The I=PAT identity is a general formula for determining the impact of a human activity. In the formula, I = Human Impact, P = population, A = affluence and T= technology. Typically, affluence is measured by GDP (Gross Domestic Product) per capita and technology is some intensity measure such as energy intensity or emission intensity. For example, the Kaya Identity is written as:
CO2 Emissions = Population (GDP/Population) (Energy/GDP) (CO2 Emissions/Energy)
where Technology = (Energy/GDP)(CO2 Emissions/Energy) = (Energy Intensity)(Emission Intensity). An important feature of ImPAcT models is that they are true by definition, that is, they are identities. If you know the ratios (sometimes called intensive variables) and the ratios are relatively stable, you can, for example, predict the impact of population growth on CO2 emissions fairly accurately, at least for a few years into the future. Another way to say this is that without decreasing energy intensity or emission intensity, population growth will increase CO2 emissions.
So, let's apply this thinking to the increases in gun ownership that are happening right now in the US.
(Mass Murder) = Population (Guns/Population) (Lunatics/Gun) (Mass Murders/Lunatic)
here the affluence measure is (Guns/Population) and the Technology measure involves the technology of mass murder (Lunatics/Gun) (Mass Murders/Lunatic) and captures increased gun ownership and the question of whether the mentally ill are more prone to violence. From this equation, we can predict that mass murders will increase in the US as gun ownership increases, other things being equal.
Now, the NRA argues (here) that mass murders would be prevented by wider gun ownership. That can be added to the equation:
(Mass Murder Prevented) = Population (Guns/Population) (Lunatics/Gun) (Mass Murders/Lunatic) (Mass Murder Prevented/Mass Murder)
Here the technology is (Lunatics/Gun) (Mass Murders/Lunatic) (Mass Murder Prevented/Mass Murder) which hinges on how many mass murders were prevented by armed citizens on the scene. There were armed citizens on the scene when Rep-Gabriele Giffords was shot in Tuscon, Arizona (here). They did not draw their weapons and take out the shooter, Jared Lee Loughner.
There are multiple reasons why in a concealed-carry State such as Arizona armed citizens did not intervene. First, if you pull your concealed weapon when you suspect mass murder is about to take place, you stand the risk of being mistaken for the shooter and taken out by some other concealed-carry citizen or by law enforcement. Second, in the chaos of an unfolding mass murder, you may not be sure you are targeting the right person as actually happened here. At the end of the day, an armed citizen at the site of a mass murder will not pull their weapon and will not prevent a mass murder for fear of either being killed themselves or killing the wrong person.
Therefore, (Mass Murder Prevented/Mass Murder) = 0 and increases in armed population will increase the number of mass murders. This suggests that the only way to reduce mass murder is to reduce the number of lunatics (increased spending on mental health), reduce lunatic's access to weapons (background checks) or reduce gun ownership. The political debate is just beginning.
NOTE: The IPCC has made extensive use of I=PAT models in climate change reports, another area that needs more simple, clear mathematical thinking.
