Are there any cases where an average value can be maximized only at the cost of increasing variance?

Are you looking for real world examples? Assuming so…

You would probably like your income from work to be as high as possible, and would like to keep the value above a certain level where you would be unable to pay your bills, in order to do that people often take a steady, guaranteed job for the security that they will always have it rather than a possibly higher paying, but less secure position.

People lower the average income in order to lower variance and avoid the possibility of running out of money (putting it as simply as possible).

This is also the key ingredient to bankroll management for professional gamblers, the “risk of ruin” basically dictates your maximum stakes regardless of your confidence in a particular win. You want to have a certain number of plays in order to ride out any likely variance without running out of chances (“chances” being whatever stake you’re playing). If you’re actually statistically winning over the long haul, you will almost certainly reduce your winnings by keeping your bets relatively low to what you could be wagering.

I’m not sure if the employment example fits the “only possible strategy” stipulation, there are just too many different variables. The gambling example probably does, since it’s a much simpler set of rules if you can identify the likelihood of winning.

Those are good examples. I wonder if there are others.

Here is what made me think of this question. Consider all the possible ways for the government to distribute money. According to Keynesian economics, there is value to the total economy if money is distributed to poorer people in order to increase demand. If we look at all possible redistribution schemes, I was wondering if the problem of average versus variance would come up. Would the scheme that maximized average income still leave some below the poverty line? I know this is all very hypothetical, but it led me to wonder if in general there might be issues of average versus variance for optimization schemes and, if so, how do you determine what is best, or does it in the end come down to a value judgment?

Yes, there are lots of examples. One example would be any gambling scenerio (stock markets, investment, anything with risks/payoffs/losses). The mean of the expected profit/loss always increases as you take more chances, but the variation also always increases.