Nothing explains the wave of contempt for government as much as the belief that politics is for sale. Across the world, the same cries echo. “They’re all in it for themselves.” “There’s one law for the rich and another for the rest.”

As Britain’s institutions totter, defenders of the status quo protest that they are clean. The odds are that they have a case. The UK comes 14th out of 175 countries on Transparency International’s global corruption index. It would be cheering if we made the top 10, but the likelihood remains that you can live your life in Britain without needing to offer a bribe to a judge or police officer. There is one obvious exception that has been there so long we are not angered by it, but too used to it for comfort. You can shake yourself out of complacency by imagining how you would explain the House of Lords to a foreigner.

“We don’t elect its members. They are appointed and stay in Parliament until they die. You don’t understand? It’s simple, really. We can’t elect lawmakers and can’t get rid of them. We can never hold them to account.

“You want to know why they’re there? Let me see – there are still hereditary peers in Parliament for the unimpeachable reason that a long-dead ancestor slept with Charles II. We’ve Anglican bishops with nothing better to do, party loyalists appointed by leaders who expect them to remain loyal, and plutocrats who have given hard cash to a party and ended up – with the help of a process no one is anxious to explain – sitting on their haunches in the legislature of a democracy.”

“Yes. Fair point. I can see why you think we aren’t much of a democracy.”

Before the libel lawyers start flapping, I must add that no one can prove with 100% certainty that rich men who paid tens of millions to the Labour, Conservative and Liberal Democrat parties were given peerages in return. If yet more donors receive peerages in David Cameron’s dissolution honours list at the end of the month, no newspaper will run the headline, “Financier gave Tories £750,000 for peerage”. But just because we cannot speak with certainty doesn’t mean we can’t look at the odds.

Andrew Mell, Simon Radford, Seth Thevoz, whose work we cover in the news pages, are model activist intellectuals. Adam Smith explained everything from the indifference to alleged corruption in the Lords to the easy ride the media give an Old Etonian PM, when he wrote 255 years ago: “We frequently see the respectful attentions of the world more strongly directed towards the rich and the great, than towards the wise and the virtuous. We see frequently the vices and follies of the powerful much less despised than the poverty and weakness of the innocent.”

Mell, Radford and Thevoz are not respectful. They have taken one of the oldest scandals in Britain, and tied it to modern concern about the power of the wealthy to manipulate government. They used the techniques of the investigative journalist to find how the parties’ nominees for the peerage donate through shell companies, holding companies, wholly owned subsidiaries, spouses and children. Then they used the techniques of the academic statisticians to analyse their findings.

They divided peers into those who had been politicians or politically active, the type of men and women who normally receive peerages to represent the interests of their parties. Beyond them were the remaining 30% whose elevation to the Lords was harder to explain. Of the 92 people nominated for peerages since 2005 who did not meet the standard criteria, 27 were big donors. Now this could just be a coincidence. So the statisticians calculated the odds. The chances of the big donors randomly ending up in Parliament, unelected and unaccountable, from the pool of available members of all the political parties, were the same as the chances of winning the lottery five weeks in a row. Or one in 73,500 decillion. Or, to spell that out for those among you who don’t count in decillions, one in 73 500 000 000 000 000 000 000 000 000 000 000 000.

I am not a lawyer, but that finding seems suggests we have a 100% right to demand that the political class open the windows and clear the stink. It’s not as if no one is drawing it to our attention. Lord Razzall looked back on his 17 years as treasurer of the Liberal Democrats, and said : “You have no idea the number of people who will suggest that, in return for a large donation, they would like to be Lord X. I would come across it once a month.” He then refused to pass their names to the police. Nigel Farage complained that Ukip was at a competitive disadvantage because his rivals could reward donors with peerages and he could not.

The old reply to all who look on with disgust is: “We’ve always done things this way.” I won’t waste your time by pointing out that “we” have never had a say in the matter, or by dwelling on the fact that “we’ve always done things this way” was an argument against abolishing slavery and giving votes to women. The important question is how to force reform. The academic investigators say that the old law against selling honours was designed to make prosecutions all but impossible because it required an explicit agreement between donor and party representative, as a relieved Tony Blair discovered when the police came after him. Few, however, have noticed that the 2010 Bribery Act is much tougher. If they’re right, anyone thinking of buying votes in parliament should consult a lawyer.

Delightful though the spectacle of the police arresting party treasurers and donors would be, it would be better if the establishment understood that abuses they have neglected because it suited their interests are no longer just an outrage, but a menace.

During the Scottish referendum, Alex Salmond argued that the union the Scots were being asked to stay in was not a democracy worthy of the name. He called on the English to have a “peasants’ revolt and get rid of the House of Lords”. If there’s another referendum, that argument will be made to Scots again, and it will be just as justified and principled.

Even our compromised leaders should realise that when Alex Salmond can mount a justified and principled criticism of a system of government, that system needs to change before it collapses.