Bentham insists that governments are not ends in themselves but instruments for securing the welfare of those they govern. The sole legitimate purpose of any government is to produce the greatest happiness of the greatest number of its members. When government fails this test — when it produces misery rather than happiness, or serves the interests of a ruling few at the expense of the many — the duty of political allegiance is weakened.
Bentham holds that the duty of obedience is contingent on utility: people ought to obey so long as the probable mischiefs of obedience are less than the probable mischiefs of resistance. This is not a permanent blank cheque to established power. When a government consistently produces more misery than happiness, the case for reform — and, in extreme circumstances, revolution — becomes compelling. The greatest happiness principle is therefore as much a standard for reform as a basis for authority.
The phrase "greatest happiness of the greatest number" appears in the Preface to the Fragment on Government; Bentham later attributed it to Priestley.