Transcript Game Theory

Game Theory
Impartial Games, Nim, Composite Games,
Optimal Play in Impartial games
Telerik Software Academy
Learning & Development Team
http://academy.telerik.com
Table of Contents (1)
1.
What is Game Theory?
2.
Types of combinatorial games and plays
 Impartial, Partisan
 Normal & misère play
3.
Game States
 Turns; P, N positions
 Optimal play
 "Scoring" game example
2
Table of Contents (2)
3.
The Game Nim
 Gameplay, winning positions
4.
Composite games
 Tweedledum & Tweedledee principle
 Sprague-Grundy Theorem
5.
Calculating Grundy Numbers
 Minimal excludant
3
WTF Game Theory?
What are The Facts on Game Theory
What is Game Theory?
 Study of decision making
 Mathematical models
 Of conflict
 Of cooperation
 Between intelligent decision makers
 Started with study of zero-sum games
 One player winning leads to other player losing
 Evolved into "decision theory"
 General decision making strategies
5
Game Theory – Concepts
 Players, Actions, Payoffs,
Information (PAPI)
 Key concepts to describing a game
 Solution strategy is based on PAPI
 Solution strategy + PAPI = predictable,
deterministic outcomes to games
 Game theory fields of application
 Politics, Economics
 Phycology, Biology, Logic
 All fields needing info on behavioral relations
6
Game Theory – Game Types
 Notable general game classifications
 Perfect vs. imperfect information
 Cooperative vs. competitive
 Symmetric vs. asymmetric
 Combinatorial
 Infinitely long
 Discrete vs. continuous, differential,
population, stochastic, metagames…
7
Combinatorial Games
Impartial, Partizan and Play Types
Combinatorial Games

Definition of a combinatorial game
•
•
•
•
•

Two players move alternatively
No chance devices
Perfect information
The game must eventually end
Winner depends on who moves last – no draws
Perfect information – players know at any time:
 Game state
 All possible player moves

No chance devices
 Actions are not random and do not depend on
random events
Combinatorial Game Examples

Examples of combinatorial games
 Chess (partly)
 Hex 7
 Checkers
 Take-away & Subtraction games
 Silver-dollar game
Partisan vs. Impartial

Partisan games
 Different moves available to different players
 Chess, Checkers, Tic-Tac-Toe… are partisan

Impartial games
 Different players have the same moves
 Examples – Nim, Quarto

Impartial games can be generalized and analyzed
 Very strong base for studying other games
 Many partisan games, but with similar principles
Combinatorial Games

Play types – determine which player wins
 Based on who moves last

Normal play
 Player which makes last possible move wins
 E.g. player who takes the last coin and wins

Misère play
 Player which makes last possible move loses
 Aim to force the other player into the last move
12
Game States & Positions
Turns, P and N Positions in Optimal Play
Game States – Chips

Game “chips” (or “pieces”)
 A chip is something a player controls
 E.g. figures in chess, stack(s) of coins, numbers
to operate on, etc.

Chips are always in some state (i.e. position)
 A position can be winning or losing
 Or, good for next or for previous player
14
Game States – Turns

Impartial games are played in turns
 Turns can be broken down like so:
Blue
moves
Blue
"thinks"
Yellow
"thinks"
Yellow
moves
Blue
moves
Blue
"thinks"
Yellow
"thinks"
 Positions are usually the "thinks" stage

Impartial game states – positions & count of chips
Game States – Player Position

Position of a player
 Depends on the positions of all the chips
 Considered in the "think" stage
 Immediately after the other player moved
 Immediately before the current player moves

Two types of positions
 Good for current player (i.e. next to move)
 Good for previous player (i.e. who just moved)
 Usually noted P (previous) and N (next/current)
16
Game States – Optimal Play

Impartial games are deterministic
 Optimal play always exists
 Player position – either winning or losing

Optimal play in impartial games
 Best actions by player, in order to win
 Optimal play + winning position = victory

So, N and P positions for the current player:
 N – Winning positions
 P – Losing position
17
How to Play Optimally

Which positions to choose to win?
 Impartial games are zero-sum
 Positions are either winning or losing
 Try to force other player into losing position
 i.e. a P-position (good for previous player -> us)
 If we are in a losing position, we can’t win
 if other player plays optimally
18
Game of Scoring

Scoring is a game where a single chip is moved
 Right to left, starting at a numbered position
 Moves have maximal step – e.g. 3 positions
 Move chip by no less than 1 and no more than 3
 0 (zero) is the leftmost position
 Can’t move from that position
 Player which cannot move is the loser
19
Game of Scoring – Demo

Let’s try to determine P and N positions
 Max step is 3
0
1
2
3
4
5
6
7
8
9
?
?
?
?
?
?
?
?
?
?
20
Game of Scoring – Positions


