Should you buy insurance at 0 EV?

[Three]

Therefore, the final conclusion should be:
For regular hands, take insurance to reduce fluctuation.
For natural BJ hand, take insurance to reduce fluctuation to non-existence.
Always take insurance when there's 0 EV!

You are failing to account for the strength of the hand. If you have a weak hand like if you have 16vA your hitting EV is -.51715, so you would win 24.1415% of the time and lose 75.8585% of the time assuming no dealer BJ.

So for taking insurance with a 16vA:
1. Lose 50% of wager 2/3 of the time when the dealer doesn't have BJ. In addition to the hands outcome.
2. Lose 100% of wager 1/3 of the time. But win it back with the BJ wager.
Obviously if the dealer has BJ you are going to lose the hand.

So the breakdown becomes:
1) Lose the insurance bet and win the wager for a 1/2 unit win, 16.0943% of the time.
2) Win the insurance wager and lose the main bet for an overall push, 33.3333% of the time.
3) Lose the insurance wager and lose the main bet for a net lose of 1.5 units, 50.5724% of the time.

Compared to not taking 0 EV insurance:
1) A 1 unit win 16.0943% of the time.
2) A 1 unit loss 83.9057% of the time.

Both sides of the comparison have the same EV, but with a weak hand I would prefer to limit the size of the downside to 2/3rds while increasing in frequency of 50% and double the size of the upside while increasing the frequency by 50% by not taking insurance. Most of the time you will lose 1.5 units if you insure this bad hand at 0 EV insurance. That is not an improvement when it comes to the certainty of BR growth with no change in EV.


But for taking insurance for a strong hand like 20vA (EV .65547 win 82.7735% of the time when dealer doesn't have BJ):
So the breakdown becomes:
1) Lose the insurance bet and win the wager for a 1/2 win, 55.1823% of the time.
2) Win the insurance wager and lose the main bet for an overall push, 33.3333% of the time.
3) Lose the insurance wager and lose the main bet for a net lose of 1.5, 11.8785% of the time.

Compared to not taking 0 EV insurance:
1) A 1 unit win 55.1823% of the time.
2) A 1 unit loss 44.8177% of the time.

But with this strong hand you rarely lose money if you insure. When you insure EV is unchanged but the certainty of BR growth is increased. This shows first a bad way to lower variance, by decreasing positive variance more than you decrease negative variance when insuring a bad hand. Versus a good way to lower variance by decreasing negative variance more than you decrease positive variance when insuring strong hands. The extreme example of a strong hand is a BJ. You totally eliminate variance and lock in a 1 unit win for 100% certain BR growth.