Hmm, I don't know : ) This whole relativistic stuff is not always immediately intuitive. It probably depends on which frame of reference, keeping in mind that the speed of light is a constant, as you well know. Analogies aren't always perfect.
For some odd reason I had always thought that clocks on orbiting satellites ran slower than ours on earth (because they were traveling faster). But it turns out that our gravity well and the acceleration of clocks in that gravity well caused our clocks to run slower.
Time DOES tick more slowly on satellites due to their high speeds, as compared to stationary clocks on the surface. But gravitational time dilation also occurs mostly strongly near the Earth's surface and would have the opposite effect. I'm pretty sure the high rate of speed wins out. GPS satellites takes these effects into account.
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From Wiki...
Note: I didn't actually read this yet, but will now. Hope it supports what I said! :)
Relativity:
Satellite clocks are slowed by their orbital speed but sped up by their distance out of the Earth's gravitational well.
A number of sources of error exist due to relativistic effects [67] that would render the system useless if uncorrected. Three relativistic effects are the time dilation, gravitational frequency shift, and eccentricity effects. For example, the relativistic time slowing due to the speed of the satellite of about 1 part in 1010, the gravitational time dilation that makes a satellite run about 5 parts in 1010 faster than an Earth based clock, and the Sagnac effect due to rotation relative to receivers on Earth. These topics are examined below, one at a time.
Special and general relativity:
According to the theory of relativity, due to their constant movement and height relative to the Earth-centered, non-rotating approximately inertial reference frame, the clocks on the satellites are affected by their speed. Special relativity predicts that the frequency of the atomic clocks moving at GPS orbital speeds will tick more slowly than stationary ground clocks by a factor of \frac{v^{2}}{2c^{2}}\approx 10 ^{-10}, or result in a delay of about 7 μs/day, where the orbital velocity is v = 4 km/s, and c = the speed of light. The time dilation effect has been measured and verified using the GPS system.
The effect of gravitational frequency shift on the GPS system due to general relativity is that a clock closer to a massive object will be slower than a clock farther away. Applied to the GPS system, the receivers are much closer to Earth than the satellites, causing the GPS clocks to be faster by a factor of 5×10^(-10), or about 45.9 μs/day. This gravitational frequency shift is also a noticeable effect.
When combining the time dilation and gravitational frequency shift, the discrepancy is about 38 microseconds per day; a difference of 4.465 parts in 1010.[68] Without correction, errors in position determination of roughly 10 km/day would accumulate. In addition, because GPS satellite orbits are not perfectly circular, their elliptical orbits cause the time dilation and gravitational frequency shift effects to vary with time. This eccentricity effect causes the clock rate difference between a GPS satellite and a receiver to increase or decrease depending on the velocity orbital altitude of the satellite.
To account for the discrepancy, the frequency standard on board each satellite is given a rate offset prior to launch, making it run slightly slower than the desired frequency on Earth; specifically, at 10.22999999543 MHz instead of 10.23 MHz.[69] Since the atomic clocks on board the GPS satellites are precisely tuned, it makes the system a practical engineering application of the scientific theory of relativity in a real-world environment.[70] Placing atomic clocks on artificial satellites to test Einstein's general theory was proposed by Friedwardt Winterberg in 1955.[71]
http://en.wikipedia.org/wiki/Global_Positioning_System#Relativity