Even if I can't list any examples, I am still sure that they are definitely technical problems solved with game mechanics. It can help on making decisions, how to react to happenings etc. And the possibilities to solve real-world problems are getting bigger and bigger. Every little thing in a game can be used.
Yes, you read that right. Many school, including my high school, have found ways to use games like Minecraft to teach students architecture and even foreign language. Many more games that are not built for education are used throughout the country. Valve, the creators of Steam, launched a campaign known as "Steam for Schools" where games like Portal 2 are available for students to play and tools to help teachers find ways to apply the games to their class.
Yes, I believe that real-world problems can be solved with game mechanics, because game mechanics can provide models for real life that people can explore. A game mechanic is no different than people creating any other kind of model to help them look at different scenarios for solutions. Games can be helpful.
A lot of real world problems are already being solved with game mechanics. Many people who had major social issues have developed proper social skills just by playing online games and interacting with the people of those worlds. Games also help a person develop critical thinking skills to help them solve every day problems in the real world.
Game mechanics can allows us to form logical predictions with regard to possibility of a given set of events. The reality represented is functionally limited by the inadequacy of the program code to emulate reality. Given this limitation, in anything but very simple simulation, the possibilities and probabilities quickly go beyond the capacity of the simulation to represent or choose likely possibilities. The larger or more complex the issue investigated the larger the errors become.
No, I do not believe that real world problems can be solved with game mechanics. A game is just that and if you make the wrong decision and die you can always restart the game and try again. This does not work in reality, if you mess up too bad you will die and there is no restart button in life.
I do not believe it possible to solve real-world problems with game mechanics. I believe in doing so, it would make it seem like the person using game mechanics is trying to manipulate the situation. Furthermore, if one side can use this strategy then the other side could utilize it as well.