'Risk management' is used in a lot of business speak, and it trickles into game designing meaning both something similar and different. It's also a term used in "Traders, Guns, and Money" - which again is more of a business sense.
Risk management is somewhat of a trick. One, it implies that there are (probably dangerous) risks out there, and that someone else can come along and manage it for you, so you can think about all the profits you'll make. Traders, Guns, and Money - risk management is supposed to limit the damage that bad trades can cause.
In games, I use it to cover both sides of a fence. If you're ahead, you want to expose yourself to the least risky course, and glide into the win. If you're behind, you're looking for someway to make up the shortfall and vault into first place. It's rare that you're at equal chances of both winning and losing, which would be the top edge of the fence. So, you use risk management one way or the other...to push the envelope, taking higher risks to win, or to find paths that let you avoid pitfalls. (Tonight's game of Mage Knight, the girl found herself behind, so she took more difficult unknowns. It didn't help.)
Several years ago, I picked up a copy of Pitchcar. To play the game, you flick your wooden disc around the track, bouncing off of guardrails, muscling past other cars, and hope to stay on the track as well. There's a lot of risk. If you flick it too hard, you go off the track, and lose your turn. You can attempt to glide along the rails, but if you miss the shot, you're off the board or rebounding back towards where you came from.
If you're in first, you skip the risky plays. If you don't go off the course, other players are going to have to push their cars hard to catch up. By flicking their cars harder, they're more likely to force their own car off the track, or in rare cases, actually catch up. If you're a half track ahead, chances are slim that they can ever catch you. (Designers of Powerboats - take note! Ugh. Really, take note.)
There's another racing game that addresses the issue, which is Mario Kart. It's fast, it's furious, you can take some risks, but you're really just trying to drive as accurately as possible in your cartoon car, at its fastest speed. There's really no risk for driving the best...but it's not exciting if you know who's going to win. The risk of being ahead in Mario Kart is that you get less options on your powerups. While others get the full range...but especially the Blue Shell of Death. It auto-hits the first place player, but only the last place players have a chance of picking one up from a box. Experienced players can complain about it, as it's a risk they can't avoid, except by not winning. Thus there's tension for 1st, even though they are winning...but will it show up in the powerups?
Other forms of risk can be demonstrated in Diplomacy, where you open yourself to dangers as treaties are formed with other players. Other players will be lying to you at some point - figuring out the how and when can limit how much damage you take, your exposure to risks.
You can also take risks when you ARE in first - to make it more interesting for you or for other players, due to perceived 'fun', or maybe because the game forces you to. The last part is particularly interesting, where you aren't allowed to just coast...you still need to make good decisions to win.
Risks are inherent in a lot of games. Eliminate the need to choose what risks a player engages in, and you're stuck playing a game of Life. Going too deep into complex ideas, while allowing the player many risks to manage, your game comes to a standstill. Creating an intuitive flow with several paths of risk management is a thing of beauty.
PS - To 'fix' Pitchcar, I mocked up some extra items to turn it into Mario Kart in some ways. (If you're interested in doing the same, it's discussed in a thread here. There are some newer threads that did a lot more work, if you liked my Mario Kart variant.)
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