I like to think of it this way.

On any given hand, there is a maximum you can lose and a maximum you can gain. The same applies to your opponent.
It's a sliding scale and a fight to swing the balance in your favor.

In your example, you are willing to make a losing bet and a losing call.

In both scenarios you (wrongly) push the balance in favor of your opponent.

Where you gain, is that given the opportunity, Villian didn't take his opportunity to swing the balance EVEN FURTHER in his favor.

So as explained by Xhad. Whether you bet and your opponent calls. Or you check\call. Still has exactly the same result on the sliding scale.

EXCEPT the bet is worse because you give your opponent a chance to swing the balance even further in his favor.

Saying that this is your gain is slightly misleading. It's villian's loss. You gain by default but the balance doesn't shift in your favor.

Another example.
Kwazzie's fundamental theorum of Murder.

Choice 1. Having my balls cut off then shot dead OR
Choice 2. Being shot dead then having my balls cut off.

I take Option 2 and I can say I'm better for it. But I haven't gained.

What your suggesting to do in your example, is to cut your own balls off then give your murderer the choice.