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Suppose there are two such groups. The First deals with the information by ignoring it (applying the indifference principle?).
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I would not call this the indifference principle. There are three Stages:
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[1] The Information. I see amount A.
[2] The Effect. ???
[3] The Decision. Switch or not. Call off bet or not.[/list]The indifference principle assumes that there is no effect. This produces a mathematical contradiction. Hence, there must be some effect, though it is unknown.
The people in your First group may very well realize this. They simply decide to not call off their bet, regardless of the information.
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The Second deals with the information by saying upon opening the envelope that the amount changes the experiment but they don't know how so they call off the $100 wager.
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Why should ignorance in Stage 2 lead one to think that calling off the bet in Stage 3 is the best decision?
What about the hypothetical group of Random Switchers who use their pocket calculator to generate an exponentially distributed random variable, and switch envelopes when A is less than the number they generated? They do better on average than the Always Switchers and the Never Switchers. This, by itself, should be convincing enough evidence that The Effect exists.