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Capture/marking phase: Capture $n_1$ animals, mark them, and release them back.

phase: Capture $n_1$ animals, mark them, and release them back. Recapture phase: A few days later, capture $n_2$ animals. Assuming there are $N$ animals overall, $n_1/N$ of them are marked. So, for each of the $n_2$ captured animals, the probability that the animal is marked is $n_1/N$ (from the capture/marking phase).

phase: A few days later, capture $n_2$ animals. Assuming there are $N$ animals overall, $n_1/N$ of them are marked. So, for each of the $n_2$ captured animals, the probability that the animal is marked is $n_1/N$ (from the capture/marking phase). Calculation: On expectation, we expect to see $n_2 \cdot \frac{n_1}{N}$ marked animals in the recapture phase. (Notice that we do not know $N$.) So, if we actually see $m$ marked animals during the recapture phase, we set $m = n_2 \cdot \frac{n_1}{N}$ and we get the estimate that:



$N = \frac{n_1 \cdot n_2}{m}$.





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First (naive) attempt



We decided to apply this technique to estimate the size of the Mechanical Turk population. We considered as "capture" period the set of surveys running over a period of 30 days. Then we considered as "recapture" period, the surveys that we ran on another 30-day period. The plot below shows the results.





In our setting we adapted the same idea, where "capture" and "recapture" correspond to participating in a demographics survey. In other words, we "capture/mark" MTurk users that complete the survey in one day. Then, in another day, we also "recapture" by surveying more workers and we see how many workers overlap in the two surveys.We decided to apply this technique to estimate the size of the Mechanical Turk population. We considered as "capture" period the set of surveys running over a period of 30 days. Then we considered as "recapture" period, the surveys that we ran on another 30-day period. The plot below shows the results.





If we focus on the black-color dots (~60 days between the surveys), we get a (naive) estimate of around 10K-15K workers. (Warning: this is incorrect.)



While we could stop here, we see some results that are not consistent with our model. Remember, that color encodes time between samples: black is for short time (~2 months) between samples, red is for long time (~2yrs) between samples. Notice that, as the time between the two periods increases, the estimates are becoming higher, and we get the "rainbow cake" effect in the plot. For example, for July 2017, our estimate is 12K workers if we compare with a capture from May 2017, but the estimate goes up to 45K workers if we compare with a sample from May 2015. Our model, though, says that the time between captures should not affect the population estimates. This indicates that there is something wrong with the model.



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Assumptions of basic model



The x-axis shows the beginning of the recapture period, and the y-axis the estimate of the number of workers. The color of each dot corresponds to the difference in time between the capture-recapture periods: black is a short time, and red is a long time.If we focus on the black-color dots (~60 days between the surveys), we get a (naive) estimate of around 10K-15K workers.While we could stop here, we see some results that are not consistent with our model. Remember, that color encodes time between samples: black is for short time (~2 months) between samples, red is for long time (~2yrs) between samples. Notice that, as the time between the two periods increases, the estimates are becoming higher, and we get the "rainbow cake" effect in the plot. For example, for July 2017, our estimate is 12K workers if we compare with a capture from May 2017, but the estimate goes up to 45K workers if we compare with a sample from May 2015. Our model, though, says that the time between captures shouldaffect the population estimates. This indicates that there is something wrong with the model.

The basic capture-recapture estimation described above relies on a couple of assumptions. Both of these assumptions are violated when applying this technique to an online environment.

Assumption of no arrivals / departures ("closed population") : The vanilla capture-recapture scheme assumes that there are no arrivals or departures of workers between the capture and recapture phase.

: The vanilla capture-recapture scheme assumes that there are no arrivals or departures of workers between the capture and recapture phase. Assumption of no selection bias ("equal catchability"): The vanilla capture-recapture scheme assumes that every worker in the population is equally likely to be captured. In ecology, the issue of closed population has been examined under many different settings (birth-death of animals, immigration, spatial patterns of movement, etc.) and there are many research papers on the topic. Catchability, by comparison, has received comparatively less attention. This is reasonable, as in ecology the assumption of closed population is problematic in many settings. By comparison, assuming that the probability of capturing an animal is uniform among similar animals is reasonable. Typically the focus is on segmenting the animals into groups (e.g., nesting females vs hunting males) and assign different catchability heterogeneity to groups (but not to individuals).

In online settings though, the assumption of equal catchability is more problematic. First we have the activity bias: Workers exhibit very different levels of activity: A worker who works every day is much more likely to see and complete a task, compared to someone who works once a month. Similarly, we have a selection bias: Some workers may like to complete surveys, while others may avoid such tasks.



So, to improve our estimates, we need to use models that alleviate these assumptions.

