The optimal way to play 1 sng against the guy is the optimal way to play against him in any, so long as you're within your bankroll. All that changes is if you know you can play him many times, you want to beat him faster.

That's not quite true. Apart from admin fee arguments...

You have 1 lifetime opportunity to score against a crazy player.

Instead of say a 55:45 edge over a normal opponent, this dude offers you 60:40.

Therefore, there's an opportunity cost if you are not slightly more cautious, because it's a one off event. If you can repeat indefinitely, then you know you take his bankroll in long run, you can rely on long run.

Hope that is clearer now.

No. If you know his strategy, the blinds determine an optimal way to play. That doesn't change however long a run. The cards you call with changes depending on the stacks, but you've worked that out in advance. In practice, with rematches you may trade some edge for an improved hourly rate.

On single trial vs. multiple trials:

When you are able to play multiple trials (or allowed multiple buy ins, ie. he/you will rebuy if he/you lose the hand) the proper strategy for maximizing EV is to play every hand that is better than an average hand. This is because your total expectation for the session is equal to the sum of all the expectations of the individual hands that you play. Therefore, any hand that has positive expectation (ie. any hand better than a random hand) should be played as it adds to your total expectation. You never have to worry about giving up future expectation because when one of the two of you goes broke, you or he simply rebuys and you get a new hand.

When you are only allowed one game or buy in (ie. as soon as you play a hand, barring a split pot, he will leave) the proper strategy for maximizing EV is to fold some hands that are slightly better than random hands even though calling with them would be +EV. This is because playing those hands prevents you from taking advantage of hands that would be more profitable against a random hand in the future due to the game ending as soon as you play one.

Clearly, it is correct to change the range of hands with which you call based on how many games you get to play. I hope this has cleared some things up.

Exactly. There is a set % of calling hands for each blind level that will maximize your return for the SNG, and it's not "any hand with an edge" that yeilds max EV for the particular SNG.

In addition, though you might widen your calling range to a still-+EV range, though not optimal, in order to increase your hourly rate (against an opponent you know will continue to play against you the same way), you STILL might not widen it to 'any hand with an edge' because what you gain in time might not be made up for by what you sacrifice in EV for the SNG.

False. The number of trials is irrelevant. Your expectation is maximized by calling with a particular range (that is tighter than any-hand-with-an-egdge). Your expectation is maximized by doing this regardless of the number of trials. The only affect the number of trails has, as fraac said, is that you might decide to sacrifice maximum expectation per SNG in order to increase $/hour rate. But if you're attempting to merely optimize expectation per each SNG, then calling with "any hand with an edge" regardless of the number of trails, will not yield optimal results per each SNG. It will yield slightly +EV results, which is a lot different than optimal.

What is so difficult about this concept? Against an opponent using a highly exploitable, bad strategy your solution is to use a strategy that just BARELY beats him, rather than one that clobbers him? WTF? If you call with 'any hand with an edge' you're doing him a huge favor.

No it is not. If you only get a fixed amount of trials, you prevent all future scenarios in which you are a larger favorite from happening because the game ends as soon as those trials end (easiest example is if you only get one trial - as soon as you call, the game is over). If you get multiple trials, you do not prevent these situations from occuring by calling, because you/opponent will rebuy. In other words, when you get multiple trials, you still get to call him with AA the same amount of times whether or not you play Q7 when he pushes. You are not giving up the situations in which you have him clobbered. You simply add situations in which you are not as much of a favorite, but are still a favorite nonetheless.

You are not trying to optimize your expectation per SNG, you are trying to maximize your total expectation. Since taking edges that are smaller than others does not prevent you from taking the larger edges as well, you maximize your total expectation by taking all of the edges. I think where you are getting confused is that you are still thinking of this in terms of limited trials. If you were only allowed ten trials, then yes you should wait for a larger edge than just playing any hand that is +EV, as you would want to maximize your expectation for each SNG, because your total expectation would be (average expectation)*(10 trials). But in this scenario, we are not limited to the amount of SNGs we get to play, so our expectation would be (average expectation)*(10 trials) + (average expectation of all other +EV hands)*(however many other trials we get).

An analogy: You are standing on the side of a conveyor belt. The conveyor belt starts, and on the belt are random bills: $1 bills, $2 bills, $5 bills, $10 bills, $20 bills, $50 bills, and $100 bills (each bill representing a hand dealt, where the denomination of the bill represents your expectation for that hand). The conveyor belt will operate for X hours (X hours representing the time you play HU). During that time, you may pick up any of the bills from the belt and toss them in a box at your feet. Once a bill has passed you, you may no longer pick it up. At the end of X, you get to keep all the money in the box. Would you only take the $100 bills because it maximized the amount that you earned per bill pick up (each time you pick up a bill representing each hand you play, and each bill passing you representing each hand you fold)? Of course not, you would pick up every bill. Of course, I did not take into account hands that are not profitable to you, because we agree that we should never play those. If you want to add those into the analogy, pretend there are also post-it notes with numbers written on them on the conveyor belt which if you pick them up, you have to pay however much the note says. Of course, you would not pick up any of the notes.

We are not only playing the hands that are barely +EV. We are also playing the hands that clobber him. You still get the same amount of money you would by playing only the hands that have him clobbered, but in addition to that money, you also get money from the hands that just barely beat his range.

In infinite trials, any fixed edge is equivalent.

All you're doing is describing the fact that you will beat him over time by calling with any hand with an edge. Which is true, but irrelevant. The question is what is the BEST way to beat him, not what is the minimum way to beat him. There are many ways to beat him, only 1 that optimizes your expectation.

Yes, you are. That's what the OP asked. How do i deal with the opponent's particular strategy. And the answer is by calling with a range that maximizes your expectation.

I like the analogy. It makes it obvious that the question needs some assumptions.

A) If you have an unlimited # of hands to play, call with all +EV hands.

B) If you have a set # of hands to play (unaffected by the results of any hand), call with all +EV hands.

C) If the # of hands to play is triggered by the loss of a hand, call with a significant +EV margin, and pass on small +EV margins.(*)

(*) Relative sizes of Stacks:Blinds & the resulting # of hands you can play after a loss should be used to figure how significant of a +EV margin you can wait for.

In the OP scenario, it's debatable which option should be applied. I'm leaning toward C, assuming this is a "one shot" game where if you lose you don't get to play again. In that case, select a range that gives you a 2:1 advantage (or so).

IF you were allowed to match his stack every hand, and this was a one-shot game with no time limit, then it would be best to wait for AA. Otherwise, the blinds will most likely destroy your chances of winning. You can (usually) wait several hands to get a 2:1 favorite hand to call with without the blinds taking very much of your stack.

What do you mean by those things I bolded?

Beat him over time? Do you mean take as much money from him after numerous games... or just the 1 SnG game?

Best way to beat him? Can you maybe give examples of how you can beat him "worse" (or "minimum") as opposed to "best"?

Only 1 optimizes expectation? What would the optimum expectation be? What measure are you using for this?

If you're 60:40 favourite, expectation is 60% of the prize times however many games you play, if you play optimally. It's never more than that for any fixed number of games (a priori). Your methods B and C are wrong.

Closer to 20% (60% of win - 40% of cost). Whether you play optimally or not. Whatever that means in the case where you either call the all-in, or fold.

Well, not if you're a 60:40 favorite every time. Of course, you won't be, so that's irrelevant.

They aren't methods. They're scenarios. And they haven't been shown to be wrong. Feel free to elaborate.