# 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

Relatived view of Decision theory:

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

See also: causal decision theory

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

- Acts are
- 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 $y$

- Defines a partial ordering of outcomes
- Utility (numerical value) using an interval scale

- Can be described using

Decision tables

- Art: providing a good formalization of a decision into a table
- 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”

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