It's commonly thought that people are more likely to accept income inequalities if they believe these arise from equality of opportunity.For example, some attribute the relative weakness of the US welfare state to the fact that Americans believe (wrongly) that the poor have plenty of chances (pdf) to get ahead.

But what exactly is the link between perceptions of fair opportunity and demand for fair outcomes? Some experiments by Eugenio Proto and colleagues at Warwick University shed some worrying light.

They got people to play an ultimatum game, where one person proposes to split ''10 between themselves and a partner. If the partner accepts the offer, both get the money agreed. If the partner rejects the offer, nobody gets anything.

The wrinkle was that the role of proposer - the advantaged position - was allocated by lottery which was either fair (giving people a 50-50 chance) or biased.

They found that when the lottery was fair, responders on average rejected offers of ''2.15 or less. But when the lottery gave people zero chance of being a proposer, responders rejected offers of ''2.96 or less.

This corroborates common sense. If people feel they have unequal chances, they'll demand more equal outcomes.

But what about intermediate levels of unfairness? When the chances of being a proposer were 20%, responders rejected offers of ''2.39 or worse. And when the chances were 1%, they rejected offers of ''2.53 or worse.

We can put this another way. The difference between a zero and one per cent chance is worth 43p - the difference between ''2.96 and ''2.53. But the difference between a 1% chance and 50% chance is only worth 38p - the difference between ''2.53 and ''2.15.

In other words, what people value most is having some chance - 1% versus nothing. Having greater opportunity - moving from a 20% to 50% chance isn't so valuable, in the sense that it doesn't so much affect bargaining behaviour.

This has (at least) two implications.

One is that it shows that people value procedural utility; having some chance to affect the outcome is what matters to them, rather than having a big chance.

The other is that, once some small chance exists, people don't greatly care about equalizing chances. They care more about having a small chance than about increasing their chances. These laboratory experiments confirm what we see in public opinion - that, like it or not, there is little demand for policies that would greatly equalize opportunity, such as abolishing private education or steep inheritance taxes.

I don't know why this is; I suspect it's related to the certainty effect. But surely lefties should be disquieted by the fact that egalitarian impulses can be bought off so cheaply even in laboratory conditions.