newton's second law experiment

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• Mechanics (Physics): The Study of Motion.

Second Law of Motion Experiments

Kinetic Energy Experiments for Kids

Sir Isaac Newton's second law of motion states that the force exerted by a moving object is equal to its mass times its acceleration in the direction from which it is pushed, stated as the formula F=ma. Because force is proportional to mass and acceleration, doubling either the mass or acceleration while leaving the other constant will double the force of impact; the force of impact increases when an object of constant weight is subject to greater acceleration. You can explore several different experiments that demonstrate this principle.

Crater Experiment

Collect a rock and a wadded up piece of paper. Because gravity's acceleration is constant, all objects fall at the same rate regardless of their mass. Test this law by dropping both items simultaneously and watching them fall at the same speed. Now place a bowl filled with powdered sugar or flour underneath the rock, and drop it from a fixed height into the powder. Set the bowl to the side, being careful not to disturb the powder in it. Drop the ball of paper from the same height into a bowl with the same amount of the same powder. Compare the craters in the powder created by each impact. Because acceleration was constant, the difference in size between the crater made by the rock and the one made by the paper illustrates that an increase in mass directly increases the force of the impact into the flour.

Softball Experiment

Screw an eyelet into a softball and another into the lintel of a door frame. Hang the softball from the door frame by a piece of string tied through the eyelets so that it hangs a few centimeters above the floor. Mark the spot directly underneath the softball's resting position. Move the hanging softball and place another softball on the marked spot. Pull the hanging softball back so it is three feet from the ground and release it so it swings and hits the softball on the floor. Measure the distance the softball on the floor travels. Repeat the experiment, substituting a plastic Wiffle ball for the softball on the floor, and measure how far it rolls after impact. This experiment illustrates that when force is held constant, the acceleration is greater in objects with less mass.

Hot Wheels Experiment

Construct a simple ramp 18 inches high and about 24 inches long using a piece of thin plywood and bricks. Place a toy car at the top of the ramp. Release it and measure how far it rolls. Tape two metal washers to the car, release it from the ramp and measure how far it rolls. Repeat the experiment with five washers taped to the top of the car. This experiment shows that as mass increases with gravity's constant acceleration, the force pushing the car along the floor increases, making heavier cars travel farther.

Wagon and String

Obtain a child's wagon, some light cotton string or thread, and two or three small volunteers. Tie the string around the wagon handle and leave 2 or 3 feet of string hanging off the handle to pull with. Begin with an empty wagon. On flat, level ground such as a sidewalk, and from a standing start, pull the string until you reach a comfortable walking speed. Note the effort it takes to pull the wagon. Next, have one of your volunteers sit in the wagon and once again pull the string until you reach walking speed. Note the effort needed to pull the wagon. The string can take only a small amount of force before it breaks; the more riders in your wagon, the more force you need to pull it, until you pass the string's breaking point. With this experiment, your acceleration is about the same each time, though you need to pull with more force due to the additional mass of each new passenger. How many passengers can you pull before the string breaks?

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About the Author

Wilhelm Schnotz has worked as a freelance writer since 1998, covering arts and entertainment, culture and financial stories for a variety of consumer publications. His work has appeared in dozens of print titles, including "TV Guide" and "The Dallas Observer." Schnotz holds a Bachelor of Arts in journalism from Colorado State University.

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5.3 Newton's Second Law

Learning objectives.

By the end of this section, you will be able to:

• Distinguish between external and internal forces
• Describe Newton's second law of motion
• Explain the dependence of acceleration on net force and mass

Newton’s second law is closely related to his first law. It mathematically gives the cause-and-effect relationship between force and changes in motion. Newton’s second law is quantitative and is used extensively to calculate what happens in situations involving a force. Before we can write down Newton’s second law as a simple equation that gives the exact relationship of force, mass, and acceleration, we need to sharpen some ideas we mentioned earlier.

Force and Acceleration

First, what do we mean by a change in motion? The answer is that a change in motion is equivalent to a change in velocity. A change in velocity means, by definition, that there is acceleration. Newton’s first law says that a net external force causes a change in motion; thus, we see that a net external force causes nonzero acceleration .

We defined external force in Forces as force acting on an object or system that originates outside of the object or system. Let’s consider this concept further. An intuitive notion of external is correct—it is outside the system of interest. For example, in Figure 5.10 (a), the system of interest is the car plus the person within it. The two forces exerted by the two students are external forces. In contrast, an internal force acts between elements of the system. Thus, the force the person in the car exerts to hang on to the steering wheel is an internal force between elements of the system of interest. Only external forces affect the motion of a system, according to Newton’s first law. (The internal forces cancel each other out, as explained in the next section.) Therefore, we must define the boundaries of the system before we can determine which forces are external. Sometimes, the system is obvious, whereas at other times, identifying the boundaries of a system is more subtle. The concept of a system is fundamental to many areas of physics, as is the correct application of Newton’s laws. This concept is revisited many times in the study of physics.

From this example, you can see that different forces exerted on the same mass produce different accelerations. In Figure 5.10 (a), the two students push a car with a driver in it. Arrows representing all external forces are shown. The system of interest is the car and its driver. The weight w → w → of the system and the support of the ground N → N → are also shown for completeness and are assumed to cancel (because there was no vertical motion and no imbalance of forces in the vertical direction to create a change in motion). The vector f → f → represents the friction acting on the car, and it acts to the left, opposing the motion of the car. (We discuss friction in more detail in the next chapter.) In Figure 5.10 (b), all external forces acting on the system add together to produce the net force F → net . F → net . The free-body diagram shows all of the forces acting on the system of interest. The dot represents the center of mass of the system. Each force vector extends from this dot. Because there are two forces acting to the right, the vectors are shown collinearly. Finally, in Figure 5.10 (c), a larger net external force produces a larger acceleration ( a ′ → > a → ) ( a ′ → > a → ) when the tow truck pulls the car.

It seems reasonable that acceleration would be directly proportional to and in the same direction as the net external force acting on a system. This assumption has been verified experimentally and is illustrated in Figure 5.10 . To obtain an equation for Newton’s second law, we first write the relationship of acceleration a → a → and net external force F → net F → net as the proportionality

where the symbol ∝ ∝ means “proportional to.” (Recall from Forces that the net external force is the vector sum of all external forces and is sometimes indicated as ∑ F → . ∑ F → . ) This proportionality shows what we have said in words—acceleration is directly proportional to net external force. Once the system of interest is chosen, identify the external forces and ignore the internal ones. It is a tremendous simplification to disregard the numerous internal forces acting between objects within the system, such as muscular forces within the students’ bodies, let alone the myriad forces between the atoms in the objects. Still, this simplification helps us solve some complex problems.

It also seems reasonable that acceleration should be inversely proportional to the mass of the system. In other words, the larger the mass (the inertia), the smaller the acceleration produced by a given force. As illustrated in Figure 5.11 , the same net external force applied to a basketball produces a much smaller acceleration when it is applied to an SUV. The proportionality is written as

where m is the mass of the system and a is the magnitude of the acceleration. Experiments have shown that acceleration is exactly inversely proportional to mass, just as it is directly proportional to net external force.

It has been found that the acceleration of an object depends only on the net external force and the mass of the object. Combining the two proportionalities just given yields Newton’s second law .

Newton’s Second Law of Motion

The acceleration of a system is directly proportional to and in the same direction as the net external force acting on the system and is inversely proportional to its mass. In equation form, Newton’s second law is

where a → a → is the acceleration, F → net F → net is the net force, and m is the mass. This is often written in the more familiar form

but the first equation gives more insight into what Newton’s second law means. When only the magnitude of force and acceleration are considered, this equation can be written in the simpler scalar form:

The law is a cause-and-effect relationship among three quantities that is not simply based on their definitions. The validity of the second law is based on experimental verification. The free-body diagram, which you will learn to draw in Drawing Free-Body Diagrams , is the basis for writing Newton’s second law.

Example 5.2

What acceleration can a person produce when pushing a lawn mower.

Substituting the unit of kilograms times meters per square second for newtons yields

Significance

Check your understanding 5.3.

At the time of its launch, the HMS Titanic was the most massive mobile object ever built, with a mass of 6.0 × 10 7 kg 6.0 × 10 7 kg . If a force of 6 MN ( 6 × 10 6 N ) ( 6 × 10 6 N ) was applied to the ship, what acceleration would it experience?

In the preceding example, we dealt with net force only for simplicity. However, several forces act on the lawn mower. The weight w → w → (discussed in detail in Mass and Weight ) pulls down on the mower, toward the center of Earth; this produces a contact force on the ground. The ground must exert an upward force on the lawn mower, known as the normal force N → N → , which we define in Common Forces . These forces are balanced and therefore do not produce vertical acceleration. In the next example, we show both of these forces. As you continue to solve problems using Newton’s second law, be sure to show multiple forces.

Example 5.3

Which force is bigger.

(b) The same car is now accelerating to the right. Which force is bigger, F → friction F → friction or F → drag ? F → drag ? Explain.

• The forces are equal. According to Newton’s first law, if the net force is zero, the velocity is constant.
• In this case, F → friction F → friction must be larger than F → drag . F → drag . According to Newton’s second law, a net force is required to cause acceleration.

Example 5.4

What rocket thrust accelerates this sled.

Calculate the magnitude of force exerted by each rocket, called its thrust T , for the four-rocket propulsion system shown in Figure 5.14 . The sled’s initial acceleration is 49 m/s 2 49 m/s 2 , the mass of the system is 2100 kg, and the force of friction opposing the motion is 650 N.

where F net F net is the net force along the horizontal direction. We can see from the figure that the engine thrusts add, whereas friction opposes the thrust. In equation form, the net external force is

Substituting this into Newton’s second law gives us

Using a little algebra, we solve for the total thrust 4 T :

Substituting known values yields

Therefore, the total thrust is

and the individual thrusts are

In this example, as in the preceding one, the system of interest is obvious. We see in later examples that choosing the system of interest is crucial—and the choice is not always obvious.

