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# Decision theory

Last updated Sep 7, 2022 Edit Source

Choices by one agent in which background conditions are independent of what other agents are doing. We usually represent these decisions with a decision matrix or decision table

Relatived view of Decision theory:

Prescriptions in decision theory are always relative to the formal problem specifications

e.g. Pascal’s Wager

God exists God does not exist
Believe Infinite reward Status quo
Don’t believe Infinite loss Status quo

Components:

• Rows are possible acts
• Acts are functions that map states to outcomes
• Columns are possible states of the world
• Probabilities are sometimes included for decisions under risk.
• Should not depend on agent action
• States should be
• Mutually exclusive
• Exhaustive: no possibility is left out
• Relevant partition: distinctions that actually have impact on probability or utility of outcomes
• Independence: (optional) each state should be causally and probabilistically independent of the acts
• Dominance principle only holds if independent holds
• Cells are outcomes.
• Can be described using
• Verbal description
• Preference ranking on an ordinal scale
• Defines a partial ordering of outcomes
• $x \succcurlyeq y$ is a weak preference
• $x \succ y$ is a strong preference
• $x \sim y$ is indifference between $x$ and $y$
• Utility (numerical value) using an interval scale

Decision tables

1. Art: providing a good formalization of a decision into a table
2. providing a justified recommendation based off of the formalization

They can also be represented using decision trees

We can transform decision tables between each other using “reasonable transformations”

1. PIR: assign equal probabilities to all states (if we have no knowledge of probabilities, aka DUI)
2. Merger: if two states yield identical columns, then we can merge them into one state and add probabilities if we know them