This is a follow-up on the question I recently posted about strategic insights that are not addressed in Wong's book (which generated some very interesting responses). I play in a weekly hybrid tournament where all of the accumulated buy-ins are split in the final round: 50% to first, 30% to second, and 20% to third. I'd estimate that the amount paid for third place is usually about 3-4 times the amount I've spent on the buy-ins to make it to the final table. When betting the last hand, my personal orientation has been to always shoot for first place (unless that would require doubling on 18 or something similarly speculative), but I've noticed that a number of other regulars will opt for locking in second or third place rather than betting aggressively and risk finishing out of the money. Until now, I've simply chalked up the different playing styles to different risk/reward preferences. But I now suspect that the allocation of the pot should generate some mathematical guidelines that supersede personal psychology. As usual, I lack any mathematical facility to determine if that hunch is correct. But here's my rudimentary thought: Assume it is the last hand, and regardless of your betting position you have a very good chance of locking third with a conservative bet, and a decent chance of taking first with an aggressive bet. Since first place pays two and a half times more than third place, it seems like the decision to try for first should depend on whether you have at least a 40% chance of winning with an aggressive bet. Coincidentally (I think), that approximates the 44% chance of beating the dealer using basic strategy, and is even closer to 40% if you're playing a "push is as bad as a loss" strategy. First hypothesis: If you are reasonably certain you can take first place by making a large bet using "normal" strategy, do so, even if losing the bet means that you finish out of the money. This should be true even if you don't have a lock on first, eg., if an opponent gets a natural or a lucky double, since you may still capture second place. In other words, the payoff odds should make someone inclined to be conservative be more aggressive. Conversely, counting on my fingers, anytime you have to double a hard 16 or higher, you have at least a 60% chance of busting, not counting that the dealer can still beat you if you don't bust. Doubling a hard 12, however, you have almost a 70% chance of not busting, although the dealer can still beat you. I don't have the math skills to figure out what hard total, when doubled, will give you a 40% chance of beating the dealer, but I suspect it's around 13 or 14. Second hypothesis: If you are reasonably certain that you can take first place only by doubling down, but you're pretty sure you can lock third with a conservative bet, only bet aggressively if you hold a hard 13(?) or less. In other words, the payoff odds should make someone inclined to be aggressive be more conservative. OK, that's very long-winded and I may be completely off-base on the specific examples I offered. Nevertheless, I think my intuition that you should play one way if the pot split is 50/30/20, and adjust your strategy if the allocation is different or if the pot is split two ways or five ways, makes some sense. What do you think? Best regards--Acercher
You're on the right track with your thinking. The actual math goes something like this: Each bet + intended strategy results in a set of probabilities that you will end up in each of the various possible finishing positions. The overall expected value of the bet + intended strategy is EV = P(1)*V(1) + P(2)*V(2) + ... where P(n) is the probability of finishing in position n and V(n) is the payout for finishing in position n. In your example above, you have n ranging from 1 to 4 with V(1)=50%, V(2)=30%, V(3)=20% and V(4)=0% What you will do on the final hand depends on your goal: If you play tournaments often, you want to make the decision which results in the highest total EV, regardless of whether it decreases your chance of winning the tournament or increases your chance of finishing out of the money. If you've never won a tournament and you would really like that feather in your cap, you want to make the decision for which P(1) is highest (i.e. the probability that you will win the tournament). If you just want to finish in the money, then you want to make the decision for which P(1)+P(2)+P(3) is highest. Note that the probabilities change as the final hand progresses and are something that you have to sometimes guess at based on your experience. This is because you do not always get to act last on the final hand. Your first decision is what to bet and, when it is your turn to bet you will have an initial set of probabilities and you should bet according to what you estimate them to be and what your goal is. By the time it's your turn to play your hand, the probabilities will have changed based on what the players to your right have already done and what the players on your left are holding. The probabilities will change with each card you take until you have finished playing your hand. For each playing decision, you need to re-evaluate. One you've finished your hand, you usually sit back and wait for the outcome, although some will continue to try to improve their chances by "coaching" the remaining players. This is the detailed idea of what is happening and how you should be thinking. In reality it's often much simpler than this. You will more typically end up thinking something like, "If I bet X I can win the tournament by winning my hand and can finish no worse than 3rd as long as I don't double or split". It's usually only when an opponent pulls a rabbit out of his hat with an unexpected natural or a 3 way split with doubles that you need to re-evaluate your bet + strategy decision.