Newton’s second law is more than a definition; it is a relationship among acceleration, force, and mass. It can help us make predictions. Each of those physical quantities can be defined independently, so the second law tells us something basic and universal about nature.

Check Your Understanding 5.4

A 550-kg sports car collides with a 2200-kg truck, and during the collision, the net force on each vehicle is the force exerted by the other. If the magnitude of the truck’s acceleration is 10 m/s 2 , 10 m/s 2 , what is the magnitude of the sports car’s acceleration?

Component Form of Newton’s Second Law

We have developed Newton’s second law and presented it as a vector equation in Equation 5.3 . This vector equation can be written as three component equations:

The second law is a description of how a body responds mechanically to its environment. The influence of the environment is the net force F → net , F → net , the body’s response is the acceleration a → , a → , and the strength of the response is inversely proportional to the mass m . The larger the mass of an object, the smaller its response (its acceleration) to the influence of the environment (a given net force). Therefore, a body’s mass is a measure of its inertia, as we explained in Newton’s First Law .

Example 5.5

Force on a soccer ball.

• We apply Newton’s second law: F → net = m a → = ( 0.400 kg ) ( 3.00 i ^ + 7.00 j ^ m/s 2 ) = 1.20 i ^ + 2.80 j ^ N . F → net = m a → = ( 0.400 kg ) ( 3.00 i ^ + 7.00 j ^ m/s 2 ) = 1.20 i ^ + 2.80 j ^ N .
• Magnitude and direction are found using the components of F → net F → net : F net = ( 1.20 N ) 2 + ( 2.80 N ) 2 = 3.05 N and θ = tan −1 ( 2.80 1.20 ) = 66.8 ° . F net = ( 1.20 N ) 2 + ( 2.80 N ) 2 = 3.05 N and θ = tan −1 ( 2.80 1.20 ) = 66.8 ° .

Example 5.6

Mass of a car, example 5.7, several forces on a particle.

Thus, the net acceleration is

which is a vector of magnitude 8.4 m/s 2 8.4 m/s 2 directed at 276 ° 276 ° to the positive x -axis.

Check Your Understanding 5.5

A car has forces acting on it, as shown below. The mass of the car is 1000.0 kg. The road is slick, so friction can be ignored. (a) What is the net force on the car? (b) What is the acceleration of the car?

Newton’s Second Law and Momentum

Newton actually stated his second law in terms of momentum: “The instantaneous rate at which a body’s momentum changes is equal to the net force acting on the body.” (“Instantaneous rate” implies that the derivative is involved.) This can be given by the vector equation

This means that Newton’s second law addresses the central question of motion: What causes a change in motion of an object? Momentum was described by Newton as “quantity of motion,” a way of combining both the velocity of an object and its mass. We devote Linear Momentum and Collisions to the study of momentum .

For now, it is sufficient to define momentum p → p → as the product of the mass of the object m and its velocity v → v → :

Since velocity is a vector, so is momentum.

It is easy to visualize momentum. A train moving at 10 m/s has more momentum than one that moves at 2 m/s. In everyday life, we speak of one sports team as “having momentum” when they score points against the opposing team.

If we substitute Equation 5.7 into Equation 5.6 , we obtain

When m is constant, we have

Thus, we see that the momentum form of Newton’s second law reduces to the form given earlier in this section.

Interactive

Explore the forces at work when pulling a cart or pushing a refrigerator, crate, or person. Put an object on a ramp and see how it affects its motion. Engage the simulation below and see how applied forces make objects move.

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In Newton's analysis of motion, the relationship between the net force acting on a body and its acceleration defines both force and mass.

Practical Activity for 14-16

Demonstration

A trolley experiences an acceleration when an external force is applied to it. The aim of this datalogging experiment is explore the relationship between the magnitudes of the external force and the resulting acceleration.

Apparatus and Materials

• Light gate, interface and computer
• Dynamics trolley
• Pulley and string
• Slotted masses, 400 g
• Double segment black card (see diagram)

Health & Safety and Technical Notes

Take care when masses fall to the floor. Use a box or tray lined with bubble wrap (or similar) under heavy objects being lifted. This will prevent toes or fingers from being in the danger zone.

Read our standard health & safety guidance

Pass a piece of string with a mass hanging on one end over a pulley. Attach the other end to the trolley so that, when the mass is released, it causes the trolley to accelerate. Choose a length of string such that the mass does not touch the ground until the trolley nearly reaches the pulley. Fix a 1 kg mass on the trolley with Blu-tack to make the total mass (trolley plus mass) of about 2 kg . This produces an acceleration which is not too aggressive when the maximum force (4 N) is applied.

The force is conveniently increased in 1 newton steps when slotted masses of 100 g are added. Place the unused slotted masses on the trolley. Transfer them to the slotted mass holder each time the accelerating force is increased. This ensures that the total mass experiencing acceleration remains constant throughout the experiment.

Fit a double segment black card on to the trolley. Clamp the light gate at a height which allows both segments of the card to interrupt the light beam when the trolley passes through the gate. Measure the width of each segment with a ruler, and enter the values into the software.

Connect the light gate via an interface to a computer running data-logging software. The program should be configured to obtain measurements of acceleration derived from the double interruptions of the light beam by the card.

The internal calculation within the program involves using the interruption times for the two segments to obtain two velocities. The difference between these, divided by the time between them, yields the acceleration.

A series of results is accumulated in a table. This should also include a column for the manual entry of values for force in newtons. It is informative to display successive measurements on a simple bar chart.

Data collection

• Select the falling mass to be 100 g . Pull the trolley back so that the mass is raised to just below the pulley. Position the light gate so that it will detect the motion of the trolley soon after it has started moving.
• Set the software to record data, then release the trolley. Observe the measurement for the acceleration of the trolley.
• Repeat this measurement from the same starting position for the trolley several times. Enter from the keyboard '1' (1 newton) in the force column of the table (see below).
• Transfer 100 g from the trolley to the slotted mass, to increase it to 200 g . Release the trolley from the same starting point as before. Repeat this several times. Enter '2' (2 newtons) in the force column of the table.
• Repeat the above procedure for slotted masses of 300 g and 400 g .
• Depending upon the software, the results may be displayed on a bar chart as the experiment proceeds. Note the relative increase in values of acceleration as the slotted mass is increased.
• The relationship between acceleration and applied force is investigated more precisely by plotting an XY graph of these two quantities. (Y axis: acceleration; X axis: force.) Use a curve-matching tool to identify the algebraic form of the relationship. This is usually of the form 'acceleration is proportional to the applied force'.
• This relationship is indicative of Newton's second law of motion.

Teaching Notes

• This is a computer-assisted version of the classic experiment. The great advantage of this version is that the software presents acceleration values instantly. This avoids preoccupation with the calculation process, and greatly assists thinking about the relationship between acceleration and force. Each repetition with the same force gives a similar acceleration. If the force is doubled, this results in a doubling of the acceleration, and so on. The uniform increases in the acceleration can be confirmed by using cursors to read off corresponding values from the graph.
• The resulting straight line fit on the graph should be scrutinized for sources of error. The quality of the fit is reduced if the suggested procedure for maintaining the total mass constant is ignored. Also, a common outcome is a very small intercept near the graph origin. The most likely cause of this is neglect of the effect of friction on the motion of the trolley.
• The gradient of the line may be correlated with 1/mass of the system (trolley and slotted masses).
• There is a variation of this experiment, in which the force is held constant but the mass of the trolley is altered by attaching further masses. This may be conducted to provide data for the complementary relationship indicated by Newton's second law: for a given applied force, the acceleration of the trolley is inversely proportional to its mass.

This experiment was safety-tested in November 2006

Download the support sheet / student worksheet for this practical.

This demonstration is rather fussy to set up, but produces good results.

Carbon dioxide cylinder (Syphon type)

• CO 2 pucks kit, including glass plate and pucks
• Dry ice attachment for cylinder
• Camera and multiflash system
• Lamp, bright, up to 500W
• Elastic cord, to accelerate puck

When using CO 2 and dry ice it is essential to have good ventilation to the room.

Remember to wear heat-insulating gloves when handling dry ice.

Read this guidance note for general hints and detail of specific methods:

Multiflash Photography

The magnetic puck is a small metal ring magnet, of the kind used as field magnets in television sets. It has a metal or card lid. When it is filled with solid carbon dioxide, the puck floats on the sublimed gas.

A kit is available from scientific suppliers. Four pucks are provided, two magnetic, two non-magnetic and made of brass. They are, however, all of the same size and mass and you can stack them one on top of the other.

Polish the glass plate using a duster and methylated spirit or window cleaning fluid. Carefully level it using the wedges supplied.

The lengths of elastic used with trolleys are not suitable because a smaller force is needed here. Instead, use a longer length, with one end attached to the top of the puck with sticky tape. The stretch on this must be kept uniform. Here is a convenient technique. Hold the end against a half-metre rule. Ensure that it is always the same distance from the puck. With practice you can produce a fairly steady force.

The start of the motion does not necessarily coincide with an exposure or with an image. A high frequency of exposure is required. This reduces any error in identifying the time of the start of the motion, relative to later images.

For further information about making dry ice see the apparatus entry:

carbon dioxide cylinder

Three-quarter length blackout is essential for good photographs.

Making the image

• Attach a bright pointer to the centre of a magnetic puck.
• Set a camera alongside the glass plate, so that it can photograph the pointer as the puck moves across the plate. Illuminate the pointer with the lamp. Align the stroboscope slit with the camera lens, as shown.

• Put 2-3 cm3 of solid CO2 underneath one of the magnetic pucks, and place the puck on the plate.
• Attach an elastic cord to the puck and apply a force.
• With the multiflash system active, open the camera shutter on the B setting. Release the puck and accelerate it with a small near-constant force.

