A model of corruption 

We have looked at how idea sets compete.  We have looked at idea sets that mutually support each other.  Surprisingly, it turned out that, for all the vitriol that sectarianism produces, the effect is merely to reinforce the sectarianism in the other camp. 

Where can we look for a set of ideas that actually feeds off another set?  We are looking for an idea set that blossoms and another idea set that starts to flourish because of the presence of the first.  As the second idea set flourishes, the initial idea set wanes. (This is unlike the symbiotic situation, in which the flourishing of the second stimulates further growth in the first.)   That is, we are looking for a predator prey situation in the world of ideas.  Before we continue we should look at the biological equivalent.

The Original Lynx effect

In the world of nature there are many predators and many prey.  The one described here is the situation in which there is one type of predator and one type of prey.  (This follows the classical mathematical approach, start simple, add more realistic assumptions later). The mathematical model describing this process was derived by Lotka and Volterra.  In the absence of predators, the rate of growth of prey is proportional to the size of the population.  The more there are the faster the rate of growth.  If N is the population and a the constant of proportionality the equation is: 


However if predators are present, the growth rate is too high.  The more predators there are the more the growth rate is too high.  The Lotka Volterra model replaces the constant term a by a term (a-bP) where b is some constant and P is the predator population.

If we now look at the predators, in the absence of prey predators would die of hunger. The more of them there are, the more that die per unit time.  If we let the constant of proportionality be d we get an initial model of .  However if we now include prey the picture for the predators is less gloomy.  The more prey the less gloomy.  The Lotka-Volterra model replaces the constant d by the term (d-Nc)  where N is the prey population and c is some positive constant.   The two equations are then:

       where N is the number of prey,  P is the number of predators, and a,b,c,d are constants.  It is easy to solve these equations numerically and obtain a plot of predator population (y axis) against prey population (x axis).  (The units are arbitrary, think of them as thousands of individuals,  if it helps.)









The point P1, on the extreme right of the diagram, is where the number of prey has reached a peak.  As the prey are plentiful the number of predators rises, and starts eating (literally) into the prey population.  This process continues until the numbers of prey are too small to sustain the predator population.  Thus the number of predators starts to fall (P2).  As the predator numbers fall, the prey population starts to recover (P3).  As the prey population rises the predator population also starts to rise (P4).  Both populations continue to rise until the number of predators causes the number of prey to reach a peak (P1). 

The diagram above shows the solution to the equations, each line represents a solution for a given set of initial conditions. We could have plotted the populations against time.  At the start there are few predators so the prey population rises.  As food becomes plentiful the predator population rises the prey population starts to fall.  The process continues till there is insufficient prey to sustain the predator population, so it falls.   It falls to a point such that the prey population starts to recover.  The peaks of the prey population precede the peaks of the predator population.


The classic example is lynx and hares as recorded by the Hudson Bay Company. 

Text Box:











This time the x axis represents time.  Note that in general there is a peak in hare population, followed by a peak in lynx population. (These are real world figures so the pattern does not follow the mathematical model exactly.)         

A predator prey model in the world of ideas

Consider the idea that it is all right to be corrupt.  Let N be the strength of this idea in some locality.  This needs to combine both of strength of an individual’s conviction and the number of people who hold the view.  Let us suppose that there are no forces opposing these ideas.  Corrupt practices grow as the corrupt person deepens and extends their scope of activities.  At the same time others around them are seduced into corrupt ideas and practices.  Thus the rate of growth of the corruption idea is (very roughly) proportional to its existing strength.

Thus a model


where a is some positive constant would seem to be reasonable.

There is evidence that, in the absence of anti corrupt forces, this is indeed the case.

The Luxow report into police corruption in New York in 1894 notes ‘nor is it strange that the contemplation of these practices by superior officers, inferior members of the force should become demoralised until the contamination has spread through the department.’2

The Helfand enquiry into police corruption in New York 1949 –1954 decided that corruption was so systematic that 336 plainclothes men were demoted and put into uniform.  ‘Thereafter all plainclothesmen attached to vice squads were to be selected from recent graduates of the police Academy’3 (that is before they had learned how to be corrupt).

