Influence in private-goods allocation

We reinterpret the ‘bossiness’ of a private-goods allocation rule (Satterthwaite and Sonnenschein, 1981) as the ability of an agent to ‘influence’ another’s welfare with no change to her own welfare. In applications where non-bossiness is not possible, we propose simple conditions on (1) which agents may have influence (acyclicity and preservation), and (2) the welfare consequences of influence (positivity and oppositeness). We apply these conditions to three well-known bossy rules: the ‘Vickrey rule’ in single-object auctions (Vickrey, 1961) (acyclic, positive), the ‘doctor-optimal stable rule’ in matching with contracts (Hatfield and Milgrom, 2005) (acyclic, positive, preserving) and ‘generalised absorbing top-trading cycles (GATTC) rules’ in housing markets with indifferences in preferences (Aziz and Keijzer, 2011) (acyclic, opposite, preserving). Under mild restrictions, we show how the nature of influence under a strategy-proof rule determines whether or not it satisfies weak group-strategy-proofness (requires acyclicity and either positivity or preservation), weak Maskin monotonicity (acyclicity and positivity) and Pareto-efficiency (acyclicity and oppositeness). In addition, we propose an influence-related generalisation of the efficiency-adjusted deferred acceptance mechanism in school choice (Kesten, 2010), and characterise influence for strategy-proof GATTC rules in housing markets.