Analyzing the image

• Measure the distances between the puck positions. Use the multiflash frequency (time = 1/frequency) to determine the time between each position.
• To find out if the acceleration was uniform, plot distance against (time)2.
• Discussion points arising from the experiment:
• The time between images is constant.
• Spacing between images of a body increases with speed.
• A constant force on a body produces constant acceleration.
• For constant acceleration, and provided that the starting velocity of the measured motion is zero, the graph will be a straight line passing through the origin.
• By stacking pucks on top of each other, up to a maximum of four, and applying the same force in each case, you can show that acceleration decreases as mass increases.

This experiment was safety-tested in April 2006

This detailed experiment involves measurement of acceleration.

For each student groups

• Dynamics trolleys, up to 3
• Rods for stacking trolleys
• Elastic cords, 3
• Ticker-tape
• Ticker-timer with power supply unit

Long runways or heavy shorter ones should be handled by two persons. In operation, ensure that a string is tied across the bottom of the runway, to prevent the trolley falling onto the floor (or someone's foot).

It might not be possible for every group to have three trolleys, and so groups may need to share.

To ensure the elastic cords (given to one group of students) all stretch by the same amount for the same force, set up a testing rig as shown in the diagram.

Oil the bearings on the trolley wheels. Do not use trolleys with bent axles (through dropping). Ensure the runway and the trolley wheels are clean.

The relationship between acceleration and force

• In this part, you will vary the force and measure different accelerations. Mass must stay the same.
• Set up the runway and compensate for friction, as in the experiment Compensating for friction.
• Set up the ticker-timer at the higher end of the runway.
• Accelerate a single trolley by a single strand of elastic cord. Use a ruler to help you to stretch the cord by a fixed amount, or extend the cords the full length of the trolley.

Finding average acceleration with a ticker-timer

• Repeat using two cords in parallel, stretched by the same amount as before. Measure and record the new acceleration.
• Repeat with three cords.
• Plot a graph of acceleration (y axis) against force (x axis). Simply use the number of cords, 1, 2 or 3, as a way of measuring force.

The relationship between acceleration and mass

Repeat steps 1 to 8, but this time apply the same force in all cases. Vary the mass by stacking up to three trolleys. (Two cords in parallel helps when pulling more trolleys.) For simplicity, you can use a trolley mass as a unit of mass (instead of mass in kilograms).

• Students gain a great deal from feeling the effect of a constant force on increasing masses, and the sluggish effect on their motion.
• The degree of necessary compensation varies with number of trolleys. Students will obtain best results if they readjust the slope of the runway when they increase the number of trolleys, so that they are still compensating for friction.
• The graphs each have three points. High precision of measurement is not possible. This can give rise to discussion. For example can a small number of measurements of modest precision yield valid conclusions? What is the nature of uncertainty and error here?
• The two investigations may take more than one lesson. You could save time by arranging for half the class to investigate the effect of force, F , and the other half to investigate the effect of mass, m . Then combine the results to arrive at a = F/m .
• How Science Works Extension: This experiment is designed specifically to avoid a pitfall present in other experiments looking at F = ma. In some experiments (such as Investigating Newton's second law of motion) the force accelerating the mass is provided by hanging masses; their weight provides the force. However, this assumes that weight is proportional to mass, and so the relationship that the experiment is designed to show is already assumed in the design of the experiment.
• You might discuss with your class how this experimental design overcomes this. The elastic cords are shown to be identical; if they are stretched the same amount, they provide the same force (although we don’t know what that force is in newtons). Similarly, the three trolleys are identical so their masses are equal.
• In principle, we can only say from this experiment that F is proportional to ma. In the SI system of units, we define the newton so that F = ma.
• For an example of some real ticker-tape charts, click here.

This experiment was safety-checked in March 2005

Class practical

Students can quickly see that force and mass have opposite effects on acceleration.

For each student group

• Stopwatch or stopclock

Long runways or heavy shorter ones should be handled by two persons. In operation ensure that a string is tied across the bottom of the runway, to prevent the trolley falling onto the floor (or someone's foot).

It might not be possible for every group to have three trolleys at all times, and so groups may need to share.

Compensating for friction

• Accelerate a single trolley using a single strand of elastic cord.
• Measure the time taken to travel a marked distance along the runway.
• Predict how this time will change if you double the force by using two elastic cords in parallel, stretched by the same amount as before. Try it out to test your prediction.

• Predict how the time will change if you double the mass (approximately) by stacking another trolley on top of the first one. Test your prediction.

• Predict how the time will change, compared with your first measurement, if you double both the force and the mass. Test your prediction.
• Predict how the time will change if you treble both the force and the mass.
• The degree of necessary compensation varies with the number of trolleys. Students will obtain the best results if they readjust the slope of the runway when they increase the number of trolleys, so that it is still compensated for friction.
• Make sure that students understand that the time is a reliable indication of acceleration. The shorter the time, the greater the acceleration.
• When the mass of a moving object is changed, students are apt to find the interpretation more difficult. For them, mass is more artificial and less familiar than force. The reciprocal relationship between F and m for a constant acceleration is itself a barrier.
• The ratio of force/mass is constant if the acceleration is kept constant.

This experiment was safety-tested in March 2005

Demonstration or Class practical

This activity demonstrates that inertia depends on mass and not on any other interpretation of size. Force, mass and acceleration are inter-related quantities.

• Mass, brass or lead, 1 kg
• Mass, aluminium, 1 kg
• Dynamics trolleys, 2
• Elastic cords for accelerating trolleys, 2
• Balance, able to measure or compare two 1kg masses
• Long weak spring or rubber thread
• Runway, if necessary

A trolley runway requires two persons to carry it and set it up on the bench.

• Use a balance to show that the two (gravitational) masses are equal even though their sizes are not.
• Put each mass in turn on a dynamics trolley and accelerate it with a standard force using elastic cord. Show that the time to travel a measured distance is the same in each case.
• Repeat this, applying a larger force.
• Place two equal trolleys far apart on a level runway. Put one of the masses on each trolley. Stretch a weak spring or long rubber thread between them to accelerate them towards each other.
• Repeat all of the previous exercises, but with unequal masses.
• You can use the activity to review ideas about inertia, and as an introduction to work on force, mass and acceleration, and their relationship. It shows that the three quantities are interdependent, and provides a basis for further work to determine the nature of the relationship.
• Step 2 : This shows that the two inertial masses are the same.
• Step 3 : The times are still equal to each other but shorter than before. Force affects acceleration, directly.
• Step 4 : When you release the trolleys, they travel equal distances to the point of collision.
• Step 5 : The two masses show different degrees of inertia. They experience different accelerations when subject to the same forces. Acceleration is inversely proportional to mass.
• At a sophisticated level, there are, in Newtonian physics, two independent definitions of mass - gravitational mass and inertial mass. Steps 1 and 2 in the above procedure relate to this and are suitable for able students only. Gravitational mass determines a body's ability to exert and experience gravitational force. Inertial mass determines a body's resistance to change in its motion (acceleration).
• The fact that mass requires two separate definitions is an unsatisfactory aspect of Newtonian physics that is resolved by Einstein's general relativity. According to Newton it is just an incredible coincidence that the two types of mass, gravitational and inertial, are equal in size. According to Einstein, this equivalence is not coincidence but fundamental. These two types of mass become one and the same.
• You could use timing techniques, such as use of ticker-timers, for quantitative work. You could then discuss good experimental practice: keeping one variable fixed, varying another, and watching the related changes in the third.

This provides an active way to vary measurement of distances, velocities and acceleration.

• Trolley, demonstration (or skateboard)
• Demonstration forcemeter, 50 N
• Additional forcemeter if measuring frictional force
• Springs, large, 3
• Card, small pieces of

Clearly, there are dangers of collisions or of students falling off skateboards. The activity should be done in a reasonably large, clear space, on a level floor or surface.

If a skateboard is used, head, knee and elbow protection should be worn by the skater.

Commercial trolleys are ideal. They have an attachment so that a wheel drives a dynamo attached to a meter, which acts as a speedometer. This is valuable, but is not essential.

If trolleys are not available, it is possible to use a skateboard.

So that trolleys can be accelerated with fairly constant force, put a strong spiral spring between the spring balance and the trolley. This smooths out jerkiness of motion.

• One student sits on the trolley. Another holds it still and releases it when ready. A third attaches the forcemeter to the trolley and pulls with a constant force of, say, 10 N. A fourth ‘catches’ the trolley to slow it down gently at the end of its run.
• The student on the trolley counts seconds, and drops a card at the same position relative to the trolley at each count. The count could be assisted by a fifth student with a watch or clock.
• Measure the distances between the cards. Use average velocity = distance/time to obtain values for average velocity.
• Use these figures to plot a velocity-time graph.
• Measure the gradient of the graph to obtain a value for the acceleration of the trolley.
• Repeat the measurement with larger applied force, and see what effect that has on the acceleration.

• Repeat the measurement with a second student on the trolley.
• Students are actively involved with this demonstration. It illustrates that an increase in applied force results in an increase in acceleration. Precision is not high, but concepts of velocity and acceleration are reinforced, and the experiment provides an introduction to the relationship between force and acceleration.
• It is possible to go further, and try to show that acceleration changes by the same proportion as net force, but this requires consideration of the influence of friction.
• Accelerate the trolley using a spring balance with a force of, say, 30 N. Plot velocity against time. Repeat this with a backward drag applied by a student pulling backward on the trolley with another spring balance and spring. This student maintains a constant force of, say, 10 N while moving forward with the trolley. Once again, produce a velocity-time graph. Repeat this with different applied backwards forces, but the same forwards force, until you obtain a graph that is as flat (zero gradient) as possible.
• For a flat graph, we know that net force on the trolley is zero, since its acceleration is zero. Frictional force is then equal to the difference between the applied forward force and the applied backward force.
• Assuming that the frictional force is always the same, you can obtain velocity-time graphs using different applied forwards force, and no applied backwards force. Then subtract the frictional force from each applied forwards force to find the net force.
• Obtain values of acceleration from the gradients of the velocity-time graphs. Plot net force (on the x axis since it is the input variable) against acceleration (on the y axis since it is the output variable). If assumptions and estimations that you have made are reasonable then this graph should be a simple shape – a straight line passing through the origin, revealing the simple nature of the relationship between force and acceleration.
• By using students of different mass, and constant force, you can also demonstrate that an increase in mass results in a decrease in acceleration.

