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@ -1,8 +1,7 @@
\chapter{Game theory} \chapter{Game theory}
In this chapter, we will focus on games where In this chapter, we will focus on two-player
two players make alternate moves and that games that do not contain random elements.
do not contain random elements.
Our goal is to find a strategy that we can Our goal is to find a strategy that we can
follow to win the game follow to win the game
no matter what the opponent does, no matter what the opponent does,
@ -11,20 +10,20 @@ if such a strategy exists.
It turns out that there is a general strategy It turns out that there is a general strategy
for all such games, for all such games,
and we can analyze the games using the \key{nim theory}. and we can analyze the games using the \key{nim theory}.
First, we analyze simple games where First, we will analyze simple games where
players remove sticks from heaps, players remove sticks from heaps,
and after this, we generalize the strategy and after this, we will generalize the strategy
for those games to all other games. used in those games to all other games.
\section{Game states} \section{Game states}
Let's consider a game where there is initially Let us consider a game where there is initially
a heap of $n$ sticks. a heap of $n$ sticks.
Players $A$ and $B$ move alternatively, Players $A$ and $B$ move alternatively,
and player $A$ begins. and player $A$ begins.
On each move, the player has to remove On each move, the player has to remove
1, 2 or 3 sticks from the heap. 1, 2 or 3 sticks from the heap,
The player who removes the last stick wins. and the player who removes the last stick wins the game.
For example, if $n=10$, the game may proceed as follows: For example, if $n=10$, the game may proceed as follows:
\begin{enumerate}[noitemsep] \begin{enumerate}[noitemsep]
@ -34,11 +33,10 @@ For example, if $n=10$, the game may proceed as follows:
\item Player $B$ removes 2 sticks (2 sticks left). \item Player $B$ removes 2 sticks (2 sticks left).
\item Player $A$ removes 2 sticks and wins. \item Player $A$ removes 2 sticks and wins.
\end{enumerate} \end{enumerate}
This game consists of states $0,1,2,\ldots,n$, This game consists of states $0,1,2,\ldots,n$,
where the number of the state corresponds to where the number of the state corresponds to
the number of sticks left. the number of sticks left.
The player must always choose how many sticks
they will remove from the heap.
\subsubsection{Winning and losing states} \subsubsection{Winning and losing states}
@ -46,17 +44,17 @@ they will remove from the heap.
\index{losing state} \index{losing state}
A \key{winning state} is a state where A \key{winning state} is a state where
the player wins the game if they the player will win the game if they
play optimally. play optimally,
Correspondingly, a \key{losing state} is a state and a \key{losing state} is a state
where the player loses if the where the player will lose the game if the
opponent plays optimally. opponent plays optimally.
It turns out that we can classify all states It turns out that we can classify all states
in a game so that each state is either of a game so that each state is either
a winning state or a losing state. a winning state or a losing state.
In the above game, state 0 is clearly a In the above game, state 0 is clearly a
losing state, because the player can't make losing state, because the player cannot make
any moves. any moves.
States 1, 2 and 3 are winning states, States 1, 2 and 3 are winning states,
because we can remove 1, 2 or 3 sticks because we can remove 1, 2 or 3 sticks
@ -70,12 +68,12 @@ from the current state to a losing state,
the current state is a winning state, the current state is a winning state,
and otherwise it is a losing state. and otherwise it is a losing state.
Using this observation, we can classify all states Using this observation, we can classify all states
in a game beginning from losing states where of a game starting with losing states where
there are no possible moves. there are no possible moves.
The states $0 \ldots 15$ of the above game The states $0 \ldots 15$ of the above game
can be classified as follows can be classified as follows
($W$ means winning state, and $L$ means losing state): ($W$ denotes a winning state and $L$ denotes a losing state):
\begin{center} \begin{center}
\begin{tikzpicture}[scale=0.7] \begin{tikzpicture}[scale=0.7]
\draw (0,0) grid (16,1); \draw (0,0) grid (16,1);
@ -117,7 +115,7 @@ can be classified as follows
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
It's easy to analyze this game: It is easy to analyze this game:
a state $k$ is a losing state if $k$ is a state $k$ is a losing state if $k$ is
divisible by 4, and otherwise it divisible by 4, and otherwise it
is winning state. is winning state.
