Given that you receive the letter, paying is indeed evidence for not having termites and winning $999,000. EDT is elegant, but still can’t be correct in my view. I wish it were, and have attempted to “fix” it.
My take is this. Either you have the termite infestation, or you don’t.
Say you do. Then
being a “payer” means you never receive the letter, as both conditions are false. As you don’t receive the letter, you don’t actually pay, and lose the $1,000,000 in damages.
being a “non-payer” means you get the letter, and you don’t pay. You lose $1,000,000.
Say you don’t. Then
payer: you get the letter, pay $1,000. You lose $1,000.
non-payer: you don’t get the letter, and don’t pay $1,000. You lose nothing.
Being a payer has the same result when you do have the termites, but is worse when you don’t. So overall, it’s worse. Being a payer or a non-payer only influences whether or not you get the letter, and this view is more coherent with the intuition that you can’t possibly influence whether or not you have a termite infestation.
“In your problem description you said you receive the letter”
True, but the problem description also specifies subjunctive dependence between the agent and the predictor. When the predictor made her prediction the letter isn’t yet sent. So the agent’s decision influences whether or not she gets the letter.
“This intuition is actually false for perfect predictors.”
I agree (and have written extensively on the subject). But it’s the prediction the agent influences, not the presence of the termite infestation.