A few weeks ago, I visited a charming wine and beer tasting room, located in a small wooden cottage, in Santa Barbara, CA. The face of the cottage is flat; except for a small wooden porch at the entryway that has a few steps on either side to invite visitors up from the sidewalk below. On each side of the entryway, about chest-high, there is one square, multi-paned window. And in the center, like an elongated nose above a smile on a person’s face, is the entryway, a pair of thick wooden double doors left wide open to welcome all passersby.
Inside the cottage there is a single rectangular room. On the left-hand side of the room there is a long waist-high bar that stretches about ¾ of the length of the room. Below the bar, on the customers’ side, is a row of wooden chairs. Above the bar, hanging from the ceiling, is a row of four lights evenly spaced along the length of the bar. In the back of the room there is a warm fire-place, some comfortable chairs, and several tall wooden wine racks filled with colorful wine bottles. Finally, on the right-hand side of the room, there are some small round waist-high tables that customers can use to rest their drinking glasses and some rectangular tables that display complementary products like meats and cheeses, cookbooks, and bottle openers.
Ordering a Flight, Receiving a Chance to Upgrade
My friends and I huddled in a space at the bar and each ordered a wine tasting. As the bartender began to fill my first glass, she used up the last of the wine in the bottle. At that point she told me about an interesting challenge – involving both risks and rewards – that they offer to whoever receives the last of the wine from a bottle.
The challenge is as follows – I am given one chance to toss the bottle’s cork into a giant wine glass (the circumference around the opening of the glass is probably similar to that of a basketball) located behind the bar at one end of the bar. I will receive a reward for successfully tossing the cork into the giant wine glass, and the value of my reward is proportional to my distance from the giant wine glass.
How Many Light Bulbs Does It Take To…?
Remember the four hanging lights evenly spaced along the length of the bar? The first light is almost directly across the bar from the giant wine glass. The second light is a bit farther away. The third light is even farther away. And the fourth light, almost at the other end of the bar, is farthest away. If I toss the cork from the first light and it goes in, I win one free tasting. If I toss the cork from the second light, I win two free tasting. From the third light, three free tastings. And from the fourth light, four free tastings. As you can see, my reward increases as my distance from the giant wine glass increases.
At the same time, the likelihood of me actually tossing the cork successfully into the giant wine glass drops precipitously as I step back from one light to the next. Let’s say (optimistically) I have a 25 percent chance of making the shot from the first light. That means I think I can make (on average) one out of every four shots from that location. Again, for sake of discussion, let’s say I have a 1 percent chance of making the shot from the fourth light, meaning I think I can make one out of every one hundred shots from that location. If I assume the likelihood of me making a shot decreases linearly, then from the second light I have a 17 percent chance of making the shot and from the third light I have a 9 percent chance.1 As you can see, my risk increases (i.e. the likelihood of a successful toss decreases) as my distance from the giant wine glass increases.
Based on all of this information, from which light should I toss the cork? Take a moment and think about the situation. How would you analyze the risks and rewards? From which light would you toss the cork?
If You Disregard Expected Value, Instead of a Flight You May Get Taken for a Ride – But It’s Your Fault, So You Can’t Whine
One way to make a decision under uncertainty is to calculate the expected value of your alternatives and select the alternative with the highest expected value. In my situation, I have four alternatives (i.e. tossing the cork from one of the four lights). The expected value of tossing the cork from the first light is 0.25.2 The expected value of tossing the cork from the second light is 0.34. The expected value from the third light is 0.27. And the expected value from the fourth light is 0.04. Based on the expected values of the alternatives, I should toss the cork from the second light because it gives me the largest expected payout.
So, did I make my decision based on expected value and toss the cork from the second light? Of course not. I was there with three friends, which means there were four of us all together. My friends were either looking at me with cute puppy dog eyes or cheering me on to encourage me to toss the cork from the fourth light so that all four of us would receive free tastings. I took a moment to visualize a successful toss from the fourth light and then let the cork fly. How did it turn out? All I’ll say is, “Next time I’m going to pay a bit more attention to the expected value of my alternatives.”
1First light: 25 percent chance; Fourth light: 1 percent chance
(Distance between first and fourth) / (Number of segments) = Distance between segments
(25 – 1) / 3 = 8
Second light: 25 – 8 = 17 percent chance
Third light: 17 – 8 = 9 percent chance
2EV = Sum (Probability * Value of Outcome) over the entire chance event. In this case, “value of outcome” equals the number of free tastings.
EVFirstLight = (0.25*1) + (0.75*0) = 0.25
EVSecondLight = (0.17*2) + (0.83*0) = 0.34
EVThirdLight = (0.09*3) + (0.91*0) = 0.27
EVFourthLight = (0.01*4) + (0.99*0) = 0.04