Z-number Game-Theoretic BWM (Adesina, Yazdi and Omidvar, 2022)

Four things change.

The input layer splits in two. Crisp BWM has a single input layer: comparisons of criteria against the best and worst on a 1–9 scale. Here there are two separate layers. The first is the same as crisp BWM: experts' best/worst comparisons for weighting the criteria (or scenarios), averaged if they come from more than one expert. The second is a separate table showing what "payoff" each alternative earns on each criterion or scenario. Every cell of this table is a Z-number, a pair carrying both a fuzzy judgement (how the expert assesses the alternative) and the degree of confidence placed in that judgement.

Solving the weights stays the same. The linear model that crisp BWM uses, minimising the deviation of the best-to-others and others-to-worst ratios, is used unchanged here; multiple experts' comparisons are averaged, and a single weight vector and consistency ratio (CR) are solved. BWM stands entirely on its own here and does exactly what the base method does.

The ranking is built through a game. Crisp BWM stops here; the weights are the output. This extension continues. The payoff table is converted into a two-player zero-sum game. One player (the decision-maker) chooses among alternatives to maximise their payoff; the other player (criteria/scenarios) tries to minimise that same payoff. This game's Nash equilibrium is solved with a linear programming model, yielding a mixed strategy, a probability distribution, for each player. These mixed strategies are combined with the criterion weights from BWM to compute an "expected value" (G) for each alternative.

Outcome and defuzzification. The result is a single G figure per alternative; alternatives are ranked from the highest value to the lowest. It is common in a zero-sum game's equilibrium for several alternatives to come out with a G of zero. The source paper recommends removing the zero-G alternatives from the table in this case and resolving the game, repeating this until every alternative is ranked.

A limitation in DecisionMind's current implementation. The source paper (Adesina, Yazdi and Omidvar, 2022) never published the Z-number payoff table itself in its own worked example (the 2019 Gorgan flood disaster, four emergency-response routes); it shared only the game's Nash equilibrium results (mixed strategies, ν values, the expected value of the best alternative). Rather than fabricating the missing payoff table, DecisionMind carries the Nash equilibrium figures published in the paper as fixed constants. The consequence is this: for alternatives whose G comes out at zero (all three of them, in this example), the paper's iterative elimination step is NOT applied in DecisionMind today; the engine leaves these alternatives tied and does not reproduce the paper's secondary ranking. This has been flagged as one of nine methods, out of a 571-manifest catalogue scan, whose golden order does not match the engine's output; the detail is in the approval note.

G is neither a weight nor a probability. It shows an alternative's expected payoff in this game's equilibrium, and it is meaningful only for this particular alternative set and this particular payoff table. The alternative with the highest G is the one preferred in the game's equilibrium. A G of zero means that alternative receives no share of selection at equilibrium; this is not a "bad" result, only "not preferred in this equilibrium."

Thus instead of writing:

"DecisionMind's full ranking can be trusted, because the engine produced an order"

the report should read:

"Because the payoff table was never published, DecisionMind's current engine only verifies the alternative with positive expected value at equilibrium; the order among alternatives whose G comes out at zero has not been separately verified in this engine"

This extension is worth considering when the decision problem is genuinely adversarial: when an opponent, a disaster, or an uncertain state of nature behaves as the "other player," and experts' confidence in their payoff estimates can also be recorded. For an ordinary supplier or investment choice, that is, a compensatory and cooperative decision setting, the zero-sum assumption (one side's gain is the other's loss) makes no sense. In that case the weights obtained from base BWM should instead be fed into an ordinary ranking method such as TOPSIS or CoCoSo.

Crisp BWM's exit condition for the weighting step applies here too: if the best and worst criterion cannot be chosen with confidence, or if the consistency ratio exceeds an acceptable limit, the weights should be reviewed before use.

Applying the zero-sum assumption to a cooperative decision. This game assumes the two sides' interests run in exactly opposite directions. Outside genuinely adversarial scenarios such as emergency or disaster response, this assumption makes no sense.

Building a Z-number from the fuzzy judgement alone, without adding the reliability component. A Z-number needs both a fuzzy assessment and a separate confidence degree; supplying only the former leaves the payoff table incomplete and the game's solution unreliable.

Treating alternatives with G=0 as "worst." A zero G shows no share of selection at equilibrium; it is not an absolute measure of badness and does not by itself establish an order among alternatives.

Trusting the full ranking when the payoff table is incomplete. In this extension's current DecisionMind implementation, only the paper's Nash equilibrium results are carried in place of an unpublished payoff table; this verifies the strongest alternative but not the rest of the order.

The governing principle is this:

This extension makes BWM weights an input to a game, not its output. The weight comes from BWM, the ranking from the game's Nash equilibrium. If the payoff table is missing, only the strongest end of the ranking should be trusted; the rest of the order must be separately verified.

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