Monday, March 21, 2011

A Nerdy Math Thing at the Airport

I was seated at an airport food court with my suitcase, backpack, a coffee and an orange juice. I wanted to walk over the diner and get some breakfast, but I already didn’t have enough hands to carry my suitcase, coffee, and orange juice (I struggled to get to the table without spilling anything to begin with), so I wouldn’t be able to carry the food, either.

I asked an old couple next to me to watch my stuff while I go get my food.

Is that rational? Is asking a stranger to guard (and not steal) your stuff any different than just leaving it?

To figure it out, I did some math.

First, we’ll need some variables and assumptions:

Variables:

Tg = Probability of a random stranger to steal a guarded item
Tu = Probability of a random stranger to steal an unguarded item
D = Probability of failing to defend an item that you own
Ds = Probability of failing to defend a stranger’s item
N = The number of strangers encountered during the risk period

Assumptions:
  • Tu > Tg. I’ll assume that the random stranger is less likely to attempt to steal something if it looks like someone is keeping track of it.

  • Ds > D. If you are guarding your own item, I assume you will more vigorously defend it than a stranger guarding your item.

  • T variables will depend on the desirability of the item. Is it a nondescript suitcase, or a laptop computer? It also obviously depends on the population of strangers.

  • D variables will vary based on the value of the item. If the thief steals a pen or a candy bar, he will probably not be pursued.


Let’s get started. We have the first base case. What if you just leave your item unguarded? In that case, each of N strangers has probability Tu of stealing the item. The probability of retaining your item is the product of the independent decisions of N strangers to not steal your item. Because we’re calculating risk of loss, we use “one minus” a lot of times, so here’s the formula:

1-((1-Tu)^N)

With our numbers, the risk of loss to 100 strangers is approximately 1%. It goes up to 10% for 1000 strangers and 63% for 10,000 strangers. Maybe this sounds reasonable. It certainly sounds like a bad idea to leave your stuff lying around.

The other base case is that I just sit there and guard my item. This has two advantages. First, an attempted theft is less likely of a guarded item. (Ok, this is circular logic, since that was one of our assumptions, but if you disagree you can change the numbers.) The second advantage is that if an attempt is made, you have the opportunity to defend your item. Again, using our numbers for 1000 strangers, the risk of loss drops to about 5% due the assumption that an attempted theft is half as likely for a guarded item, and then to 0.5% for the 9 out of 10 likelihood of defending your item. The full formula is:

(1-(1-Tg)^N)*D

So what about my decision to trust a stranger to watch my stuff? For this we have a small probability tree. On one branch, we have an honest stranger, and on the other branch I actually chose the thief to guard my item (oops).

If you choose the thief to guard your item, you are sunk. In that branch, there is a 100% probability of loss. However, that branch only occurs with probability Tu. (Obviously, once you hand your item to the thief, we consider it to be unguarded.)

If an honest stranger is chosen, then the math is the same as when you guarded your own item, except that we assume the stranger will less vigorously defend your stuff than you would your own, so we use Ds instead of D:

(1-(1-Tg)^(N-1))*Ds

Predictably, the risk of loss in that case is double (again because that was our assumption). (While it’s only relevant when N is small, we use N-1 because we’ve already picked one of the strangers as the guardian, so his probability of being a thief is already in the other branch.) The answer has to be multiplied by the probably of entering that branch (choosing an honest stranger to begin with) and then we add the probability from the “I chose the thief” branch:

Tu+((1-Tu)* (1-(1-Tg)^(N-1))*Ds)

That looks ugly, but it’s just iteratively applying what we’ve been doing. The first instance of Tu is implicitly multiplied by the 100% probability that the chosen thief will take your stuff.

Here’s a handy table for different values of N (using our initial numbers):

NUnguardedSelf-guardedStranger-guarded
100.10%0.00%0.02%
1001.00%0.05%0.11%
1,0009.52%0.49%0.98%
10,00063.21%3.93%7.88%
100,000100.00%9.93%19.87%


Conclusions:

The decision about whether or not to trust a stranger depends mostly on your assumption that the stranger will adequately defend your item, and that a guarded item is less likely to be stolen. Since the individual probability that the stranger chosen is a thief is so small, it doesn’t factor into the decision much at all.

In all cases, trusting a single stranger is overwhelmingly better than leaving the item unguarded, because that is equivalent to trusting EVERY stranger. This should also be intuitively true.

It would be neat if there is any data out there for T and D. Has anyone done a study like this?

In real life, choosing a stranger to trust is not done randomly. We expect that we can improve on random chance by profiling. I chose an old couple that looked like they have enough of their own stuff. Even though there were two of them, they know each other and thus are not independent random samples. In fact, I expect choosing a couple is better because of social pressure to be honest. The fact that they were old probably reduces their ability to defend when compared with myself, but I didn’t think of that at the time.

The meaningful difference in probabilities between self-guarding and stranger-guarding must be up to you. Is not trying to wrestle with my suitcase, coffee, orange juice, and backpack , and a plate of scrambled eggs with hash browns worth the additional 0.11% risk of theft? I suppose it was, because that’s what I decided to do.

I told you it would be nerdy.


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3 comments:

Lisa Ann Homic, M.Ed. D.C. said...

that was interesting. i sped read through it b/c it's night time and i can't think that deeply. good conclusion. so do you still have your stuff?

Chuck Homic said...

As far as I know!

jaberwok said...

Thanks, a very Interesting analysis numerically. There has been work in social psychology that backs this up empirically but points out more factors. "Influence- The psychology of persuasion" by Cialdini is a superb pop review of this work. For example There has been systematic work by Levine on acts of kindnesses to strangers (using a dropped pen, hurt leg or blind person crossing the road model) showing big cross national differences (Rio did well and NY badly) on these measures. There have also been studies on the "looking after your bag" model suggesting that giving a good reason for the request "Please would you be kind enough because I have to .." is very helpful. Even more interestingly if the person you ask has been made to feel guilty recently they are even more likely to help (ie have not given to a charity collector)! For losing a wallet Seattle is the place! I have found this out personally on one occasion and my daughter had both her passport and separately a large check returned to her. But there have been systematic studies-
If there is such a thing as a good place to lose your wallet, Seattle is it, an experiment by Reader's Digest has found.The magazine left a trail of 120 "lost" wallets in 12 communities around the nation and kept track of how many were returned with the $50 contents intact. Seattle had the best rate of return, 9 of 10.
Despite the large role that chance could play in such a small experiment, Mayor Norm Rice of Seattle was ecstatic.
Three small cities were the next best, with a score of 8 in 10: Meadville, Pa.; Concord, N.H., and Cheyenne, Wyo.Of the two other big cities tested, St. Louis returned 7 in 10 and Atlanta trailed with 5 in 10. Readers Digest 1995