======================================================================
SOURCE FILE:     hillinger_t3_arbitrariness.yaml
TABULATED FILE:  hillinger_t3_arbitrariness_tabulated.txt
======================================================================

election_title: "Hillinger Table 3 — one approval result, two opposite Borda winners"

scenario_description: |-
  Claude Hillinger, "Voting and the Cardinal Aggregation of Judgments" (Munich
  Discussion Paper 2004-9), section 10, Table 3 — his answer to the
  Saari-van Newenhizen (1988) criticism of Approval Voting.

  SVN's argument: a voter with the strict preference a > b > c cannot express it
  on an approval ballot. They must arbitrarily pick (1,0,0) or (1,1,0), and one
  can construct profiles where that arbitrary choice decides the election. So
  Approval - and, SVN generalize, cardinal voting at large - is indeterminate.

  Hillinger's inversion: the argument only bites if the STRICT ORDERINGS are the
  true preferences. Assume instead that the approval marks are what the voters
  actually mean, and it is the rankings that become arbitrary - because a coarse
  score under-determines the ranking in exactly the same way.

  These are the seven approval ballots of his Table 3. Ada 5, Ben 2, Cora 4;
  Ada wins. Now complete each ballot to a strict ranking. TWO completions are
  consistent with these very marks:

    reading 1   3x Ada>Ben>Cora   2x Ben>Cora>Ada   2x Ada>Cora>Ben
                Borda: Ada 10, Ben 7, Cora 4   ->  ADA wins
    reading 2   3x Ada>Cora>Ben   2x Ben>Cora>Ada   2x Cora>Ada>Ben
                Borda: Ada  8, Ben 2, Cora 11  ->  CORA wins

  Same ballots, same voters, opposite Borda winners. The ordinal formalism is as
  under-determined by the marks as the marks are by the formalism - which is
  Hillinger's point, and it is a genuine standoff rather than a refutation.

  Teaching page: 04_Approval/01_Learn/approval_indeterminacy.md
  Concept page:  07_Concepts/topics/cardinal_utility.md

voting_method: Approval
num_winners: 1

# Hillinger (2004), Table 3, the AV row: (1,0,0) x3, (0,1,1) x2, (1,0,1) x2.
# Written one row per voter; collapse_ballots regroups them as the paper does.
ballots: |-
  Ada,Ben,Cora
  1,0,0    # approves Ada only
  1,0,0    # approves Ada only
  1,0,0    # approves Ada only
  0,1,1    # approves Ben and Cora
  0,1,1    # approves Ben and Cora
  1,0,1    # approves Ada and Cora
  1,0,1    # approves Ada and Cora

expected_winners:
  - Ada

# file: hillinger_t3_arbitrariness.yaml

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TABULATION RESULTS
======================================================================

--- Approval Voting (single winner) ---
 Tabulating 7 ballots (any non-zero score = approval).

Ballots:
   columns = Ada, Ben, Cora      (1 = approve; 0 = not approved)
     3 × 1,0,0
     2 × 0,1,1
     2 × 1,0,1

   Ada  -- 5 (71%) -- Elected
   Cora -- 4 (57%)
   Ben  -- 2 (29%)

[Approval Distribution] (how many candidates each ballot approved)
   11 approvals across 7 ballots — average 1.6 of 3 (range 1–2).
     approved 1: 3 ballots
     approved 2: 4 ballots

[Co-Approval Matrix]
 Of the voters who approved the ROW candidate, the % who ALSO approved the COLUMN candidate.
         |  Ada   |  Cora  |  Ben   |
   ----------------------------------
   Ada   |   --   |  40%   |   0%   |
   Cora  |  50%   |   --   |  50%   |
   Ben   |   0%   |  100%  |   --   |

Winner — Approval Voting (single winner)
  Ada