The Knapp report into corruption in New York Police stated in 1972 that ‘...the rookie who comes into the department is faced with the situation when it is easier for him to become corrupt than to remain honest’ Knapp P4.  ‘Patrolman Droge described how department pressures gradually converted an idealistic rookie into an increasingly bold finder of bribes and payoffs’4. 

‘What is alarming is that plainclothes units serve as an important breeding ground for large scale corruption in the areas of the department.  Some officers who have managed to stay honest before being assigned to plainclothes are indicted into corrupt practices when in plainclothes and go on to practice what they learned there for the rest of their tenure in the department.  Others who have indulged in minor corruption before assignment to plainclothes learn how to expand their activities’.5

 ‘Logan testified that his PEP [Preventive Enforcement Patrol] squad sergeant taught him the various techniques of scoring and that such scoring was standard police procedure among his fellow officers’6.

The Mollen report into police corruption in New York in1992 states

‘In the 73rd and 75th precincts raids often began as a means of accomplishing a law enforcement end….Over time, however, the raids assumed a different purpose, to steal drugs money and handguns from the drug dealers. And as their purpose changed, the raids became more frequent and highly organised’7. 

Within the business world if all around are corrupt it can seem reasonable to be corrupt.

Foster Winans (of Wall Street Journal’s Heard it on the Street column who had been jailed for leaking the contents of his column to a stockbroker so that they could benefit from stock trading in response to his column) said that everyone cheated on Wall St.8

‘Something deep inside me forced me to try to catch up with the pack of wheeler dealers who always raced in front of me’ 9

‘The judge at the Guinness trial said of Saunders, ‘I am satisfied that you would not have been sucked into dishonesty but for the ethos of those days’10

            Now consider the idea that corrupt practices need to be exposed and the individuals concerned punished.  Let us call the strength of this idea P.  In the absence of the corruption idea and corrupt individuals, journalists would have no one to expose, politicians would have no one to expose and courts would have no one to prosecute.  Thus the strength of the anti-corruption idea will decline.

Thus a model


where d is some positive constant would seem reasonable.  There is evidence from the real world that this is indeed the case.

Knapp reported that there had been graft as early as 1844 there had been enquiries into police corruption, followed by a purge.  Knapp continues ‘after the heat was off it was largely back to business as usual.’11 

After the Knapp commission extensive structures and attitudes were put in place to oppose corruption.  The Mollen report found that these had broken down. 

‘The assumption was simply that the top brass would be responsible for overseeing the IAD [Internal Affairs Department] and the anti corruption apparatus.  That worked fine for a time…Once public attention and departmental priorities shifted elsewhere the department’s interest in corruption control began to wane So too did any oversight of and commitment to IAD or to any component of the anti corruption apparatus. Over time the system began to decay.’12

‘The integrity film that was shown to thousands of recruits for years all to clearly epitomised the department’s lack of commitment to anti corruption: a tattered black and white film from the 1970s that focussed on issues of little relevance to cops on the streets today.’13

The collapse of the anti corruption apparatus does more than minimise the likelihood of uncovering corruption it fuels it.  ‘It sends a powerful message to officers of all ranks that the department does not care about corruption’. 14

Following a number of cases of corruption in the year 2000 David Wood, who had run the Metropolitan Police’s anti corruption squad said, ‘The Met relaxed its grip on corruption for a decade and it flourished ...it always will when officers think that they can get away with it.’15

Now let us consider the two ideas in conflict.

Every time that someone is exposed as corrupt, there are consequences.  The first is that the individual is punished and is presumably less inclined and able to engage in and spread corrupt practices and ideas. 

Ivan Boesky was found guilty of four counts of securities fraud, perjury, and income tax evasion, below is a report of the proceedings prior to sentencing.

‘Once Wall St’s most aggressive speculator in take over stocks, Boesky was a picture of contrition in court. "I am deeply ashamed," he said.  "I have spent the last year trying to understand how I veered off course"16.  His lawyer described how he had taken up rabbinical studies and helped the homeless.’17

The second consequence is that all those around, who see a person exposed and punished, are more likely to reject the idea that it is all right to be corrupt. 

