SMAA-2 (Stochastic Multicriteria Acceptability Analysis 2)

SMAA-2 is a ranking approach, a direct continuation of SMAA, developed for situations where you hold a decision table but do not know the criterion weights with certainty. SMAA looks only at "what is the probability that this alternative comes first"; yet between two alternatives, one that never comes first but always comes second may still be valuable. SMAA-2 captures this detail: it calculates how often each alternative lands in each rank, first, second, third and so on (the full rank distribution), then weights these ranks by the logic that "being first is worth more than being second, and second more than third," combining them into a single holistic acceptability score. Lahdelma and Salminen proposed it in 2001 to extend SMAA in this direction for group decisions.

SMAA-2's underlying idea is that "looking only at first place loses information." An alternative that never comes first under any weight combination but comes second under almost every combination is far stronger than one that "always finishes last"; yet SMAA's form, which counts only the first rank, does not show this difference. SMAA-2 takes the whole set of ranks into account and combines them using a decreasing sequence of coefficients, called "meta-weights," that values first place the most and last place the least. In this way, an alternative that is "almost always second" can receive a higher holistic score than one that is "half the time first, half the time last."

This idea carries a consequence: SMAA-2 preserves SMAA's consensus-seeking philosophy but is a more detailed tool, able to distinguish being "consistently good" from being "sometimes excellent, sometimes poor." In group decisions, where different stakeholders argue for different weights and the search is for an alternative that "condemns no one to the worst outcome," this distinction is especially valuable.

The method proceeds through six steps.

First, defining the uncertainty and the meta-weights. The uncertainty in the measurement values and any prior information about the weights are defined; in addition, a meta-weight sequence, decreasing from first place to last, is chosen to answer the question "how much more valuable is first place than second?" If there is no prior information, a standard decreasing sequence (central weights) is used by default.

Second, drawing thousands of samples. A large number of random samples (typically tens of thousands) are drawn from the measurement uncertainty and the weight distribution.

Third, calculating a total score for every sample. In each sample, every alternative's weighted total score is calculated.

Fourth, extracting the full rank distribution. Unlike SMAA, it is not just the first rank but how often each alternative lands in each possible rank (first, second, third…) that is counted separately. This is called the "rank acceptability matrix."

Fifth, calculating the holistic score. Every alternative's frequency at each rank is multiplied by the meta-weights from step one and summed; this is the alternative's holistic acceptability score. The central weight vectors are also calculated here.

Sixth, ranking and the confidence factor. Alternatives are ranked by their holistic score; the calculation is redone with each alternative's central weight to produce a confidence factor.

The formulas behind each step, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.

The holistic acceptability score is a summary, using a weighting that values first place more highly, of an alternative's performance across not just first place but every rank. This score is not a percentage or a probability; it is an index used to compare the relative strength of different alternatives. A high holistic score means "either the alternative comes first often, or it almost never falls into a poor rank"; the difference between these two cannot be understood without looking at the rank acceptability matrix. The holistic score should not be confused with the rank-1 index (the rate of coming first alone); an alternative can have a low rank-1 index and still receive a high holistic score.

Thus instead of writing:

"SMAA-2 found the best alternative"

the report should read:

"Taking every rank into account, with the extra value given to first place, this alternative has received the highest holistic score; this alternative's rate of coming first alone is a different figure and should be reported separately"

SMAA-2 works with probabilistic (stochastic) data: measurement values are treated either as exact or with an uncertainty distribution, and weights too are treated as a probability distribution rather than exact numbers. DecisionMind currently has no separate extension of SMAA-2; its close relative SMAA is the simpler form that considers only the first rank.

You need alternatives in rows, criteria in columns, a measurement value in every cell (optionally with a margin of uncertainty), and direction information for every criterion. SMAA-2 asks for no exact external set of weights. In addition, a meta-weight sequence must be defined (how much more valuable first place is judged to be than second, and second than third); if there is no preference, a standard decreasing sequence is used. As the number of alternatives grows, the size of the rank acceptability matrix (a number of cells equal to the square of the number of alternatives) grows with it; the number of samples should be kept in the tens of thousands.

Where there is no consensus on the weights and the question is not just "who is first" but also "who is consistently good," SMAA-2 is suitable, particularly in group decisions and multi-stakeholder processes. Its typical territory includes major infrastructure projects, public policy, and decisions requiring group consensus.

It should not be used in choice situations where only first place matters and "coming second or third" carries no value; in that case, SMAA's simpler form is enough and SMAA-2's extra complexity is unnecessary. Likewise, if stakeholders cannot agree on how the meta-weight sequence should be chosen, this extra source of uncertainty can make the result even more contentious.

Weights uncertain, only first place matters → SMAA

Weights uncertain, consistent goodness also matters, a group decision → SMAA-2

Weights already clear, a single exact result is wanted → TOPSIS, SAW, VIKOR

No consensus on the meta-weights either → the meta-weight discussion first, then SMAA-2

SMAA-2's greatest strength is that it recovers the information lost by SMAA's focus on first place alone; it can distinguish an alternative that is "consistently good" from one that is "sometimes excellent, sometimes poor." In group decisions, it is especially suited to finding a consensus alternative that does not entirely exclude any stakeholder group. The meta-weight sequence is an adjustable parameter; the preference between "only first place matters" and "consistent goodness matters" can be expressed explicitly.

Its limitations stem from the method's added complexity. First, the choice of meta-weight sequence (how much more valuable first place is judged than second) directly affects the result, and there is no single "correct" answer to this choice (Lahdelma and Salminen, 2001, §4.2). Second, the size of the rank acceptability matrix grows with the square of the number of alternatives; with many alternatives, interpreting the whole matrix becomes difficult, and usually only the holistic score is examined while the detail is missed. Third, in its basic form, because it uses an additive score function, it carries the same compensatory logic as SMAA. Fourth, if the number of samples is not kept large enough (under ten thousand), both the rank acceptability matrix and the holistic score can come out unstable (Tervonen and Figueira, 2008).

The most common mistake is treating the meta-weight sequence as "a probability distribution summing to one," whereas the standard definition fixes the coefficient of first place at 1 rather than normalising the sum to 1. A second mistake is confusing the holistic score with the rank-1 (first place only) index; an alternative can come out differently on the two, and both should be reported separately. A third is reading only the holistic score without ever looking at the rank acceptability matrix; this hides the answer to the question "why did this alternative come out ahead?" A fourth is keeping the number of samples low and treating the result as stable.

The governing principle is this:

SMAA-2's holistic score is a summary, using the meta-weights, of an alternative's performance across every possible rank; it is not the same thing as the rate of coming first alone, and the two should be read together.