If we are considering it from an individual perspective, then we need to hold the other individuals fixed, that is, we assume everyone else sticks to the tails plan:

In this case: 10% chance of becoming a decider with heads and causing a $1000 donation 90% chance of becoming a decider with tails and causing a $100 donation

That is 0.1 * 1000 + 0.9 * 100 = $190, which is a pretty bad deal.

If we want to allow everyone to switch, the difficulty is that the other people haven't chosen their action yet (or even a set of actions with fixed probability), so we can't really calculate expected values.

One way to approach this is to imagine that a random decider will be given a button and if they press it, everyone will change their view going with heads. The problem is that when heads comes up, the single decider will have 9 times the probability of getting the button than any of the individual deciders when tails comes up. So we get:

50% chance of causing a $1000 donation and 50% of causing a $100 donation. This is very similar to what Saturn was saying. However, there are subtle, but important differences.

Let's suppose now that each decider gets a button and that they all press it or don't press it due to being identical. If one presses a button and there is one decider, then they get the full effect. If there are 9 deciders, then the easiest way to model this situation will be to give each button one 1/9th of the effect. If we don't do this, then we are clearly counting the benefits of everyone switching 9 times.