When, in 2015, Times columnist Giles Coren’s daughter Kitty failed to get into the local primary 200 yards away from where he had lived for 20 years, Coren accused his neighbours of being “liars, cheats and hypocrites”.

Sharp-elbowed parents had exploited a loophole in the admissions rules by renting a place nearer the Ofsted “outstanding” school, pretending this was their permanent home. “You are doing poor kids out of school places and homes,” Giles Coren wrote in a column entitled: “Thanks for nothing, you middle-class scum”. “You are making local schools selective on income by the back door,” he argued. Many of his neighbours agreed.

Teachers had only noticed the scam when a stream of new families notified the school office of “a change in home address” soon after their children had started at the school.

Council officials vowed to clamp down on abuses of the school admissions system. But rumours were already swirling that the sharp-elbowed warriors had identified a new deceit: paying private doctors to provide evidence for “exceptional medical or social reasons” to catapult their children to the front of the school queue. These were ailments that, just like addresses, could mysteriously disappear once term had begun.

Coren’s daughter ended up leaving the state system and going to a private school. The real losers, he acknowledged, were children from poorer homes who had been pushed out of their local school and had no other option but to attend a state school much further away.

Like many aspects of the ever-escalating education arms race, it seems school admission battles are just not fair. The signs of the arms race are all around us: inflated house prices in the neighbourhoods of the most sought-after state schools, the boom in private tutoring outside school hours, and the increasing lengths schools and universities have to go to select their intakes as they struggle to distinguish between too many applicants.

We cling on to the hope that education can act as the great social leveller, enabling children from poorer backgrounds to overcome the circumstances they are born into. But in our book Social Mobility and Its Enemies, Stephen Machin and I report evidence, gathered over a number of decades and for a range of countries, showing that, for most children, education has failed to live up to these lofty expectations. It is an impossible task when the gap between the haves and have-nots is so wide outside the school gates. And one way the middle classes gain advantage is by ensuring their sons and daughters get the “best” school and university places.

Parental wealth can price poorer pupils out of school catchment areas. One study, by the London School of Economics, found that local houses cost £45,700 more in catchment areas of popular comprehensives in England compared with elsewhere. Those from higher social classes are more likely to deploy strategies that cost money, including moving home or hiring a private tutor to help their children. The Sutton Trust found that one in three (32 per cent) professional parents with school-aged children had moved to an area they thought had good schools.

Parents are reluctant to admit to any wrongdoing themselves but are happy to report that “others” will resort to cheating to secure that cherished school place: not just renting a property nearby, but providing the address of a relative living closer to a favoured school. Many councils have established crack units to investigate fraudulent applications.

These admissions over admissions are likely to be the tip of the iceberg. “Nothing causes parents, particularly middle-class parents, so much angst as secondary school admissions,” observed the journalist Peter Wilby. “Go to a dinner party in Islington or Edgbaston, and they will talk of little else.”

University admissions

Dinner party chatter increasingly turns to university admissions as well. Previous generations of parents packed their teenagers off to campuses expecting little say in their choice of degree. The era of fees has changed all that: armies of mums and dads have invaded university open days up and down the country. A bog-standard degree no longer counts enough; many want their sons and daughters distinguished from the rest of the pack by getting an elite degree from a prestigious university and/or a postgraduate degree.

The fierce competition for university places has fuelled the boom in private tutoring outside normal schooling hours. The percentage of children aged between 11 and 16 in England receiving private or home tuition rose by over a third in a decade, according to the Sutton Trust, increasing from 18 per cent in 2005 to 25 per cent by 2016. Studies indicate a weakening link between A-level results and intelligence test scores from earlier in children’s lives. School examination results have become as much a signal of the support children receive at home as their “natural academic ability”.

Meanwhile, bombarded by thousands of A-grade candidates, the most sought after universities are resorting to “hyper selectivity” - ever more refined but unreliable ways of selecting the “very best” academic talent. The battery of admissions criteria designed to distinguish between equally well-qualified candidates expands with each year: personal statements, teacher recommendations, school exam grades, university admissions tests, interviews, “contextual offers” and much more. In the past year, the murky practice of offering unconditional offers has come on to the market, as universities strive to fill their degree courses.

At Oxford and Cambridge, the most sought after of all the academic elites, the complexity of admissions goes to new levels: involving earlier deadlines, interviews with specialist tutors at one of 69 different independent colleges (with their own preferences and traditions), as well as an increasing battery of bespoke subject tests. For those not in the know, higher education admissions is a bewilderingly complex process to navigate.

The outcome of this time-consuming exercise is a highly socially segregated education system. More than 85 per cent of topperforming state schools take in fewer disadvantaged pupils than they should for their catchment area. Intakes of the grammar and faith schools are particularly skewed on social grounds.