‘Securities lawyer David Martin, a former SEC special counsel, says that the deterrent effect of a case like the Boesky scandal can be enormous: "One guy goes down, and ten people clean up their act-that’s the real benefit"’18

Or as Trenchard, a Police Commissioner in the 1930s put it ‘I had time enough to deal only with the flagrant cases and to hope that the uncaught people would profit by the example I made of those to mend their ways’.19

Thus if the two ideas coexist, the rate of growth of corruption, a in equation 1, needs to be reduced.  The stronger the anti corruption forces, the more the a needs to be reduced.  The simplest model that does this replaces a by  a-bP where b is some positive constant.  Thus equation 1 becomes


The third consequence, when corrupt individuals are exposed, is that the anti corruption forces become strengthened.  Journalists write about the case, politicians denounce the caught one, and judges sentence them.  These actions in turn encourage others to see if there is similar wrongdoing in their locality.

The Lexow Commission started after newspaper reports of police corruption in New York.

The District Attorney was actively engaged in exposing corruption in New York in1912.  A gambler called Rosenthal had written an article on police corruption.  (He paid his protection money to the police but they raided anyway.)  He was due to appear before the District Attorney on July 16 1912.  That was the day he was killed with apparent police complicity.  This gave a huge spur to anti-corruption forces.

            ‘The murder of Rosenthal caused intense public excitement and indignation’20

A series of articles in the newspapers led to both the Knapp Report and the Helfand investigations.

 The same growth of anti corruption forces can be seen in business.  Thus following the exposure of Boesky others took a look at his business associates.

‘A Sunday newspaper wrote of Saunders that he ‘can now stand beside the modern legends of British commercial success’ ’21

However, he did have an association with Boesky.  ‘Saunders might have evaded attention if Boesky had not fallen from grace and mentioned Guinness’22. 

Saunders was reduced from legend status to criminal when he was found guilty of insider dealing. 

Thus if both ideas co-exist, the rate of decay of anti-corruption ideas, d in equation 2, needs to be reduced.  The stronger the corrupt ideas and practices the slower the decline in anti corruption ideas, the more we need to reduce d.  The simplest way to do this is to replace d by d-cN, where c is some positive constant.  Equation 2 thus becomes

Thus our conflicting corruption and anti-corruption ideas satisfy the Lotka-Volterra predator prey model.

where a ,b, c and d are positive constants.  These are the same equations as the Lotka Volterra above.

A plot of corruption and anti corruption ideas against time should look identical to the biological equivalent.




The peaks of anti corruption forces lag behind those of corruption. 

There are a similar number of problems with this ideological model as with its biological counterpart.  The key qualitative conclusion is the same.  From the model we would expect that corruption and anti-corruption follow a cyclic pattern.  That is we would expect corrupt practices to grow followed by a purge followed by further growths and further purges.  Is there any evidence that this is the case? 

To get quantitative data on activities of corruption is by its nature very difficult.  However the Seabury Commission of 1932 into police corruption in New York gives a hint.  One form of corruption at the time was as follows

‘The picture of the ring is complete.  The stool pigeon or the officer framed the woman, the officer arrested her, the bondsman bailed her out at an exorbitant charge and usually recommended a lawyer, the lawyer gouged her savings and either himself or through the bondsman “fixed” the arresting officer and the District Attorney’23.

Thus there was a huge gap between those arrested and those who were actually tried.  In the case of bookmaking in 1929 4,238 were arrested and only 161 were tried24.  Thus a measure of corruption would be the number of arraignments.  The diagram below shows arraignments for prostitution in the city of New York just before the Seabury Commission reported.

[The figures are from the Seabury  Commission25 with the exception that the figure given in the report for 1931 was 513 for 11 months, I have multiplied this by 12/11 to get a comparable annual figure].

It is interesting that the decline of arraignments coincides with the rise of anti corruption forces.  It was in August 1930 that Franklin D. Roosevelt set up the Seabury Commission.


While it is, in general, difficult to measure corruption forces, anti corruption forces are very happy to publicise their activities.  Indeed publishing their successes gives a major boost to their activities.  The timing of peaks of anti corruption activity is given by the dates of the official reports.

According to the model, these should be regularly spaced.

Gabriel Chin collected the major reports into police corruption in New York.  The dates are shown below.

