Decision rules when the agent is ignorant of all probabilities

## 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: 1. Avoid dominated acts and prefer dominant acts 2. Can only use dominance principle if states are independent of acts 3. 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

This rule is 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 ($α=0$ is pessimistic, $α=1$ is optimistic): $V_{α}(A)=αmax(A)+(1−α)min(A)$

Objections:

- Requires interval scale instead of ordinal scale

### Minimax regret

Basis: it is 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, $1/n$)

- Assign each of the $n$ states probability $1/n$ and maximize the expected value
- For an act $A$, calculate $∑_{i=1}n1 value(A,S_{i})$

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 $[21 A1,21 A2]$
- Presupposition that the agent has a neutral attitude to risk

- Eliminates Maximin (as above)
- Eliminates Minimax Regret (as above)
- Eliminates Optimism-pessimism rule with $α=21 $
- 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 $[21 A1,21 A2]$
- Independence of Irrelevant Alternatives
- A rational agent’s choice will be invariant under an irrelevant expansion B: if $A1≥A2$ before adding option B, then $A1≥A2$ 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