When three players are in a pot in a tournament situation where one of the players is all in and there is little or no side pot, it is often to the non-all-in players' advantage to just "check it down", that is, to allow the hand to play to the showdown without any further betting.

Certainly, it is very much against a player's interest to bluff here, or to bet with only a draw, or even to bet with a medium-strength hand. In fact, experienced tournament players will often check down even strong hands in this situation, betting only once they have a virtual lock on the nuts. The reasoning is clear:

  • The all in player cannot be bluffed out of the hand; they will see the showdown of the hand regardless of any betting by the other players.
  • If there is no pot except for the main pot (or if the side pot is minuscule in size), there is nothing for a bluffing player to win with a bluff; the all-in player is still in play for the main pot.
  • It is to every player's advantage for a short stacked player to be eliminated: it gets all players one slot closer to the money (or, when in the money, it moves everyone up a notch on the money list).
  • Bluffing or betting with a draw could possibly (even probably) push out a better hand by the other non-all-in player. If doing so pushes out that player but the all-in player ends up winning the main pot, it would be a disaster if the pushed-out player would have won: the bluffer wins nothing, and the shortstack player triples up. If the bluffer had not bluffed, they still would have won nothing, but at least the shortstack would have been eliminated.

An example might help here:

A no-limit hold 'em tournament, with 100/200 blinds.

Player A goes all-in preflop in middle position with their T500 shortstack and holding KQ suited, knowing that with blinds at 100/200, they have only one or two rounds to go before they are blinded off anyway.
Player B calls from the button with A4 offsuit, since the call is only 1/20th of their T10,000 stack, and they know that they get added EV when knocking out a player.
Player C calls from the big blind, since it's only T300 more out of his T8900 stack, so his suited 97 looks good. Also, he calls because of the added EV when knocking out a player, and calling might help eliminate the shortstack.
The flop comes 6h, 8d, 2c
Player C might be tempted to bet here, since he's flopped an open-end straight draw. But what would he achieve?
Player B would probably fold, having flopped nothing. But player B actually has the best hand right now! If C's straight doesn't come, and he doesn't pair, then player A wins the pot and has tripled up to T1500. If C's straight does come, he wins the T1500 in the main pot -- but he would have won that anyway if his straight came and B was still around.If C leaves B in the pot, then if the straight doesn't come, B wins the pot and A is eliminated, moving everyone up on the money list.
If the straight eventually comes, C may indeed bet, since C would not be concerned that A's unknown all-in hand might be ahead, in which case in the unlikely event that B calls C's bet, C wins that much more money, at little to no risk of losing the main pot to A.
The key here is to ensure at all costs that the main pot does not get awarded to the all-in player. If it does, the all-in player triples up and regains playing power in the tournament.

Of course, none of these considerations apply if the all-in player is so shortstacked that even tripling up would make no difference. In our example above, if the shortstacked player A had only 25 chips instead of 500, players B and C can effectively ignore A since even if they triple up, they still have less than even one small blind, so it doesn't really matter if the player triples up or not.

Another example: This is the same scenario as above, except this time player A has QQ, player B has KK, player C has AA. The preflop betting is the same and the flop is the same. Suppose now (to simplify the calculations), the flop is checked and the turn is another blank, say the 5s. There is T1500 in the main pot. Player A has a 2/42 chance of winning the pot, player B has a 2/42 chance of winning the pot and player C has a 38/42 chance of winning the pot.

If player C bets and forces player B to fold, player C increases his equity in the pot from T$1357 to T$1428. Notice that player A still has only a 2/42 chance of winning the pot, so player C strictly increases his tournament EV by betting. (He has the same chance of eliminating A, but player B's 2 outs now become outs for player C.)

Of course, in a real tournament with real players, player C probably would not have such a sure read on player B and A to be 100% certain of their holdings, making a scenario like this less likely. Still, when a player has a hand that is a big favorite but not a lock, it might pay to bet in this situation. For example, if player C flopped top set, but does not want to lose to a runner-runner flush or straight by player B, it might pay to bet the flop because player C is probably way ahead of the all-in player A.

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