Answer by Dan Munro:
Earlier this month I wrote an answer to a question that also has some applicability here.
The applicability is that the only real criteria I find among all the really "efficient" healthcare systems around the world is Universal Health Coverage (UHC). How UHC is paid for isn't as important to efficiency as just having UHC.
Bloomberg built an "efficiency" score based on three criteria:
- Life Expectancy (60%)
- Relative Per Capita Cost (30%)
- Absolute Per Capita Cost (10%)
They then applied these to countries with at least 5 million in population, GDP per capita of at least $5,000 and a life expectancy of at least 70 years.
Countries were scored on each criterion and the scores were weighted and summed to obtain their efficiency scores. Relative cost is health cost per capita as a percentage of GDP per capita. Absolute cost is total health expenditure, which covers preventive and curative health services, family planning, nutrition activities and emergency aid. 
I was curious to see how the top ten countries fared relative to type of Universal Health Coverage (UHC) so I added two columns (UHC Type and Adoption) .
Single payer systems represented 4 of the top 10 countries – so 40% – but the top 2 countries were both "two tier." Here are the top 10 countries:
Hong Kong and Singapore are literally neck-and-neck – and Japan isn't far behind. Japan scores extra points for population size.
… and here are the bottom 10 countries:
There we are – between Iran and Serbia (in terms of efficiency).
Clearly our Life Expectancy is much better – and even though it represents 60% of the weighting – it's not enough to offset the HUGE cost (both relative and absolute).
For those wondering, the 3 types of UHC referenced are:
Single Payer: The government provides insurance for all residents (or citizens) and pays all health care expenses except for copays and coinsurance. Providers may be public, private, or a combination of both.
Two-Tier: The government provides or mandates catrastrophic or minimum insurance coverage for all residents (or citizens), while allowing the purchase of additional voluntary insurance or fee-for service care when desired. In Singapore all residents receive a catastrophic policy from the government coupled with a health savings account that they use to pay for routine care. In other countries like Ireland and Israel, the government provides a core policy which the majority of the population supplement with private insurance.
Insurance Mandate: The government mandates that all citizens purchase insurance, whether from private, public, or non-profit insurers. In some cases the insurer list is quite restrictive, while in others a healthy private market for insurance is simply regulated and standardized by the government. In this kind of system insurers are barred from rejecting sick individuals, and individuals are required to purchase insurance, in order to prevent typical health care market failures from arising. 
How countries elect to pay for Universal Health Coverage is variable. What isn't, however, is UHC.
 (last updated August, 2013 – but I can tell by some of the numbers that they're probably using data from 2010 – possibly earlier)