An Introduction to Game Theory
Math 480: Mathematics
Seminar
Dr. Sylvester
Game Theory - What is it?
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For us, a game is an interactive situation
where players are believed to be rational.
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Game theory is the study of the strategies,
actions, and interactions among the game’s
players.
General Categories for Games
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Deterministic
─ Economic Applications
─ Military Applications
─ Applications to Political Theory
Probabilistic
─ Games of Chance
─ Randomizing Strategies in Repeated Games
Behavioral
─ Coalitional Strategies
─ Auctions
Brief History
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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.
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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
A
n
 A.
i 1
i
(3)
A sequence of payoff functions f1(a1,....,an),
...., fn(a1,....,an) for each player.
A Starting Point: Easy Games
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We start with the simplest games to analyze: Two-Person
Zero-Sum Impartial Games.
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Impartial - Both players have the same “information” set.
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Zero-sum - player one’s gains are player two’s losses (and vice
versa), i.e. Σfi = 0.
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These games do not necessarily represent “fair games” in a
probabilistic sense.
A Collection of Simple Games
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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.
2.
3.
4.
5.
Two players.
We have a pile of n chips in the center of the table.
A move consists of removing 1, 2, or 3 chips from
the pile.
Players alternate with player 1 starting.
The player who removes the last chip wins (for
now.) That is, we assume the “normal play rule.”
Play the Game!
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Let’s play the game with n = 21 chips.
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Game Analysis I
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P-positions (Previous Player) and Npositions (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
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This is a simple game, but can be
confounding!
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Solution Technique - backward induction!
This is a common technique for solving lots
of games.
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Note: 0 chips is a “P-position.” Why?
Game Analysis III
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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
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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
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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 Pposition.
(3) From every P-position, any legal move goes to an N-position.
Some Exercises
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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.)
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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 Ppositions assuming normal play rules.
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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
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What are P-positions for games with pile
sizes (x1,x2,x3)?
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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
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But what if all three piles are still non-empty?
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We can easily show that some standard positions are Npositions 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 Pposition. 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
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Since (0,0,0) is clearly a P-position in normal play, we have the
following proposition:
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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.
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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
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Exercise 4: Find the P-positions for threepile nim using the misère play rule.