Ihor Kendiukhov

The strongest case for independence.

Let's steelman the argument for the independence axiom.

The best argument does not come from raw intuition, of course irrelevant alternatives shouldn't matter, but, in my view, from a 1988 result by Peter Hammond, and it goes like this.

Consider an agent facing a decision that unfolds over time, in stages.

At stage 1, some uncertainty is resolved, say, a coin is flipped.

Depending on the result, the agent proceeds to stage 2, where they must choose between options.

Before any uncertainty is resolved, the agent can form a plan, if the coin comes up heads, I will do X. If tails, I will do Y.

Hammond showed that if you accept two properties of sequential decision-making, then you are logically forced to satisfy the independent axiom.

The first property is dynamic consistency.

Whatever plan you make before the uncertainty is resolved, you actually follow through on once you arrive at the decision node.

Your ex-antiplan and your ex-post-choice agree.

The second property is consequentialism in the decision-theoretic sense, not the ethical one.

When you arrive at a decision node, your choice depends only on what is still possible from that node forward.

If you accept both properties and you violate independence, you can be money-pumped.

Here is how it works, concretely.

Suppose your preference between gambles A and B depends on what the common component C is, as the independence axiom says it shouldn't.

Before the uncertainty resolves, you evaluate the compound lottery holistically and prefer the plan involving B because, in combination with the C branch, B produces a better overall distribution.

But then the coin comes up heads, the C branch is now off the table, and you find yourself choosing between A and B in isolation.

Consequentialism says you should evaluate based on what's still possible.

And in isolation, you prefer A.