Stochastic Dominance - Statewise Dominance

The simplest case of stochastic dominance is statewise dominance (also known as state-by-state dominance), defined as follows: gamble A is statewise dominant over gamble B if A gives a better outcome than B in every possible future state (more precisely, at least as good an outcome in every state, with strict inequality in at least one state). For example, if a dollar is added to one or more prizes in a lottery, the new lottery statewise dominates the old one. Similarly, if a risk insurance policy has a lower premium and a better coverage than another policy, then with or without damage, the outcome is better. Anyone who prefers more to less (in the standard terminology, anyone who has monotonically increasing preferences) will always prefer a statewise dominant gamble.