Choosing Random vs. Random Choice

A game design issue that I've been working and struggling with at my job lately is the difference between choosing to take a gamble that involves some random chance in it vs. having some outcome of the game be randomly selected for you.  The end result is going to be the same regardless of which way the game is designed, but it feels incredibly different from a player's point of view because in one case they get to choose to take a risk and in the other case, they have no choice and whether the end result is good or not, it doesn't feel as satisfying.

I'll give an example to make this a little clearer.  In the game I'm currently working on, there's a lot of hidden information.  So what information each player gets is very important and what they do with that information is the real crux of the game.  In the original design, one player moved pawns around and each turn, had a random chance of seeing what the other player did at those spots.  Having multiple pawns on the same spot would increase that random chance.  So, the "interesting" choice they had was how to allot their resources and hope for the best.  It turns out (unsurprisingly now that I look back at it) that this is really not fun to play.

The new design is to instead give the first player a bunch of false information along with information about what the other player is doing.  They can't tell the difference between false and real information, so the choice is still seemingly random, but the player gets to choose which information they think is real.  Eventually, the player should have enough information to start figuring out what the other player is doing and deduce their strategy and come up with a good counter strategy.  This makes it so the interesting choice is how to deal with the information you have and not how to hopefully get the right information.

A similar example of random choice is battle in the game of Risk.  You do get to choose where to attack, but the outcome is pretty much completely random, so even if your strategy is amazing or you have 99 times as many units as the other player, you can still lose because the dice hate you.  It's still important for the defender to have some method of surviving an attack, but at some point, the better strategy should be rewarded over the luckier player.  Of course, this is why the combat in my first board game took a long time to play because I eliminated random die rolling and had straight up math (attack vs. defense), which taught me a whole other lesson.  Still a balance I have yet to completely figure out.