Theoretical problem about coinflips

By grouping terms in just the right way, I was able to extend my proof to cover tournaments that pay 3 places. Under the independent chip model, when at least 3 remain in the tournament, it is ALWAYS negative EV in terms of tournament winnings to take part in a bet with one other player for which your chip EV is 0 (or less). Without mathematical typesetting available, I don't think I can convey any details in this forum. (The algebra is sufficiently involved that I used a symbolic manipulation package to double-check my work.)

Hopefully, more to come. If successful, I'll probably eventually write it all up in a format I could send out at some point, although 1st semester calculus and the idea of partial fractions might be a prerequisite for reading it.


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Here's what I know so far.

1. Assume you have the option to call (but not raise) the final bet by the
only other player still in the pot. Imagine a bet putting one of you
all-in or a bet on the river.
2. There may be dead money in the pot from blinds or earlier bets.
However, assume the bet is "fair," i.e. the pot odds equal the odds against
your winning the bet.
3. And so far, unfortunately, assume the tournament pays only two places with at least three still in it.

The ICM then implies that calling the bet has negative EV.

Sketch of the proof. Your total EV is comprised of several terms. It is
fairly easy to check that your expected value for finishing first or for
finishing second when anyone except the bettor finishes first is the same
whether or not you call the bet. The only remaining term is when the
bettor finishes first and you finish second. Here your EV decreases if you
call the bet. In a nutshell, this is because f(z)=(y+z)(x-z)/(1-y-z) is
concave down (x represents your percentage of the total chips, y the
bettor's, and z your loss, which could be negative).

This same proof shows that raising on the river if you are certain your
opponent will call is worse than calling.

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No because of the declining value of chips in a tournament.

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No because of the declining value of chips in a tournament.

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This is a pretty vague statement. For instance, it's easy to come up with an example where, in a single hand, your chips have increased AND your tournament $ EV per chip has increased, i.e. the average value of your chips has gone up as your stack has increased. (My example is an all-in where you are short-stacked, win the main pot while the big stack busts several others out.) Can you define "value" and "declining value" precisely? (Even better would be a reference that has a proof.)

In the context of the OP, I think I can now show that $EV can only go down in tournaments that pay up to 4 palces. The jump from 3 to 4 wasn't as easy as I had hoped at the time of my earlier posts.