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Table of Contents (1)
• What is Game Theory?
• Types of combinatorial games and plays• Impartial, Partisan• Normal & misère play
• Game States• Turns; P, N positions• Optimal play
• "Scoring" game example
3
Table of Contents (2)
• The Game Nim• Gameplay, winning positions
• Composite games• Tweedledum & Tweedledee principle• Sprague-Grundy Theorem
• Calculating Grundy Numbers• Sprague-Grundy function
5
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
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
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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
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Combinatorial Games
• Definition of a combinatorial game
• 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
• Two players move alternatively• No chance devices• Perfect information• The game must eventually end• Winner depends on who moves last – no draws
10
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
11
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 wins
• Misère play• Player which makes last possible move loses• Aim to force the other player into the last move
13
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• In other words, good for next or for previous player
14
Game States - Turns
• Impartial games are played in turns• Positions are usually the "thinks" stage
• Impartial game states – positions & count of chips
Blue "thinks"
Yellow"thinks"
Blue"thinks"
Yellow"thinks"
Blue moves
Yellowmoves
Bluemoves
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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
• 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
P N N N P N N N P N
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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
23
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
24
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 – 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
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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
28
Nim – General Observations
• We could go on generating positions• That would take too much (exponential) time
• Another pattern can be noticeda) Even same-sized heap count – bad positionb) Odd same-sized heap count – good positionc) 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?
29
Nim-Sum
• Nim-sum (aka XOR, ^, addition modulo two)• The Nim-sum of a position (n0, …, nk) isn0 ^ n1 ^ … ^ nk
• 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
A non-zero Nim-sum denotes a winning positionA zero Nim-sum denotes a losing position
30
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
31
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-sumFinds 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 bitNullify that number NCalculate the Nim-sum againSet the nullified number N to the new Num-sum
33
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:
• I.e. every impartial game is a Nim in disguise or some variant of it
Every impartial game, under normal play is equivalent to a nimber
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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
• Seem familiar?
0 1 2 3 4 5 6 7 8 9
35
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
36
More General Nim?
• In standard Nim, we can take any number of elements from 1 pile
• We can generalize Nim for K piles• Take any number of elements from at least 1 pile and at most K piles
• This is called Moore's Nim• More formally, for K piles it is Nimk
• So normal Nim is actually Nim1
• Can you think of a way to edit the Nim solution to work for Moore's Num?
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More General Nim (Moore's Nim) – Solution
• We need to change the Nim-sum operator
• For Nim1 (Standard Nim) we convert piles to binary and use "sum modulo 2"
• So for Nimk we again convert piles to binary, but instead use "sum modulo k+1"• E.g. if we can take from at most 2 piles, Nim-sum is "sum modulo 3"
• The remaining part of the solution is the same• Positions with Nim-sum > 0 are winning• Positions with Nim-sum < 0 are losing
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Composite Games
• Generally, composite games can be broken down into subgames
• Usually composite games can be formed from any game• By adding more chips
• Examples:• Game of Scoring with N Chips• Nim with N groups of piles• Etc.
40
Game of Scoring – Demo 2
• Let’s play Scoring again• This time with 2 chips• Max step is 3• Are the marked P/N 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
41
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
42
Composite Games – Solving
• Composite impartial games• Equivalent to nimbers (as other impartial games)
• 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?
44
Calculating SG function
• The nimbers we needed to describe a composite game's state can be calculated with the Sprague Grundy function• Yep, those guys again
• Sprague-Grundy (SG) function• Recursively generates "Grundy" values/numbers• Numbers correspond to heap sizes in Nim
• SG function a has "helper" functions to adapt to a specific game• The "Follower" function
45
Calculating SG function
• Sprague-Grundy function – formal description:
• g – Sprague-Grundy function• F – follower function• Mex – minimal excludant• p and q – two game positions
g(p) = Mex(g(q)); q ∈ F(p)
46
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
47
Calculating SG function
• So, thisreads 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
g(p) = Mex(g(q)); q ∈ F(p)
48
Calculating SG – example
• Grundy values with Scoring• Losing
positions ina game haveg() = 0
g(0) = 0, by definition.g(1) = 1 since g(F(1)) = {0}.g(2) = 2 since g(F(2)) = {0, 1}.g(3) = 3 since g(F(3)) = {0, 1, 2}.
g(4) = 0 since g(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 g(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
49
Calculating SG – pseudocode
• Grundy values pseudocode• Walk positions
recursively• Generate set
of Grundy values for followers
• Take the Mex of the Grundyvalues of the followers
int GrundyNumber(position pos) {
moves[] = Followers(pos);set s; for (all x in moves){
s.insert(GrundyNumber(x));}//Mex – return smallest non-zero //integer, not in s; int mexCandidate=0; while (s.contains(mexCandidate)) {
mexCandidate++; }return mexCandidate;
}
50
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
51
Solving 2-Chip Scoring with Sprague-Grundy and Nim
• Say we have the following position, with SG numbers below
• We have to 2 chips at position 7 and g(7) = 3
• So, this is equivalent to a Nim with heaps (g(7), g(7)) = (3, 3)• which is a losing position because 3 xor 3 = 0
0 1 2 3 4 5 6 7 8 9
0 1 2 3 0 1 2 3 0 1
53
Northcott's Game – (click to play on cut-the-knot.org)
• Rectangular board, 2 checkers per row, 1 per player
• Move rule: move exactly 1 checker anywhere on it's row, without jumping over an opponent's checker
• Lose rule: first player which cannot move loses
• Video: Solution of a task with this game @ Telerik Algo Academy
1 2 3 4 5 6 7 8 9 10 11123456
54
Corner the Lady – (click solution on cut-the-knot.org)
• Rectangular board, 1 checker, players take turns moving
• Move rule: move left, down or left-down diagonallyby any amount of cells. You can't move from the bottom-left position (1, 1)
• Lose rule: first player which cannot move loses
* This is actually a "visual adaptation" version of Wythoff's Nim – read the description at the cut-the-knot link above
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9
55
Other Impartial Combinatorial Games
• Northcott's Game with moving chips in more than 1 row• Spolier: This is a variant of Moore's Nim
• Corner the Lady with more than 1 chip• Spoiler: Composite game – calculate SG values and Nim-sum to solve
• Dots and Boxes, with "first player to make a box wins" rule• See this article on composite mathematical games at Oxford
• The Knights game in this TopCoder article
• A lot of games on www.cut-the-knot.org
56
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
57
Conclusion
• What we haven’t covered• Misère play• Partisan 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.