election_title: "Hillinger Table 4, rescaled — what 'cardinal' actually guarantees"

scenario_description: |-
  The SAME 30 voters and the SAME opinions as hillinger_t4_ev3.yaml, with every
  score put through the positive affine transformation u' = 2u + 1. Hillinger's
  EV-3 marks (0, 1, 2) become (1, 3, 5).

  This is the definition of a cardinal scale made runnable. A utility scale is
  called CARDINAL when it is unique up to a positive affine transformation —
  meaning you may slide the origin and stretch the unit freely, and every
  question the scale is entitled to answer keeps the same answer. Hillinger
  makes the point in §5 ("The choice of x0 does not affect the outcome") and
  again in §6, where he notes that the scales of the physical sciences are
  arbitrary up to a linear transformation in exactly this way — "Only, when
  adding different measurements, we must use the same scale!"

  So compare the two runs. The totals move: Ana 39 -> 108, Bruno 29 -> 88,
  Chloe 22 -> 74 (each total is 2x the old one plus 30, one point per ballot).
  Nothing that decides the election moves: the same two finalists, the same
  winner, the same runoff margin, the same ordering.

  The caveat is the one Hillinger flags with an exclamation mark, and it is the
  live objection to every score method: the invariance holds because EVERY voter
  was rescaled together. Let two voters use different scales and the sum is no
  longer a rescaling of anything — it is a different election. That is the
  interpersonal-comparability problem, and it is not solved by this file.

  Concept page: 07_Concepts/topics/cardinal_utility.md

voting_method: STAR
num_winners: 1

# u' = 2u + 1 applied to every mark of hillinger_t4_ev3.yaml: 0->1, 1->3, 2->5.
ballots: |-
  Count:Ana,Bruno,Chloe
  9:5,3,1    # Ana > Bruno > Chloe
  10:3,5,1   # Bruno > Ana > Chloe
  11:3,1,5   # Chloe > Ana > Bruno

expected_winners:
  - Ana

# file: hillinger_t4_affine.yaml