1. A control device, comprising: a memory that stores: a pre-change scale; and a computer-executable scale change program; a display device that displays a map; and a processor that, when executing the stored scale change program: specifies an operation amount of a user operation that changes a scale of the displayed map; accesses the stored pre-change scale that is a scale of the displayed map before the user operation; and based on the accessed pre-change scale and the specified operation amount of the user operation, sets a final scale that is a value of the scale after the user operation by changing the scale of the displayed map in a stepwise manner from the pre-change scale to the final scale within a prescribed scale range, a number of steps of the stepwise change being based on a difference between the pre-change scale and the final scale, the final scale being set by: specifying a tentative final scale that is shifted from the pre-change scale by a shift amount that is based on the specified operation amount of the user; comparing the tentative final scale and a scale limit, the scale limit comprising at least one of an upper limit and a lower limit; setting the final scale equal to the scale limit when the tentative final scale exceeds the scale limit; and setting the final scale equal to the tentative final scale when the tentative final scale does not exceed the scale limit; identifying a plurality of settable scales between the upper limit and the lower limit; calculating a number of the settable scales included between the pre-change scale and the final scale, the number of steps being set in proportion to the calculated number of the settable scales; while changing the scale, calculating a number of the settable scales included between a current scale and the final scale; and repeatedly changing the scale by a shift amount corresponding to a number that multiplies the number of settable scales included between a current scale and the final scale by a prescribed percent to change the scale in a stepwise manner until the final scale is reached.