Is using a paper bag effective if someone is fainting? What is the rationale behind it?

It depends on why the person faints- an act known as syncope. The causes for syncope are many;

Table 1. Causes of syncope
Cause	Prevalence
Cardiac
Arrhythmia	14%
Mechanical*	4%
Neurologic	10%
Neurally mediated
Vasovagal	18%
"Situational"	5%
Orthostatic	8%
Psychiatric	2%
Idiopathic	34%
*Obstruction to cardiac outflow, decreased ejection fraction, and decreased ventricular filling (ie, pulmonary embolus, pulmonary hypertension, and cardiac tamponade).

When a person is hyperventilating, they excrete more CO2 than usual and their drive for breathing decreases (CO2 mediates the urge for brething in normality). The person stops breathing temporarily and this will cause them to faint. Once unconcious CO2 builds up again and they will start breathing again and regain conciousness. By using a paper bag (or any bag- it's just that paper looks better than plastic) in a person who is hyperventialting, you capture their expired CO2 and they then rebreath this- therefore their CO2 levels don't decrease as quick and they won't faint. It is important to try then and get the person to breath normally.

Also see this explanation;

Hyperventilation causes various symptoms such as numbness or tingling in the hands, feet and lips, lightheadedness, dizziness, headache, chest pain, slurred speech and sometimes fainting, particularly when accompanied by the Valsalva Maneuvre.

In normal breathing, both the depth and frequency of breaths is varied by the neural system primarily in order to maintain normal amounts of carbon dioxide but also to supply appropriate levels of oxygen to the body's tissues. This is mainly done by measuring the carbon dioxide content of the blood; normally, a high carbon dioxide concentration signals a low oxygen concentration, as we breathe in oxygen and breathe out carbon dioxide at the same time, and the body's cells use oxygen to burn fuel molecules to carbon dioxide.

The gases in the alveoli of the lungs are nearly in equilibrium with the gases in the blood. Normally, less than 10% of the gas in the alveoli is replaced each breath. Deeper or quicker breaths exchange more of the alveolar gas with air and have the net effect of drawing more carbon dioxide out of the body, since the carbon dioxide concentration in normal air is very low.

The resulting low concentration of carbon dioxide in the blood is known as hypocapnea. Since carbon dioxide is held in the blood mostly in the form of carbonic acid, hypocapnia results in the blood becoming alkaline, i.e. the blood pH value rises. (In the normal person, this alkalosis would automatically be countered by reduced breathing, but for various reasons this doesn't happen when the neural control is not present.)

If carbon dioxide levels are high, the body assumes that oxygen levels are low, and accordingly the brain's blood vessels dilate, to assure sufficient blood flow and supply of oxygen. Conversely, low carbon dioxide levels (e.g. from hyperventilation) cause the brain's blood vessels to constrict, resulting in reduced blood flow to the brain and lightheadedness. Doctors sometimes artificially induce hyperventilation after head injury to reduce the pressure in the skull, though the treatment has obvious risks.

The high pH value resulting from hyperventilation also reduces the level of available calcium (hypocalcemia), which affects the nerves and causes the numbness or tingling of the hands. This occurs because alkalinization of the serum proteins (mainly albumin) increases their calcium affinity.