The tontine puzzle
研究了唐提式养老金与传统终身年金在现金流分布上的优劣，发现设计合理的唐提式养老金在参与者达到一定数量后，对任何偏好递增且连续的退休者都更具吸引力，但现实中此类市场却发展缓慢。
Retirement income tontines pool idiosyncratic and systematic longevity risk among their participants. This paper studies conditions under which the cash flow distribution of a tontine dominates that of a traditional life annuity. Building on the almost first order stochastic dominance (AFSD) criterion of Leshno and Levy (2002), we show that properly designed tontines dominate equally priced constant annuities whenever the latter embed actuarial loadings and/or the retiree underestimates the survival probabilities assumed in annuity pricing. In large pools the AFSD relation converges to classical first order dominance. Hence for any preference that is increasing and continuous in consumption, including but not limited to generalized expected utility (GEU) and cumulative prospect theory (CPT) frameworks, retirees prefer a tontine once the pool reaches a modest size. We quantify the minimum number of participants required for dominance under both normative (GEU) and descriptive (CPT) benchmarks and find that one or two digits are typically sufficient. These findings constitute what we label the tontine puzzle: despite their theoretical appeal to retirees, real world tontine markets remain nascent. 1