Consider the following type of circumstance: you have only a few big blinds left and want to shove soon, but you know you will get multiple callers.

Do you have a better chance of winning with two middle cards, ideally suited and connected, than a weakish King or Ace?


The thinking being that although the King or Ace with a middle kicker might be a better hand in a vacuum, you are more likely to be going against Aces and Kings with better kickers in an all in hand with multiple players than 6s, 7s, 8s or 9s. So your best chance is to have unique cards at the flop and hope the board comes in low and you have the only pair. It also gives you more ability to hit a straight.

Is this crazy?

Use an equity calculator and find out? Guessing high cards are better than low cards even against a range that contains hands that dominate yours a lot.

Also holding high cards mean you block some of the hands villain's will call with meaning you get slightly more folds.

It depends a lot on players calling ranges.

Don't forget that holding an ace acts as a blocker against others calling with one which can't be solved for in an equity calculator. The king has a much smaller effect in terms of blockers or being dominated.

I believe the blocker/high card value is going to benefit you greater than the odds of being dominated will hurt you but it's not that straight forward a problem to solve.

I think it's inherently solved in range-based calculators like pokerstove and propokertools. When I wrote a similar simulate, it definitely was, because each iteration I'd randomly select a hand from the possible hands. If there were cards dead then the number of Ax hands for your opponent to select from would be lower in comparison to the other hands in the range

Card removal is accounted for when calculating equity but they can't account for the increased odds of players behind folding when the all-in player is holding an ace blocker.

The tighter the calling range, the more holding an ace would reduce the odds of a caller since the percentage of hands in a tighter range that include an ace would also go up.

The problem is difficult to answer in the abstract due to the issue of opponents' unknown calling ranges. So there have been two general approaches followed.

First, you can assume that everyone behind you calls in which case the problem becomes a straightforward equity calculation.

Second, you can assume "optimal" play by your opponents and create/consult push-fold charts.

Under the first approach of N-player all-in equities, you will find that K7o is preferred to 87o when there are "few" opponents and 87o is preferred to K7o when there are "many" opponents. I believe the switch-over point may be going from 6 to 7 opponents though I am not positive.

Under the second approach of push-fold charts with few big blinds, the push-fold charts in Kill Everyone recommends shoving K7o from button or small blind with only 3 big blinds and no antes but only shoving 87o from the small blind.

I've run a few sample cases just to get an idea of how they do for boundary conditions to test some worst case ranges against A7.

Example: I ran EV for one tight call where A7 was completely dominated without being ahead of any hand and 87 was live vs all unpaired hands:
88+, A9s+, ATo+, KQs

Despite the bias against A7 the blocker pretty much nullified the domination factor. Blocker turned 25% of the unpaired range and half of the AA combos into folds so A7 EV > 87 EV vs any single opponent range which I'd speculate is the most common scenario except for the push is so small, which shouldn't happen very often.

Anytime you get more than one caller, the blocker stops being nearly as effective since blind steels are no longer possible using the blocker..

I'm struggling over how to weight the blocker effect over two different ranges right now, I suppose it just involves weighting it proportional to the Ace/range ratios.

Also finding a way to weight the likelihood of any scenario over another is a real pain I thought I could find a shortcut for this but I keep coming back to creating 5 hit/miss ranges.

I tried some other shortcuts with some success but it's getting to be more work than it's worth and I'm limited on time. Meh.

For a real result I think you'd need to test every permutation of outcomes which is just not realistic.

I think push bot charts always just assumed 0-1 calls which is a reasonable estimate in most cases unless players are ganging up or your very short as in this case.

caro did some work on this, ranking hands for heads up and multiway, in his book on hold'em.