Deal or No Deal Banker's Formula

UPDATED 8:20 p.m.: Another episode of data were added, and the formula was updated. Scatterplot updated.

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NBC's Deal or No Deal is one of the most popular programs on television. Even though the show is in summer re-runs, Nielsen Media Research says the show was No. 3 for the week of June 18, 2007.

I was first introduced to the show by former Ohio State master's student Tim Laubacher. Although I am far from a regular viewer, I do find the show interesting.

Here is the basic premise: the contestant begins with 26 cases, each of which represents a monetary amount ranging from $0.01 to $1,000,000.


The contestant selects one case, which then becomes "their" cases. The contestant is entitled to whatever monetary amount is inside.

At this point, 25 cases remain on stage. The contestant must "open" six cases (in the first round), which reveals the amount inside each case. As more cases are opened, we have more information about what amount might be inside the contestant's case.

After each round, a silhouetted "banker" makes an offer to buy the case. Host Howie Mandel repeatedly reminds the audience that the banker wants to buy the case for "as little as possible."

If the contestant takes the "deal," then the game is over. If the contestant refuses the deal (i.e., "no deal"), then more cases are opened. If the contestant has bad luck, the opened cases have large amounts, which means that large amount was not in the contestant's case. The opposite is true with small amounts. After each round, a new offer is made. And each round requires the contestant to open fewer cases.


As a statistically oriented person, I immediately wondered how the banker came up with the "offer."

At any moment, we can estimate the expected value of the contestant's case. This is a simple part of probability theory, and it is intuitive to most people. Imagine that there is one case left on stage. The contestant has one case. Now assume that the amounts $100,000 and $200,000 remain.

What is the expected value of the case? If you were to play the game in those exact same circumstances many times, the long-term average value of the contestant's case would be $150,000. That is the expected value.

This expected value is the most logical offer for the banker. However, it does not take long to realize that although the offer is usually close to the expected value, it is not a perfect match.

So last night, I watched two episodes and wrote down the amounts remaining and the offer. This morning I used hierarchical regression to figure out the banker's formula. Although there is still some error in the formula, it is 99% accurate in predicting the bank's offer (see scatterplot comparing offers and predictions below).


According to my (latest) calculations, the formula is:

Banker's offer =

$12,275.30 +

(.748 * expected value) +

(-2714.74 * number of cases left) +

( -.040 * maximum value left ) +

(.0000006986 * expected value squared ) +

( 32.623 * number of cases left squared ).

Together these values explain 99% of the variance in the banker's offer. Admittedly, this is based upon a small sample of only (now 31) offers. When I get bored enough to chart some more data, I will update the formula.

Negativity Bias on Television

A growing group of media researchers are guided by a notion that human emotion is guided by two underlying motivational systems, one appetitive (e.g., approach) and one aversive (e.g., avoid).

To grossly oversimplify the underlying theory, we believe that at low levels of intensity, the appetitive system wins out, and you explore. However, as things become intense, the aversive system wins, and you cut and run.

This system allows you to explore during peaceful times and stay alive during chaotic times.

Much of the related theories were developed by studying how people respond to affective photographs (specifically the International Affective Picture System). When people have been asked to rate many pictures, you can plot the average arousal ratings and the average valence ratings for each picture.

When you do this, as unpleasant pictures become arousing, they become more unpleasant relatively quickly. Pleasant pictures, however, are slower to become extremely positive as they become more arousing. If you draw a trend line for both pleasant and unpleasant pictures, the slope of the line is "steeper" for unpleasant pictures. This is called the negativity bias.

Recently I became curious how such a scatterplot would look for television clips, so I gathered descriptive statistics for 68 clips I have used in various experiments (my friend James also sent some descriptive statistics that I did not have time to work in here).

As you will see here, we see the exact same effect with short 30-second television clips. As unpleasant clips become more arousing, they become much more unpleasant. However, as pleasant clips become more arousing, they are not much more pleasant.

Cool.

