# Monthly Archives: January 2012

## are squirrels optimizers or satisficers?

Last month, I had the pleasure of meeting Yakov Ben-Haim and talking with him at length about info-gap decision theory. He used an example of squirrels foraging for nuts to illustrate the types of problems for which info-gap decision theory models are useful.

A squirrel needs calories to survive, and nuts provide the perfect source of calories. The squirrel has a decision to make: where should the squirrel go to forage for nuts? Different foraging locations have different potentials for nut payoffs. They also have risks (not enough food). Foraging in a new location may carry highly uncertain risks that are impossible for the squirrel to estimate (being hit by a car, eaten by a wolf, etc.)

The squirrel has two options: the squirrel can hunt in the usual area where he can obtain n nuts with certainty or he can try a new location where he has a probability P of obtaining N nuts (with N > n) and a probability (1-P) of obtaining zero nuts. Let’s say that N and P are wild guesses.

Let’s say that the squirrel is an optimizer and decides to build a decision tree to maximize the number of nuts he can collect. Using basic decision analysis, he devices that he should choose the new location if PN>n.

The squirrel's decision tree. Squirrels don't really make decision trees, do they?

If the squirrel needs to collect n nuts to survive, then maximizing is nuts (pun intended. Sorry!) Staying with the status quo guarantees survival, even if P and N are large. The payoff for the new location may be greater, but there is a 1-P chance that the squirrel would starve.  The traditional decision tree is not robust to the squirrel’s desire to survive (neither is darting in front of cars on the highway, but I digress).

On the other hand, if the squirrel needs to collect N nuts to survive, then staying with the status quo guarantees the squirrel’s demise.  The new location is worth a look no matter how risky.

In both of these scenarios, the squirrel isn’t really maximizing the subjective expected nuts that he can collect–he really wants to maximize the probability of meeting his nut threshold (the one that guarantees survival). This is a satisficing strategy (although not dissimilar from an optimizing strategy with a moving threshold). The satisficing strategy is a better bet for the squirrel than the optimization strategy in this decision context. The squirrel doesn’t always need to know the exact probabilistic information to make a good decision, as illustrated above.  In fact, he can have absolutely no idea what N and P would be to find an effective nut foraging strategy–even when there is severe uncertainty.

The idea of a squirrel building a decision tree is, of course, ludicrous. But it makes the point that what we should rethink our traditional optimization models so make sure they fit the real decision criteria on hand.  Info-gap decision theory thus focuses on satisfying a given acceptable level of what is traditionally considered the objective function value and instead optimizing robustness.  It also has philosophical implications for how one views certainty.

I’ve been looking more closely at robustness lately.  I won’t abandon my optimization models, but I will acknowledge that including robustness in certain scenarios leads to decisions that more accurately reflect the criteria at hand and decisions that could be counter-intuitive.

Yakov Ben-Haim can explain this much better than I can, so I’ll refer you to his blog about info-gap decision theory and his article about foragers in the American Naturalist if you want to learn more.

## what are the odds of winning the lottery two times?

A Chicago area man won the lottery for the second time. The Chicago Tribune reports:

Scott Anetsberger duplicated his \$1 million win of nine years ago in the same instant Merry Millionaire game, lottery spokesman Mike Lang said.

Despite long odds, Anetsberger isn’t the first two-time \$1 million instant winner. Kimberly Pleticha of Villa Park won \$1 million twice in the instant Cash Jackpot game–the first time in August 2010 and the second only six months later in February.

Lottery officials could not instantly compute the odds against multiple winners, but did note there have been a dozen or more two-time Little Lotto winners over the years.

What would the odds of winning the lottery twice would be? Well, it depends on how frequently one plays the lottery.

Winning the Illinois Lottery requires picking six correct numbers, where the numbers range from 1 to 52. The odds of getting all six numbers correct is 1 in 20,358,520.  It costs \$0.50 to play the lottery, and there are three lotteries per week. Assuming that each lottery is independent (a reasonable assumption), one would have to play the lottery 20,358,520 times, over average, to win (using the geometric distribution). If one plays the lottery three times per week, then it would take 130,500 years to win the lottery once at a cost of more than \$10M.

