Whetting Your Appetite: The Game of Concentration 5
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1.3 Whetting Your Appetite: The Game of Concentration THE VALUE OF GAME THEORY in exploring strategic situations is its delivery of a better understanding of human behavior. When a question is posed, the tools of game theory are wielded to address it. If we apply these tools appropriately, we’ll learn something new and insightful. It’ll take time to develop the tools so that you can see how that insight is derived—and, more importantly, so that you can derive it yourself—but you are certain to catch on before this course is over. Here, I simply offer a glimpse of the kind of insight game theory has to offer.
Game theory can uncover subtly clever forms of strategic behavior. To see what I mean, let’s consider the common card game of Concentration that many of you undoubtedly have played. Through your own experience, you may already have stumbled across the strategic insight we’ll soon describe. The beauty of game theory is that it can provide insight into a situation before you’ve ever faced it.
The rules of Concentration are simple. All 52 cards are laid face down on a table. Each player takes turns selecting 2 cards. If they match (e.g., if both are Kings), then the player keeps the pair and continues with her turn. If they do not match, then the cards are returned face down and the turn goes to the next player. The game is played until all the cards are off the table— 26 matched pairs have been collected—and the player with the most pairs wins.
What does it take to win at Concentration? A bit of luck helps. Early in the game, players have little choice but to choose randomly. Of course, the first player to move is totally in the dark and, in fact, has less than a 6% chance of making a match. But once the game gets rolling, luck is trumped by a good memory. As cards fail to be matched and are turned back over, remembering where those cards are will lead to future matches. So memory and luck are two valuable traits to possess (to the extent that one can possess luck). And then there is, of course, strategy. Strategy, I say? Where is the role for strategy in Concentration?
To focus on the strategic dimension to Concentration, we’ll neutralize the role of memory by assuming that players have perfect memory.2 For those of you who, like me, lack anything approaching such an enviable trait, consider instead the following modification to the game: When two cards are turned up and don’t match, leave them on the table turned up. So as not to confuse our- selves, we’ll now speak of a player “choosing” a card, and that card may al- ready be turned up (so that all know what card it is), or it may be turned down (in which case the card is yet to be revealed).
6 CHAPTER 1: INTRODUCTION TO STRATEGIC REASONING
Suppose two players—Angela and Zack—are playing Concentration and face the following array of cards on the board:
Board 1
There are six remaining cards, of which one is known to be a queen. Of the five unknown cards, one is another queen; assume that the others are two kings and two 10’s.
It’s Angela’s turn, and suppose she chooses one of the unknown cards, which proves to be a king. The board now looks as follows, with the selected card noted.
Board 2
What many people are inclined to do at this point is choose one of the four unknown cards with the hope of getting another king, rather than select the card known to be a queen. But let’s not be so hasty and instead explore the pos- sible ramifications of that move. If Angela flips over one of the other four un- known cards, there is a one-in-four chance that it is the other king, because, of those four cards, one is a king, one is a queen, and two are 10’s. Similarly, there is a one-in-four chance that the card is a queen and a one-in-two chance that it is a 10.
What happens if it is a king? Then Angela gets a match and gets to choose again. If it is instead a queen, then Angela doesn’t get a match, in which case it is Zack’s turn and he faces this board:
Board 3
1.3 Whetting Your Appetite: The Game of Concentration 7
Notice that Zack is sure to acquire one pair by choosing the two Queens; he could get more if he’s lucky. Finally, suppose the second card Angela selects turns out to be a 10. Then Zack inherits this board:
Now Zack gets all three remaining pairs! If he chooses any of the three re- maining unknown cards, he’ll know which other card to select to make a match. For example, if he chooses the first card and it is a king, then he just needs to choose the fourth card to have a pair of kings. Continuing in this manner, he’ll obtain all three pairs.
TABLE 1.1 summarizes the possibilities when Angela has Board 2—having just gotten a king—and chooses one of the four remaining unknown cards as her second card. She has a 25% chance of getting a pair (by getting a king), a 25% chance of Zack getting at least one pair (by Angela’s getting a queen), and a 50% chance of Zack getting all three remaining pairs (by Angela’s getting a 10).
Board 4
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