Decision theory

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. Sometimes called evidential decision theory.

See also: causal decision theory

e.g. Pascal’s Wager

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