Winning the lottery twice can be modeled as a negative binomial random variable. Assuming that our lottery winner plays the lottery three times per week before and after winning the lottery, then it takes ~261,000 years, on average, to win twice.

Since it is only newsworthy to report additional wins by those who have already won the lottery, then we are really only interested in the odds that a lottery winner would win the lottery again. This is a different question. Assuming that our lottery winner continues to play the lottery three times per week, then the odds of winning again are same as the odds of someone else winning the lottery for the first time: 1 in 20,358,520 per lottery. That is, it would take our lottery winner an additional 130,500 years to win the lottery.

If someone plays the lottery more than three times per week, then the odds of winning go up.

Of course, many people play the lottery, so the odds that someone wins the lottery twice over their lifetime is much, much higher. I tell my students every semester, “Someone will win the lottery. Just not you.” If 130,500 people buy one lottery ticket per game, then there would be a two-time winner every 2 years, on average.

Little Lotto involves picking five correct numbers, where the numbers range from 1 to 39.  It is easier to win, but it has a lower payout. The odds of winning are 1 in 575,757, which means that one is 35 times as likely to win the Little Lotto than the regular lottery. It would take 3691 years to win Little Lotto once (by playing three times per week) and 7382 years to win it twice.

Given that there have been 12 two-time winners in Little Lotto in its 23 years of existence, there there is approximately one two-time winner every two years. Given my assumptions, this would suggest that ~3691 people buy a Little Lotto ticket every time. That seems a bit low to me. But I have a head cold and maybe it has temporarily impaired my mathematical abilities.

A seven-time lottery winner’s advice for winning the lottery is to invest more (not less!) of one’s money into buying lottery tickets, as long as one can afford it. He also recommends treating the lottery as a job: the lottery is a skill, and one can improve at it after investing a lot of time. While skill plays a role in playing the lottery (identifying which numbers to pick and identifying which games have the best payoff), I’m pretty sure that this is bad advice. The expected payoff for the lottery is negative, meaning that on average, you are guaranteed to come out behind. The variance in earnings is large, meaning that over many attempts, it is possible that you can come out ahead. But given that one comes out ahead, it would be foolish to attribute one’s success to skill. But maybe I’m missing something.

For the record, I do not recommend gambling or routinely playing the lottery.

For more, read Mike Trick’s post on conditional probabilities and March Madness odds.

Related post:

## slideshare: what’s not to love?

I am due for another teaching with technology post. This post is on slideshare, a social networking tool for uploading and sharing your presentations. There is a lot to like about slideshare and not much to dislike. Here is my list of likes:

2. You can upload your presentations once to share them with many colleagues and viewers. You can allow viewers to download your slides (or not).  Likewise, you can designate some slides as “private.”
3. You can embed slides into blog posts or other html code. See my posts here and here. This way, viewers do not have to visit another site to view slides.
4. I cannot say enough about the slidecasts. Here, one can upload the mp3 file for a presentation and sync the presentation to the slides for an easy to make webinar. I have only done this once for my slides on financial advice. This was incredibly simple to do, and it truly makes the slides come alive. In retrospect, I wish I had recorded all of my presentations.
5. Slideshare keeps good statistics. I know exactly how many times someone has viewed each presentation, either through slideshare or through a blog post.
6. People find my slides without me having to advertise them. Granted it helps if I advertise my slides, but 107 people have viewed my slides on technical writing tips presumably through google searches and word of mouth.

I haven’t used slideshare frequently for my teaching in the classroom.In the future, I may require students to maintain presentations for a course on slideshare.

I do use slideshare for teaching outside of the classroom. I shared three presentations that I gave in seminars to a broad audience (on finances, technical writing, and applying to graduate school). I use slideshare to share these slides with students who did not attend the seminar but who may be interested. People seem to find the presentations using google.

I put several of my INFORMS talks on slideshare (see these slides about blogging for operations research) and shared them on twitter using the conference hashtag (#informs2011 this year). That way, other people attending the conference could easily find my slides. I noticed a sharp increase in my presentation feed hits after sharing via twitter.

My slideshare presentations are here.

Related posts on teaching with technology:

## punk rock OR goes punk

I’ve maintained this blog for almost five years. I have also been waiting for almost five years for someone to call me out for not being “punk” enough for a Punk Rock OR blog. After all, I do even have a single tattoo.

