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What makes a voter a critical voter in a voting coalition?

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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, a voter is considered a critical voter within a voting coalition if their support is necessary for a measure to pass. Specifically, a voter is critical if their presence transforms a losing coalition into a winning one, or conversely, if their defection from a minimal winning coalition renders that coalition a losing one.1

The concept of a critical voter is central to measuring voting power, such as the Banzhaf value. In this framework, voting power is not determined by the number of votes a member casts, but rather by the frequency with which that member acts as a critical voter across all possible winning combinations. If a voter is never critical, they are referred to as a "dummy," meaning their vote has no influence on the outcome, regardless of how many votes they possess.1