Yes, there are - but it isn't fantastically wide spread.
My basis for this answer relies on my professional experience. I work for my state government (in the United States) as an auditor, and my major claim is that I use computer simulation to analyze government programs.
In general? Yes
I have personally observed many different simulations being used. Here is a quick list of some cases where simulation is used in government:
- The United States Forest Service has a number of simulation tools for simulation the occurrence and spread of wildfires. These programs are used to plan for wildfires and to help deal with fires that are occurring. For an example, check out the FARSITE software This tools are used at the federal and state level.
- There are prison population simulation tools that corrections agencies can use to forecast the prison population in the coming year. This is used for financial and managerial planning. Here is an example of a report from the state of Kansas which uses such a simulation.
- Combat modelling is an entire discipline. The Winter Simulation Conference (the premiere venue for simulation research) usually has an entire track of presentations dedicated to simulating combat. Westpoint (a military training institution) has a website that describes some of their educational simulations. Sometimes these tools include more than loss of life indicators, such as environmental damage.
- Although I don't have a link for an example, there are an array of ecological simulations used by the United States federal government. They include meteorological, climatological, predator/prey, ecological, and other models. These are often used to analyze trends in the natural environment, which is valuable for agencies that work in those fields. For example, the Environmental Protection Agency uses simple Monte Carlo methods enough that they have published guidance on the subject.
- You specifically excluded economic analyses from the question, but of course economic simulations are relatively common. For example, the Congressional Budget Office used a simulation when analyzing the Affordable Care Act in December.
- Governments at all levels use simulation to analyze traffic patterns. This is one case where simulation is actually common. City planners and traffic engineers often use simulation to understand how proposed changes to roads (including adding roads, changing stop light configurations, etc.) will impact traffic patterns. It's so common that the Federal Highway Administration even published some guidance on simulation projects.
- The European Union has developed a simulation tool for evaluating the safety of nuclear reactors (thanks to ohwilleke for pointing out this one).
- The National Aeronautics and Space Administration uses simulated extraterrestial environments to help develop technology for their missions. (thanks again to ohwilleke)
Like Democracy? No.
Democracy attempts to analyze an entire country in a relatively holistic way. That is too general to be useful to good government in most instances. They typically want to deal with a particular program with particular goals. For this purpose, you should really have a simulation tailored to the program.
Simulations also get exponentially more complex as you add more concerns. A simulation which tried to be that holistic would likely be a very large project. Not impossible - but difficult.
Why not more?
I don't know of any specific religious simulations (which was part of the question). Most simulations are focused on "hard" problems which have measurable outcomes.
Why not use simulation more? And why not include more subjects?
- Simulation requires specialist knowledge. Not just anyone can create a simulation with a reasonable degree of validity. You can't even hire a computer programmer or statistician to do it - it really requires a unique combination of skills and experience unique to computer simulation. These people can be hard to find sometimes.
- Simulation is expensive. Creating a simulation typically requires contracting experts - and they aren't cheap. The project is also going to be far longer than an ordinary statistical analysis. It's more of an engineering project than pure analysis. Expect to spend several months (at least) on a simulation of even relatively small scale. Meanwhile, if you are tracking that data in your agency's systems you could perform a reasonable statistical test far quicker. Or you could make a judgment call and accept the risk in no time at all.
- Simulation is still fairly obscure. It comes from the world of operations research and industrial engineering; fields which don't generally penetrate public administration. Public administration professionals aren't taught about these methods, so they are largely unknown. Professionals from those fields are unlikely to move to public service, so there is no way for this approach to really get to the public service world.
Some people in industry are trying to create tools specifically for public policy use. One such company is Forio. Although I am unfamiliar with their products, they claim to have several customers in the public policy space.