Watermelon Trebuchet

A decision tree provides a rational method for choice in the presence of uncertainty.

Making Intelligent Bets

Decision trees provide a rational method for making choices when uncertainty about the outcomes can be quantified.  For example, suppose someone offers us a raffle ticket, at a cost of $1, for a chance to win a car that is valued at $100,000.  A prudent person purchases the proffered paper posterior to predicting a positive payback.  Should you buy the ticket?  Here’s how to find out!

We need to assign values to our variables.  We assign a value of $1 to cost, represented by c, otherwise we have chosen the “DO NOT” branch.  There is no “TRY” in this type of analysis.  The nerdy among you might catch the sci-fi reference.  The true geeks caught the reference in the sentence pointing out the more obvious reference two sentences back.  But, I digress, recursively . . .

The car value of $100,000 must be adjusted, since you know that a certain bankrupt uncle will force you to sell it in order to pay the tax bill.  By the time the state and city get their shares, a paltry $50,000 remains.  To most people that represents a lot of money, and a lot of people buy the ticket anyway without performing a formal analysis.  We keep our wallets in our pockets, and instead assign $50,000 to v.

Finally, you need to know the probability of winning.  Assuming a fair game and a one-ticket purchase, the probability of winning becomes the reciprocal of the number of tickets in the hopper at the time of the drawing.  Symbolically, p = 1/N.

Unfortunately, we cannot know N unless we obtain a count of the tickets in the hopper immediately before the drawing occurs.  How do we overcome this information deficit?  Estimate it.

•  If we can find the number of tickets currently sold, divide this number by the amount of time that has elapsed, then multiply the result by the total length of time that tickets are being sold.  This method extrapolates the average rate of sales to the close of the sale period.
• If the history of previous raffles indicates an almost certain chance of selling all tickets, then just find out how many tickets exist.
• If the history of previous raffles indicates that a fixed number of tickets sell every year, use that number.  Various types of regression techniques could be used to extrapolate trends from previous years.

For the sake of argument, let us pretend that a generous dealer donated the car (also for tax purposes) to a poor rural community school, who raffles it off to buy iPads and install wireless networks on their campus.  Due to the rich prize, people came from miles around to buy, so our estimated ticket count is 50oo.  This means that p = 1/5000 = 0.0002.

Should we buy?  Our expected value is the sum of the outcomes multiplied by their probability.  We know that probabilities sum to one, so we have E = p(V-C)-(1-p)C=p V-C=0.0002*50000-1=$9.  Our expected payoff is $9, so we gladly buy a ticket.

Epilogue

You didn’t win, but you still gained more than $1 worth of warm fuzzies for helping the youth of your community compete in a globally connected world.  Now, fast-forward one year.  The new technology enabled the director of fundraising to learn that she needs to raise the price of tickets to maximize the profits.  The tax laws did not change, so next year the same dealer donates another car of equal value to the school.  What price should the director set for the tickets?

Naturally, the cost should lead to a neutral expected payout.  People buy raffle tickets for the warm fuzzies anyway, right?  With E=0, we rearrange the equation for expected value to p V = c.  Thus, the cost threshold for a rational purchase can be computed by multiplying the prize value by the probability of winning.  Our director assumes that she can sell 5000 tickets again, and sets the raffle ticket price at 0.0002*50000=$10.

As a matter of insurance, the director only prints 5000 tickets and announces this fact.  Neglecting inflation, the expected payout will never g0 negative, and thus any rational person of legal age with $10 to invest and no religious restrictions on gambling should buy a ticket.  How can this be?  This final exercise is left for the readers!

Tschüß!

Granovetter’s model for collective action uses thresholds to determine whether agents in a population “join the movement.”  The number of an agent’s peers who must participate in the action in question before said agent also begins to participate defines the thresholds.  The model was published in the following academic paper: Threshold Models of Collective Behavior, Granovetter, M.  The American Journal of Sociology, Vol. 83, No. 6 (May, 1978), 1420-1443.

What follows represents a simple example of the model’s operation, as presented in the paper cited above.  Let’s define the collective action as a fad: listening to dubstep music.  I won’t listen to dubstep until ten of my peers are listening to dubstep.  My “threshold” is therefore defined to be ten.  Furthermore, nine of my peers won’t listen to dubstep unless one of us also listens to dubstep.  Their thresholds are all one.  As a result, nobody listens to dubstep.  Now, imagine that a new person joins our peer group, and that person likes dubstep.  This first person’s threshold was zero.  Seeing this development, my nine impressionable peers begin listening to dubstep.  Now that ten of my peers are listening to dubstep, I download some dubstep and discover what I’m missing.  This example, while contrived, represents the type of group dynamics Granovetter’s model seeks to describe.  It can describe fads, riots and other collective actions.  As a matter of terminology, people with zero thresholds are called “instigators.”