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Endowing workers with survival probabilities





We can extend the model, allowing each worker to have a certain survival probability, to allow workers to "disappear" from the platform. If we see the plot above, we can see that the population estimate increases as the time between two samples increases. This hints that workers leave the platform, and the intersection of capture-recapture becomes smaller over time.





If we account for that, we can get an estimate that the "half-life" of a Mechanical Turk worker is between 12-18 months. In other words, approximately 50% of the Mechanical Turk population changes every 12-18 months.





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Endowing workers with propensity to participate





propensity for each worker. The propensity is the probability that a worker is active and willing to participate in a task, at any given time.



In our work, we assumed that the underlying "propensity to participate" follows a Beta distribution across the worker population, and the parameters of the Beta distribution are unknown. When we assume that follow a Beta distribution, then the probability that a worker participates in the survey k times, follows a



Notice that we had to depart from the simple "two occasion" model above, and instead use multiple capturing periods over time. Intuitively, workers that have high propensity to participate will appear many times in our results, while inactive workers will appear only a few times. We can also extend the model by associating a certainfor each worker. The propensity is the probability that a worker is active and willing to participate in a task, at any given time.In our work, we assumed that the underlying "propensity to participate" follows a Beta distribution across the worker population, and the parameters of the Beta distribution are unknown. When we assume that follow a Beta distribution, then the probability that a worker participates in the survey k times, follows a Beta Binomial distribution . Since we know how many workers participated k times in our surveys, it is then easy to estimate the underlying parameters of the Beta distribution.Notice that we had to depart from the simple "two occasion" model above, and instead use multiple capturing periods over time. Intuitively, workers that have high propensity to participate will appear many times in our results, while inactive workers will appear only a few times.





By doing this analysis, we can observe that (as expected) the distribution of activity is highly skewed: A few workers are very active in the platform, while others are largely inactive. A nice property of the Beta distribution is its flexibility: Its shape can be pretty much anything: uniform, Gaussian-like, bimodal, heavy-tailed... you name it.













In our analysis, we estimated that the propensity distribution follows a Beta(0.3,20) distribution. We plot above the "inverse CDF" of the distribution (Inverse CDF: "what percentage of the workers have propensity higher than x").



As you can see, the propensity follows a familiar (and expected) pattern. Only 0.1% of the workers have propensity higher than 0.2, and only 10% have propensity higher than 0.05.



Intuitively, a propensity of 0.2 means that the worker is active and willing to participate 20% of their time (this is roughly equivalent to full-time level of activity; full-timer employees work around 2000 hrs per year, out of 24*365 available hours in a year). A propensity of 0.05 means that the worker is active and available approximately 24 hr * 0.05 ~ 1 hour per day.



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How big is the platform?





So, how many workers are there? Under such highly skewed distributions, giving an exact number for the number of workers is rather futile. The best that you can do is give a ballpark estimate, and hope to be roughly correct on the order of magnitude. What our estimates are showing is that there are round 180K distinct workers in the MTurk platform. This is good news for anyone who is trying to reach a large number of distinct workers through the platform.





Our analysis also allows us to estimate how many workers are active and willing to participate in our task at any given time. For that, we estimate that around 2K to 5K workers are available, at any given time. If we want to convert this number to full-time employee equivalence, this is equivalent to 10K-25K full-time workers.





The latter part also allows us to give some low and high estimates on the transaction volume of MTurk.

Lower bound : Assuming 2K workers active at any given time, this is 2000*24*365=17,520,000 work hours in a year. If we assume that the median wage is \$2/hr, this is roughly \$35M/yr transaction volume on Amazon Mechanical Turk (with Amazon netting ~\$7M in fees).

: Assuming 2K workers active at any given time, this is 2000*24*365=17,520,000 work hours in a year. If we assume that the median wage is \$2/hr, this is roughly transaction volume on Amazon Mechanical Turk (with Amazon netting ~\$7M in fees). Upper bound: Assuming 5K workers active at any given time, this is 5000*24*365=43,800,000 work hours in a year. If we assume average wage of \$12/hr, this is around \$525M/yr transaction volume (with Amazon netting ~$100M in fees). I understand that a range of \$35M to \$500M may not be very helpful, but these are very rough estimates. If someone wanted my own educated guess, I would put it somewhere in the middle of the two, i.e., transaction volume of a few hundreds of millions of dollars.







A topic that frequently comes up when discussing Mechanical Turk is "how many workers are there on the platform"?In general, this is a question that is very easy for Amazon to answer, but much harder for outsiders. Amazon claims that there are 500,000 workers on the platform. How can we check the validity of this statement?A common technique for this problem is thetechnique, that is widely used in the field of ecology, to measure the population of a species.The simplest possible technique is the following: