Step 1: Connect wattage to filament temperature.
A hot filament radiates by the Stefan-Boltzmann law, so its power goes as the fourth power of temperature, $W \propto T^4$.
Step 2: Invert to get temperature.
Turning that around, the temperature climbs only slowly with wattage, \[ T \propto W^{1/4}. \]
Step 3: Recall what sets collision frequency.
The collision frequency of an evaporated tungsten atom with argon atoms is proportional to the average speed of the moving atoms.
Step 4: Link speed to temperature.
From kinetic theory the typical thermal speed grows as the square root of temperature, $v \propto \sqrt{T} = T^{1/2}$.
Step 5: Chain the two relations.
Since frequency $\propto v \propto T^{1/2}$ and $T \propto W^{1/4}$, we combine the exponents, \[ \text{frequency} \propto (W^{1/4})^{1/2} = W^{1/8}. \]
Step 6: Read off the exponent.
So the power $\alpha$ that links collision frequency to wattage is one eighth. \[ \boxed{\alpha = \dfrac{1}{8}} \]