# Decisions under ignorance (DUI)

Decision rules when the agent is ignorant of all probabilities

## # Background Points

- No information about probabilities
- A value function on outcomes a&s where a is an act and s is a state
- Most of the rules respect dominance

## # Rules

- Dominance
- Weak Dominance: act $a$ is as good or better than $b$ for each possible state and there is at least one state where it is strictly better
- Strong Dominance: act $a$ is strictly better than $b$ for all possible states
- Principle:
- Avoid dominated acts and prefer dominant acts
- Can only use dominance principle if states are independent of acts
- Gold standard, use this whenever possible

- Maximin/Leximin
- Find the minimum value of each act
- Choose the act with the least bad worst-case outcome
- Note: can violate dominance if there are rows with the same minimum
- Leximin can help resolves ties by removing the minimum value in case of ties. Note that this violates dominance!
- Leximin* only strikes out a
*single*minimum value in case of ties. This does*not*violate dominance - Extremely conservative, avoids the worst-case scenario

- Optimism/Pessimism and “Best Average”
- Maximax (pick the best of the best-case outcomes)
- Best Average: take the best of each row and worst of each row and average it, pick the act with best average
- Optimism/Pessimism: Uses a weighted average (a linear combination) of the minimum and maximum values ($\alpha = 0$ is pessimistic, $\alpha = 1$ is optimistic): $V_\alpha(A) = \alpha \max(A) + (1-\alpha) \min(A)$
- Objections
- Requires interval scale instead of ordinal scale

- Minimax regret
- Irrational to reject an act with a chance of a great gain, where the cost is slight.
- Regret value for each outcome = value of the outcome - maximum value in that column
- Max regret for each act is the most negative regret for each row A
- Choose the act with the minimum max regret
- Objections
- Requires interval scale instead of ordinal scale
- Adding irrelevant alternative acts potentially affects recommended acts

- Principle of Insufficient Reason (PIR)
- If there are n possible states and you have no reason to believe any of them more likely than any other, then it is rational to assign each state equal probability (namely, 1/n)
- So, we assign each of the $n$ states probability $1/n$ and maximize the expected value
- For an act $A$, calculate $\sum_{i=1}^n \frac{1}{n} value(A, S_i)$
- This turns the problem into a DUR
- Objections
- Requires interval scale instead of ordinal scale
- Arbitrary partitions of states (can result in incoherence)
- Doesn’t apply outside games of chance (e.g. Pascal’s Wager)

## # Rationality Constraints

Find criteria that any rational decision rule should satisfy. Use these to rule out one or more decision principles

Milnor proposes a few axioms for rules under DUI:

- Mixture condition (randomization)
- If a rational agent is indifferent between A1 and A2, then the agent must be indifferent between A1, A2 and the mixed strategy $[\frac 1 2 A1, \frac 1 2 A2]$
- Presupposition that the agent has a neutral attitude to risk

- Eliminates Maximin (as above)
- Eliminates Minimax Regret (as above)
- Eliminates Optimism-pessimism rule with $\alpha \neq \frac 1 2$
- Eliminates Best Avg if we allow other mixtures
- Only PIR survives

- If a rational agent is indifferent between A1 and A2, then the agent must be indifferent between A1, A2 and the mixed strategy $[\frac 1 2 A1, \frac 1 2 A2]$
- Independence of Irrelevant Alternatives
- A rational agent’s choice will be invariant under an irrelevant expansion B: if $A1 \geq A2$ before adding option B, then $A1 \geq A2$ after adding B.
- Eliminates Minimax Regret

Perhaps we should use different rules for DUI in different situations. Can we be systematic?

For decisions under partial ignorance, see: Precautionary Principle