Seabury 18932


Curran 1912


Mollen 1992


Helfand 1949-1954


Knapp 1972






Lexow 1894


The Metropolitan Police

The same pattern emerges from a look at the Metropolitan Police.  The famous saying, ‘If you want to know the time ask a policeman’ had its origins in the ability of Victorian police to separate drunks from their watches.

In 1877 came the first of the great police corruption scandals-the Turf Fraud, in which five senior officers from the detective division were arrested and appeared at the Old Bailey.  In turn this led to the foundation of the Criminal Investigation Department (C.I.D).  However, in accepting bribes, suppressing evidence and giving advanced notice of raids, the behaviour of the officers involved was to provide a blueprint for behaviour by corrupt policemen in that CID and other detective forces for the next 115 years’.26

In 1931Trenchard was appointed Metropolitan Commissioner.  ‘He came ‘to the firm belief that corruption was endemic’27 and tried to stamp it out. 

On 17th November 1955The Mail carried a report revealing, ‘a vast amount of bribery and corruption among certain uniformed officers attached to the West End Station’.28  

In 1972 Sir Robert Mark became Commissioner.  He made the observation, ‘that the CID were the most routinely corrupt organisation in London’.29  In his book The Office of Constable he boasts that during his spell as Commissioner some 478 officers resigned in anticipation of disciplinary action or criminal proceedings.30

Nearly two decades later the situation was again not good.

In 1993 soon after his appointment of police Commissioner, Sir Paul Condon, had decided to find out just how serious a problem police corruption was.  The results were profoundly shocking...The first investigation speculated that the number of seriously corrupt officers could be as high as 250.31

There is a similar pattern in business corruption.  ‘Reisman … for instance, discerns a pattern of ‘crusades’ in relation to commercial bribery in which periodic displays of ‘sound and fury’ [by anti corruption forces]are succeeded by business as usual’.32

A search for the number of occurrences of the word corruption in the journal The Economist shows the following.  The overall trend is upwards, perhaps due to more content.  There does however seem to be a cyclical element.

It is worth pointing out that the assumption of one predator and one prey is unreasonable.  Corruption activities by businessmen are likely to stimulate anti corruption forces in the press, law courts, and the police.  Stimulated anti corruption forces are likely to impact on corruption in other spheres of activity,



1                                      (http://www.sfws.auburn.edu/ditchkoff/images/Lecture%20Images/Carnivores/lynx-hare_cycle.gif) Accessed May 5th 2007

2                                      Luxow P49 (The Luxow, Curren, Seabury, Helfand, Knapp, and Mollen Reports were gathered together by Gabriel Chin Ed. ‘New YorkCity Police Corruption, Investigation Commissions 1894-1994 (S. Hein Co. Inc. Buffalo, New York 1997)

3                                      Helfand P65

4                                      Knapp P9

5                                      Knapp P61

6                                      Knapp P100

7                                      Mollen P25

1.                                          Newsweek, 22 September, 1986:44, Quoted in Punch, M.  Dirty Busines. (London 1999): Sage Publications P148

2.                                          Levine, 1991, p. 390; Quoted in Punch, 1996, p.148

3.                                          The Times, 29 August, 1990, p.1; Punch, 1996, p. 176

4.                                          Knapp P61

5.                                          Mollen P75

6.                                          Mollen P108

7.                                          Mollen P108

8.                                          Electronic Telegraph, 2000, 25th Aug

9.                                          Time, 21 December, 1987, 37

10.                                      Punch, 1996, p. 144

11.                                      Newsweek, 1December, 1986, p.36; Punch, 1996, p.144

12.                                      Morton,J. (1994) Bent Coppers:Warner Books 1993, p. 88

13.                                      Curren 1912 P1

14.                                      Punch, 1996, P 167-168

15.                                      Punch, 1996, P.174

16.                                      Seabury P86

17.                                      Seabury P63

18.                                      Seabury P.101

19.                                      Morton, 1993, P. 45

20.                                      Morton, 1993, P. 87

21.                                      Morton, 1993, P. 92

22.                                      Morton, 1993, P. 139

23.                                      Morton, 1993, P. 371

24.                                      Electronic Telegraph, 25/8/2000, Tainted Love

25.                                      Punch, 1996 P71