That day has come!

David Baughman, a web developer for Boeing in Seattle, stumbled upon my blog. He later sent me the photoshopped picture (see below), showing me as a punk rocker.

He writes:

I found the article interesting, and your picture just yelled “conservative” so I had to say something.  My buddy Matt gets all credit for the photoshop genius, and I’ll just take credit for the idea.

This is a photoshopped version of me.

This photo is as punk rock as I am willing to be, at least on days other than Halloween. I will continue to enjoy life as a fake punk rocker.  In my Science of Better podcast interview with Barry List, I discuss the blog name and my thoughts on tattoos.

## no confidence in confidence intervals

With all of the upcoming primaries, I have been reading a little bit about polling data.  Nate Silver of the NY Times discusses how frequently a candidate’s vote total falls in the margin of error (based on poll data) .  Usually, 95% confidence intervals are reported, so you would expect a candidate’s numbers to be outside the confidence interval ~5% of the time.

FiveThirtyEight has a database consisting of thousands of primary and caucus polls dating back to the 1970s. Each poll contains numbers for several candidates, so there are a total of about 17,000 observations. How often does a candidate’s actual vote total fall within the theoretical margin of error?

The answer is, not very often. In theory, a candidate’s actual vote total should fall outside the margin of error only 5 percent of the time. In reality, the candidate’s vote total was outside the margin of error 65 percent of the time! Part of this is because the database includes some polls conducted months before the actual voting took place. But even if you restrict the analysis to polls conducted within the final week of the campaign, about 40 percent of the vote totals fell outside the margin of error — eight times more often than is supposed to happen if you could take the margin of error at face value. [emphasis added]

Silver argues that it is important to recalibrate the polling data based on the accuracy of past polls. To make predictions about election/primary results based on polling data, he (b) adjusts the results based on how recent the polls are (more recent = more accurate), (c) accounts for undecided voters, and (c) accounts for “momentum.” Silver’s methodology can be found here and his prediction for the New Hampshire primary can be found here.

Related post:

## New Years Resolutions for 2012

In the spirit of OR bloggers Mike Trick, Paul Rubin, John Poppelaars, and Thaddeus Sim, here are a few New Year’s Resolutions:

1. Punk Rock OR will try to gain weight: I blogged 74 times this past year. I’d like to average about two blog posts per week (>100 per year). It’s hard with twitter: would-be short blog posts tend to turn into tweets. I have achieved 3% of this goal so far.
2. Figure out a way to reduce the amount of academic spam I receive.  Most of the spam I get is to attend conferences that are not in my area. I can simply “unsubscribe” to get off of these lists (I’ve been deleting lately). I’m not sure what to do about the emails that advertise “microarray technologies” and “maximize cell viability with uniform ultra-low storage.” I don’t even open these suspicious emails. Your feedback is welcome.
3. Reduce the amount of service I perform at my university to make more time for research, professional service, and blogging. This is easier said than done, especially since I went up for tenure this year and the service load should only go up from here. See the PhD comic below to understand what I mean.
4. Do some international travel. This is my sixth year as a faculty member and I haven’t gone an international trip yet. I have cabin fever. Please send me recommendations and invitations!
5. Get a hobby. After having my third daughter in March, I’ve essentially stopped all of my hobbies. I don’t know if I’ll have the energy to train for another marathon. I need to run a half-marathon, read a few books, or plan a big sewing project or I will go crazy.

## I grew up near a fake city

I recently learned that map makers intentionally put fake roads, etc., into their maps to identify copyright infringements (when maps are stolen, even the errors are reproduced).  These are called copyright traps. I grew up outside of Chicago near Ohare International Airport. I heard that Rand McNally once put a fake town near Ohare in their maps. I checked up on it and and found that it is true. The fake town is called Westdale and it appeared in the 1981-1983 editions of Rand McNally’s maps. Westdale is near a “Westdale Park” and a “Westdale Elementary School” (see it on google maps here).

Here is a link to more on copyright traps from the Straight Dope (the 1991 edition). This author was skeptical about the existence of the fake Chicago suburb, but it was written before google. I found the rest of the information to be interesting.