The Three Doors of Serendip: Experimental Understanding
Door images from Woodstone
Welcome to Three Doors of Serendip (Part 2). This is the place to be if you've had some experience with the Three Doors of Serendip (Part 1) game, or if you've thought a bit about and/or puzzled over the "Monty Hall Dilemma" or the "Let's Make a Deal" game or the "Three Door Problem". One way or another, you've acquired some intuitions or beliefs about how to best play the game (whose rules are the same in all versions). This is the place to test your intuitions/beliefs, to see whether you're right that your way is in fact better than some other way.
The Rules and How to Play
Let's briefly review the rules, to be sure we're on the same wavelength and to introduce you to the details of the game implementation available here for your experiments. There are three doors, with a prize (in this case, one of Serendip's logos) hidden randomly behind one of them. You pick one of the three doors, after which Serendip will show you that one of the two remaining doors is NOT where the prize is. At this key point in the game, you need to decide whether you want to stay with your original choice or switch to the remaining unopened door. After you make your choice, Serendip will show you whether you've won or not, recording both that and your choice, and you can continue playing as many times as you like. A cumulative total of your choices and the results will be kept by Serendip, and displayed when you click the summary button. You can click the 'start new game' button to reset the game and to start a new collection of observations. To start playing, click on the door below. The applet will launch in a new window which also includes detailed instructions for playing the game.
Playing Around With Experiments
Got some intuitons/ideas that you want to test? If so, go to it. Remember though that you need to have not one but two "strategies" in mind for a good experiment, one strategy that you think is better and one that you think is less good. Play the first strategy for a number of games and then see how well you did by using the monitor that tracks the percentage of wins for each strategy. Reset and then play your second strategy for a number of games, and then see whether you did better or worse the second time. Remember too, though, that there's some randomness in the game, so "a number of times" should be pretty big number (ten or twenty at least), and you shouldn't take small differences in the outcomes too seriously (do the comparison several times to see whether any differences you observe are reliable).
To stay or not to stay, that's a question
Not sure what to test? Then here's a suggestion. Since the prize is randomly placed behind any one of the three doors at the beginning of each game, your first choice in each game probably doesn't make any difference, and it probably also doesn't make any difference what you did or what happened during the previous game. If this is true, then all that matters is whether you stay or switch with your second choice on each game. Hence the problem boils down to three possibilities:
1) staying is better,
2) switching is better,
3) or it doesn't matter whether you stay or switch.
You can explore all three possibilities by comparing two strategies: always stay with your second choice and always switch with your second choice. If one reliably does better, you've established that that one is the better strategy; if neither reliably does better, then it doesn't matter whether you switch or stay. Either, as well as choosing randomly each time, is an equally good strategy.
You can test whether stay or switch is the best strategy by playing a series of games using one strategy, finding out the percentage of times you win, then doing the same with the other strategy, and comparing the two percentages. Serendip also provides a "short cut" for this particular comparison of strategies. If you play series of games, sometimes using the stay strategy and sometimes using the switch strategy Serendip stores and displays the results separately for the two strategies. So you can intermingle stay and switch strategies and still see whether either has a higher winning percentage.
It turns out that... (click for spoiler). Is that what you expected? Are you surprised? No, Serendip is not "rigged". The prize is actually put behind a randomly selected door at the outset of each game, and stays there through that game. Serendip knows which door this is, and hence knows which door it can safely open (choosing randomly between two possibilities in the case that you first select the door with the prize). If you're still skeptical about Serendip cheating, you can play the game with some home-made equipment and a friend; the answer will come out the same.
But, but, but ... and beyondMaybe the answer you got from experimental observations fits your intuition (this might particularly be true if you've played Three Doors of Serendip (Part 1). More likely, it doesn't. In either case, though, you're probably still puzzled. How COULD the answer come out that way? It doesn't make LOGICAL sense. Now that's pretty interesting, and says something pretty important for "understanding understanding". If you like, you can go in that direction. If you're impatient though to have a logical explanation of what you've found, you should go through Door 3, Broader understanding.
Hands on understanding