The Standing Ovation Problem

by johnmccreery

Another book has caught my eye, Complex Adaptive Systems: An Introduction to Computational Models of Social Life; John H. Miller and Scott E. Page (2007).  Oh, yes, another of those books that attempts to introduce people who remember a little math to what people who construct mathematical models are up to. This one seems pretty good.

The writing is readable and the attitude pleasantly modest. The authors write, for example,

Suppose we want to construct a model of a standing ovation. There is no set method or means by which to do so. To model such a phenomenon we could employ a variety of mathematical, computational, or even literary devices. The actual choice of modeling approach depends on our whims, needs, and even social pressure emanating from professional fields. (2007:11)

A standing ovation? Why a standing ovation? Turns out to be a good first example for introducing what modeling is about. First, everyone knows what we’re talking about. At the end of a performance, the applause begins. Some people rise to their feet. Others may join them. Then others may join as well….Or they may not. Sometimes only a few people stand. Sometimes everyone stands. Often there is a ripple effect, with wave after wave of people joining in. Seen from this perspective, a standing ovation is a good starter model for thinking about all sorts of things, social movements, political campaigns, lynch mobs, or epidemics, for instance. But how do you construct the model?

The authors lead us through a series of steps that show how rapidly complexities enter in.

Let N be the number of people in the audience and q the quality of the performance. Assume that each member of the audience responds to a signal defined as Si(q). The strength of the signal can vary, reflecting different personal estimates of q. To work that in, add a random variable e and define, Si(q) = q+e. Finally, assume that there is a threshold T, such that if Si (q)>T the individual rises. We now have a simple prediction. The number of people who join in is simply the number of people for whom Si(q)>T.

Notice, however, that this model fails to account for things like new waves of people joining in once the first people who stand have done so. Now we start thinking about things like the effects of people sitting in different places; those who sit in the front are seen by everybody but can’t see anyone behind them. Or the fact that people often come to concerts in couples or groups of friends or may know other people in the audience; if one member of a set stands up, will the others follow suit? What are the spatial distributions of individuals, groups and lines of sight that will make members of an audience more or less likely to join the standing ovation?

Practical applications? The authors suggest a number of possibilities:

Do performances that attract more groups lead to more ovations? How does changing the design of the theater by, say, adding balconies, influence ovations? If you want to start an ovation, where should you place your shills? If people are seated based on the preferences for the performance, say left or right of the aisle or more expensive seats up front, do you see different patterns of ovations?

Turns out to be difficult to write a simple formula like the one we started with. We need to look at simulations involve interacting agents, for which computer modeling works better than conventional assumption, theorem and proof mathematics. How does that work?

(To be continued)

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7 Comments to “The Standing Ovation Problem”

  1. It seems to me that the “boids” algorithm might also be useful here.

  2. Do tell us more! I am just looking into this stuff. Have no idea what the “boids” algorithm does.

  3. Asher Kay beat me to it! The boid algorithm is an attempt to model flocking behavior in birds or fish. It’s remarkably successful. Variations of it are used in CGI effects for the film industry. Every flock has a leader but if you just had all the rest of the flock looking to the leader for what to do you get chaos. So what you do is you have each member look to it’s neighbor and adjust his flight behavior accordingly.

    Boids

  4. Thanks, now it clicks. I knew about the phenomenon, didn’t know that it was called the boid algorithm.

  5. Oh yeah, that flocking boid algorithm. For when the question is what the flock.

    I’m thinking of doing the wave at a sporting event (why there’s no waving at choral performances is a matter of some interest for another discussion). There’s a division of attention – a sort of general anticipation from watching the whole thing go around, then a much more focused attention as it gets close, with a specific action-trigger from the immediate neighbors.

  6. I wrote a boids app for the iPhone not too long ago, which is why it came to mind. The key to it, as noen already mentioned, is proximity.

    In this case, you’d probably need to expand it to multiple flocks to cover the grouping phenomenon. There’s also some individuality at work — for example, I’m not one to give anything a standing ovation (unless the performance truly blew me away), but I’m probably medium likely to go along if everyone else is standing up. A true leader would be likely to initiate, but unlikely to follow suit. It would be interesting to know if that aspect would affect the simulation.

    The idea of “simple formulas” is the really interesting thing to me. In your view, what’s the difference between a sort of cellular automaton simulation which follows simple formulaic rules and a set of simple formulas that describe the system?

  7. Multiple flocks? Trying to think about that raises a question for me. If the the boid algorithm is dependent on interactions between immediate neighbors, it doesn’t fit the standing ovation case, where someone in back can be moved to rise by someone rising several rows in front or off to the side, even if the immediate neighbors stay seated.

    Simple formulas versus stimulation using simple formulaic rules? That’s tricky. I think of simple formulas as equations with single solutions and formulaic rules as generating multiple branching outcomes via recursion. But then I remember that even simple looking equations can have multiple answers (the square root of 1 can be either +1 or -1 for example) and the logistic map x at time t+1 = R*x at time t* (1-x at time t) is the mother of all chaos generators. Then things don’t look so simple.

    How would you draw the difference?

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