Why is the winning a bet 58% of the time not the same as splitting the bet up 58%/42%, every time? So this question came about from fantasy football, but has turned into a statistics question and I'm deeply curious as to why my intuition is massively incorrect here.
So the total pot is $\$1000$ and I have a $58\%$ chance of winning the game. So, assuming "winner takes all" over say $100$ games, the expected outcome would be that I get $\$58,000$ and my opponent gets $\$42,000$, obviously.
Now, what if, before the game, my opponent and I agree to split the pot up to minimize some risk and ensure we both make a bit of cash. Intuitively, the split that makes the most sense is $58/42$ - winner would get $\$580$, loser would get $\$420$. Any more even (say $50/50$) seems like it would be unfair to me, because I have a $58\%$ chance of winning but only would get $\$500$. Any further apart, I would assume, would be unfair to my opponent due to the same logic.
But, when I do the math on a $58/42$ split, it seems massively unfavorable to me. Over $100$ games, I would get ($58$ wins * $\$580$ per win) + ($42$ losses * $\$420$ per loss) = $\$51,280$. I would expect this number to also be $\$58,000$, but clearly I am wrong.

*

*Where does my reasoning go wrong here?

*Is there a split that would be fair to both sides?

 A: The total pot over all 100 games is \$100,000. With a 58% chance of winning, you would therefore expect an average gain of \$58,000 if the winner of a round gets the whole pot. In some sense, you might consider this to be fair.
On the other hand, if you get half of the pot whether you win or not, you'd have an average gain of $50,000. You might consider this to be fair instead, and your friend would be more likely to agree with you.
Your proposed payout is 58% to the round winner, which is between 100% (winner takes all) and 50%. You'd therefore expect an average gain somewhere between \$58,000 and \$50,000, so the expected gain you calculated is not surprising.
This is obviously unfavourable to you compared to your expected winnings under winner-takes-all. If by "fair" you want your expected winnings to be half of the total money involved, then you need $58 \times p + 42 \times q = 50,000$, where $p$ is the amount you get when you win, and $p$ is the amount you get when you lose.
If these payouts are the same for both players, you're under the constraint $q = 1000 - p$, so the only solution is $p = 500$, i.e. winner and loser get the same amount. This is rather dull, so to make a fair-but-interesting game you need the payout to depend on who won the round.
To do this, you can simply make the payout to the winner depend on who the winner is: if you get \$500 + \$x when you win, you get \$500 - \$ $\frac{58}{42}$ x when you lose, so you losses are more punishing than your friend's, to make up for you winning more often.
Alternatively, you can look at more complex payout schemes, where, e.g., the stronger player needs to win several times in a row to get decent payouts.
A: 
But, when I do the math on a $58/42$ split, it seems massively unfavorable to me. Over $100$ games, I would get ($58$ wins * $\$580$ per win) + ($42$ losses * $\$420$ per loss) = $\$51,280$. I would expect this number to also be $\$58,000$, but clearly I am wrong.

Effectively this is like making a smaller bet.
You win \$420 no matter what the outcome is. You win an additional \$160 in the case that you are a winner.
So effectively the pot is not \$1000 bit instead it is \$160. So your net winning will be \$51280-\$50000 = \$1280 instead of \$58000-\$50000 = \$8000. This difference 1280 versus 8000 is just like the ratio in the difference in the pot 160 versus 1000.
Some issues

*

*In your question you see to assume that you are winning \$58000 and this goes down to \$51280. However, I guess that this is not a situation where you get this \$58000 for free and you probably needed to invest \$50000 in the pot. So you are not 'winning' less. It is just that the stakes are smaller. On average you will be winning \$1280.


*The 58000-50000=8000 or 51280-50000=1280 are 'expected' profits or average profits. Going down from 8000 to 1280 means that on average you will have less winnings but also your risks are reduced. You are not always gonna win that 8000 or 1280. This person with the 42% might have some chance to win more often than you do.
(But on 100 games with 58% win probability it is a very small overall probability to have a big risk. The consideration to choose a smaller betting amount would be a meta-consideration, outside the scope of the mathematics. E.g. matters like risk-aversion versus risk-seeking. Or possibly there's some game theory, like you need to keep your opponent interested enough to make their stupid 42% bet)
