File:Cyclic preferences after Schumm 1987.gif

The example assumes that more red colors^{[note 1]} are preferred^{[note 2]} over more blue colors,^{[note 3]} but 9 and 5 are indistiguishable for the human eye,^{[note 4][note 5]} as are 5 and 1. Furthermore it assumes that a triple (a,b,c) is preferred^{[note 6]} over a triple (p,q,r) e.g. if a>p and b~q and c~r.^{[note 7]}

But this implies the following paradoxical preference cycle: (9,5,1) > (1,9,5) > (5,1,9) > (9,5,1).

This cycle is illustrated in the picture, with each triple depicted as a column.

• The leftmost column is preferred over the 2nd from left since its top color is clearly more red, while their middle and their bottom colors are indistinguishable.
• Similarly, the 2nd column from left is preferred over the 2nd from right since its middle color is more red while top and bottom colors are indistinguishable.
• And the 2nd column from right is preferred over the rightmost column since its bottom color is more red, while top and middle colors are indistinguishable.

However, the leftmost column is identical to rhe rightmost one, so the preferences form a cycle.

Notes:

References: