There are basically two types of bridge pin, those with a 3 degree taper and those with a 5 degree taper.

Ideally, the hole in the bridge should be reamed out to suit the taper of the pin ... it has long been a bone of contention that Martin do not do this operation ....they drill the holes to 3/16" parallel ...and then fit the tapered pins , the result being that the ball end of the string has a vastly increased propensity to chew the underside of the bridge plate, with consequent tuning stability problems.

Even more ideally, the bridge should be slotted so that the ball end of the string rests firmly against the underside of the bridge plate instead of being sandwiched between the pin and the edge of the hole in the bridgeplate.

For an alternate view of how strings should be fitted , Google "alternate string anchoring".