In the interest of full disclosure, this is because we kept it "clean." For the photographs, the really pleasant pictures show naked couples in erotic poses. Since we're not studying pornography, the only sex scenes among these clips were subtle.

Commcognition: Family Friendly Since 2005

Those who know me might get a laugh out of this.

As a journalist, I was taught to keep "naughty" words out of my writing.

This test suggests that I passed.

The problem is like a hydraulic system. The pressure leaks out somewhere. In my case, at lab meeting, which probably is not G rated.

Arousal Takes Time: Insight from Latencies

Instead of doing the work I had to do Saturday, I spent some of the day working on old data. I love to work on data, so this was a much better way to spend the late afternoon.

These data kept bringing me back to time. Time matters a lot.

In a recently published article, Dynamic, embodied limited-capacity attention and memory: Modeling cognitive processing of mediated stimuli (read PDF here), I make an argument that time matters.

But I do not really do time justice in the model. Time matters a lot more than even I have acknowledged.

Consider the current data. We showed participants 30-second television clips varying in valence and arousal. Thus, positive and negative clips were shown that varied in arousal. In total, there were 5 levels of arousal.

After each clip, we asked participants how arousing the clip was, how negative the clip was, and how positive the clips was. These were 7-point scales ranging from "very" to "not at all."

The question is, how do you make that decision? How do you decide how arousing/exciting a clip was? Do you search your memory? And, of course, the most interesting question to me is: how do you search your memory?

If something was really arousing, then judging its arousal value should be easy, right?

Wrong, it turns out.


You see, arousal is not just another variable. It is exactly what I argue it is in the Media Psychology article cited above: a dynamic, embodied variable.

Imagine if I ask you how often you see a co-worker. Presumably, you see her five days a week. So as soon as you attempt to search your memory, you are going to immediately retrieve many exemplars of that co-worker.

So it's an easy question. It's also an easy question if I ask you how often you see someone that you rarely see.

But what about someone you see somewhat regularly?

In several data sets that we have (and the data of others, such as Ohio State's Russ Fazio), those moderate decisions take more time to make.

But arousal, as I stated above, is not just another variable. Instead, physiological arousal gets you going. It helps you move. It pumps the heart to get oxygen to muscles all over the body. Fight-or-flight is driven by arousal. When you watch something arousing on television, it reaches in and grabs a hold of your sympathetic nervous system and shouts: do something!

So when I ask you how arousing something was, you should begin to think about that thing. And if the stimulus was really arousing and you try to access that memory -- especially the memory of how arousing it was -- the memory should trigger the earlier arousal.

And that triggered arousal should invoke automatic cognitive processes that attempt to divert limited capacity attentional resources away from the arousal-judging task and to the stimulus itself.

If this is the case, then arousal judgments should get slower as the original stimulus became more arousing. If the subsequent judgments are only a matter of "counting" how many arousal-related features were in the original message, then those judgments should speed up as a function of arousal.

Our data support the first interpretation. See the figure above. As the content became more arousing (i.e., as the label on the X-axis increases from 1 to 5), the latency of the response increases from 2,300 ms to 2,700 ms. Thus, it took these participants almost a half second longer to make arousal judgments about arousing stimuli. And those seem like the easier decisions.

Although the figure is far from perfectly linear, the linear trend is strongest, F(1,53) = 10.82, p = .002, eta-squared = .17 (sorry non-stats readers). And this is for raw response latencies. That is, these data include trials where participants started daydreaming and took 30 seconds to answer the question.

Time matters. Sure, if you look at the self-report data, the arousing content was rated as more arousing (also a linear trend). But the answers to the questions do not tell the entire story. I need to know what you responded, and how look it took you to do it. The response latency data help tell the story.

Emotion is not just something we manipulate in the lab, as Peter J. Lang once argued during an address to the Society for Psychophysiological Research. Instead, emotion is a property of the individual. Emotion is a survival-oriented trait that helps keep our eyes focused when attention is crucial and allows them to wander when surveying the landscape is more adaptive.

I've tried to keep a focus on time, but I am realizing now that I have not done enough. Time is on my mind. I hope it's on your mind, too.

