Improbable Pigeons

I have bad news for you – a pigeon can probably outperform you in the area of probability and statistics.  Yes, that’s right, a pigeon.

The Problem:

Consider the classic “Monty Hall” problem, named after the original host of the Let’s Make a Deal game show:

Suppose you’re on a game show and are given the choice of three doors. Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show. Before opening the door you’ve picked, the host, who knows what’s behind the doors, must open one of the remaining doors and make you an offer. Accordingly, he opens a door, reveals a goat, and asks you whether you want to stay with your first choice or switch to the last remaining door.

Assuming you want a car and not a playful goat, should you stick with your first choice or go for the remaining door?

The Answer:

This may sound counterintuitive (unless you’re a pigeon), but you actually have twice the chance of winning the car if you change your selection and pick the remaining door.  Why is this?  Well, the relevant Wikipedia1 entry includes the following table, which shows the three possible arrangements of one car and two goats behind three doors and the result of switching or staying after initially picking Door 1 in each case:

Door 1 Door 2 Door 3 Result if switching Result if staying
Car Goat Goat Goat Car
Goat Car Goat Car Goat
Goat Goat Car Car Goat

As shown above, a player who stays with the initial Door 1 choice wins in only one out of three of these equally likely possibilities, while a player who switches wins in two out of three.

How Do People Perform?

In a word, poorly.

Most people will stay with their initial choice or, at best, express no preference either way. In one high profile case, Marilyn Vos Savant (she of the world’s highest IQ) published the answer to the puzzle in Parade magazine and approximately 10,000 readers, including nearly 1,000 with Ph.D.’s, wrote in to vehemently claim she was wrong. The New York Times2 published a fuller explanation of the Monty Hall problem as well as an entertaining account of the Vos Savant incident and how a large number of mathematicians and other well-educated people refused to accept the correct answer, even after being shown multiple proofs of its accuracy.

How Do Pigeons Perform?

Much better!

As published in the Journal of Comparative Psychology3, researchers Walter Herbranson and Julia Schroeder designed a series of experiments in which six pigeons were tested to see how well they would do at solving the Monty Hall problem, and how their performance would compare to that of university undergraduate students. Discover Magazine’s Not Exactly Rocket Science4 blog describes the experiments and the results:

Each pigeon was faced with three lit keys, one of which could be pecked for food. At the first peck, all three keys switched off and after a second, two came back on including the bird’s first choice. The computer, playing the part of Monty Hall, had selected one of the unpecked keys to deactivate. If the pigeon pecked the right key of the remaining two, it earned some grain. On the first day of testing, the pigeons switched on just a third of the trials. But after a month, all six birds switched almost every time, earning virtually the maximum grainy reward.

Every tasty reward would reinforce the pigeon’s behaviour, so if it got a meal twice as often when it switched, you’d expect it to soon learn to switch. Hebranson and Schroder demonstrated this with a cunning variant of the Monty Hall Dilemma, where the best strategy would be to stick every time. With these altered probabilities, the pigeons eventually learned the topsy-turvy tactic.

It may seem obvious that one should choose the strategy that would yield the most frequent rewards and even the dimmest pigeon should pick up the right tactic after a month of training. But try telling that to students. Hebranson and Schroder presented 13 students with a similar set-up to the pigeons. There were limited instructions and no framing storyline – just three lit keys and a goal to earn as many points as possible. They had to work out what was going on through trial and error and they had 200 goes at guessing the right key over the course of a month.

At first, they were equally likely to switch or stay. By the final trial, they were still only switching on two thirds of the trials. They had edged towards the right strategy but they were a long way from the ideal approach of the pigeons. And by the end of the study, they were showing no signs of further improvement.

In their article, Herbranson and Schroeder summarized the results even more succinctly: “The surprising implication is that pigeons seem to solve the puzzle, arriving at the optimal solution while most humans do not.”

Conclusion

While we will accept the view of the researchers that this doesn’t prove that pigeons are smarter than humans, we still think that, if you ever have a chance to appear on Let’s Make a Deal, you should consider bringing a real bird rather than a friend dressed up in a giant bird costume.

_____

1Wikipedia, Monty Hall Problem, visited on June 27, 2011.

2The New York Times, “Behind Monty Hall’s Doors: Puzzle, Debate and Answer?,” July 21, 1991.

3Herbranson WT, Schroeder J. Are birds smarter than mathematicians? Pigeons (Columba livia) perform optimally on a version of the Monty Hall Dilemma. J Comp Psychol. 2010 Feb;124(1):1-13.

4Discover Magazine, Not Exactly Rocket Science Blog, “Pigeons outperform humans at the Monty Hall Dilemma,” April 2, 2010.

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