Conditional Acceptance in School Choice

We provide a systematic analysis of the conditional acceptance mechanism in the standard school-choice model. Students play a strategic game in which they may submit preference lists of length at most κ (the cap). Equilibrium sets are nested in the cap (Theorem 1) and coincide with the set of stable matchings for κ ≤ 2 (Propositions 1 and 2), whereas unstable equilibrium outcomes can arise for every κ ≥ 3 (Example 2 and Corollary 1). Our main results compare conditional acceptance with deferred acceptance. Ergin-acyclicity is sufficient (Theorem 2), but not necessary (Example 3), for conditional acceptance to implement the set of stable matchings, so conditional acceptance implements the set of stable matchings whenever deferred acceptance does (Corollary 2). Strikingly, under acceptable-only reports, every unstable conditional-acceptance equilibrium outcome is also a deferred-acceptance equilibrium outcome (Theorem 3). Overall, conditional acceptance may outperform deferred acceptance in producing stable equilibrium outcomes, but cumulative removal makes strategic mistakes more costly.