Why were some members powerless in the European Economic Community voting system?

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A colorful payoff matrix for a normal form game in game theory, showing strategy options-support, oppose, evade-for both party A and party B. Each cell contains cartoon figures representing each party's action and lists percentage payoffs for A and B. The bottom-right cell is labeled saddlepoint, highlighting the equilibrium. — game theory Table 1
The normal-form table illustrates the concept of a saddlepoint, or entry, in a payoff matrix at which the expected gain of each participant (row or column) has the highest guaranteed payoff.

In the context of game theory, some members of the European Economic Community (EEC) voting system were powerless because their assigned voting weight did not grant them the ability to act as a "critical voter" in any minimal winning coalition.1

From 1958 to 1973, Luxembourg was considered a "dummy" because its single vote could never change the outcome of a Council of Ministers decision, which required a qualified majority of 12 out of 17 votes. Because the votes of the other five member countries were all even numbers, a winning coalition of exactly 12 votes could never include Luxembourg’s odd-numbered vote. Furthermore, while a 13-vote coalition could theoretically include Luxembourg, the country's defection from such a group would never render the coalition a losing one.1

This illustrates a fundamental principle in voting power analysis: voting weight is not synonymous with actual power. A member's influence is determined by the frequency with which they can cast a critical vote, rather than the number of votes they possess.1