The expansion of universities, meanwhile, has widened, not narrowed, the graduation gap between the rich and poor. The proportion of poorest young people earning degrees grew from 6 per cent in 1981 to 18 per cent in 2013; yet the proportion of the richest young people earning degrees went up from 20 per cent to 55 per cent.

We mount brave rearguard attempts to level the academic playing field: tightening school admissions rules, prioritising poorer pupils, lowering degree offers for poorer students. But tinkering with selection criteria can only go so far. Sharp-elbowed parents will always find ways to game the rules.

Chance - the great leveller

The only way of sweeping away this middle-class advantage is deploying lotteries alongside other selection criteria. Undeniably the most equitable way to allocate places to equally deserving candidates at oversubscribed institutions is to pick them randomly. As Phillip Collins, the Times journalist has quipped: “A lottery, of course, is fair. That is why people hate them.”

Politicians grimace at the thought of using ballots in education, but there is growing support among schools themselves. Increasing numbers of academies are deploying random allocation as part of their oversubscription criteria because they are seen as means of creating more balanced school intakes. And when explained clearly there is more support for the idea among the general public than might be imagined. The Sutton Trust found that nearly half of parents (47 per cent) would support making ballots a part of school admissions rather than only prioritising how close parents live to a school.

Opponents to random allocation fail to recognise that these are not pure ballots but tie-breakers deployed when other selection criteria are met. They might be used, for example, for children who live in a school catchment area slightly further away from a school than those living on neighbouring streets. For faith schools, meanwhile, lotteries would be used alongside binary criteria to determine religious belief (you either believe or you do not). It would help to address the cynical middle-class practice of attending church services purely to demonstrate their strength of religious belief in the run-up to admissions decisions.

Lotteries when schools are oversubscribed would also enable grammar schools to help rather than hinder social mobility. Instead of trying to distinguish between equally impressive academic performers by the slimmest of margins, the schools could agree a threshold of academic excellence deemed good enough for entry. Those who pass the grade would then be randomly picked.

Deploying lotteries also offers a potential solution to the highly pressurised world of university admissions. A university selection system that was not only fair but fit for purpose would identify a threshold of academic excellence and then select students through a ballot. Again, this would not be a pure lottery; no one wants to set young people up to fail in an intense and demanding academic environment. At the same time, we must also remind ourselves what universities are selecting for: most graduates pursue non-academic careers after university, requiring only a certain level of academic achievement.

A weighted lottery system would allow poorer students to be entered with slightly lower grades than their more privileged counterparts. To make the exercise more palatable you might compensate losers in the lottery in some way - perhaps guaranteeing a place at another similar academic institution.

Another model, adopted by Dutch medical schools, is to select the highest academic performers on academic grades, and enter lower achievers into a lottery. Deploying random allocation alongside academic criteria would have the added benefit of cutting down on the escalating costs of admissions. The evidence suggests academic standards would be upheld.

If ballots are a step too far then an alternative option is to create a British “per cent” scheme following the model trialled by universities in Texas and California. This would guarantee a university place to, say, the top 10 per cent of academic-performing pupils in each state school in a local region or across the country - irrespective of the actual grades they achieved. This would recognise the achievement of children in the context in which they grow up: getting to the top of the class in difficult circumstances means more than doing so in a highly supportive environment.

The biggest impact of a per cent system could be to attract more middle-class pupils into state schools - a likely boon for social mobility in itself. As Peter Wilby has argued, the scheme “would have the wondrous effect of encouraging the middle classes to spread their children through the school system, instead of continuing to shovel them into social ghettos”.

If ministers are true to their word about wanting to improve social mobility through education, ballots offer a powerful but pragmatic measure that would make a genuine difference. And schools and universities would be empowered to help the pupils and students who need them the most. They could be more of the social mobility engines we yearn for. Prime minister Theresa May has vowed to be driven by the interests of “ordinary, working class people”. The government’s promised review of school admissions policy and its post-18 review in England are opportunities to stand up for those interests.

Talking social mobility is easy but addressing it is hard. When urged to encourage schools to deploy random allocation to help decide admissions, generations of politicians have been frightened off by the reaction in the media, which is all too keen to paint the idea as children’s futures being determined by “a roll of the dice”.

The irony is that many children’s life prospects are damaged precisely because of the lottery of where they happen to be born and grow up. As the Nobel laureate James Heckman has argued: “Some kids win the lottery at birth, far too many don’t - and most people have a hard time catching up over the rest of their lives.” Ballots in school and university admissions can offer a second chance to roll the dice of life.

Lee Elliot Major is chief executive of the Sutton Trust. His and Stephen Machin’s book Social Mobility And Its Enemies is published by Penguin