Plugging the preceding situation into my handy nash calculator it tells me to shove and this was my decision in game (no one called). There are only four players behind me, since one is sitting out, but I do have around 20x the BB and there is only one short stack behind me so I do not think this is cut and dried. One benefit of shoving is that it prevents someone from reraising three betting me. I certainly do not want a hand like 98s to three bet me to 1500 and then be getting just under 2:1 if I shove, unless of course they are three bet folding. There is 600 to win immediately, or 1/8th of my stack and I likely have the best hand most of the time. Also my opponents can and perhaps should fold 55-99 to my shove. The calculator suggests this :
as a calling range for the player immediately after me in the CO. I excluded the high jack from the simulation since he is sitting out. One advantage to raise folding is that I can get away from the hand if it goes raise, reraise behind me, as this will make it pretty likely (though not a lock) that someone behind me has a bigger pair, although with only four people left to act this scenario is somewhat unlikely.
I'm not sure what the cut off is for push / fold decisions, in terms of BB's in your stack. My 19 bb stack looks somewhat smaller if you adjust for antes (12BB stack adjusted). My guess is that this is right on the border of where we can play push fold and where we should simply open for 2x. The tournament is also a knockout, so there is a premium for eliminating players, but I do not think that is a significant factor in this particular situation, with only one player shorter than me still to act. It would typically suggest slightly looser play vs stacks which are shorter than you.
The following chart from Wizard Of Odds gives the probability of a pair being beaten
http://wizardofodds.com/ask-the-wizard/texas-hold-em/probability-pairs/
Probability Pair Beaten by Higher Pair
| PAIR | 2 PL. | 3 PL. | 4 PL. | 5 PL. | 6 PL. | 7 PL. | 8 PL. | 9 PL. | 10 PL. |
|---|
| KK | 0.49% | 0.977% | 1.462% | 1.945% | 2.425% | 2.903% | 3.379% | 3.852% | 4.323% |
| QQ | 0.98% | 1.95% | 2.91% | 3.861% | 4.803% | 5.735% | 6.659% | 7.573% | 8.479% |
| JJ | 1.469% | 2.917% | 4.344% | 5.749% | 7.134% | 8.499% | 9.843% | 11.168% | 12.473% |
| TT | 1.959% | 3.88% | 5.763% | 7.609% | 9.42% | 11.194% | 12.934% | 14.64% | 16.312% |
| 99 | 2.449% | 4.838% | 7.168% | 9.442% | 11.66% | 13.823% | 15.934% | 17.992% | 20.001% |
| 88 | 2.939% | 5.791% | 8.56% | 11.247% | 13.855% | 16.387% | 18.844% | 21.229% | 23.544% |
| 77 | 3.429% | 6.74% | 9.937% | 13.025% | 16.007% | 18.887% | 21.668% | 24.353% | 26.947% |
| 66 | 3.918% | 7.683% | 11.301% | 14.776% | 18.115% | 21.324% | 24.407% | 27.369% | 30.215% |
| 55 | 4.408% | 8.622% | 12.65% | 16.501% | 20.181% | 23.7% | 27.063% | 30.279% | 33.352% |
| 44 | 4.898% | 9.556% | 13.986% | 18.199% | 22.205% | 26.016% | 29.64% | 33.086% | 36.363% |
| 33 | 5.388% | 10.485% | 15.308% | 19.871% | 24.188% | 28.273% | 32.137% | 35.794% | 39.253% |
| 22 | 5.878% | 11.41% | 16.617% | 21.517% | 26.13% | 30.472% | 34.559% | 38.405% | 42.025% |
13.986% is the number for our situation, 44 beaten by 4 players. That makes us a 6:1 favourite to take down the pot and a 1:6 underdog to lose 60% of our stack (our opponent has a 60% edge on us as an 80:20 favourite). Actually that's slightly facile analysis, we also have a 20% share of the current pot, so 60% of our stack but gain 100 from the pot. So 1 time in 6 we lose roughly 3k (slightly less) and 5 times in 6 we win 600. So it's break evenish. Also 1 time in 4 we lose less, vs the sb. and 55-88 might fold.
So shoving is almost certainly +EV. But what about raise folding or raise getting it in? This is a more complicated analysis. If we raise fold, we might get bluffed, or might induce a hand like 8s9s which actually has us beat 52.75% : 47.25% to play. There is also a chance a might three bet fold vs us, a real coup but a fairly unlikely scenario at these stack sizes. One nice approach to taking a minimum raise with this hand is that we can take the same approach with bigger pairs, which we would like. The strength of the nash approach, that you will take it down frequently, becomes a problem when you have a stronger hand like AA. We should take the same approach with AA as we do with 44, whatever we decide, lest we leave ourselves exploitable to our observant opponents. Even if this consideration is lesser in our present scenario it is important to build a consistent strategy for other games where this consideration becomes predominant. We need to build healthy poker habits now.
Our stack size does seem slightly too large for the nash approach. I'm not sure if there is a hard and fast rule for these scenarios. Certainly 10x the BB or less qualifies and 20x the BB seems to large. In between perhaps there is a grey area. This is a regular tournament, not a turbo, and that urges a slightly more patient approach. We could also raise call vs sb and raise fold vs the other players, or raise fold vs co but raise get it in vs other players, especially BB, who has some incentive to push (no players behind him and money already in the pot). I suspect it may be better to simply shove at this stack size and this position but that any deeper and I must abandon Nash but I am certainly unsure of this conclusion.
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