The Games We Play

by Terje Norderhaug

Terje is taking a masters degree at the School of Communications as part of his graduate work in Media Systems Design at the University of Oslo. His current goal is to build a company in San Diego.

Instructional Objective The students will get hands-on experience in a section of game-theory that models many common situations in which two parties make a decision with an outcome that depends on the other parties choice. They will understand the four cell matrix often used to model games in game theory, and learn how different configurations of the scores in this matrix affect the tactics.

Learners/Context The learners are students in university-level courses in such fields as Communications, Philosophy, Political Science, Mathematics, and Game Design.

The card game can be played without further introduction to game theory. However, to increase learning, the students should have been exposed to the simple matrix often used in game theory for modeling games. This will help the students to understand the connection between game theory and the card game, thus facilitating the learning of the game as well as the understanding of the theory.

During the game, the students will encounter such situations as "The Prisoners Dilemma" (Watzlawick, 1976). In its original version, an attorney holds two suspects, but has not enough evidence. If none of the suspects confesses, each will only get a half year sentence. The attorney tells each of the suspects that if they both confess, they will get the sentence of two years. If only one confesses, he will go free, while the other will get twenty years. Then he has them locked up in separate cells, hoping that their fear that the other will confess brings both to confess.

This situation is often modeled by using a four-cell matrix, where two players (A and B) each has two choices: left and right. The players make their decision simultaneously. This is one possible configuration:

This matrix shows that if A chose left and B chose left, they will each get five points each. If B instead chose right, A will lose five points and B will get eight. The opposite happens if A chose right while B chose left. Finally, if both chose right, they both lose three points. This model contains the essence of the prisoners dilemma.

After playing, a debriefing should present the players for different configuration of the four-cell matrix, and be explained how game theory suggests different tactics. The debriefing should also map game theory to daily life examples, such as disarmament talks.

The game itself is fun, and can be played as recreation. It invites advanced tactical play, and is thus a potential general card game, especially since it can be played with a normal card stock.

Rationale A game is perfect for demonstrating models from game theory. The models used in game theory are abstractions of general games, and a card came is an efficient way of making these models concrete while at the same time keeping close to the structure of the game theory models.

Rules Two or four people may play at the same time. If four people are playing, they should sit on all four sides of the table. If two people are playing, they should sit on adjacent sides (i.e. on a 90 degrees angel from each other).

Use a standard 52 card pack, or a special pack designed for the game. Each card has a value from -10 to +10. If you use a standard card stock, each number card is worth its face value, Aces are worth one point, and courts (face cards) are worth ten points, or 11, 12, and 13 if so agreed upon before dealing. Every red card has its negative value. Spades and hearts symbolize "left column", and clubs and diamonds symbolize "right column".

The aim of the game is to gain as many points as possible during the game. The score can be compared both to the other players and to own highest score in previous games.

The game is played in the following manner:

  1. Deal seven cards to each player. Use two cards face up to form a cross on the table, first putting one card vertically (directed toward the dealer) and another horizontally. Going clockwise, make three more crosses to build a 4 cell matrix on the table (four two-card crosses as corners in a square). Stack the rest of the cards face down.
  2. Two adjacent players disclose a card from their hand simultaneously. Depending on if the card symbolizes a left or a right column, each player puts their card vertically on the bottom of the left or the right column of their side of the matrix. The cell where these columns intersects is the selected cell.
  3. Each player collects the card in the selected cell that are in a vertical position from the player. This card are put into each players pile of score cards on the table.
  4. The two previously played cards are used to form a new cross, replacing the cards just collected. Each card should be positioned vertically towards the person that previously had that card on the hand.
  5. If playing with only two players, then the player that collected the lowest card has the option to rotate one of the cells in the matrix. This is done by turning one of the crosses 90 degrees, so that the cards changes who they are directed towards. No cells are rotated in the case of a tie.
  6. Each player draw a card from the stock (unless the stock is empty), so that each player has seven cards on their hand. If the game is played with four players, the game now continues with the two other players.
  7. Repeat until the stock and all hands are empty. The value of each of the cards in each players score pile is then added together. Hint: Counting can be speeded up by using that a negative card eliminates a positive card of the same value.