This experiment was safety-tested in December 2004

This experiment illustrates a fundamental point about the nature of measurement, as well as providing a way of measuring the speed of a trolley.

• Demonstration trolley
• Meter attachment, including wheel contact, small DC dynamo, and millivoltmeter

Clearly there are dangers of collisions. The activity should be done in a reasonably large, clear space, on a level floor or surface.

The meter attachment is a special device, recommended not for general use but for the learning involved in setting it up and calibrating it.

• One wheel of the trolley drives a small DC dynamo. Connect the dynamo to a millivoltmeter which will show a reading that depends on velocity.
• Calibrate the system. Move the trolley at constant velocity, as closely as you can. Drop cards at 1-second intervals and use the distances between them to calculate the velocity of the trolley. Do this at several speeds. Match these measured velocities with the readings on the millivoltmeter.
• Once calibrated, you can now use the system for further investigations on velocity and hence also on acceleration. Measurements taken from the meter at 1-second intervals can, for example, substitute for the cards used in the previous experiment.
• Measuring acceleration: Instead of connecting directly to the millivoltmeter, you can feed the output to the primary coil of a small transformer; the secondary of the transformer is connected to the millivoltmeter. The voltage across the secondary coil will be roughly proportional to the rate-of-change of the primary current, so that the meter gives a measure of acceleration. Once again, students could test whether the meter measures this quantity and can attempt a rough calibration.
• Since this activity is linked with work on acceleration, we refer to velocity rather than speed. It would, however, be correct to say that the meter measures speed.

You can use the formula F = ma to calibrate a forcemeter without using gravity. It is difficult to do with precision.

• Lightweight forcemeter, 0-10 N, with scale markings concealed by paper (this is to be written on)
• Balance, 5 kg

Ensure that a string is tied across the bottom of the runway to prevent the trolley falling onto anyone.

It is important that the balance is lightweight, so that it does not add to the mass. It is also better if the balance is not too precise.

• Load the trolley, so that its total mass is a whole number of kilograms. Check this by putting the trolley and load on the balance.

• Attach the forcemeter to the trolley by a length of string.
• With the forcemeter, apply a force to the trolley to accelerate it from rest.

• Measure the time, t , for the motion over a measured distance, x .
• Use the formula x = 1/2 at 2 to calculate the acceleration, a .
• Use F = ma , where m is the measured mass, to find force F in absolute units.
• Mark the paper over the forcemeter scale with this force. For the forcemeter, you also know where to mark zero.
• Assume that the meter acts in a linear way. That is, that equal changes in force are represented on the scale by equal distances.
• Make a complete scale for the forcemeter, on the paper.
• Compare this with the scale provided by the forcemeter manufacturer.
• Newton's laws of motion define the concept of force. A scale of forces in newtons is essentially derived from measurements of mass and acceleration.
• Students often suppose that the manufacturer has access to some unknown means of providing a scale of absolute reliability. They may have little awareness that calibration is an important process of inherent uncertainty, however it is done. Invite them to discuss which is more reliable, the scale made by themselves or by the manufacturer.
• It is essential to keep the logic straight: force is calculated from Newton's second law once the acceleration produced by the forcemeter has been found experimentally.

You can use the formula F = ma to calibrate a forcemeter without using gravity.

• Leightweight forcemeter, 0-10 N, with scale markings concealed by paper (this is to be written on)
• Bathroom scales, calibrated in kg (or N)

• Measure the mass of a student and add it to the mass of the trolley.
• Seat the student safely on the trolley. Attach the forcemeter to the trolley by a length of string.
• With the forcemeter, apply a constant force to the trolley to accelerate it from rest.
• Use F = ma , where m is the measured mass, to find force F .
• To allow for friction, first use the spring balance to pull the trolley with whatever force is needed to maintain constant speed. Mark the reading of the pointer for constant speed on the paper that covers the scale. Then do an acceleration experiment, keeping the pointer at another mark for the larger force that is used. Remove the paper and read off the places of both marks on the scale. Compare the force calculated by using F = ma with the difference between those scale readings.
• Students often suppose that the manufacturer has access to some unknown means of providing a scale of absolute reliability. They may have little awareness that calibration is an important process of inherent uncertainty however it is done. Invite them to discuss which is more reliable, the scale made by themselves or by the manufacturer.

Teaching Guidance for 14-16

Multiflash photography creates successive images at regular time intervals on a single frame.

Method 1: Using a digital camera in multiflash mode

You can transfer the image produced direct to a computer.

Method 2: Using a video camera

Play back the video frame by frame and place a transparent acetate sheet over the TV screen to record object positions.

Method 3: Using a camera and motor-driven disc stroboscope

You need a camera that will focus on images for objects as near as 1 metre away. The camera will need a B setting, which holds the shutter open, for continuous exposure. Use a large aperture setting, such as f3.5. Digital cameras provide an immediate image for analysis. With some cameras it may be necessary to cover the photocell to keep the shutter open.

Set up the stroboscope in front of the camera so that slits in the disc allow light from the object to reach the camera lens at regular intervals as the disc rotates.

Lens to disc distance could be as little as 1 cm . The slotted disc should be motor-driven, using a synchronous motor, so that the time intervals between exposures are constant.

You can vary the frequency of ‘exposure’ by covering unwanted slits with black tape. Do this symmetrically. For example, a disc with 2 slits open running at 300 rpm gives 10 exposures per second.

The narrower the slit, the sharper but dimmer the image. Strongly illuminating the objects, or using a light source as the moving object, allows a narrower slit to be used.

Illuminate the object as brightly as possible, but the matt black background as little as possible. A slide projector is a good light source for this purpose.

Method 4: Using a xenon stroboscope

This provides sharper pictures than with a disc stroboscope, provided that you have a good blackout. General guidance is as for Method 3. Direct the light from the stroboscope along the pathway of the object.

In multiflash photography, avoid flash frequencies in the range 15-20 Hz, and avoid red flickering light. Some people can feel unwell as a result of the flicker. Rarely, some people have photosensitive epilepsy.

General hints for success

You need to arrange partial blackout. See guidance note

Classroom management in semi-darkness

Use a white or silver object, such as a large, highly polished steel ball or a golf ball, against a dark background. Alternatively, use a moving source of light such as a lamp fixed to a cell, with suitable electrical connections. In this case, place cushioning on the floor to prevent breakage.

Use the viewfinder to check that the object is in focus throughout its motion, and that a sufficient range of its motion is within the camera’s field of view.

Place a measured grid in the background to allow measurement. A black card with strips of white insulating tape at, say, 10 cm spacing provides strong contrast and allows the illuminated moving object to stand out.

As an alternative to the grid, you can use a metre rule. Its scale will not usually be visible on the final image, but you can project a photograph onto a screen. Move the projector until the metre rule in the image is the same size as a metre rule held alongside the screen. You can then make measurements directly from the screen.

Use a tripod and/or a system of clamps and stands to hold the equipment. Make sure that any system is as rigid and stable as possible.

Teamwork matters, especially in Method 3. One person could control the camera, another the stroboscope system as necessary, and a third the object to be photographed.

• Switch on lamp and darken room.
• Check camera focus, f 3.5, B setting.
• Check field of view to ensure that whole experiment will be recorded.
• Line up stroboscope.
• Count down 3-2-1-0. Open shutter just before experiment starts and close it as experiment ends.

First and second laws

If you are considering the forces acting on just one body, either law I or law II will apply.

The first law describes what happens when the forces acting on a body are balanced (no resultant force acts) – the body remains at rest or continues to move at constant velocity (constant speed in a straight line).

If a book is placed on a table, it stays at rest. This is an example of Newton’s first law. There are two forces on the book and they happen to balance owing to the elastic properties of the table. The table is slightly squashed by the book and it exerts an elastic force upwards equal to the weight of the book. You can show this by placing a thick piece of foam rubber on a table and placing a book on top of it. The foam rubber squashes.

Galileo was the first person to challenge the common sense notion that steady motion requires a steady force. He looked beyond the obvious and was able to say if there was no friction then an object would continue to move at constant velocity. In other words, he put forward a hypothesis. He could see that a motive force is generally needed to keep an object moving in order to balance frictional forces opposing the motion.

The motion of air molecules is a good example to consider with students. When air temperature is constant, no force is applied to keep air molecules moving, yet they do not slow down. If they did, in a matter of minutes the air would condense into a liquid.

The second law describes what happens when the forces acting on a body are unbalanced (a resultant force acts). The body changes its velocity, v , in the direction of the force, F , at a rate proportional to the force and inversely proportional to its mass, m . The rate of change of v is proportional to F / m . And rate of change of velocity is acceleration, a .

So if the table mentioned above were in an upwardly accelerated lift, an outside observer would see that the two forces acting on the book were unequal. The resultant force would be sufficient to give the book the same upward acceleration as the lift. Put some bathroom scales between the book and the table. If the book is accelerating downwards, its weight would be greater than the reaction force from the table. The book would, however, appear to be weightless.

Mass is measured in kilograms and acceleration in m /s 2 . With an appropriate choice of unit for force, then the constant of proportionality, k, in the equation F = k ma is 1. This is how the newton is defined, giving F = ma or a = F / m .

This can also be expressed as F = rate of change of momentum or F = Δ p / Δ t .

