on their state. Such systems are effectively modeled in a framework of constrained multi-agent Markov decision processes MDPs.
A single-agent MDP model captures the uncertainty in the outcome of the actions chosen by a specific agent. These rewards incorporate the action costs in addition to any prizes or penalties that may be awarded. Negative rewards are called punishments. Indefinite horizon problems can be modeled using a stopping state. A stopping state or absorbing state is a state in which all actions have no effect; that is, when the agent is in that state, all actions immediately return to that state with a zero reward.
Goal achievement can be modeled by having a reward for entering such a stopping state. A Markov decision process can be seen as a Markov chain augmented with actions and rewards or as a decision network extended in time.
Resource-constrained Multi-agent Markov Decision Processes
At each stage, the agent decides which action to perform; the reward and the resulting state depend on both the previous state and the action performed. We only consider stationary models where the state transitions and the rewards do not depend on the time. A Markov decision process or an MDP consists of. S , a set of states of the world. A , a set of actions.
Markov Decision Processes in Practice
A finite part of a Markov decision process can be depicted using a decision network as in Figure 9. Suppose Sam wanted to make an informed decision about whether to party or relax over the weekend. Sam prefers to party, but is worried about getting sick.
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If Sam is healthy and relaxes, Sam will more likely remain healthy. Sam estimates the immediate rewards to be:. Thus, Sam always enjoys partying more than relaxing. However, Sam feels much better overall when healthy, and partying results in being sick more than relaxing does. The problem is to determine what Sam should do each weekend. A grid world is an idealization of a robot in an environment. At each time, the robot is at some location and can move to neighboring locations, collecting rewards and punishments.
Suppose that the actions are stochastic, so that there is a probability distribution over the resulting states given the action and the state.
Figure 9. If the agent carries out one of these actions, it has a 0. If it bumps into the outside wall i.
In each of these states, the agent gets the reward after it carries out an action in that state, not when it enters the state. Note that, in this example, the reward is a function of both the initial state and the final state.
Value Function for MRPs
The agent bumped into the wall, and so received a reward of - 1 , if and only if the agent remains is the same state. Knowing just the initial state and the action, or just the final state and the action, does not provide enough information to infer the reward. As with decision networks , the designer also has to consider what information is available to the agent when it decides what to do.
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There are two common variations:. In a fully observable Markov decision process MDP , the agent gets to observe the current state when deciding what to do. At each time, the agent gets to make some ambiguous and possibly noisy observations that depend on the state.