You could hash the file contents plus a salt that is contained within the application binary. Most people wouldn’t know how to extract the salt from the binary so they could get the hash right. Though I guess if people are determined enough to hack your game, one hacker might just publish the salt or a small program to rehash files for you. If a user has enough time on their hands, there’s nothing you can do to stop them from running your software in a VM with a debugger and finding out its secrets; the best you can do is making that hard enough that people won’t bother.
If you wanted more security, you could keep a secret that you don't include in the saves but do include when you calculate the hash, so that anyone who doesn't have the key is going to get the wrong answer. That's about as far as I'd consider going for relatively trivial data like save games. Though that's, in principle, discoverable if someone's sufficiently interested.
After that point, it becomes much simpler for someone to watch the memory associated with your program and extract/alter the values there. (Programs to do that to games are generally called Trainers.) That's not a complicated thing to do unless someone's tried to stop you doing it.
There are some techniques to provide some degree of security there. Changing where in memory you place your information each time springs to mind, thus making it more difficult for people to find out where the values are and then share their locations. However, even that's not perfect. Depending on how sure you want to be that no-one's going to alter the values, you're potentially looking at requiring very deep knowledge of security there.
After that point the next easiest target may be the program file itself.
That said, if you want to get around that sort of problem and you're really serious about it, then running your encryption in an environment that hostile may be making things more difficult than they need to be. You might use a trusted platform module, to try to make the environment you were in less hostile, if one were present on the user's machine. But, honestly, I'd want the information not to be stored or calculated on the user's machine if it were that valuable. Have the user's end be the input, encrypt their signals with your public key, and do the calculations that you needed to be sure of remotely.
Though then the user has to trust you. I wouldn't usually advocate that my users trust me that much - not unless we were dealing with a situation where the information we were talking about was entangled with others in some way such that a reasonable argument could be made that they didn't own it, and I was just the best common arbiter I could think of.
-she shrugs awkwardly-
You can get yourself into a situation where it's probable that the amount of effort someone would have to invest is vastly greater than the likely value of the information fairly easily. But ultimately it's a question of how expensive you want to make things and what that's worth to you. Against a sufficiently dedicated adversary, with a sufficiently valuable target, there are so many unknowns in computer security that I wouldn't even be sure that storing the data on your server would be sufficient ^^;
