The cartogram, or value-by-area map, is a popular technique for cartographically representing social data. Such maps visually equalize a basemap before mapping a social variable by adjusting the size of each enumeration unit by a second, related variable. However, to scale the basemap units according to an equalizing variable, cartograms must distort the shape and/or topology of the original geography. Such compromises reduce the effectiveness of the visualisation for elemental and general map-reading tasks. Here we describe a new kind of representation, termed a value-by-alpha map, which visually equalizes the basemap by adjusting the alpha channel, rather than the size, of each enumeration unit. Although not without its own limitations, the value-by-alpha map is able to circumvent the compromise inherent to the cartogram form, perfectly equalizing the basemap while preserving both shape and topology.