Newton wanted to understand what moves the planets. He realized that a planet requires no force along its orbit to move at constant speed, but it does require a force at right angles to its motion (gravitational attraction to the Sun) to constantly change direction.

The third law

Newton’s third law can be stated as ‘interactions involve pairs of forces’. Be careful in talking about third law pairs (often misleadingly called ‘action’ and ‘reaction’). Many students find this law the most difficult one to understand.

Returning to the book on a table, there are three bodies involved: the Earth, the book, and the table. In this example, the interaction pairs of forces are:

• The weight of the book and the pull of the book on the Earth (gravitational forces)
• The push of the book on the table and the push of the table on the book (contact forces)

In general, action and reaction pairs can be characterized as follows:

• They act on two different bodies
• They are equal in magnitude but opposite in direction
• They are the same type of force (e.g. gravitational, magnetic, or contact)

Students will have discovered that:

• a constant force accelerates a given mass with constant acceleration;
• doubling the force doubles the acceleration, i.e. the acceleration is directly proportional to the force for a given mass. F is proportional to a ;
• the force, F , needed for a given acceleration is inversely proportional to the mass, m
• for a given force, F , the acceleration, a , is inversely proportional to the mass, m .

(many students find inverse proportion a problem).

Considering these points together leads to F is proportional to ma or F = a constant x ma.

Mass is measured in kg and acceleration in m/s/s but what of force? If the constant is equated to unity, then we are defining a unit of force. In the SI system the force is measured in newtons (symbol N), leading to F = ma .

Solid carbon dioxide is known as dry ice. It sublimes at –78°C becoming an extremely cold gas. It is often used in theatres or nightclubs to produce clouds (looking a bit like smoke). Because it is denser than the air, it stays low. It cools the air and causes water vapour in the air to condense into tiny droplets – hence the clouds.

It is also useful in the physics (and chemistry) laboratory.

The Institute of Physics has kindly produced this video to explain how dry ice is formed.

Dry ice can be dangerous if it is not handled properly. Wear eye protection and gauntlet-style leather gloves when making or handling solid carbon dioxide.

Dry ice has many uses. As well as simply watching it sublime, you could also use it for cloud chambers, dry ice pucks, and cooling thermistors and metal wire resistors in resistance experiments. It can also be used in experiments related to the gas laws.

Obtaining dry ice

There are two main methods of getting dry ice.

1. Using a cylinder of CO 2

It is possible to make the solid snow by expansion before the lesson begins and to store it in a wide-necked Thermos flask.

Remember that the first production of solid carbon dioxide from the cylinder may not produce very much, because the cylinder and its attachments have to cool down.

What type of cylinder, where do I get CO 2 , and what will it cost?

A CO 2 gas cylinder should be fitted with a dip tube (this is also called a ‘siphon type’ cylinder). This enables you to extract from the cylinder bottom so that you get CO 2 in its liquid form, not the vapour.

NOTE: A plain black finish to the cylinder indicates that it will supply vapour from above the liquid. A cylinder with two white stripes, diametrically opposite, indicates it has a siphon tube and is suitable for making dry ice. A cylinder from British Oxygen will cost about £80 per year for cylinder hire and about £40 each time you need to get it filled up. (The refill charge can be reduced by having your chemistry department cylinders filled up at the same time.)

Don't be tempted to get a small cylinder, it will run out too quickly.

If the school has its own CO 2 cylinder there will be no hire charge, but you will need to have it checked from time to time (along with fire extinguisher checks). Your local fire station or their suppliers may prove a good source for refills.

CLEAPSS leaflet PS45 Refilling CO 2 cylinders provides a list of suppliers of CO 2 .

A dry ice attachment for the cylinder

Dry ice disks can be made using an attachment that fits directly on to a carbon dioxide cylinder with a siphon tube. Section 13.3.1 of the CLEAPSS Laboratory Handbook explains the use of this attachment (sometimes called Snowpacks or Jetfreezers). This form is most useful for continuous cloud chambers and low-friction pucks.

You can buy a Snowpack dry ice maker from Scientific and Chemical. The product number is GFT070010.

2. Buying blocks or pellets

Blocks of solid carbon dioxide or granulated versions of it can be obtained fairly easily with a search on the Internet. Local stage supply shops or Universities may be able to help. It usually comes in expanded foam packing; you can keep it in this packing in a deep freeze for a few days.

The dry ice pellets come in quite large batches. However, they have a number of uses in science lessons so it is worth trying to co-ordinate the activities of different teachers to make best use of your bulk purchase.

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Newton’s second law.

Experiment #9 from Physics with Vernier

Introduction

How does a cart change its motion when you push and pull on it? You might think that the harder you push on a cart, the faster it goes. Is the cart’s velocity related to the force you apply? Or, is the force related to something else? Also, what does the mass of the cart have to do with how the motion changes? We know that it takes a much harder push to get a heavy cart moving than a lighter one.

A Force Sensor and an Accelerometer will let you measure the force on a cart simultaneously with the cart’s acceleration. The total mass of the cart is easy to vary by adding masses. Using these tools, you can determine how the net force on the cart, its mass, and its acceleration are related. This relationship is Newton’s second law of motion.

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Newton’s Cradle

Newton's cradle.

Newton's cradle consists of five iron balls, each hanging on two threads to prevent the ball from spinning.

Originally Newton's cradle was created to demonstrate Newton's third law. If you collide a ball from one side, the exact impact returns through the other.

The process of movement

If you drop one marble, the fallen ball hits other balls and stops completely. One ball on the other side immediately moves up at the same speed as the ball from which it fell, and the upward ball moves to the same height that the ball was initially pulled. This indicates that the bounced ball has inherited most of the energy and momentum of the falling ball. The balls in the middle remain still and only transmit waves created by compression. During the propagation of the wave, some of the energy is lost as heat.

Assuming no energy loss, the number of balls that bounce always equals the number of dropped balls. If you drop two balls, two balls bounce on the other side, and if you drop three, three balls bounce on the other side.

When dropping a ball

Newton's cradle is described by the law of conservation of momentum (= mv) and kinetic energy (= 1/2 mv 2 ) of an elastic body. The collision can be explained simply if the two balls have the same mass. The impacted object completely takes over the momentum and kinetic energy of the impacted object. In the case of an utterly elastic body, no loss occurs due to heat and sound energy. A hard iron ball does not compress well but is elastic, so it does not cause energy loss and transfers energy efficiently.

When two or more balls are dropped

Let's think about lifting two balls and then dropping them toward the third. With a subtle difference, the second ball hits the third ball first. According to the law of collision of an elastic body, the third ball takes over momentum and kinetic energy and moves, and the second ball that has fallen stops. The first ball hits the second ball that has stopped. This process is repeated so that if you lift and drop two balls, the two opposite balls will bounce at the same speed.

Phenomena caused by friction

In Newton's cradle, friction reduces momentum and kinetic energy gradually. A ball falling from Newton's cradle is pushed back slightly by conflict immediately after impact. When these effects accumulate, after time, all the balls turn into shaking little by little at the same time, eventually stopping the whole movement as well.

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• > Journals
• > Judgment and Decision Making
• > Volume 4 Issue 4
• > The mean, the median, and the St. Petersburg paradox

Article contents

• Introduction

The mean, the median, and the St. Petersburg paradox

Published online by Cambridge University Press:  01 January 2023

• Supplementary materials

The St. Petersburg Paradox is a famous economic and philosophical puzzle that has generated numerous conflicting explanations. To shed empirical light on this phenomenon, we examined subjects’ bids for one St. Petersburg gamble with a real monetary payment. We found that bids were typically lower than twice the smallest payoff, and thus much lower than is generally supposed. We also examined bids offered for several hypothetical variants of the St. Petersburg Paradox. We found that bids were weakly affected by truncating the gamble, were strongly affected by repeats of the gamble, and depended linearly on the initial “seed” value of the gamble. One explanation, which we call the median heuristic , strongly predicts these data. Subjects following this strategy evaluate a gamble as if they were taking the median rather than the mean of the payoff distribution. Finally, we argue that the distribution of outcomes embodied in the St. Petersburg paradox is so divergent from the Gaussian form that the statistical mean is a poor estimator of expected value, so that the expected value of the St. Petersburg gamble is undefined. These results suggest that this classic paradox has a straightforward explanation rooted in the use of a statistical heuristic.

1 Introduction

In the St. Petersburg paradox, originally proposed in 1738, the house offers to flip a coin until it comes up heads. The house pays $1 if heads appears on the first trial; otherwise the payoff doubles each time tails appears, with this compounding stopping and payment being given at the first heads (Bernoulli, 1738; shown in Figure 1 ). By conventional definitions, the St. Petersburg gamble has an infinite expected value; nonetheless, most people share the intuition that they should not offer more than a few dollars to play. Explaining why people offer such small sums to play a gamble with infinite expected value is an important question in economics and philosophy ( Reference Datson Datson, 1988 ; Reference Samuelson Samuelson, 1977 ; Reference Martin Martin, 2008 ; Reference Gigerenzer and Selten Gigerenzer & Selten, 2002 ).

Figure 1: The St. Petersburg paradox.

A. Outcome tree for St. Petersburg gamble. The St. Petersburg gamble consists of a series of coin flips offering a 50% chance of $1, a 25% chance of $2, a 12.5% chance of $4, and so on. The gamble may continue indefinitely.

B. The probability of each possible outcome decreases as a function of outcome size. The probability of large outcomes is very low, but not zero.