Figure A: According to Granovetter's model, the equilibrium number of participants in a riot is given by the leftmost intersection of the cumulative distribution of thresholds (blue line) with the diagonal, shown here in red. (mean=25, sigma=10)

Granovetter sought to demonstrate that unanticipated collective actions can occur without changes in individual preferences.  Furthermore, the dynamics could be highly unstable.  To demonstrate this, he defined a simple crowd of 100 people, with the cumulative distribution of thresholds (CDT) to participate in a riot described by a normal distribution with mean=25 and variable sigma (Figure A).

Figure B: Close-up of equilibrium point for (25,10). Since the distribution intersects the diagonal at 0.769551, no riot occurs for a system described by this distribution.

Once an instigator acts, higher and higher thresholds are activated until reaching an equilibrium when the number of people rioting equals the CDT.  Thus, Figure B demonstrates that for (mean,sigma)==(25,10), no riot occurs because the equilibrium falls below a single rioter.  If something causes the width of the CDT to increase such that at least one person with zero threshold exists, then a riot will ensue.  The threshold for the appearance of an instigator is approximately sigma=10.3166 for a mean of 25.

Figure C: Increasing sigma to twelve causes results in an easily kettled riot of 4.02111 protestors.

Figure D: At sigma=12.22208, the CDT just touches the diagonal to give an equilibrium riot size of 6.18741.

Figure C demonstrates the small riot that results from increasing sigma to twelve.  Normally, we dismiss misbehavior by a small number of people in a crowd as reflecting poorly on the individual participants.  Granovetter’s model demonstrates the weakness of this conclusion.  Consider what happens when we increase sigma to 12.22208?  Figure D shows that for this value of sigma called the “critical point” the CDT just touches the diagonal, and the riot participation rate increases to 6.18741 rioters.  If sigma were instead just a tiny bit greater, say 12.22209, the leftmost intersection jumps to the far right of the graph and everyone joins in.  For any value of sigma greater than or equal to the critical sigma, the CDT does not cross the diagonal until after the mean (Figure E).  Thus, one population has six “bad eggs” and another becomes an angry mob despite possessing nearly identical CDTs.

Figure E: The lower "knee" of the sigmoidal CDT acts like a floodgate. If it doesn't cross the diagonal before the mean, the situation changes from a bad news segment to a full-blown riot.

We can think of the lower “knee” of the sigmoidal CDT as a sort of floodgate.  When this floodgate opens by rising above the diagonal, a full-blown riot ensues.  A plot of riot size vs. sigma is presented as Figure F.  These two graphs (E+F) present Granovetter’s key results:  the switch from minor disturbance to full-blown riot is quite sensitive to the distribution width, and occurs far below the mean riot threshold of the group.

Figure F: Riot size as a function of standard deviation of the CDT for a mean threshold of 25.

Oddly enough, further increases of sigma cause the number of rioters to go down.  This results from the same phenomenon that caused the sudden jump; specifically, the symmetrical knees of the normal CDF (Figure G).  In this figure, the green trace represents the case where everyone in the crowd possess approximately the same propensity to riot, and no riot occurs unless 25 or more people spontaneously riot.  The odds of this many instigators occurring for such a narrow distribution is infinitesimally low, so no riot is predicted.  The blue trace shows the critical case, where there are enough agents activated to cause a disturbance, but cool heads still prevail.  The orange line demonstrates how the riot size decreases as the width of the CDT is increased.

Figure G: Comparisons of various sigma values demonstrating three key results from Granovetter's normal CDT. (Key: green, sigma=1; blue, sigma=12.22; orange, sigma=50)

The statistically erudite reader may balk at the range mismatch between the normal distribution on infinite limits and our discrete basis set.  We’ll leave that observation as barroom argument fodder for pedants, and instead accept it for the moment and move on to the more important question:  What does all this mean?  For starters, it means that we must consider the distribution of propensities for collective action independently of how they are formed.  Granovetter presents cases where random fluctuations in the sample from the same distribution results in no riot, minor disturbance, and full-blown riot.  The “take-home message” states that the observation of a riot does not necessarily indicate that the participants are more riot-prone than the population at large.  It merely means that the distribution of thresholds were such that the actions of relatively few instigators started an unfortunate chain reaction in a metastable dynamical system.

Certainly, how one attains a certain threshold stands as an important question.  One could argue that understanding how thresholds are formed holds great promise as a method for shaping the distribution of thresholds, as the width of the distribution seems to be of greater import than the mean in the presence of instigators.  Nevertheless, Granovetter’s work shows that individual attitude formation and collective action participation should be treated as distinct, yet related, phenomena.

I will close this post by mentioning that I used Mathematica 8 Home Edition to solve the difference equations and construct the plots.  An archive of the notebook can be downloaded here.