@ -137,17 +135,17 @@ they play optimally.
\subsubsection{State graph} \subsubsection{State graph}
Let's now consider another stick game, Let us now consider another stick game,
where in state $k$, it is allowed to remove where in each state $k$, it is allowed to remove
any number $x$ of sticks such that $x$ any number $x$ of sticks such that $x$
is smaller than $x$ and divides $x$. is smaller than $k$ and divides $k$.
For example, in state 8 we may remove For example, in state 8 we may remove
1, 2 or 4 sticks, but in state 7 the only 1, 2 or 4 sticks, but in state 7 the only
allowed move is to remove 1 stick. allowed move is to remove 1 stick.
The following picture shows the states The following picture shows the states
$1 \ldots 9$ of the game as a \key{state graph}, $1 \ldots 9$ of the game as a \key{state graph},
where nodes are states and edges are moves between them: whose nodes are the states and edges are the moves between them:
\begin{center} \begin{center}
\begin{tikzpicture}[scale=0.9] \begin{tikzpicture}[scale=0.9]
@ -221,7 +219,7 @@ and all odd-numbered states are losing states.
The \key{nim game} is a simple game that The \key{nim game} is a simple game that
has an important role in game theory, has an important role in game theory,
because many games can be played using because many other games can be played using
the same strategy. the same strategy.
First, we focus on nim, First, we focus on nim,
and then we generalize the strategy and then we generalize the strategy
@ -241,17 +239,17 @@ where $x_k$ denotes the number of sticks in heap $k$.
For example, $[10,12,5]$ is a game where For example, $[10,12,5]$ is a game where
there are three heaps with 10, 12 and 5 sticks. there are three heaps with 10, 12 and 5 sticks.
The state $[0,0,\ldots,0]$ is a losing state, The state $[0,0,\ldots,0]$ is a losing state,
because it's not possible to remove any sticks, because it is not possible to remove any sticks,
and this is always the final state. and this is always the final state.
\subsubsection{Analysis} \subsubsection{Analysis}
\index{nim sum} \index{nim sum}
It turns out that we can easily find out It turns out that we can easily classify
the type of any nim state by calculating any nim state by calculating
a \key{nim sum} $x_1 \oplus x_2 \oplus \cdots \oplus x_n$, the \key{nim sum} $x_1 \oplus x_2 \oplus \cdots \oplus x_n$,
where $\oplus$ is the xor operation. where $\oplus$ is the xor operation.
The states with nim sum 0 are losing states, The states whose nim sum is 0 are losing states,
and all other states are winning states. and all other states are winning states.
For example, the nim sum for For example, the nim sum for
$[10,12,5]$ is $10 \oplus 12 \oplus 5 = 3$, $[10,12,5]$ is $10 \oplus 12 \oplus 5 = 3$,
@ -280,12 +278,12 @@ In this case, we can remove sticks from
heap $k$ so that it will contain $x_k \oplus s$ sticks, heap $k$ so that it will contain $x_k \oplus s$ sticks,
which will lead to a losing state. which will lead to a losing state.
There is always such a heap, where $x_k$ There is always such a heap, where $x_k$
has a one bit in position of the leftmost has a one bit at the position of the leftmost
one bit in $s$. one bit in $s$.
~\\ ~\\
\noindent \noindent
As an example, let's consider the state $[10,2,5]$. As an example, consider the state $[10,2,5]$.
This state is a winning state, This state is a winning state,
because its nim sum is 3. because its nim sum is 3.
Thus, there has to be a move which Thus, there has to be a move which
@ -308,7 +306,7 @@ The nim sum of the state is as follows:
In this case, the heap with 10 sticks In this case, the heap with 10 sticks
is the only heap that has a one bit is the only heap that has a one bit
in the position of the leftmost at the position of the leftmost
one bit in the nim sum: one bit in the nim sum:
\begin{center} \begin{center}
@ -350,18 +348,19 @@ optimally played almost like the standard nim game.
The idea is to first play the misère game The idea is to first play the misère game
like a standard game, but change the strategy like a standard game, but change the strategy
at the end of the game. at the end of the game.
The new strategy will be used when after the next move, The new strategy will be introduced in a situation
each heap would contain at most one stick. where each heap would contain at most one stick
after the next move.
