# Telephone Game With Functions

In the old telephone game each person would pass on a phrase to the next person in the chain; the final phrase might little resemble the first.  An interesting variation appeared in the Phil. Trans. Royal Society last November:

Here each row is a chain of people passing along a function relating X to Y.  Each person first guesses and is corrected on 50 (X,Y) cases, then just guesses on 100 more cases.  The final guesses of the last person become data for the next person.  The final relations are all basically lines, 7/8 with a positive slope, 1/8 with a negative slope.

The lesson?  When we are mainly rewarded for predicting what others will say on a topic, rather than predicting a more basic reality, our answers become dominated by typical prior expectations; reality has little influence.  HT to Jef Allbright. More from that paper:

In situations in which each person’s response is used to determine the data seen by the next person, people converge on concepts consistent with their inductive biases irrespective of the information seen by the first member of the chain. …

In this experiment, each generation of par ticipants received 50 trials of training on a single function. On each trial, the value of the stimulus was presented as a visual magnitude, being the width of a horizontal bar on a computer screen.  Participants responded by adjusting the height of a vertical bar and then received corrective feedback. … After training, participants responded to 100 stimuli that covered the entire possible range of magnitudes without receiving feedback. …

Participants were arranged into eight families of nine generations, for each of four conditions. The conditions differed with respect to the function used to generate the training data seen by the ﬁrst generation of participants: those initial values were drawn either from a positive linear, negative linear or quadratic function, or entirely at random. … The responses of each participant on 50 of the test trials were taken as the data used to train the participant in the next generation of that family. …

Across generations all of the initial functions gradually disappeared and transited into only one of two stable functions: positive linear (28 out of 32 families) and negative linear (4 out of 32), both with approximately unit slope.

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• ac

Cell b(iii) looks suspicious. How much noise does their procedure introduce so that someone with 50 interpolation points can end up coloring in the opposite corners of the graph? Or were they wasted? I need to read the paper, that’s just a reaction to the plot. The other cells all seem plausible. Maybe the experiment just measured how much people suck at estimating the length of a horizontal bar?

• Arthur B.

It seems a weak link can be clearly identified in each row. It’s not so much gradual decay but the eventual intervention of a moron

b-iii is a dumbass… or there’s an error in the protocol
c-iii and c-iv share the blame
d-v is a moron

• http://denisbider.blogspot.com denis bider

50 corrected and 100 blind guesses for each function seems like an awful lot of work. What incentive did the subjects have to do this diligently? Fascinating study though.

• Tom Breton (Tehom)

On each trial, the value of the stimulus was presented as a visual magnitude, being the width of a horizontal bar on a computer screen. Participants responded by adjusting the height of a vertical bar and then received corrective feedback

To me, this seems like a pretty strong cue that the respective magnitudes should correspond. That cue alone might explain why everything tends towards x=y

• http://pancrit.org Chris Hibbert

That’s a hilarious result. non-mathematicians (I assume) plausibly believe that all functions are positive, smoothly monotonic, and roughly 1:1.

Though of course, as Arthur B. points out, each of the rows in the summary shows a gradual decoherence interrupted by someone who believes that the best guess when you don’t understand the data is f(x) = x. And that one turns out to be easy to recognize, remember, and transmit.

Brilliant experiment.

There must be some shared structure in the brain or brain plus visual system that is processing the input.

Does this make x=y a platonic form?

Open loops are unstable! Feedback is crucial to accurate signal measurement.

• Tyrrell McAllister

Fascinating experiment. This is one that I’ll remember to bring up in future conversations. However, I see Tom Breton (Tehom)’s point. I’d be very interested to see what happens when the subjects are asked to encode function values in other ways.

• Unnamed

People aren’t very good at conveying uncertainty – for instance a lot of times they’ll say 50-50 to mean “I don’t know” even when the correct probability is nowhere near .5 (e.g. footnote 10 of this study implies that over one fifth of respondents said they had a 50% chance of being hurt in a terrorist attack in the next year). As Tom and others have suggested, in this design some people may have felt that the way to express uncertainty is by matching the length of the horizontal bar, and you only need one or two of those people in a chain to converge on y=x.

• James Andrix

I’d say that comparing a horizontal bar to a vertical bar is a terrible way to receive a function. We see an obvious pattern in (c) immediately because it is plotted for us. It’s easy to remember this plot once you have it and reference it to output.

This would be a good candidate for an online game or mechanical turk implementation.