When designing the combat system in Civilization: Revolution, Sid Meier found himself up against some interesting design problems. His players didn't understand math. In Civ Rev, the strength of units were displayed up front to players before battle to show the odds of victory. For example, an attacking unit might be rated at 1.5 with the defending unit at 0.5. This is a 3-to-1 situation.
Unfortunately, the testers expected to win this battle every time despite there being a 25% chance of losing each time. Sid tweaked the math to make the player win more in this situation. Next, the reverse case was tested. The player had 1-to-3 odds. If they won, the math was functioning properly. They had a slim chance to win and they did.
Sid identified a few cases of interest. When the player was presented with 3-to-1 or 4-to-1 odds, they expect to win. With 2-to-1 odds, the player will accept losing some of the time, but expect to win at 20-to-10, which is just a larger expression of 2-to-1. When the numbers get larger, the perceived advantage grows.
To adjust for this, Sid actually changed the math again so that the outcomes of previous battles are taken into account. He found that if a player lost too many 2-to-1 battles in a row, they would get frustrated. Instead of risking a player shutting the game down, Sid changed the math.
Switching over to Blizzard, Rob Pardo discussed the rest experience system in World of Warcraft and the perception of bonuses versus punishments. One of the original design goals for WoW was to discourage marathon play sessions and avoid grinding. In the beta, to this end, the experience system worked as such:
When a player first logged in, they would earn 100% experience. After a set amount of time in-game, they would only earn 50% experience. This way, they would take a break. This felt like a punishment to players and it was universally hated among the testers. To fix this, Rob decided to do the exact same thing, but play off of player psychology.
Now, when a player first logs in, they receive 200% experience. After a set amount of time they drop down to 100% experience. To complete to ruse, Rob doubled the amount of experience required to level up across the board. The math ended up being exactly the same, but players loved this new system, which would become Rest XP. It was perceived as a bonus, not a punishment.
Furthermore, Rob talked briefly about drop rates for quest items. Apparently, players would get into cold streaks where they went a long time without getting a quest item drop. Instead of correctly attributing this to randomness, they would blame the math or random number generator. To get around this, Rob and his team actually changed the code to increase the drop rate after each kill until it hits 100% and then reset it for the next one.
It's interesting that players don't want to accept math when it doesn't work out for them, but are more than happy to accept it when it rewards them. Purists might dislike Sid Meier changing the match to appease players, but the game became more fun. Blizzard opts to package systems as a reward over penalties and takes a little randomness out of the equation.