I noticed the other time a friend of mine went to a casino and bought in 100 dollars for a 1-2 table. Other players had heavier buy ins. I have received two opposing arguments. One says that buying in more (like say $1000) is more optimal because it gives you more leverage if you were to get a monster hand, but on the downside you are unable to all-in on a "strong" hand if you don't want to take on too much downside risk. Are these truly strong tradeoffs which nullify the utility of having a higher buy in as opposed to a smaller one or not?
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It is very hard to give a clear mathematical answer to this question because there are so many factors involved, such as your skill level, the dynamics of the table you are playing and the amount of money others have on the table.
I'll try and give some intuitive pointers on how to decide how much to buy in for.
1) If you are comfortable playing an aggressive style (calling down hands with drawing potential based on implied odds, bluffing and semi-bluffing in the right spots and so on) and the rest of the table isn't going to match you in aggression, you should buy in higher than the table average.
2) If the rest of the table is also showing aggression, you can bring higher than the average buy in and try to be more patient and pick your spots for showing aggression more carefully.
3) If you are not very aggressive in general and like to wait for killer hands before making a move you can buy in somewhat low. It is usually advisable to buy in at least 25-30 big blinds, so that you will have some pay off when you hit the big hands.
Some additional clarifications:
In 1) and 2) I would advise relatively high buy ins for a couple of reasons. If you play as in 1), your stack will be subject to random swings because you are taking more risk and you will lose a few pots every now and then. However, since the rest of the table isn't showing as much aggression, you can recover easily if you have a big stack.
In 2) you need a stack bigger than the table average because you want to capitalize on the big hands when they come your way and extract as much as possible from the other players. Having a big stack also means you can extend the betting till the river and therefore increase bet sizes in a natural progression each round. In both cases, there is no benefit to buying in more than the table leader because you can't win any more money than what he has.
In 3), you are the novice player and therefore don't want to have prolonged betting through to the river card. If you have a small stack and wait for good hands, chances are you are all in either pre-flop or on the flop and you have a good chance of doubling up when the odds are in your favor. You won't have to make difficult decisions on later streets and if you bust out, you can buy in again for a small amount and continue.
As I mentioned before all of this is based more on intuitive than mathematical reasoning. I'm not aware of any mathematical treatment of this question and this seems to the next best thing.
The optimal and maximally exploitative strategies are different for different stack sizes, so it is possible that one player's personal strategy might be more successful at one stack size and not at another. For example, if someone is always willing to put all their chips in with just a good one pair hand, he is better off playing with a shorter stack than a larger one, simply because he will therefore lose less money the times he has one good pair and his opponent has something better. With a sufficiently short stack size, the strategy of never folding a good pair is good; but with lots of chips left to bet, it is a disasterous strategy.
It is also true that in multiway pots, it is theoretically more desirable to have a shorter stack than the rest of the table if everyone is equally skilled. The reason is that the shorter stack will often be able to profitably remain in a hand and realize his full pot equity, while a larger stack might be forced to fold and forfeit his share of the pot's equity because of the threat of another large stack's future betting. In practice, however, this is rarely a major concern.
My understanding is that if you are playing in a game of weak players who are deviating far from the optimal strategy and have very large stacks of chips in front of them, then in practice, you will be able to exploit their mistakes for larger amounts in absolute dollars by playing a reasonably good strategy with a large buy-in that matches theirs than by playing an absolutely maximally exploitative strategy with a much smaller buy-in.
The basic idea is that if you buy-in for \$100 and are good enough to increase your stack 10% per hour, on average, then you would win roughly \$10/hour. Now suppose you buy-in for \$1000 but are only good enough to increase your stack 3% per hour, on average. Then you would still win roughly \$30/hour. Of course, your risk would be much higher and you would have to have a larger bankroll to handle the variance in the game.
In practice it is rare to have a situation where one players has an edge over others with X money on the table, (atleast for x>30BB) but not with say 2X. However some players are somewhat good with 20BB (which usually is the minimum you can have), but totally crap at say 100BB and they run off once they double their money. 20BB does not depend on postflop skills. Mathematically it doesnt have an answer, because with perfect strategy its a breakeven game.
Generally one can say, the deeper your stack is, the more postflop play and more room for strategy and n'th level thinking. This is because you can put in more money in the pot without mathemathically comitting your entire stack. You almost never see a 3bet river or 5bet preflop with normal buyins.