The Spread That Informed Flow Demands

You are the sole market maker in a one-off contract. Everyone agrees the contract will settle at either $\$90$ or $\$110$, each equally likely, so its unconditional fair value is $\$100$.

Orders arrive one at a time, in unit size. Each order comes from an informed counterparty with probability $\alpha = 0.2$ -- that trader already knows the settlement value, and will buy from you whenever your offer sits below it and sell to you whenever your bid sits above it. Otherwise (probability $0.8$) the order comes from a noise trader who buys or sells with probability $\tfrac{1}{2}$ each, regardless of price.

(a) Why is quoting $\$100$ bid at $\$100$ offered a losing strategy, even though $\$100$ is the honest fair value? (b) Find the bid and offer at which you break even on the next order. (c) What happens to your quote if informed flow rises to $\alpha = 0.4$?