Looking Back on 2 Years with a Ph.D.

Two years ago this week, I defended my doctoral dissertation in telecommunications (technically mass communications) and cognitive science at Indiana University.

As with so many things in the past, this "two year" figure is mind-blowing for the fact that it both seems far more than and far less than two years ago.

So much has happened since then. We moved to Columbus, Ohio, and spent a year at The Ohio State University. Then for reasons that are largely family oriented, we moved to Lubbock, Texas, where I spent the past year at Texas Tech University.

I built a psychophysiology lab at both places, and I trained graduate students to work in each place. In so many ways, that must have taken more than two years.

At the same time, June 2005 could be yesterday. I can remember holing away at the education library at Indiana to proofread yet one more draft of the dissertation before handing it over to my advisor, Annie Lang.

All this time my kids are two years older and my hair is two years, well, less dark brown.

Two years. Seven hundred and thirty days.

One third of the "tenure track" has passed by. We accomplished a lot, but we could have done more. That's how I always feel. There's always the one additional data set that we did not get collected or the one additional manuscript that we did not get out the door.

We'll keep trying. I'll keep you posted on the progress. In a week or so, this interminable summer class will be finished, and the research will begin in earnest. Look for some cool figures here in the near future.

Quizzing Effect Impacts Memory Interpretations

Every once and a while you read something that changes everything.


The other day I was reading Rob Potter's weblog, and I came across this post about his summer class. In that post, Dr. Potter mentions a Chronicle of Higher Education article on research about the relationship between quizzing and learning.

The underlying study found that the act of quizzing itself impacted learning. It was not that students studied for the quizzes -- it was the thought processes that went on while students tried to answer the quizzes, especially short answer quizzes.

"... Every time you test someone, you change what they know," Washington University psychology professor Henry L. (Roddy) Roediger III told the Chronicle.

The implications are huge.

First, I will likely never teach another undergraduate class that does not employ quizzing. I used to think that it was mean. Now I know that it works.

Secondly, the implications for my research are immediate and direct.

Consider our standard memory testing paradigm. First, we test free recall. That is, we ask participants to recall everything they can remember about the stimulus.

Oops. We already changed them.

Then, sometimes, we test cued recall. We changed them again.

Finally, we test recognition. But we've already changed the memory a lot.

"In the process of retrieving Fact A," said Washington psychology professor Kathleen B. McDermott, "if it takes you a minute to get there, you think, Hmm — what did I learn about this general topic? So in a sense, you're also retrieving Fact B and Fact C, even though that's not what you were directly asked to do."

To his credit, Texas Tech master's student Wes Wise proposed looking at some of his own data to see whether recall predicted later recognition months before the Chronicle story.

I guess that we'll have to run the data now.

Lab Is Moving Soon ... To the Basement

It was about this time last year that the Communication and Cognition lab moved to Texas Tech.

Shouldn't it be time to move again?

Yes ... but. We are moving within the next 10 days. But we're staying at Tech. We're moving to a much better -- and quieter -- space in the basement of the mass communications building.

I'm excited about the new space. But not about the move. I toured the new space today, and it will be most excellent.

It'll be a busy year. We have three master's theses to run in addition to whatever studies we run to follow up out glut of data.

The Bimodal Attitude World of Wal-Mart

Numerous surveys have shown that Wal-Mart is the most hated and the most loved company in America.

Some people love Wal-Mart for their cheap prices. Some people hate them for their corporate politics and harsh treatment of employees.

For the advertising practitioner, this split opinion provides an interesting dilemma: how do you keep your "base" happy while courting your detractors?

I offer no solutions here, but I do have what I believe to be an interesting observation.

We ran an experiment this year where we collected physiological responses to advertising brands. Although we're still hard at work analyzing the physiological data, I have peeked at the self-report data.

After we recorded physiological responses to each brand, we asked participants several questions about the brand. Most of these question dealt with how positive or how negative their attitudes were toward the brand.

