I've been told that "80% of success is showing up." If you're selected to play DoND and you show up, your expected profit is $131.477.54. The rest is simple math.
That's enough silliness; let's get down to business. I've been hearing two recurring complaints about the banker's offers on DoND, and I want to take this opportunity to address these complaints.
Before we begin, we're going to play a simple game. I'm going to give you a lump sum of cash. It's yours to keep. No strings attached. Then I'm going to give you the opportunity to gamble with that cash. You can keep your cash or you can gamble if you prefer. If you'd like to gamble, you can bet your lump sum on a coin flip. If the coin shows heads, you win $1 million. If it comes up tails, you win nothing. A 50/50 chance at $1 million and a 50/50 chance at zilch. This coin flip has an expected value of $500K. Expected value simply means that you'll win an average of $500K per coin flip if you choose to gamble many many many times.
If the lump sum is small, you're likely to gamble on the coin flip. If the lump sum is large, you'll likely keep the lump sum instead of gambling. Somewhere in the middle is a happy medium where you struggle to make a decision. This decision point is different for different people, and the lump sum associated with your decision point indicates your risk profile. Are you risk-averse, risk-neutral or risk-seeking?
If I gave you $250K, would you gamble or keep the cash? What about $300K? $400K? $450K? $500K? $550K? $600K?
What if I change the coin flip? Heads = $510K and Tails = $490K. The expected value hasn't changed — it's still $500K, but the variability of outcomes has been dramatically reduced. If I gave you $500K, would you gamble on the coin flip now? (Most of you probably chose $500K cash instead of the coin flip for the first game.)
That's the essence of DoND. The gameshow provides a stage and the means so that we can watch contestants struggle with this decision over large sums of cash. DoND provides a simple game that everybody can play since the game is purely a game of chance, which means that each viewer can imagine himself on stage as the contestant. Plus DoND provides immediate feedback so that we can watch the instant euphoria or despair that follows a contestant's decision to gamble.
Now that we've defined the entertainment objective of the game, we must also consider one more goal. The production company has a simple objective — generate profit. Since the production company plays for the long run, they can play the odds. The company knows up front that each contestant is likely to have an average payout of $131.477.54 over the long haul if every contestant selects their suitcase in lieu of any of the banker's offers. Each episode also has a fixed production cost. That means that the company must generate revenue through advertising, which simply means the show must attract viewers by presenting a viable dramatic situation. The company can also slightly reduce costs by using the banker effectively.
OK. It's finally time to address the complaints.
Complaint #1: The banker's offers are initially too small. True. The offers are small relative to the expected value in the first few rounds, as shown below in Figure 1, but these offers are small for a reason. The banker/producer wants to make sure that even the most risk-averse person plays at least a few rounds. Can you imagine a game show where the contestant picks a suitcase, the banker offers the contestant $100K for that suitcase, and the contestant accepts the offer before playing the first round? Not very dramatic, is it? If it's not dramatic, it doesn't meet the producer's objectives for suspense.
(Click on the image to see a larger version.)
Figure 1. The effect of number of remaining cases on the banker's offer relative to the expected value. The ratio of the banker's offer and the expected value is plotted as a function of the number of cases remaining. The number of remaining cases includes the contestant's case. Data is missing for some contestants from the early rounds of Monday's and Tuesday's shows.
Complaint #2: The ratio of the banker's offer and the expected value vary significantly in the later rounds for different contestants. Also true. But the distribution of cash values in the suitcases also varies significantly in the later rounds for different contestants, and this variation affects both the expected value as well as the variability of outcomes. At this point in the game the banker/producer often has an opportunity to accomplish both of its objectives simultaneously. Think back to the simple game you played earlier. Didn't your answer change for the coin flip when the payouts changed? That simply means that your decision point often strongly depends on how different the outcomes really are. As variability increases, the cash value associated with your decision point decreases. If there is little variability in the outcomes, the banker/producer simply offers the expected value. If there is significant variability, the banker/producer can reduce its offer to save some cash and increase the drama since the lower offer is closer to the decision point. Your tolerance for risk also depends on the amount of cash. You're probably more willing to gamble $30 than $300,000 on a 50/50 chance of tripling your money and a 50/50 chance of losing your entire bet. In Figure 2, you can see how outcome variability and expected value affect the ratio of the banker's offer and expected value.
(Click on the image to see a larger version.)
Figure 2. The effect of expected value and outcome variability on the ratio of the banker's offer to the expected value when four cases remain. As you move up (higher expected value) and to the right (greater variability), this ratio decreases. The expected value is plotted against the standard deviation. To compare similar situations, data are used only when a contestant has four cases remaining, including his/her own case. The size of the symbol is proportional to the ratio of the banker's offer and the expected value. Larger symbols, such as those for Daryl and Karen, depict a ratio of unity. Tracy's symbol represents a ratio of 0.73. Venus is not included since she still has 15 cases left at the time I posted this entry.