Impartial selection with additive guarantees via iterated deletion

Javier Cembrano, Felix A. Fischer, David Hannon, Max Klimm

Abstract: Impartial selection is the selection of an individual from a group based on nominations by other members of the group, in such a way that individuals cannot influence their own chance of selection. For this problem, we give a deterministic mechanism with an additive performance guarantee of O(n(1+κ)/2)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mi is="true">κ</mi><mo stretchy="false" is="true">)</mo><mo stretchy="false" is="true">/</mo><mn is="true">2</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math> in a setting with n individuals where each individual casts O(nκ)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi is="true">κ</mi></mrow></msup><mo stretchy="false" is="true">)</mo></math> nominations, where κ∈[0,1]<math><mi is="true">κ</mi><mo is="true">∈</mo><mo stretchy="false" is="true">[</mo><mn is="true">0</mn><mo is="true">,</mo><mn is="true">1</mn><mo stretchy="false" is="true">]</mo></math>. This bound is O(n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msqrt is="true"><mrow is="true"><mi is="true">n</mi></mrow></msqrt><mo stretchy="false" is="true">)</mo></math> for κ=0<math><mi is="true">κ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">0</mn></math> and O(n)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo stretchy="false" is="true">)</mo></math> for κ=1<math><mi is="true">κ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">1</mn></math>. The latter is trivial, as even a mechanism that never selects provides an additive guarantee of n−1<math><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn></math>. We show, however, that it is also best possible: for every deterministic impartial mechanism there exists a situation in which some individual is nominated by every other individual and the mechanism either does not select or selects an individual not nominated by anyone.