Does it have something to do with keeping it in tune, since the smaller strings might be more prone to be stretched? (And therefore they spread out the impact to more than one string).
Or maybe just to help keep them from breaking?
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The fatter bass strings move a lot more air when they're hit with the hammers in the piano, so they produce more volume of sound.
The short thin strings at the top do not, so having more of them compensates. Also, they sound richer when more are played. Think of an orchestra - not many double basses, but quite a few violins.
With one thin string or ten, it wouldn't make any difference to their propensity to break as they would individually be under the same tension,but it could make the stress on the frame higher,causing other problems. Three appears to be the optimum number, brought about by years of practice.
In addition to Tim's answer, a correctly tuned piano actually de-tunes the group of strings on each note a teensy bit. This leads to resonant energy transfer back and forth among the strings, which improves sustain as well as sounding more pleasing to (most) ears.
Keep in mind that "volume of air moved" does not translate linearly to "volume of sound," due to the ear's frequency response curve and because perception of sound depends more on delta pressure than delta mass (think current vs. voltage).
The pitch of a string is roughly inversely proportional of its length, the square root of its mass per unit length, and the square root of its "average" tension (the term "average" referring to a complicated weighted average, as opposed to an arithmetic mean). In order for the pitch of a string to remain reasonably constant as the vibrational amplitude changes, the average tension must remain reasonably constant as well. In practical terms, this means that the tension of the string at rest must not be too much less than the tension when it's maximally displaced.
If one were to use a small-gauge string for the lower notes in a piano, the only ways to get the proper pitch would be to either have the strings be impractically long, or else have them be so loose that flexing them would cause the tension to increase substantially. This would cause loud notes to start sharp and go flat. The only practical way to make strings play the lower notes on a piano is to increase the mass per unit length. This will increase the amount of tension that will be required for the proper pitch, and thus reduce the relative difference between minimum and maximum tension the string will see during each vibrational cycle. It has the natural consequence, however, of making the strings be physically quite "fat".
The reason that pianos don't have three strings for each of the lower notes is very simple: there's no room. While it might be possible to build a piano in which the hammers for lower notes were spaced wider than the keys, it would probably be difficult to do so without the lower keys having a heavier "feel". Nearly all pianos have the hammer spacing essentially match the key spacing, which in turn requires that the strings associated with each key must, collectively, fit within a certain width. As strings get larger, fewer of them will fit in the fixed width allotted to each key.
Not sure how much this is a factor: The bass strings are not proportionally longer in length, to the higher pitched strings. In other words, a note that's an octave below another is not twice the length, which I believe has to do with the practicality of being able to make the whole piano not too long, and to not have a string deflect so far that it bumps into other strings. So then the other way to lower pitch is to increase the mass of the string (thus an increase in thickness); since the thickness is increased, the factors mentioned in the other posts above become relevant. https://en.wikipedia.org/wiki/Piano_acoustics