The St. Petersburg paradox has attracted explanations from many well-known thinkers, including Daniel and Niklaus Bernoulli, Cramer, de Morgan, Condorcet, Euler, Poisson, and Gibbon, and economists including Marschack, Cournot, Arrow, Keynes, Stigler, Samuelson, von Mises, Ramsey and Aumann ( Reference Arrow Arrow, 1951 ; Reference Aumann Aumann, 1977 ; Reference Dutka Dutka, 1988 ; Reference Keynes Keynes, 1921 ; Reference Samuelson Samuelson, 1960 ). The earliest discussions of the St. Petersburg paradox led to the idea of utility curves, now a central concept in economics ( Reference Dutka Dutka, 1988 ). Moreover, Menger’s discussion of the paradox sparked von Neumann’s interest in utility, which in turn influenced Von Neumann and Morgenstern’s foundational book Theory of Games and Economic Behavior ( Reference Morgenstern Morgenstern, 1976 ; Reference Von Neumann and Morgenstern Von Neumann & Morgenstern, 1944 ). Despite the importance of the St. Petersburg paradox, there is no widely accepted explanation for the low values most people place on this theoretically priceless gamble ( Reference Martin Martin, 2008 ; Reference Samuelson Samuelson, 1977 ). Moreover, there is precious little empirical data on the St. Petersburg Paradox (but see Reference Bottom, Bontempo and Holtgrave Bottom, Bontempo, and Holtgrave, 1989 ; Rivero, Holtgrave, Bontempo, and Bottom, 1990; Reference Kroll and Vogt Kroll and Vogt, 2009 ; Cox, Sadiraj, and Vogt). In the present study, we review several major classes of explanations, and then provide empirical data designed to test these explanations.

1.1 Diminishing marginal utility

Daniel Bernoulli was the first to argue, in his explanations of the St. Petersburg paradox, that the marginal value of money to an individual diminishes as his wealth rises ( Reference Bernoulli Bernoulli, 1738 ). This hypothetical concept is now known as utility ( Reference Friedman and Savage Friedman & Savage, 1948 ; Reference Von Neumann and Morgenstern Von Neumann & Morgenstern, 1944 ). The concavity of the utility function guarantees that, while the expected value of the gamble is infinite, its expected utility is finite. Thus, “any fairly reasonable man would sell his chance, with great pleasure, for twenty ducats” (Daniel Bernoulli, quoted in Baron, 1998). In some more recent discussions, a requirement that the utility curve have finite bounds has been added ( Reference Aumann Aumann, 1977 ; Reference Martin Martin, 2008 ; Reference Menger Menger, 1934 ; Reference Samuelson Samuelson, 1977 ; Reference Vickrey Vickrey, 1960 ). This requirement deals with super-St. Petersburg paradoxes (see below) and is psychologically plausible ( Reference Samuelson Samuelson, 1977 ).

The utility curve explanation is widely held up as the explanation for risk aversion in the St. Petersburg paradox (e.g. Aumann, 1977; Reference Friedman and Savage Friedman & Savage, 1948 ; Reference Real Real, 1996 ; Reference Schoemaker Schoemaker, 1982 ; Reference Von Neumann and Morgenstern Von Neumann & Morgenstern, 1944 ). Indeed, the St. Petersburg paradox is often used in introductory courses to motivate the idea of utility functions (e.g. Baron, 1998; Reference Mas-Colell, Whinston and Green Mas-Colell, Whinston, & Green, 1995 ; Reference Camerer Camerer, 2005 ). Despite its wide acceptance, the utility curve explanation has been discredited several times, mainly because it over-predicts bids ( Reference Lopes Lopes, 1981 ; Reference Martin Martin, 2008 ; Reference Menger Menger, 1934 ; Reference Moritz Moritz, 1923 ; Samuelson, 1960, 1977). (This is not to say that marginal utility does not diminish, just that this factor is not sufficient to explain the paradox.) Replacing expected utility with more modern variants, such as cumulative prospect theory (CPT) does not help either, as empirically fit values strongly over-predict bids in the St. Petersburg paradox ( Reference Blavatskyy Blavatskyy, 2005 ; Rieger & Wang, 2006; Reference Camerer Camerer, 2005 ).

An early strong critique of the utility explanation came from Karl Menger, who introduced the idea of the Super St. Petersburg paradoxes ( Reference Menger Menger, 1934 ). In these variants, the value of the gamble increases much faster than doubling on each round. The change function can be chosen so as to rise fast enough to overtake any reasonable utility function. Despite these richer payoffs, people intuitively feel that they would not pay more than a modest sum for any of these super-St. Petersburg gambles ( Reference Menger Menger, 1934 ; Reference Samuelson Samuelson, 1977 ).

An elegant demonstration of the weakness of this explanation comes from a recent experiment involving manipulations of delays instead of monetary rewards. People, like animals, are risk-seeking for delays, and so presumably have convex utility curves for time (Bateson & Kacelnik, 1996). Nonetheless, human subjects choose low values in a variant of the St. Petersburg paradox in which it is potential delays, not money, that doubles ( Reference Kroll and Vogt Kroll & Vogt, 2009 ). This pattern reveals that something other than a concave utility function explains how people value the gamble.

1.2 Finitude of resources

Another well-known explanation is that since the amount of money in the world is finite, the gambler must doubt the ability of the house to pay the large outcomes of the gamble. A related argument is that time is finite, and the gambler, knowing he or she cannot play the gamble forever, bids less than the expected value of the gamble. This argument has been expressed, in various forms, by Poisson, Catalan, Pringsheim, von Mises, Czuber, Buffon ( Reference Dutka Dutka, 1988 ) and others ( Reference Savage Savage, 1954 ; Reference Tversky and Bar-Hillel Tversky & Bar-Hillel, 1983 ; Reference Vickrey Vickrey, 1960 ). Camerer proposes that belief in finite resources, when combined with loss aversion, provides a parsimonious explanation for risk-averse behavior in the St. Petersburg paradox ( Reference Camerer Camerer, 2005 ). In this particularly well developed treatment, Camerer argues that if the maximal payoff is $1 billion, and the loss aversion coefficient is 2, the maximal buying price will be $17.55 (assuming a seed value of $2). Even if the maximal payout is raised to $1 trillion, the bid will rise only to $22.71. These values are close to the 20 ducats that Bernoulli thought was reasonable.

Several critics have noted the weaknesses in these arguments. Bertrand argues that even if the house cannot pay the money, units of currency can be replaced with something that is much more plentiful, such as grains of sand, inches, or molecules of hydrogen, and the risk aversion remains ( Reference Dutka Dutka, 1988 ). By similar logic, the payment may even be purely hypothetical or it may be psychological ( Reference Aumann Aumann, 1977 ; Reference Martin Martin, 2008 ). Furthermore, the house’s payment can remain fixed while the gambler’s is cut in half at each round ( Reference Keynes Keynes, 1921 ; Samuelson, 1960, 1977). Finally, common sense shows that the bids are much lower than they would be if the maximal payout of the gamble was limited only by the supply of money in the world. Indeed, even though Bernoulli felt comfortable offering 20 ducats, our intuition is that a lower value would be more reasonable. Nonetheless, this hypothesis remains to be tested empirically.

1.3 Ignoring low probabilities and risk aversion

D’Alembert argued that probabilities less than 1/10,000 are essentially equal to zero. Buffon and Condorcet appear to have agreed ( Reference Samuelson Samuelson, 1977 ). Niklaus Bernoulli set the cutoff at a more conservative 1/100,000 ( Reference Dutka Dutka, 1988 ). The arbitrariness of these cutoffs was criticized by contemporaries, including Condorcet ( Reference Dutka Dutka, 1988 ), and by more recent thinkers ( Reference Arrow Arrow, 1951 ). Regarding D’Alembert’s view, Gibbon said, “if a public lottery were drawn for the choice of an immediate victim, if our name were inscribed on one of the 10,000 tickets, should we be perfectly easy?” (quoted in Samuelson, 1977). Contemporary thinkers recognize that small probabilities may be discounted psychologically ( Reference Menger Menger, 1934 ; Reference Sennetti Sennetti, 1976 ; Reference Weirich Weirich, 1984 ).

This de minimis argument is subject to a counterargument that is similar to Menger’s super-St. Petersburg approach. If low probabilities are systematically discounted, then payoff can be increased to counterbalance this effect. Nonetheless, people will still presumably offer finite sums for the St. Petersburg gamble ( Reference Martin Martin, 2008 ). Moreover, small probabilities are often exaggerated ( Reference Blavatskyy Blavatskyy, 2005 ; Reference Kahneman and Tversky Kahneman & Tversky, 1979 ), rather than being discounted, making this explanation unlikely.

1.4 Is the expected payoff of the St. Petersburg gamble infinite?

In a rare application of empirical methods, Buffon in 1777 hired a child to flip a coin until it came up heads, and to do so 2048 times. In 1838, Augustus De Morgan added another 2048 data points ( Reference Moritz Moritz, 1923 ). De Morgan and Buffon both argued that actual experience demonstrates that the true expected value of the gamble is quite low, justifying the low value placed on the gamble. In more recent times, computers have made it possible to simulate coin flips more rapidly, and although estimated values are higher, the fundamental result does not change ( Reference Ceasar Ceasar, 1984 ; Reference Hinners-Tobraegel Hinners-Tobraegel, 2003 ; Reference Liebovitch and Scheurle Liebovitch & Scheurle, 2000 ; Reference Lopes Lopes, 1981 ). Importantly, the results of these empirical studies are generally greater than bids people intuitively offer.

A few writers have suggested that the root of the paradox lies in the definition of expected value ( Reference Gigerenzer and Selten Gigerenzer & Selten, 2002 ; Reference Liebovitch and Scheurle Liebovitch & Scheurle, 2000 ; Reference Lopes Lopes, 1981 ). Statistically, expected value is the central tendency of the distribution embodied in a risky gamble. For highly non-gaussian distributions the mean is not considered a valid estimator ( Reference Hinners-Tobraegel Hinners-Tobraegel, 2003 ; Reference Liebovitch and Scheurle Liebovitch & Scheurle, 2000 ; Reference Vivian Vivian, 2003 ). The set of outcomes for the St. Petersburg gamble is infinitely positively-skewed, and thus highly deviant from this assumption or normality. Because the statistical mean is considered invalid for distributions highly non-normal distributions, the true expected value of the St. Petersburg gamble is undefined, not infinite ( Reference Liebovitch and Scheurle Liebovitch & Scheurle, 2000 ; Reference Lopes Lopes, 1981 ).