0 – losing (P-position)
1, 2, 3 – winning positions (N-positions)
 We can immediately move the chip to 0

4 – losing position (P-position)
 We place chip at 1, 2 or 3 and other player wins

5, 6, 7 – winning positions (N-positions)
 Can place other player in a losing position (4)

Scoring has a direct formula for P-positions
21
Generalization of Positions

For all impartial games
 A position is winning if it can lead to at least one
losing position
 A position is losing, if it leads only to winning
positions
 This regards single-chip games
P-position
N-position
Every option leads to
an N-position
There is always at least
one option that leads to
a P-position
22
Solving Scoring in C++
Live Demo
The Game Nim
Gameplay, Positions, Nimbers and
Nim-sum
Nim

Ancient, but fundamental game
 Impartial, Many variations
 Nim theory developed early 20th century
 Important related terms – nimbers, Nim-sum

Rules
 Several heaps (piles) of stuff (coins/stones/...)
 Player takes 1 or more elements from 1 pile
 Normal play – player taking last element wins
25
Nim – formal notation

Nim state is easy to express formally
 If the number of heaps is k
 Then (n0, n1, …, nk) is the state
 n0 is the number of items in the first heap, etc.

State is often referred to as position
 If we have 3 heaps of sizes 3, 4 and 2
 Then we are at
position (3, 4, 2)

Position (0) is losing
 In normal play
26
Nim – Demo

Let’s play Nim
index
0
1
2
3
4
5
27
Nim – Observations

What happens for Nim with 1-element heaps?
a) (1) – obviously, first player wins (N)
b) (1, 1) – first player takes a heap, second wins (P)
c) (1, 1, 1) – first player forces second player to case
b), so second player will lose (N)
d) (1, 1, 1, 1) – first player goes into case c), so
second player will win (P)

Nim with k 1-element heaps
 Winning position if k is odd
28
Nim – More Observations

What happens for Nim with 1-element heaps
mixed with 2-element heaps?
a) (1, 2) – first player wins (winning position)
 first player can reduce 2 to 1
 player 2 forced into (1, 1), which is losing
b) (2, 2) – other player wins (losing position)
 First player’s moves are to (1, 2) or (0, 2) *
 First option leads second player to case a)
 Second option – second player takes the heap
*(0, 2) and (2, 1) are also possible, but with the same outcome, so we will not consider them
29
Nim – General Observations

We could go on generating positions
 That would take too much (exponential) time

Another pattern can be noticed
a) Even same-sized heap count – bad position
b) Odd same-sized heap count – good position
c) b) + one larger heap – good position
 can move to position a) by reducing larger heap
d) Other similar even-odd considerations

Can we easily filter through good/bad positions?
30
Nim-sum

Nim-sum (aka XOR, ^, addition modulo two)
 The Nim-sum of a position (n0, …, nk) is
n0 ^ n1 ^ … ^ nk
A non-zero Nim-sum denotes a winning position
A zero Nim-sum denotes a losing position
 Accredited to Charles L. Bouton as part of the
solution of the game
 Mathematical notation typically:
n0 ⊕ n1 ⊕ … ⊕ nk
 Heap sizes are usually called nimbers
31
Nim-sum – Why it Works

Position (0) has a Nim-sum = 0
 No moves can be made – position is losing

Position (k) has a Nim-sum > 0
(k > 0)
 Player takes the entire heap – position is winning

Changes to a position with Nim-sum = 0
 Always lead to positions with Nim-sum > 0

Changes to a position with Nim-sum > 0
 At least one change leading to Nim-sum = 0

So, we can always force a losing position
 From a winning position
32
Nim-sum – Finding Positions

Considering we are in a winning position
 Need to search for positions with Nim-sum zero

Finding a zero Nim-sum position
 Decrease numbers, which make the Nim-sum > 0
 Achieve even number of 1’s in each bit column
Calculate the current Nim-sum
Finds its leftmost bit equal to 1
(denotes a column with an odd number of 1’s)
Find any number N (heap size) having a 1 at that bit
Nullify that number N
Calculate the Nim-sum again
Set the nullified number N to the new Num-sum
33
Solving Nim in C++
Live Demo
Nim Importance

What’s the big deal with Nim?
 The Sprague-Grundy theorem

Who?
 R. P. Sprague & P. M. Grundy in 1935 & 1939
 Independently discovered the theorem

The Sprague-Grundy theorem states:
Every impartial game, under normal play
is equivalent to a nimber
 Layman’s terms: every impartial game is a Nim
in disguise or some variant of it
35
Nim Importance

Consider the game Nimble
 Several coins, at some positions
 Must move exactly one coin at least 1 position
 Coins move only right to left (towards zero)
 Player who moves the last coin to 0 wins
0
1
2
3
4
5
6
7
8
9
 Seem familiar?
36
Nim Importance

