Great news for people who hate doing dishes: if you have a dishwasher, washing your dishes by hand is a colossal waste of water, energy, time, and money, and you can prove it with maths.
Most people drastically underestimate how much water their taps put out. Energy Star-certified dishwashers must use less than 270 kWh per year and 15l per cycle, but most kitchen faucets move one to 7.5 litres a minute. In other words, running the tap for just four or five minutes can use more water than an entire dishwasher cycle.
Older dishwashers do use more water, but it's still probably less than you think; unless yours is more than 10 years old (and somehow still working), there's a good chance that it's more efficient than hand-washing a sink full of dishes. The exact amount of water and energy you can save depends on a number of factors, so whoever pays the utilities should probably look up the usage stats for your dishwasher and faucet to see what they're working with.
What they find might just be the push they need to finally spring for a new dishwasher.
Switching over to a fully-automated dishwashing lifestyle may take a load off your energy bills — not to mention your feet, back, and cuticles — but it won't save the world. Certain items will always need a hand wash, and anyway, your shower and washing machine use more water than anything in the kitchen. Still, it's an easy way to save water with basically zero drawbacks. I think they call that a "win-win."