The median, an alternative estimator of central tendency, is robust to noise and is often favored for highly skewed distributions (like the St. Petersburg). The median of the distribution associated with the St. Petersburg gamble is between $1 and $2, and is set by convention at $1.50 ( Reference Weissstein Weissstein, 2008 ). Theoretical considerations therefore support the idea that gambles in the St. Petersburg paradox reflect the application of the median. The median is mathematically similar to the expectation heuristic, by which people estimate the trial on which heads is most likely to appear, and bid accordingly ( Reference Treisman Treisman, 1983 ). Importantly, the expectation heuristic is supported by empirical data, although these data may also support a median heuristic ( Reference Bottom, Bontempo and Holtgrave Bottom, Bontempo, & Holtgrave, 1989 ; Rivero, Holtgrave, Bontempo, & Bottom, 1990).

To distinguish between these two models, as well as to contribute more empirical data on this phenomenon, we collected responses of 200 individuals to several hypothetical variants of the St. Petersburg gamble and responses of 20 individuals to a real-stakes version of the St. Petersburg gamble. Our results support the idea that people estimate the value of the gamble using the median, thus endorsing this simple explanation of the St. Petersburg Paradox.

We performed one study using real stakes with twenty subjects. These subjects were recruited from the undergraduate and graduate student population at Duke University. Subjects were provided with either $10 or $5 (randomly chosen) one week before the survey took place. We introduced this delay in order to increase the feeling of ownership and thus to prevent subjects from feeling like they were gambling with “house money.” We used a variant of the Becker-DeGroot-Marschak auction (Becker & Brownson, 1964) to obtain valuations. In our variant, an outcome value from the St. Petersburg distribution was chosen at random, and then was augmented or decreased by one cent (randomly chosen, to reduce the likelihood of a tie). The subject’s bid was compared to this random number. If the bid was greater, the subject paid the value of the number and the gamble commenced; if the bid was lower, the gamble did not occur. We explained the details of the gamble and the auction carefully (see Supplement for details).

We performed a second study investigating hypothetical responses of 200 subjects to several variants of the St. Petersburg Paradox. An article by John Tierney about Dan Ariely in the New York Times offered readers the option to submit their email addresses and receive future surveys. We sent invitations to take web-based surveys to 700 of these email addresses. Of these recipients, 200 responded within two weeks, at which point we closed the survey. The population of subjects thus consists of people who were sufficiently motivated to provide their email addresses for future studies, and to then respond to a subsequent request for an additional survey. The first 100 respondents took survey 1 while subsequent respondents took survey 2, both available in the Supplement. Footnote 1 The survey consisted of a simple series of html documents using PHP for dynamic content and access to a database to store the results.

A preamble to the survey stated that “The probability of heads on the first toss is 50%; the probability of heads on the second toss is 25%; the probability of heads on the third toss is 12.5%, and so on.” The purpose of the preamble was to orient the subject to the general format of the questions. The first question was as follows: “Consider the following gamble: You may flip a coin. If it comes up heads, you receive $1. If it comes up tails, you will flip it again. If it comes up heads, you receive $2. If it comes up tails, you will flip it again. If it comes up heads, you receive $4. This process repeats until the coin comes up heads, with the payoff for heads doubling each time it comes up tails. If the opportunity to play this gamble was on sale, what is the maximum amount of money that you would pay for it?” By combining the preamble with the questions, we believe that we presented the puzzle as clearly as possible and removed, as much as possible, the potential for confusion. Twenty additional respondents answered the first question only. These subjects were undergraduate students in an introductory psychology class offered at Duke University. Survey recipients received a URL that linked to a webpage containing instructions for the survey.

In one analysis, we showed that the median payoff of the St. Petersburg gamble depends systematically on the number of times the gamble is repeated. We did this by use of a bootstrap simulation ( Figure 3 D). Each point on the graph represents the mean outcome per repetition that a gambler can expect (ordinate) in a string of St. Petersburg gambles repeated r times (abscissa). To determine the ordinate for each value of r, we repeated the St. Petersburg r times, and took the mean payoff. We then repeated this process 10,000 times, and took the median of these 10,000 generated numbers. Thus each point on the line represents the outcome of (10,000 · r) individual St. Petersburg gambles. The mean of these numbers does not depend on the number of times the gamble is repeated (see Reference Samuelson Samuelson, 1963 ).

3.1 The standard St. Petersburg gamble

The median bid of subjects (n=20) faced with the real-world version of the St. Petersburg gamble was $1.75. The median bid of subjects faced with a hypothetical St. Petersburg gamble was $1.50 (n=220). These bids did not differ significantly (Wilcoxon rank-sum test, p>0.4). This match between these two groups confirms that differences in the method of elicitation do not strongly influence choices. This first group of 20 subjects then answered the first survey (see Appendix). These subjects’ choices did not differ significantly from those made by our internet subjects, confirming that these two groups had overlapping preferences (Wilcoxon rank-sum test, p>0.15 in all cases).

Bids for the hypothetical St. Petersburg gamble ranged from zero to $50,000 ( Figure 2 ). The median bid was $1.50, and the distribution of bids had strong modes at $1 and $2. Of the 220 responses, 14 (6.4%) refused to gamble, 26 (11.8%) offered positive values less than $1, and 68 (30.9%) offered $1, the modal bid. 13 (5.9%) offered between $1 and $2 and 7 (3.18%) offered the median bid ($1.50). 43 (19.6%) people offered $2, and 56 (25.5%) offered more than $2.

Figure 2: Histogram of bids offered for the standard St. Petersburg paradox. Although the expected value of the gamble is infinite, all bids were finite. The median bid was $1.50. The distribution was bimodal, with large modes at $1 and $2.

We made every effort to ensure that our subjects understood the task. To verify this, we asked a probe question of the 20 human subjects who performed the real Petersburg gamble. We asked them to indicate what they believed the probabilities were of obtaining $1, $2, $4, $8, $16, $32, and $64. These subjects all performed correctly, indicating that they understood the probabilities, and suggesting that the majority of the other subjects also understood the probabilities.

On all questions, subjects were offered the option to explain their rationale. We received 254 comments (median bid for these subjects, $1.50). We coded all responses based on which explanation best matched. A strong plurality related to the median argument ( Table 1 ). Explanations in this category referred to the probabilities of obtaining specific outcomes as being the critical factor influencing options (as also found by Lopes, 1996). Collectively, these comments support the notion that subjects make use of the median in estimating the value of the St. Petersburg gamble.

Table 1 Explanations for gamble offers. We coded responses into 6 categories, corresponding to the major theories explaining St. Petersburg offers. The most common class of explanations focused on the chance of a given outcome occurring. Actual examples are given for each category.

3.2 Changing the seed value

To probe the robustness of the median heuristic, we examined responses in several variants of the standard St. Petersburg gamble. In one set of questions, we manipulated the gamble’s “seed value,” but kept the structure of the gamble the same ( Figure 3 A). We tested four seeds: $0.01 (1 cent), $0.50, $1 (the standard gamble), and $4 ( Figure 3 A). If the median outcome is a driving factor in valuations, then bids should follow the average of the first two outcomes (i.e. the median). If instead concave utility curves are a major factor, gambles with smaller stakes would elicit larger bids relative to the seed value (since possible outcomes are lower and must be less discounted; see Reference Samuelson Samuelson, 1960 ). Similarly, if the gambler believes the house’s wealth has a particular limit, then gambles with lower stakes must elicit greater relative bids. However, relative bids did not vary with seed, and were nearly identical to the median in all cases (Wilcoxon rank sum test, p>0.2).

Figure 3: Bids in variants of the St. Petersburg paradox.

A. When the stakes are varied, bids (blue dashed line) closely track the median (red solid line). Expected value (mean) is infinite for all variants.

B. When the St. Petersburg gamble is truncated, the median remains $1.50 (gray shadow), but the mean grows with the truncated value. Bids remain low, close to the median. The standard St. Petersburg gamble has an infinite expected value, and is the rightmost point on the graph.

C. As the St. Petersburg gamble is repeated, offered bids per gamble grow. The offer bids closely tracks, but over-estimates, the expected median value of the series of gambles. Means do not change as the gamble is repeated.

D. Median outcome per gamble of repeated St. Petersburg gamble increases as a function of number of repeats. The more repeats the gambler faces, the greater the value of each gamble. Although the function is concave, it is unbounded.

3.3 The truncated St. Petersburg gamble

In the truncated variant of the St. Petersburg gamble, coin-flipping stops if some predetermined maximum payout is achieved ( Figure 3 B). This finite variant circumvents any question about the ability of the house ability to pay, as well as any arguments concerning the strangeness of infinity ( Reference Martin Martin, 2008 ). We examined bids for the St. Petersburg gamble truncated at 3 flips (maximum $8, EV: $2.50), 5 flips (maximum $32, EV: $3.50), 8 flips (maximum $256, EV: $5), 10 flips (maximum $1024, EV: $6) and 15 times ($32,768, EV: $8.50). In all cases, the median value of the gamble was identical (between $1 and $2). Bids for these gambles were either $1.50 or $2, and did not covary with expected value (Wilcoxon rank-sum test, p>0.05 for all pairs).

3.4 The repeated St. Petersburg gamble

According to the Law of Large Numbers, the average value of a stochastic variable repeatedly drawn from almost any distribution will eventually converge on a single value. This property applies to almost all gambles we encounter, a fact that allows us to assign an expected value to any gamble. A simple consequence of this fact is that casinos and insurance agencies cannot predict whether a single gamble or car trip will produce a profit or a loss, but can reliably predict the profitability of a group of gambles or drives. However, a few distributions deviate so strongly from the assumptions of the Law of Large Numbers that repeated samples do not converge on any value. The St. Petersburg Paradox is one such distribution. The expected value of such distributions is not defined. It is fallacious therefore to argue that the St. Petersburg paradox has an infinite expected value.