Card Design

This is an example of two cards in a cross. The card in the bottom have the value of 4, and symbolizes "left column". The card on the top has a negative value of 2, and symbolizes "right column".

The value of the card is in the uppermost right corner, so it is easy to see when the card is on the hand. The value consist of the number and a sign, denoting if it has a positive or negative value. Following the tradition from bookkeeping, negative numbers are colored red to make them stick out.

To facilitate that the players see what value the card has when it is on the table, the top of the card uses a graphical bar. This bar consist of the same number of lines as the value of the card, using the same color coding as for the numbers. The lines are made thin so they are not confused as symbols for cards.

An icon is used to show whether the card selects "left column" or "right column" in the matrix. This icon is placed on the left corner of the card so it is easy to see when the card is on the hand. The icon is placed under the value number, both because this follows the tradition from normal playing cards, and because this icon does not need to be seen when the card is on the table.

The middle of the card shows a diagram of the four cell matrix on the table. This is done to make the setting up of the game as self-explanatory as possible. At the same time, it does not favor the player that is collecting the card on top by displaying more information about the content of the card.

Design Process
I started by choosing game theory as my topic for the game. Most other educational card-games have the content to be learned on the playing cards. I chose to instead to design the structure of the game close to the subject area, and make the cards so they just facilitated playing the game. That the game could be played with normal playing cards was also an important consideration. I chose to use both negative and positive values, since this is used in the example from game theory.

I tried to find a way to arrange cards on the table so they resembled the four cell matrix used in game theory. The natural way was to make four pairs of cards. I quickly found it most appropriate to make each pair of cards into a cross, because this allows each player to read their card and thus determine which is theirs. The adjacent seating of people was chosen to imitate the way the matrix is presented in game theory.

I tried a prototype of the game with three cards to each player, but after trying it out on a two-player session, I found that the players often ended up with only "left column" or "right column" cards, making it impossible for them to make a choice by using the cards. After several experiments, I found that seven cards were sufficient.

I chose the simultaneously disclosure of the cards because it simulates the way choices are done in game theory. It was difficult to find an easy way for the players to select a cell in the matrix. The approach where each player chooses a column was selected because it is the one that comes most closely to the presentation practice in game theory.

I suggested that each player puts the card in the bottom of the column that it represents, because this works as an efficient signal to the other player, and helps both players to find the selected cell without having to look closer at the card that the other player put on the table. The vertical position of this card were suggested because it makes the player think of it as column , and not a row as a horizontal card may suggest.

It was difficult to describe the cell selection process in an easy manner, because I could not follow any way used in traditional card games. However, players that are familiar with game theory should be able to fast understand how it works, and others seems to be able to understand it after a few attempts.

To keep the game simple and stable, I decided that the score matrix was going to be rebuilt during the game. The cards discarded from hand was a natural choice for material for building a new cross. The rule that says that each card in the cross should be positioned towards the originator was added because a determination of this is necessary so the players don't try to build the cross in a way that benefit them.

A problem that occurred when testing this format with two players, was that it turns out to be deterministic. Each player ended up with the same cards in the score-pile as they previously had on the hand. To make it into a strategic game, I added that cells were rotated during the game. First I tested out the game with a rule allowing rotation only when certain cards were played, but I found this to be difficult and little intuitive. The solution was that one or both of the players should be able to rotate. I chose that the one that collected the lowest card should be the rotator, both because this seemed to be "fair" and because it opens up for tactics such as to collect a lower card for later benefit.

That each player draw from the stock to keep a constant number of cards on their hand, follows from tradition in card games.

A number of exits from the game were possible. I tried different ways in test-games, and ended up with that all cards should have been played, because this felt most intuitive for the players.

The scoring mechanism was designed to be as easy as possible. Instead on relying on writing down points won in each round, I chose to let the user collect cards with an value into a pile with earlier scores. This way it is easy to summarize the numbers in the end. Another advantage is that half of the cards are negative. This makes it even easier to calculate the score by that negative scores eliminates positive scores with the same value.

References Watzlawick, P. (1976). How Real is Real? New York, NY: Random House.