Depositors and banker are risk-neutral. The banker observes the return costlessly, but the depositors must pay information cost I in order to know which happened . Depositors can get outside safe return of S on 1 unit invested. Otherwise pays contracted return P to depositors. Consumers decide whether to buy info for I. Consumers decide whether to force liquidation.
Banker decides whether to pay or abscond.
Information costs are exactly balanced by information benefits. No, so long as all runs are justified by bad info. No, unless the government has lower info costs than private institutions. The three models yield quite different views of bank runs and bank insurance. Bank insurance is just a subsidy to risky investments.