The Shapley value averages over your marginal contribution to utilities of sub coalitions. The guy who votes against Hitler would be involved in some sub coalitions in which he is the marginal vote that defeats Hitler, and thus would have a positive Shapley value, where the guy who voted for Hitler would be involved in some sub coalitions where he is the marginal vote that elects Hitler, and thus would have a negative Shapley value.
I think Yvain is right and you’re wrong. The Shapley value takes as input the whole game, not a certain play of the game, so it doesn’t know that you actually voted for Hitler and the other guy didn’t.
The formula for the Shapley value (from the wiki article):
What this means is you take all sub coalition S of the total coalition N, excluding sub coalition that include yourself. Then average over the difference in value of the sub coalition S plus yourself and just the sub coalition S. (The first term in the sum makes it a weighted average depending on the sizes of S and N.) These sub coalitions S, and S plus yourself, did not actually happen, you are considering the counterfactual value of those being the actual coalitions.
The point is that the formula knows how your inclusion in a coalition changes its value.
The Shapley value averages over your marginal contribution to utilities of sub coalitions. The guy who votes against Hitler would be involved in some sub coalitions in which he is the marginal vote that defeats Hitler, and thus would have a positive Shapley value, where the guy who voted for Hitler would be involved in some sub coalitions where he is the marginal vote that elects Hitler, and thus would have a negative Shapley value.
I think Yvain is right and you’re wrong. The Shapley value takes as input the whole game, not a certain play of the game, so it doesn’t know that you actually voted for Hitler and the other guy didn’t.
The formula for the Shapley value (from the wiki article):
What this means is you take all sub coalition S of the total coalition N, excluding sub coalition that include yourself. Then average over the difference in value of the sub coalition S plus yourself and just the sub coalition S. (The first term in the sum makes it a weighted average depending on the sizes of S and N.) These sub coalitions S, and S plus yourself, did not actually happen, you are considering the counterfactual value of those being the actual coalitions.
The point is that the formula knows how your inclusion in a coalition changes its value.