In the standard game, we should choose a move In the standard game, we should choose a move
after which there is an even number of heaps with one stick. after which there is an even number of heaps with one stick.
However, in the misère game, we choose a move so that However, in the misère game, we choose a move so that
there is an odd number of heaps with one stick. there is an odd number of heaps with one stick.
This strategy works because the state where the This strategy works because a state where the
strategy changes always appears in a game, strategy changes always appears in the game,
and this state is a winning state, because and this state is a winning state, because
it contains exactly one heap that has more than one stick, it contains exactly one heap that has more than one stick
so the nim sum is not 0. so the nim sum is not 0.
\section{Sprague–Grundy theorem} \section{Sprague–Grundy theorem}
@ -369,23 +368,23 @@ so the nim sum is not 0.
\index{Sprague–Grundy theorem} \index{Sprague–Grundy theorem}
The \key{Sprague–Grundy theorem} generalizes the The \key{Sprague–Grundy theorem} generalizes the
strategy used in nim for all games that fulfil strategy used in nim to all games that fulfil
the following requirements: the following requirements:
\begin{itemize}[noitemsep] \begin{itemize}[noitemsep]
\item There are two players who move alternatively. \item There are two players who move alternatively.
\item The game consists of states, and the possible moves \item The game consists of states, and the possible moves
in a state don't depend on whose turn it is. in a state do not depend on whose turn it is.
\item The game ends when a player can't make a move. \item The game ends when a player cannot make a move.
\item The game surely ends sooner or later. \item The game surely ends sooner or later.
\item The players have complete information about \item The players have complete information about
states and moves, and there is no randomness in the game. the states and allowed moves, and there is no randomness in the game.
\end{itemize} \end{itemize}
The idea is to calculate for each game state The idea is to calculate for each game state
a Grundy number that corresponds to the number of a Grundy number that corresponds to the number of
sticks in a nim heap. sticks in a nim heap.
When we know the Grundy numbers for all states, When we know the Grundy numbers for all states,
we can play the game like a nim game. we can play the game like the nim game.
\subsubsection{Grundy number} \subsubsection{Grundy number}
@ -395,7 +394,7 @@ we can play the game like a nim game.
The \key{Grundy number} for a game state is The \key{Grundy number} for a game state is
\[\textrm{mex}(\{g_1,g_2,\ldots,g_n\}),\] \[\textrm{mex}(\{g_1,g_2,\ldots,g_n\}),\]
where $g_1,g_2,\ldots,g_n$ are Grundy numbers for where $g_1,g_2,\ldots,g_n$ are Grundy numbers for
states to which we can move from the current state, states to which we can move from the state,
and the mex function returns the smallest and the mex function returns the smallest
nonnegative number that is not in the set. nonnegative number that is not in the set.
For example, $\textrm{mex}(\{0,1,3\})=2$. For example, $\textrm{mex}(\{0,1,3\})=2$.
@ -447,8 +446,8 @@ The Grundy number of a losing state is 0,
and the Grundy number of a winning state is and the Grundy number of a winning state is
a positive number. a positive number.
The Grundy number corresponds to a number of sticks The Grundy number of a state corresponds to
in a nim heap. a number of sticks in a nim heap.
If the Grundy number is 0, we can only move to If the Grundy number is 0, we can only move to
states whose Grundy numbers are positive, states whose Grundy numbers are positive,
and if the Grundy number is $x>0$, we can move and if the Grundy number is $x>0$, we can move
@ -457,12 +456,12 @@ $0,1,\ldots,x-1$.
~\\ ~\\
\noindent \noindent
As an example, let's consider a game where As an example, let us consider a game where
the players move a figure in a maze. the players move a figure in a maze.
Each square in the maze is either floor or wall. Each square in the maze is either floor or wall.
On each turn, the player has to move On each turn, the player has to move
the figure some number the figure some number
of steps either left or up. of steps left or up.
The winner of the game is the player who The winner of the game is the player who
makes the last move. makes the last move.
@ -497,7 +496,7 @@ denotes a square where it can move.
The states of the game are all floor squares The states of the game are all floor squares
in the maze. in the maze.