One consequence of this degenerate behavior of the St. Petersburg distribution is that the median payoff of the gamble depends on the number of times the gamble is repeated. This strange property of the median is a consequence of the fact that the repeated gamble may occasionally lead to long strings of tails that give extremely large rewards ( Reference Liebovitch and Scheurle Liebovitch & Scheurle, 2000 ). To confirm this counterintuitive result, we ran a simple Monte Carlo simulation ( Figure 3 D). Each point on the abscissa corresponds to a number of repeats of the gamble, while each point on the ordinate shows the expected payoff per gamble obtained from 10,000 independent simulations.

We hypothesized that subjects would intuitively recognize these facts, and bid more as more gambles are offered. In contrast, the utility curve explanation, and most of its variants, necessarily predict decreasing valuations as the gamble is repeated (since marginal utility declines, the multiplying of payouts associated with repeated gambles would dampen enthusiasm for multiple gamble). Similarly, the finitude argument predicts decreasing valuations as more gambles are offered (since the house’s limit is more likely to be reached with more gambles). We offered subjects 1, 10, 100, and 1000 repeats of the St. Petersburg gamble ( Figure 3 C). We found systematically greater offers with greater number of repeats, with bids closely tracking the expected median (these bids differed significantly, Wilcoxon rank-sum test, p<0.01 in all cases). Overbidding for the 100 and 1000 repeat condition may reflect failures to perform the complex calculations needed to estimate the expected median, or they may reflect the contribution of other factors.

4 Discussion

The St. Petersburg paradox has generated abundant speculation but very few empirical data. (Martin, 2008, provides the best summary of the theories.) This paucity of data has obscured the fact that most proffered explanations are poor predictors of bids for this gamble. Our data set, consisting of 20 real bids and 220 hypothetical bids for each of several variants, addresses this lack of data. We present three main findings.

1. Observed bids for the original form of the St. Petersburg gamble are much lower than is commonly supposed. Specifically they are lower than twice the smallest payoff. This finding is inconsistent with the majority of proffered explanations, but is consistent with a median heuristic.

2. Bids are only weakly affected by truncating the gamble, are strongly affected by repeating the gamble, and depended linearly on the initial ‘seed’ value of the gamble. These findings are inconsistent with the majority of proffered explanations, but are consistent with a median heuristic.

3. Contrary to popular belief, the expected value of the St. Petersburg gamble is not infinite, but is instead undefined. The St. Petersburg gamble’s infinite variance, a consequence of its fractal nature, means that it violates a critical assumption of the law of large numbers.

Because the St. Petersburg gamble has no expected value, we conjecture subjects instead use the median, which provides a well-defined, robust, estimator of central tendency, instead. The median is an appealing tool for decision-makers. It is simple, cognitively plausible, and is less susceptible to noise, error, and outliers than the mean. These data are not explained by any of the major alternative approaches outlined above, although other psychologically plausible mechanisms may also explain the data.

We thus argue that true root of the St. Petersburg Paradox lies in the definition of expected value ( Reference Gigerenzer and Selten Gigerenzer & Selten, 2002 ; Reference Liebovitch and Scheurle Liebovitch & Scheurle, 2000 ; Reference Lopes Lopes, 1981 ). Expected value is the central tendency of the distribution embodied in a given gamble. For a Gaussian distribution, the central tendency is given by its mean. For highly non-gaussian distributions, such as the St. Petersburg gamble, the mean provides a poor estimate ( Reference Hinners-Tobraegel Hinners-Tobraegel, 2003 ; Reference Liebovitch and Scheurle Liebovitch & Scheurle, 2000 ). Consequently, the true expected value of the St. Petersburg gamble is not infinite, but is undefined, so no bid is inconsistent with theory, and there is no paradox ( Reference Liebovitch and Scheurle Liebovitch & Scheurle, 2000 ; Reference Lopes Lopes, 1981 ). Indeed, misapplication of the mean to non-Gaussian disrtibutions may cause several other fascinating paradoxes ( Reference Blyth Blyth, 1972 ; Reference Hacking Hacking, 1980 ; Reference Schwitzgebel and Dever Schwitzgebel & Dever, 2008 ).

A few previous studies have used empirical methods to investigate the St. Petersburg paradox ( Reference Bottom, Bontempo and Holtgrave Bottom, Bontempo, and Holtgrave, 1989 ; Rivero, Holtgrave, Bontempo, and Bottom, 1990; Cox, Sadiraj, and Vogt, 2008; Reference Kroll and Vogt Kroll and Vogt, 2009 ). In two closely-related studies, subjects chose hypothetical bids and sale prices in accordance with the “expectation heuristic” ( Reference Bottom, Bontempo and Holtgrave Bottom, Bontempo, & Holtgrave, 1989 ; Rivero, Holtgrave, Bontempo, & Bottom, 1990). The expectation heuristic argues that subjects simplify the problem by assuming that the coin will come up heads on the second flip (i.e., the expected flip), which is mathematically similar to the median heuristic ( Reference Treisman Treisman, 1983 ). Various results in their data sets both supported and contradicted this theory, although most supported it ( Reference Bottom, Bontempo and Holtgrave Bottom, Bontempo, & Holtgrave, 1989 ). Although similar to the median heuristic, the expectation heuristic predicts bids of $2 in our statement of the St. Petersburg paradox, while the median heuristic predicts bids in the range of $1-$2. Thus our observation that bids are $1.50 (for hypothetical gambles) and $1.75 (for real gambles) are slightly lower than the expectation heuristic, and the diversity of responses with strong modes at $1 and $2 is more consistent with the median heuristic. Moreover, the expectation heuristic predicts that the prospect of repeating the gamble should not affect bids, which is inconsistent with our observed results. Nonetheless, due to the similarities between the expectation heuristic and the median heuristic, we believe that these earlier data provide complementary evidence to that provided here. Thus, the present results build on and extend these earlier results, showing that preference for median bids is preserved with additional variants of the paradox, and is preserved when bids are made with real stakes.

We do not believe that risky decision-making consists solely of identifying the median outcome, but we think that the present data argue that estimating the median, or some other ordering statistic, is a critical step in the decision-making process in this task. Instead, we suspect that people choose from several available strategies, and perhaps combine multiple strategies ( Reference Gigerenzer and Selten Gigerenzer & Selten, 2002 ; Reference Payne, Bettman and Johnson Payne, Bettman, & Johnson, 1993 ) based on specific task demands ( Reference Payne, Bettman and Johnson Payne, Bettman, & Johnson, 1993 ). Moreover, we acknowledge that many other models of risky decision-making may explain these results as well, although we wish to emphasize that observed bids are much lower than those predicted by most standard models in behavioral economics. Thus, we are not certain that any of the major models can explain all the results shown here without substantial modifications. In any case, the median provides a parsimonious and compelling model for risky decision-making.

The present results do not implicate the median heuristic in gambles outside the context of the St. Petersburg paradox; subsequent studies will be needed to determine whether this strategy is applicable in a wider array of decisions. We note that use of the median in non-St. Petersburg gambles has some empirical support ( Reference Lopes Lopes, 1996 ; Reference Wedell and Boeckenholt Wedell & Boeckenholt, 1994 ). We would predict that the median would be especially likely to be used for highly non-gaussian distributions, including exponential distributions like the St. Petersburg gamble, where the mean is a poor estimator of central tendency. Interestingly, another familiar non-Gaussian distribution is the standard two-outcome gamble. Several writers have argued that expected values are irrelevant for such gambles ( Reference Knight Knight, 1921 ; Reference Lopes Lopes, 1981 ; Reference Weaver Weaver, 1963 ). As Moritz says, “the mathematical expectation of one chance out of a thousand to secure a billion dollars is a million dollars, but this does not mean that anyone in his senses would pay a million dollars for a single chance of winning a billion dollars” ( Reference Moritz Moritz, 1923 ).

We thank Dan Ariely for useful discussions and help with materials, and Charles Hayden for help with webpage design. We thank Rachel Croson, Jon Baron, and Sarah Heilbronner for invaluable help with the manuscript. This work was supported by the National Institute of Health (5R01EY013496) and by a Kirschstein NRSA to BYH (DA023338).

1 The Supplement is included with the present article in this issue of the journal: http://journal.sjdm.org/vol4.4.html .

Figure 1: The St. Petersburg paradox. A. Outcome tree for St. Petersburg gamble. The St. Petersburg gamble consists of a series of coin flips offering a 50% chance of $1, a 25% chance of $2, a 12.5% chance of $4, and so on. The gamble may continue indefinitely. B. The probability of each possible outcome decreases as a function of outcome size. The probability of large outcomes is very low, but not zero.

Figure 3: Bids in variants of the St. Petersburg paradox. A. When the stakes are varied, bids (blue dashed line) closely track the median (red solid line). Expected value (mean) is infinite for all variants. B. When the St. Petersburg gamble is truncated, the median remains $1.50 (gray shadow), but the mean grows with the truncated value. Bids remain low, close to the median. The standard St. Petersburg gamble has an infinite expected value, and is the rightmost point on the graph. C. As the St. Petersburg gamble is repeated, offered bids per gamble grow. The offer bids closely tracks, but over-estimates, the expected median value of the series of gambles. Means do not change as the gamble is repeated. D. Median outcome per gamble of repeated St. Petersburg gamble increases as a function of number of repeats. The more repeats the gambler faces, the greater the value of each gamble. Although the function is concave, it is unbounded.

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• Volume 4, Issue 4
• Benjamin Y. Hayden (a1) (a2) and Michael L. Platt (a1) (a2) (a3)
• DOI: https://doi.org/10.1017/S1930297500003831

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