How about now?
 The solutions of Nimble and Nim are equivalent
 After indices are turned into heap sizes
index
0
1
2
3
4
5
37
Composite Games
Dividing into Subgames,
Tweedledum & Tweedledee
Principle, Sprague-Grundy
Game of Scoring – Demo

Let’s play Scoring again
 This time with 2 chips
 Max step is 3
 Are the marked P/N (make them win/lose if you
want) positions correct in this case?
0
1
2
3
4
5
6
7
8
9
P
N
N
N
P
N
N
N
P
N
39
Composite Games – 2 Chips

Tweedledum & Tweedledee Principle
 2-chip game, the second player can mimic moves
 If the two chips are in the same position
 Leading to victory for the second player
 In positions which would be winning for the first
player in a single-chip game
 Losing positions remain losing, problem is with
winning ones

Similar cases occur in games with more chips
 And more equivalent positions
40
Composite Games – Solving

Composite impartial games
 Equivalent to nimbers (as other impartial games)
 As per the Sprague-Grundy theorem
 Yep, these guys again

A game has k chips, each with a position,
 Game position – set of k numbers (n0, …, nk)
 Equivalent to a position in Nim, hence
 Nim-sum of the position = 0 -> position is losing

How do we get those numbers/nimbers?
41
Calculating the
Sprague-Grundy Function
Follower positions,
Minimal excludent
Calculating SG function

Sprague-Grundy function
 Recursively generates "Grundy" values/numbers
 Numbers correspond to heap sizes in Nim
 Adapt to specific game through
 Follower function
 Mex function
43
Calculating SG function

Sprague-Grundy function – formal description:
g(p) = Mex(g(q)); q ∈ F(p)
 g – Sprague-Grundy function
 F – follower function
 Mex – minimal excludant
 p and q – two game positions
44
Calculating SG function

Follower function F(position)
 Returns positions, reachable in 1 move from
given position
 i.e. "followers" of that position

Minimal excludant Mex(numbers)
 Returns the minimal non-negative number
 Not belonging to a given set
 i.e. first number different than any of the given
numbers
45
Calculating SG function

So, this g(p) = Mex(g(q)); q ∈ F(p)
reads as follows:
 The Grundy value of position p
 Is the minimal non-negative integer
 Which is NOT a Grundy value of
 Any of the followers of position p
46
Calculating SG – example

Grundy values
with Scoring
 Losing
positions in
a game have
g() = 0
 Here, can’t
move from 0,
so we lose
there and
g(0) = 0
g(0) = 0, by definition.
g(1) = 1 since F(1) = {0}.
g(2) = 2 since F(2) = {0, 1}.
g(3) = 3 since F(3) = {0, 1, 2}.
g(4) = 0 since F(4) = {1, 2, 3}.
- The values of g on F(4) are {1, 2, 3} and
the minimum that does not appear there is 0.
g(5) = 1 since F(5) = {2, 3, 4}.
- The values of g on F(5) are {2, 3, 0} and
the minimum that does not appear there is
1...
Index
0
1
2
3
4
5
6
7
8
9
SG value
0
1
2
3
0
1
2
3
0
1
47
Calculating SG – pseudocode

Grundy values
pseudocode
 Walk positions
recursively
 Generate set
of Grundy
values for
followers
int GrundyNumber(position pos)
{
moves[] = Followers(pos);
set s;
for (all x in moves)
{
insert into s GrundyNumber(x);
}
//Mex – return smallest non-zero
//integer, not in s;
int mexCandidate=0;
while (s.contains(mexCandidate))
{
mexCandidate++;
}
return mexCandidate;
}
 Take the Mex
of the Grundy
values of the followers
48
Solving Composite Games
Using SG & Nim

Determining a position as winning or losing
 We have the positions of the chips in the game
 We have the Grundy values of those positions
 Hence, we have the Nim position:
 (g(p0), g(p1), … g(pk))
 Where p0 … p1 are the chip positions
 So, we can compute the Nim-sum
 If it is zero, we are in a losing position
 If it is non-zero, we are in a winning position
49
Solving
Multi-Chip Scoring
in C++
Live Demo
Conclusion

What we covered
 Impartial games
 Winning and losing positions (N and P)
 Nim and representing impartial games as Nim
 Sprague-Grundy theorem & function
 Solving composite games
51
Conclusion

What we haven’t covered
 Misère play
 Partizan games
 Non-deterministic games

The above share some characteristics with
impartial games
Most games are describable by mathematical models,
which are based on similar concepts to those in the
lecture. Just take your time to unravel the model –
here, programming is the easy part once you’ve
understood the logical base of the problem.
52
Game Theory
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