The histogram shows attitudes toward Wal-Mart for the 54 experimental participants. This is the average of 6 attitude-related questions, and the scales range from 1 to 7. Here 7 would represent the most positive attitude possible.

As you can see from the superimposed normal curve, these data are not normally distributed. Instead they appear to be bimodal. That is, they appear to have two most-frequent responses. Just as surveys suggest, we appear to have a group that loves Wal-Mart, and we appear to have a group that hates Wal-Mart.

In West Texas, the pro-Wal-Mart crowd appears larger, but these data are hardly representative. They are from a convenience sample used for an experiment. Nonetheless, I still find it interesting that the split-attitude trends appears present even with the small non-representative sample.

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See our (small non-representative sample) most-loved and least-loved brands here.

Branded: Love and Hate in West Texas

We continue our work into the cognitive processing of brands in the communication and cognition lab. We're just beginning to analyze the data, but in the interim, I thought that I would bring you the most and least loved brands on the South Plains.

Most Loved Brands
1. Disney
2. Google
3. Starbucks
4. Dr Pepper
5. Target (go figure)

Least Loved Brands
28. McDonald's
29. Citibank
30. Abercrombie & Fitch
31. Camel (cigarettes)
32. Marlboro

Most Arousing (Exciting) Brands
1. 20th Century Fox
2. BMW
3. Bacardi
4. Smirnoff
5. Nike

Least Arousing (Exciting) Brands
28. Dell
29. Microsoft
30. Gap
31. Citibank
32. Maxwell House

Sopranos Fuss Highlights TV's Power

I don't watch the Sopranos. There's no good reason, really. When the series started, I had Showtime rather than HBO. And I was in graduate school, so there was not much spare time for television.

Having several obsessive-compulsive quirks, I usually get on a bandwagon early or I don't get on at all.

Sure, I've seen a few episodes. I understand why the show is both highly rated and critically acclaimed. I just haven't had the time.

Nonetheless, I have followed the saga of the show's ending closely and with great interest.

I have a high need for closure. As someone who did not watch the show, I think the open-ended ending was brilliant. If I had been a viewer, I probably would be angry.

But the debate (more than 80,000 people logged in to vote on an ESPN Radio poll) captures why I study television: stories fascinate me.

As I have indicated before, I never set out to study narrative. If I had continued my early methodology of showing television in 30-second clips, I might never have gotten it.

But I started showing entire episodes. For my dissertation (and subsequent work), participants watched an entire episode of ER. And I watched them watching.

The story draws you in. When participants find out that they will be watching ER, some are happy and some are annoyed. One young lady did a little dance.

Once the story starts, however, they get reeled in. We like stories. Homer figured this out several hundred years ago. And it fascinates me today that we're still so hooked on these stories.
Indeed, many old women call soap operas their "stories."

And since we were children, we were taught that stories have endings. We seem to get mad when that's violated, even though it's not so life-like.

There's no definitive answer about the Sopranos. The ESPN Radio vote was 50.8% in favor of the ending and 49.2% against. You cannot get much more split than that.

In the end, creator David Chase wrote the series ending that lets the story live on.

This seems more appropriate in a drama than a situation comedy, where we beg for closure. Just a few days ago, I asked my wife, "Did Ross and Rachel end up getting back together in the Friends finale?"

Pictures: So Much More Than 1,000 Words

In my home office, I have a bunch of the clear plastic storage containers.

For whatever reason, this picture fell against the edge of one of the containers.

So I've been seeing it for several weeks.

As photographs go, it is pretty bad. I took an inexpensive disposable camera to San Diego that May 2003 weekend for the annual meeting of the International Communication Association.

Nonetheless, this was the view from the hotel room I shared with Byungho Park, now an assistant professor at the National University of Singapore. The great view made it a great room, and we hung out there at night.

Of course, I did not know about San Diego's "may grey." So other than a blazer, I did not even pack a jacket.

The point is that this photograph triggers so many memories. Although that conference is now more that four years ago, this photograph triggers memories of fish tacos, my friend Nancy's craziness, our jaunt to La Jolla.

