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In the realm of game theory, a dummy player in a cooperative game is one whose presence or absence does not impact whether a coalition can achieve its goal. Think of a group working together to complete a project. A person who contributes no unique or critical skill to the team would be considered a 'dummy player.' In terms of power, this player does not possess any voting power that would sway decisions from failing to passing or vice-versa.

If you consider any coalition (group of players) in a game that is successful or winning, if it includes a dummy player, it remains successful even if the dummy player is removed. This is because the dummy player was never needed to reach the success initially. The player’s removal does not make or break the winning condition of the coalition.

Therefore, if a player is identified as a dummy, any coalition's status, whether winning or losing, remains unchanged by their participation. This highlights the importance of recognizing a dummy player when analyzing power dynamics in cooperative games.

The Banzhaf power index is a way to measure the power or influence of a player in a decision-making body, like a coalition or voting game. It looks at how often a player can change the outcome of a group decision by being a 'swing vote.'

When we talk about swing votes, we mean those situations where the vote of a single player can decide whether a proposal passes or fails. If a player never finds themselves in this crucial position, they would have a Banzhaf power index of zero.

For a dummy player, the Banzhaf power index captures their lack of influence. Since a dummy player cannot alter the outcome, they are never a swing vote, and therefore, their Banzhaf power index is zero. This zero index confirms their lack of significance in changing the coalition's outcome, reaffirming their status as a dummy in the coalition.

The Shapley-Shubik power index is another tool used to evaluate a player's influence in a cooperative game. This index is particularly interesting because it considers all possible orders in which players might join a coalition. It analyzes how each player can contribute to achieving a successful outcome as they join in various sequences.

For a dummy player, since they do not add any unique value to the coalition, their joining sequence is irrelevant to the coalition's success. Their influence remains constant, regardless of when they join, and this is reflected in a Shapley-Shubik power index of zero.

This index is useful because it not only considers permutations within a coalition's formation but also provides insight into each player's potential contribution throughout different stages. When a dummy player's Shapley-Shubik index is zero, it affirms that their participation has no impact, aligning perfectly with their established role as a non-critical member in coalition scenarios.