How many miles of riding require the same effort as one mile of running? A co-worker and myself are having a friendly bike / run contest. What is a good ratio for miles ridden vs. miles run? I know it depends on pace. But I'm curious what a generally accepted range of ratios might be.
The short answer to your question is "probably between 2.5 and 3.5 miles; the faster the runner the closer to 2.5 while the slower the runner the closer to 3.5." The long answer is quite long. 

As you already noted in your question, the full answer depends on pace; but it also depends on the surface being run or ridden on, whether it is up hill, down hill, or flat, whether it is windy or calm, and how "aero" the cyclist is. That said, if we are talking about a road bike on a firm flat surface under calm conditions, it is possible to make some estimates of the “equivalent” distance. The simple answer is given in the chart below, which relates the distance one must cycle in a given amount of time to equal the effort in running a mile at a given pace.



In this chart, the x-axis is the run pace for a mile and the y-axis shows the distance one must cycle in the same amount of time. For example, if you can run a mile in 4 minutes, an equivalent cycling distance if you were riding "in the drops" would be to ride about 2 miles in the same 4 minutes (that is, to average 30 mph). If you ran a mile at a 10 minute/mile pace, it would take you 10 minutes to run a mile and an equivalent cycling distance would be to ride about 3.5 miles in the same 10 minutes (that is, to average about 21 mph). 

How was the chart calculated? In running, the power needed to run X meters/second is approximately X watts/kg of body weight (cf. Hall, Figueroa, Fernhall, and Kanaley, 2004 or McArdle, Katch &amp; Katch, 2007), while the power needed to ride at Y meters/second varies with Y^3 (cf. Wilson and Papadopoulos, 2007). Thus, if we know the amount of time it takes to run a mile, the weight of the individual and the bike, and some other simplifying assumptions we can make the conversion. An exact estimate of running power will depend on the individual's gross metabolic efficiency, the runner's energetic economy (that is, how economical in an energy sense is the individual's running style), and the runner's resting metabolic rate, but it is commonly observed that for a normal-sized individual without unusual abilities or disabilities, energy expenditure while running on a firm flat surface will range between about 0.8 to 1.1 kcal/kg/km. Thus, a common rule of thumb is that running requires about 1 kcal/kg/km. There are 4.2 joules per kilocalorie but gross metabolic efficiency usually lies in the range of 19-25%; if we assume a value of 23.8% for GME, we can derive the happy simplification that (running speed in meters per second)*(body weight in kg) is a reasonable estimate of power in watts; or speed in meters/second ~ watts/kg.

For this chart, I first calculated the speed in meters per second for various running paces along the x-axis, from a 4 minute mile (= ~6.7 m/s) to a 10 minute mile (= ~2.7 m/s). By the running rule of thumb, this requires roughly 6.7 watts/kg down to about 2.7 watts/kg. I then presumed “average” values for the coefficient of rolling resistance (Crr) of .005, flat ground, no wind, and a CdA (drag area) of the cyclist of 0.25, 0.33, and 0.42 sq. meters (typical of a road bike rider "on aerobars", "in the drops", and "on the hoods") and calculated the cycling speed for a cyclist able to produce between 6.7 watts/kg and 2.7 watts/kg. The running pace defines a total amount of time needed to run a mile so the question becomes "how far must a cyclist producing this much power ride in the same amount of elapsed time?" Knowing the speed and the elapsed time lets us calculate the distance. That is what is shown on the y-axis. Note that for the fastest runners, the equivalent cycling distance approaches 2 miles while for the slowest runners competing against the most aerodynamic cyclists, the equivalent cycling distance approaches 4 miles. Thus, for a friendly race between you and your co-worker, if neither of you is a world-class athlete, a reasonable range is 2.5x - 3.5x the running distance to handicap a race so that you both finish in approximately the same time. 

If the cyclist were a bit more aerodynamic (for example, if the cyclist used aerobar extensions and subsequent CdA was below 0.25 m^2) the curve would move upward and the rider would have to ride farther in the same amount of time (that is, faster) to match the runner's energy expenditure. If the cyclist were a bit less aerodynamic (for example, if the cyclist's position was more upright, or was wearing loose and flappy clothing) the rider would be expending energy at a faster rate so the curve would move down.

A rough validation of this approximation is that, anecdotally, people who both run and cycle say that running 10 km in about 42 minutes is roughly about as hard as riding 40 km in an hour. That's a running pace of about 6:45 per mile, and the chart equates that to cycling about 2.75 miles in 6:45, or about 24.5 mph -- which is 39 km/h. Our rule of thumb for equivalence is "between 2.5 and 3.5 times as far; closer to 2.5 times if the runner is fast and closer to 3.5 for a slower runner." Running a mile at a 6:45 pace is moderately fast, so we would expect the equivalent multiplier for cycling distance to be closer to 2.5 than to 3.5 -- as you can see, the predicted multiplier is 2.75 so the rule of thumb appears to work. Further evidence can be gathered from duathlons or triathlons. Below you can see data from the 2010 Ironman World Championship in Kailua-Kona, Hawaii. The Ironman comprises a 2.3-mile ocean swim, a 112-mile bike ride, and a 26.2-mile marathon. The scatterplot matrix below shows the swim, bike, and run times (in hours) for each of the finishers. 



If we ignore the swim leg and concentrate only on the relationship between the run and the bike, we can prorate them to equivalent 10k run times and 40k bike times, as is done here:



What these latter two charts clearly show is that equivalent bike distance will depend heavily on run pace but also on the individual. There is a great deal of "scatter" in the scatterplot, which indicates that while the rule of thumb is reasonable it is not precise.