OUTCOMES WHEN ANGELA CHOOSES AN UNKNOWN CARD AFTER GETTING A KING

Identity of Second Number of Pairs for Number of Pairs for Card Chosen Chances Angela on This Round Zack on Next Round

King 25% 1 (maybe more) 0 (maybe more)

Queen 25% 0 (for sure) 1 (maybe more)

10 50% 0 (for sure) 3 (for sure)

Having randomly chosen her first card and found it to be a king, what, then, should Angela select as her second card? Game theory has proven that the best move is not for her to choose one of the four remaining unknown cards, but instead to choose the card that is known to be a queen! It will take us too far afield for me to prove to you why that is the best move, but it is easy to explain how it could be the best move. Although selecting the queen means that Angela doesn’t get a pair (because she’ll have a king and a queen), it also means that she doesn’t deliver as attractive a board to Zack. Instead, Zack would receive the following board:

Board 5

8 CHAPTER 1: INTRODUCTION TO STRATEGIC REASONING

Notice that Zack is no longer assured of getting a pair. If, instead, Angela had chosen one of the four unknown cards, there is a 25% chance that she’d have gotten a pair, but a 75% chance that Zack would have gotten at least one pair.

What this analysis highlights is that choosing an unknown card has bene- fits and costs. The benefit is that it may allow a player to make a match— something that is, obviously, well known. The cost is that, when a player chooses a card that does not make a match (so that the revealed card remains on the board), valuable information is delivered to the opponent. Contrary to accepted wisdom, under certain circumstances it is optimal to choose a card that will knowingly not produce a match in order to strategically restrict the information your opponent will have and thereby reduce his chances of col- lecting pairs in the next round.

Generally, the value of game theory is in delivering insights of that sort. Even when we analyze a decidedly unrealistic model—as we just did with players who have perfect memory—a general lesson can be derived. In the game of Concentration, the insight is that you should think not only about trying to make a match, but also about the information that your play might reveal to the other player—a useful tip even if players’ memories are imperfect.

1.4 Psychological Profile of a Player I think that God in creating Man somewhat overestimated his ability. —OSCAR WILDE

A STRATEGIC SITUATION IS described by an environment and the people who inter- act in that environment. Before going any further, it is worth discussing what defines a person for the purposes of our analysis. If you are asked to describe someone you know, many details would come to your mind, including the per- son’s personality, intelligence, knowledge, hair color, gender, ethnicity, family history, political affiliation, health, hygiene, musical tastes, and so on. In game theory, however, we can ignore almost all of those details because, in most sit- uations, understanding or predicting behavior requires knowing j