Is it totally dead, like you flipped off the switch, or still on and come circuits die like a tilt? And once it does this, can you power it back up right away with the main switch?

Also, do you have a multimeter?

Following up deviman... assuming the game is acting like you turned it off and then back on immediately, there is no single "simple fix" but there is a lot of info available for this common issue.

http://www.pinwiki.com/wiki/index.php?title=Williams_WPC#Game_resets

Any pattern to this? Consistent time from bootup to shutdown or totally random (something that would indicate ties to heating up)? Or are there any events that provoke it (hitting flippers, other solenoids acting?). Do the red LEDs on the mainboard go out as well? When you reset it how long does it stay on before going off again? Does it only fail when you play or also when it's just sitting idle but on? Any credit dot or errors on startup?

There are a lot of possible explanations detailed on the pinwiki site above but if you can narrow it down by observing the behavior it might help you know where to go next.

Before digging into or replacing a board I would start tracing where you lose the voltage with your dmm once it happens, if you are able. The more you can isolate the location of the failure the easier it will be to diagnose.

When it goes dead where no leds are lit up on the boards, I would start with checking the power before and after the switch.

Firstly devlman makes a good point about observing what causes the game to shut down. Also I would not start replacing stuff at this point since there's no way of even guessing where the problem is.

After the game shuts down, check the led's on the cpu as suggested. I would guess they will be off, but definitely worth checking. Since the game is turning off and staying off, the pinwiki page will be of little value (it's focused on resets rather than shutdowns).

Get out your meter and the power diagram from the manual and wait for the machine to shut down and then start checking voltages. I would start at the transformer and make your way upstream towards the boards. Of course you could start at the boards and work downstream if you prefer.

The good news is this should be much easier to troubleshoot than a reset problem.

I'm interested in hearing the outcome. I've not come across a circuit or other element in the wpc that would cause a shutdown until the power is cycled. But just because I haven't seen it does not mean it does not exist.