Don’t Loot Unless Your Bad (#Clickbait)

LSV posed the following hypothetical:

The Looter Problem

This is my answer.

1) Looting in this situation is functionally equivalent to milling yourself 1 card.

The only difference is if we would keep the card we draw off the loot. Which we would only do if the top card of the library is better than lightning blast.

But if we knew the top card was better than lightning blast, in this situation we would prefer to not loot anyways. So lets consider another less controversial equivalent question:

2) In the middle of a game of limited if you were given the opportunity to mill the top card of your library would you?

2a) Decking
There is a small chance that you deck earlier and this ends up being relevant.

Verdict: Very Small Negative.

2b) Information
Harder to evaluate but I think generally symmetrical information is bad for player 1. The unknowns in your hand are the hardest part for an opponent to play around optimally. Knowing that player 1 has milled their bomb/sweeper/combat trick, is almost certainly going to be more valuable for their opponent than it is for Player 1. In some sense, the most valuable information for you is when some # of outs have been eliminated. But again we have a case where in order for the information to be valuable our deck composition has become worse.

Verdict: Net Small Negative.

2c) Variance

This is the most important factor for the decision. If you choose to mill you are increasing the variance of your draw. At the moment you mill each card has some value to you. And your draw step has the expected value of those cards. But if you remove a card at random you are changing the EV of your draw step. If your remove a good card then you are decreasing it and if you mill a bad card you are increasing it.

However the EV changes of all possible mills is net zero. This should hopefully be intuitive.

So milling the card increases the variance of your draw step (it is now higher or lower EV than before), but that change was net neutral in EV gained. So you have just generated variance with no gain in EV.

Verdict: Negative if adding variance is bad.

3)  So when do want more variance?
As usual we want more variance in the situations where we are behind or if we think our opponent is better than us.

In this particular example, there is no evidence we are currently behind. And I would hope for most people that they think they are better than their random opponent.

Verdict: In the middle of limited game you normally wouldn’t want to mill your top card.

Final: Thus you don’t loot.