An Introduction to Game Theory

Math 480: Mathematics Seminar

Dr. Sylvester

Game Theory - What is it?

For us, a game is an interactive situation where players are believed to be rational.

Game theory is the study of the strategies, actions, and interactions among the game’s players.

General Categories for Games

Deterministic ─ Economic Applications ─ Military Applications ─ Applications to Political Theory

Probabilistic ─ Games of Chance ─ Randomizing Strategies in Repeated Games

Behavioral ─ Coalitional Strategies ─ Auctions

Brief History

Games and their strategies have been of interest for centuries!

Mathematical Study, however, is very young! Major results started in the early 1900’s, but the field became popular in the mid 1900’s.– Von Neumann (Matrix Games)– Nash (and his truly “Beautiful Mind!”)

Characterizations of the Game

(1)Extensive form - which gives the most detail about the play of the game. This is typically expressed as a “game tree.”

(2)Normal (or strategic) form - is generally given as a tableau, i.e. it is a “matrix game.”

(3)Coalitional form - Here the emphasis is on the basis of the rational players and their interactions.

Components of a Simple Game

(1) A set of N players {1,2,...,N}

(2) A sequence of strategy sets (or action profiles) for each player: A1,. . .,An. The strategy set is given by

(3) A sequence of payoff functions f1(a1,....,an), ...., fn(a1,....,an) for each player.

A Ai

n

i1

.

A Starting Point: Easy Games

We start with the simplest games to analyze: Two-Person Zero-Sum Impartial Games.

Impartial - Both players have the same “information” set.

Zero-sum - player one’s gains are player two’s losses (and vice versa), i.e. Σfi = 0.

These games do not necessarily represent “fair games” in a probabilistic sense.

A Collection of Simple Games

A Class of Simple Two-Player Impartial Zero-Sum Games - Take-Away Games

Take-Away Games fall in a class of games where two players take turns, in succession, removing chips from piles. The winner is determined to be the last person to take a chip. (This version is known as the “normal play rule.” We can vary the game by changing the ending condition to be the last player taking a chip is the loser. This is called the “misère play rule.”)

A Take-Away Game’s Rules

1. Two players.

2. We have a pile of n chips in the center of the table.

3. A move consists of removing 1, 2, or 3 chips from the pile.

4. Players alternate with player 1 starting.

5. The player who removes the last chip wins (for now.) That is, we assume the “normal play rule.”

Play the Game!

Let’s play the game with n = 21 chips.

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Game Analysis I

P-positions (Previous Player) and N-positions (Next Player) - A P-position is one in which at the end of the player’s turn, his move is a winning strategy. An N-Position is one in which at the end of the player’s turn, his move is a “losing” strategy; i.e. it allows the next player (under perfect strategy) for force a win.

Game Analysis II

This is a simple game, but can be confounding!

Solution Technique - backward induction! This is a common technique for solving lots of games.

Note: 0 chips is a “P-position.” Why?

Game Analysis III

1 chip is an “N-position” 2 chips is an “N-position” 3 chips is an “N-position” 4 chips is a “P-position” 5 chips is an “N-position” 6 chips is an “N-Position” 7 chips is an “N-Position” 8 chips is a “P-Position”

Game Analysis IV

What is the set of all P-positions? (That is, if you are player 1, what is your winning strategy?)

Theory of P- and N-positions

What we should learn from studying the P- and N-positions of games:

(1)Under “normal play rules,” all terminal positions are P-positions. (Under “misère rules” all terminal positions are N-positions.)

(2)From every N-position, there is at least one move to a P-position.

(3)From every P-position, any legal move goes to an N-position.

Some Exercises

Exercise 1: Solve the same game using the misère condition. (i.e. All of the rules of the game are identical, with the exception that the last person to take a chip loses.)

Exercise 2: Consider a take-away game where you start with n chips, and you can subtract away {1, 2, 4, 8, . . .} chips. (That is, you can remove chips in powers of two.) Find the P-positions assuming normal play rules.

Exercise 3: Revisit the game of exercise 2. If n = 100 chips to start, who can guarantee a win (player 1 or player 2)? Justify your answer.

Nim: An Interesting Take-Away Game

Basic Description - There are three piles of chips. On any given turn, a player selects exactly one of the piles and chooses 1 (or more) chips from that pile. The winner is the player who:

(a) Chooses the last chip (normal play rule)

(b) Forces his/her opponent to choose the last chip (misère play)

Nim: An Interesting Take-Away Game

Historical Information: There is some discrepancy over the origin of Nim.

Some scholars believe it may have originated in China.

Early written work of the game is dated to the 1500’s in Europe. The German verb nimm means “take.”

Charles Bouton (1901, Harvard Univ.) provided a comprehensive analysis on the game.

Versions have emerged from many cultures.

An Interesting Nim Configuration

A simple, but interesting, Nim Setup is (3,4,5).

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Looking for Mathematical Patterns

What are P-positions for games with pile sizes (x1,x2,x3)?

Certainly under normal play rules, the terminal node (0,0,0) is a P-position. (0,0,x) is an N-position. It is easy to see that positions of the form (0,x,x) are P-positions.

Mathematical Patterns II

But what if all three piles are still non-empty?

We can easily show that some standard positions are N-positions and should be avoided as points of completion for a player’s turn: (1,1,1), (1,1,2), (1,1,3),(1,2,2)..... Why? Because they can be converted into (1,1,0) which is the next player’s P-position. Note (1,2,3) is probably the next simplest position. It must be a P-position, because the only possibilities are to move it to a previously stated N-position. We can continue the game analysis using this form of reverse induction.

A Clever Method to Solve Nim

Properties of the Nim-Sum

Strategy for Nim with the Nim-Sum

Since (0,0,0) is clearly a P-position in normal play, we have the following proposition:

Proposition: A position (x1,x2,x3) is a P-position if and only if the nim-sum of the number of chips in each pile is zero.

This tells us when solving the three-pile nim game, we simply need to ensure that the nim-sum of the piles after our turn is zero. We can experiment solving this game using web applets. One particular version can be found at:

http://www.cut-the-knot.org/nim_st.shtml

A Final Exercise

Exercise 4: Find the P-positions for three-pile nim using the misère play rule.