Unfortunately, the photo also reminds me that my grandmother died just as I was leaving for the conference.

If I were to describe all of those memories, the resulting text would span far more than 1,000 words. That's the power of photography and human visual memory.

I'm grateful that my father inspired my love for photography at a very young age.

Much Ado About Nothing ... Paris

Why in the world do we care about Paris Hilton?

I love that it has become "campy" to track her jail tribulations. The faculty of the college of mass communications now routinely send e-mail updates, and they plaster news all over Facebook.

It may be satire, but we're following her every move nonetheless.

I am a bit embarrassed.

Coach Donovan, I Understand Leaving Angst

When I am at work, decisions come easily. Working on deadline at the New Mexico State student newspaper, someone would shout a question. The answers always came easily. The same was true running the sports department at the Las Cruces Sun-News. And to a large extent, it's true in the lab today.

Personal decisions, however, are another matter.

Plenty of people are vilifying University of Florida/Orlando Magic/? coach Billy Donovan because of his waffling on the NBA job.

To quickly catch you up, Donovan turned down powerhouse Kentucky to stay with the Gators earlier in 2007. That alone won Donovan a place in my heart. But less than two months later, the winner of the last two NCAA championships bolted for the NBA and a $5 million+ annual salary.

He seemed excited.

Then it came time to say goodbye. Not so easy.

Donovan changed his mind. He wanted to stay at Florida. However, he actually inked the Magic deal (unlike his Florida contract upgrade). So now Donovan dances for a release.

I've heard some pretty harsh words for Donovan. You won't hear them from me. Such decisions tear me apart.

Just about a year ago, I left Ohio State for Texas Tech (read here). In many ways, this was an easy decision driven by family considerations. However, in many other ways, this was a gut wrenching decision.

My kids had to move schools ... but they'd be closer to their grandparents. There was one Ph.D. student willing to move to Tech, but there was another very talented incoming Ph.D. student staying behind. Here we have the opportunity to build a Ph.D. program, but building is hard work.

It's been about 56 weeks since I was offered the job here. And I'm pretty sure that there has not been a single week go by where I did not wonder -- at least for a moment -- whether I made the correct decision. More often than not, the answer has been "yes." But the vote has been far from unanimous.

And that's not a property of Ohio State or Texas Tech. Instead, it's about the fact that there are many great things about each place. Choosing either place meant leaving a lot of great things on the table. And as I am sure is the case with the Orlando Magic and the Florida Gators, the great things are not the same at each place.

Thus one is left to decide which great things matter the most.

My decision was a difficult one. And I had spent only a year at Ohio State. Although I made some great relationships -- which were very difficult to leave behind -- I had hardly won two national championships with another finals appearance over the past 11 years.

Leaving "home" always should be difficult. It's still easy for me to recall how I felt when my dad and I drove the moving truck out of Las Cruces on the way to Albuquerque in November 1998. Sure, my career was expanding. But I left a lot of good things behind in Las Cruces.

And I have not forgotten those good things almost nine years later.

My guess is that Donovan realized that he never would have forgotten those good things in Gainesville.

And I, for one, get it.

How Are You Going to Secure the Transducer?

I've been a bit of a slouch posting lately due to the trip to San Francisco (see pictures on Dr. Rob Potter's Web site).

In the interim, I wanted to relay a funny story.

We are in the process of purchasing some new equipment for the psychophysiology lab. Specifically, we're trying to choose between a couple of new measures that I have never used before.

The impetus behind this is a study idea of Wes Wise, a master's student here at Texas Tech. Since Wes knows the study best, I asked him to call Coulbourn Instruments to see what we needed.

Now, Wes is a top notch master's students, and I would clone him if I could (hello, Dolly), but he is new to the psychophysiology game. And the poor guy was felled by the second question:

"How are you going to secure the transducer?" he was asked.

To listen to him explain it still makes me laugh out loud. He mumbles something about the flux capacitor, vaguely mentions armed guards, and generally panics.

I cannot do Wes's sense of humor justice here, obviously. But he forever has a catch phrase in my mind.