In this paper, we empirically analyze a market for annuity contracts. To this end, we use multi-stage and multi-attribute auctions, where life insurance companies that have private information about their annuitization costs compete for retirees' savings by offering pensions and bequests, and retirees choose \winners" that maximize their expected present discounted utilities. We establish the identification of our model parameters and estimate them using rich administrative data from Chile. Our estimates suggest that (i) retirees with lowest savings quintiles value firms' risk ratings (e.g., AAA, AA) the most, while the high savers do not; (ii) almost half the retirees who choose an annuity do not value bequest, and (iii) firms are more likely to have low annuitization cost for retirees in the top-two savings quintiles. Counterfactuals show that private information about costs harms only these high savers, and simplifying the current mechanism by using English auctions and \shutting-down" risk ratings lead to higher pensions in equilibrium, but only for the high savers.