In this case, the Grundy numbers In this situation, the Grundy numbers
are as follows: are as follows:
\begin{center} \begin{center}
@ -544,7 +543,7 @@ are as follows:
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
Thus, a state in the maze game Thus, each state of the maze game
corresponds to a heap in the nim game. corresponds to a heap in the nim game.
For example, the Grundy number for For example, the Grundy number for
the lower-right square is 2, the lower-right square is 2,
@ -564,22 +563,22 @@ to escape from a losing state.
\subsubsection{Subgames} \subsubsection{Subgames}
Next we will assume that the game consists Next we will assume that our game consists
of subgames, and on each turn, the player of subgames, and on each turn, the player
first chooses a subgame and then a move in the subgame. first chooses a subgame and then a move in the subgame.
The game ends when it's not possible to make any move The game ends when it is not possible to make any move
in any subgame. in any subgame.
In this case, the Grundy number of a game In this case, the Grundy number of a game
is the nim sum of the Grundy numbers of the subgames. is the nim sum of the Grundy numbers of the subgames.
The game can be played like a nim game by calculating The game can be played like a nim game by calculating
all Grundy numbers for subgames, and then their nim sum. all Grundy numbers for subgames and then their nim sum.
~\\ ~\\
\noindent \noindent
As an example, let's consider a game that consists As an example, consider a game that consists
of three mazes. of three mazes.
In thsi game, on each turn the player chooses one In this game, on each turn the player chooses one
of the mazes and then moves the figure in the maze. of the mazes and then moves the figure in the maze.
Assume that the initial state of the game is as follows: Assume that the initial state of the game is as follows:
@ -755,7 +754,7 @@ In the initial state, the nim sum of the Grundy numbers
is $2 \oplus 3 \oplus 3 = 2$, so is $2 \oplus 3 \oplus 3 = 2$, so
the first player can win the game. the first player can win the game.
An optimal move is to move two steps up An optimal move is to move two steps up
in the left maze, which produces the nim sum in the first maze, which produces the nim sum
$0 \oplus 3 \oplus 3 = 0$. $0 \oplus 3 \oplus 3 = 0$.
\subsubsection{Grundy's game} \subsubsection{Grundy's game}
@ -775,8 +774,8 @@ Grundy numbers $a_{k,1},a_{k,2},\ldots,a_{k,m}$.
An example of such a game is \key{Grundy's game}. An example of such a game is \key{Grundy's game}.
Initially, there is a single heap that contains $n$ sticks. Initially, there is a single heap that contains $n$ sticks.
On each turn, the player chooses a heap and divides On each turn, the player chooses a heap and divides
it into two nonempty heaps such that the numbers of it into two nonempty heaps such that the heaps
sticks in the heaps are different. are of different size.
The player who makes the last move wins the game. The player who makes the last move wins the game.
Let $f(n)$ be the Grundy number of a heap Let $f(n)$ be the Grundy number of a heap
@ -785,11 +784,11 @@ The Grundy number can be calculated by going
through all ways to divide the heap into through all ways to divide the heap into
two heaps. two heaps.
For example, when $n=8$, the possibilities For example, when $n=8$, the possibilities
are $1+7$, $2+6$ ja $3+5$, so are $1+7$, $2+6$ and $3+5$, so
\[f(8)=\textrm{mex}(\{f(1) \oplus f(7), f(2) \oplus f(6), f(3) \oplus f(5)\}).\] \[f(8)=\textrm{mex}(\{f(1) \oplus f(7), f(2) \oplus f(6), f(3) \oplus f(5)\}).\]
In this game, the value $f(n)$ is based on values In this game, the value of $f(n)$ is based on values
$f(1),\ldots,f(n-1)$. of $f(1),\ldots,f(n-1)$.
The base cases are $f(1)=f(2)=0$, The base cases are $f(1)=f(2)=0$,
because it is not possible to divide heaps because it is not possible to divide heaps
of 1 and 2 sticks. of 1 and 2 sticks.
@ -807,7 +806,7 @@ f(8) & = & 2 \\
\end{array} \end{array}
\] \]
The Grundy number for $n=8$ is 2, The Grundy number for $n=8$ is 2,
so it's possible to win the game. so it is possible to win the game.
The winning move is to create heaps The winning move is to create heaps
$1+7$, because $f(1) \oplus f(7) = 0$. $1+7$, because $f(1) \oplus f(7) = 0$.