Stable Marriage

• Basic tools in probability • Expectations • High probability events • Deviations from expectation Coupon collecting. • n coupon types. Get a random one each round. How long to get all coupons? • general example of waiting for combinations of events to happen. • expected case analysis: – after get k coupons, each sample has 1 − k/n chance for new coupon – so wait for (k + 1) st coupon has geometric distribution. – expected value of geo dist w/param p is 1/p – so get harmonic sum – what standard tools did we use? using conditional expectation to study on phase; used linearity of expectation to add – expected time for all coupons: n ln n + O(n). Problem: • complete preference lists • stable if no two unmarried (to each other) people prefer each other. • med school • always exists. Proof by proposal algorithm: • rank men arbitrarily • lowest unmarried man proposes in order of preference • woman accepts if unmarried or prefers new proposal to current mate.