Equal opportunity: a thought experiment

This report from the Sutton Trust shows that poor but bright kids are slipping behind richer stupider ones, showing that parental background strongly influences life-chances. How worried should we be by this?
A little thought experiment will help. Imagine two equally rich societies, A and B, both with GDP of 30.
In society A, nine people have incomes of 1 and the other of 21. This society has equal opportunity, so everyone has a 1 in 10 chance of having 21 and a 9 in 10 chance of getting 1. Expected income for everyone is 3: ( 0.1 x 21) + (0.9 x 1).
Now, in society B, nine people have incomes of 2 and one has 12. However, one person has a 30% per chance of getting 12, whilst the other nine each have only a 7.8% chance. Opportunity is unequal.
In this society, the nine people have expected income of just under 2.8: a 7.8% chance of 12 and a 92.2% chance of 2.
Which society is better - A or B? It depends in large part upon risk aversion. If they were indifferent to risk, the nine less favoured people in B would prefer society A. Their expected income would rise from 2.8 to 3.
However, if they had Rawlsian maximin preferences, they'd clearly prefer B: the worst-case outcome is twice as good in B as in A.
For more plausible risk preferences - modest risk aversion - there's little in it. The choice between equal opportunity with big inequality of income versus unequal opportunity with greater equality of outcome is therefore a close call.
One consideration here is how big incomes of 1 and 2 are. If an income of 1 is a lot, people might prefer equal opportunity, as they have a lot to fall back on. But if 1 means grinding poverty and 2 means a decent life, they'll prefer B.
There's another consideration - other-regarding preferences. These enter in two conflicting ways.
First, people might object to society B because they hate the thought of someone doing well through a rigged lottery. Such people might prefer A, even though they are risk averse and so would prefer society B for themselves.
This, though, raises the question of what status other-regarding preferences have in politics? Should we pander to envy?
Second, assuming that societies A and B are equally rich begs the question. If people work hard in order to give their kids better opportunities, society B - which gives them superior chances - will be richer. If this effect is strong enough, and taxation sufficiently redistributive, the worst-case income in B might be greater than the expected income in A.
So, why is equal opportunity so valuable? 

Comments

[Matthew]

"For more plausible risk preferences - modest risk aversion - there's little in it."

Really? Aren't you better off 90% of the time in the second scenario. I think even mild risk seekers would prefer it.

[reason]

And what does this have to do with the real world? Why does increasing equality of opportunity also increase inequality of outcome?

[chris]

Reason - I've no reason to suppose it does. I constructed the example to pose the contrast between an inegalitarian society with equal opportunity vs a more equal one without equal opportunity. The point being to ask why one would prefer equality of opportunity to more equality of outcome.

[Jay]

Personally, I would choose equal opportunity with inequality in income simply because it gives everyone a fair shot at the prize. If you're one of those who got the higher income, then that means you've earned your way to it. If you're included in the lot who are at the bottom, that means you only have to try harder. My decision sounds a little too simple, I know. Still, I'm all for equal opportunity. If the situation were the reverse, what would those without the proper opportunity do?

[reason]

However the example was cleverly crafted to confuse the issue. Most of the people who want equality of opportunity also want a public safety net, because the rich already have a private one.

[Company Formation India]

Of course the dumbest of rich people have access to those resources which the intelligent poor cannot even think of.

[smally lerned]

Jay:
"If you're one of those who got the higher income, then that means you've earned your way to it."

That doesn't follow. A lottery might offer all contestants an equal shot at the prize, but that doesn't mean the winner "earned" it.