Game Theory: Race to 15 and Knight’s Lines (Tues-Friday)

Game theory is the study of how and why people make decisions. Specifically, it is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers". It helps people understand parts of science and politics. An alternative term suggested "as a more descriptive name for the discipline" is interactive decision theory.

In the Cold War period (1947-1991), the strategic decisions of the United States and the Soviet Union were sometimes viewed as an exercise in game theory. In that case the "players" being studied were the United States and the Soviet Union. Game theory is not just about games, but how and why businesses make decisions, and just about any decision based on valuing likely outcomes. In game theory, all of these situations are "games" since the people involved make choices based on how they value the possible outcomes of the choices. This is true even of cases where the decisions of a single person only affect that one person. Game theory is found in the financial choices people make, and is found in the study of economics.

I devised two games for the Mathletes to illustrate competitive game theory: Race to 15 and Knight’s Lines.

Race to 15:

• there are 9 different cards numbered 1-9 (a. you can use the ace, 2, 3, 4, 5, 6, 7, 8 and 9 from a deck of cards; b. you can cut out the numbers 1-9 below; c. or just write-out  numbers 1-9).
• place the 9 cards face up.
• two players take turns picking up cards one at a time until one player has three cards (or different numbers) that add up to exactly 15. 
• the player with three cards that add up to 15 has won the game (two, four, or five cards that add up to 15 will not win; getting three cards that add up to more than 15 also will not win).
• is it possible for two good players to ever have a winner?

What is Race to 15 really? Tic-tac-toe with numbers.

• Here are all 8 combinations of three different numbers from 1-9 that add up to 15 (horizontally, vertically, and diagonally).
• Can you see that this is really the game of tic-tac-toe with Xs and Os? If you are able to block a player from getting three in a row, you can block a player from winning Race to 15.

Knight’s Lines:

This game is played on a square grid, using two colored markers. 

• Each set of opposite sides of the square have the same color. 
• Flip a coin or do rock-paper-scissors to determine who goes first. 
• Player 1 tries to connect the red sides of the game board, while player 2 tries to connect the blue sides. 
• All moves are knight's moves using straight lines from one corner (vertex) to another. 
• The routes are not permitted to cross and players 1 and 2 may not share a vertex (however, one player may share his/her own vertex). 
• Remember, knight's move two squares in any direction and then turns right or left one square or one square in any direction and then turns right or left two squares. 

Now the challenge is for the children to see that game theory applies to their everyday decision making. When they are negotiating with you to stay up later or to go to an event with a friend, they could employ cooperative game theory so everyone wins. We will explore cooperative game theory next week.