Impartial achievement and avoidance games for generating finite groups

In this talk, we will explore two impartial games introduced by Anderson and Harary. Both games are played by two players who alternately select previously unselected elements of a finite group. The first player who builds a generating set from the jointly selected elements wins the first game (achievement). The first player who cannot select an element without building a generating set loses the second game (avoidance). After the development of some general results, we determine the nim-numbers of both games for abelian and dihedral groups. In addition, we present a criteria on the maximal subgroups that determines the nim-numbers of avoidance games. Lastly, we apply our criteria to compute the nim-numbers of avoidance games for several families of groups, including nilpotent, generalized dihedral, generalized quaternion, and Coxeter groups. This is joint work with Bret Benesh and Nandor Sieben.

This talk was given on April 7, 2015 in the Northern Arizona University Department of Mathematics and Statistics Colloquium.

Transcript

1. impartial achievement & avoidance games for
   generating finite groups
   NAU Department of Mathematics & Statistics Colloquium
   Dana C. Ernst
   Northern Arizona University
   April 7, 2015
   Joint work with Bret Benesh and Nándor Sieben
   

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2. combinatorial game theory
   Intuitive Definition
   Combinatorial Game Theory (CGT) is the study of two-person games
   satisfying:
   ∙ Two players alternate making moves.
   ∙ No hidden information.
   ∙ No random moves.
   Note
   CGT should not be confused with the branch of economics called
   game theory.
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3. combinatorial game theory
   Combinatorial games
   ∙ Chess
   ∙ Go
   ∙ Connect Four
   ∙ Nim
   ∙ Tic-Tac-Toe
   ∙ X-Only Tic-Tac-Toe
   Non-combinatorial games
   ∙ Battleship (hidden information)
   ∙ Rock-Paper-Scissors (non-alternating and random)
   ∙ Poker (hidden information and random)
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4. impartial vs partizan
   Definition
   A combinatorial game is called impartial if the move options are the
   same for both players. Otherwise, the game is called partizan.
   Partizan
   ∙ Chess
   ∙ Go
   ∙ Connect Four
   ∙ Tic-Tac-Toe
   Impartial
   ∙ Nim
   ∙ X-Only Tic-Tac-Toe
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5. our setup
   Comments
   ∙ We are interested in impartial games.
   ∙ We will require that game sequence is finite and there are no ties.
   ∙ When analyzing games, we will assume that both players make
   optimal moves.
   ∙ Player that moves first is called α and second player is called β.
   ∙ Normal Play: The last player to move wins.
   ∙ Misère Play: The last player to move loses.
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6. nim
   Single-pile Nim
   Start with a pile of n stones. Each player
   chooses at least one stone from the pile. The
   player that takes the last stone wins. Game is
   denoted ∗n (called a nimber).
   .
   .
   .
   n
   Question
   Is there an optimal strategy for either player?
   Answer
   Boring: α always wins; just take the whole pile.
   Multi-pile Nim
   Start with k piles consisting of n1, . . . , nk
   stones, respectively. Each
   player chooses at least one stone from a single pile. The player that
   takes the last stone wins. Denoted ∗n1 + · · · + ∗nk
   .
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7. nim
   Example
   Let’s play ∗1 + ∗2 + ∗2. Here’s a possible sequence.
   (1, 2, 2)
   α
   →
   (0, 2, 2)
   β
   →
   (0, 1, 2)
   α
   →
   (0, 1, 1)
   β
   →
   (0, 1, 0)
   α
   → Yay!
   (0, 0, 0)
   In this case, α wins.
   Question
   Is there an optimal strategy for either player?
   Answer
   Short answer is yes: whittle down to an even number of piles with a
   single stone. If players make optimal moves, this is only possible for
   one of the players.
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8. x-only tic-tac-toe
   Start with a single ordinary Tic-Tac-Toe board. Place a single X in any
   empty square. The first player to get 3 in a row wins.
   Example
   α
   →
   X
   β
   →
   X
   X
   α
   →
   X
   X
   X
   β
   →
   X
   X
   X X
   α
   →
   X X
   X
   X X
   Boom, α wins.
   Optimal Play
   If α plays in the middle square, the game is over quickly.
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9. x-only tic-tac-toe
   Multi-Board X-Only Tic-Tac-Toe
   Suppose there are k boards.
   ∙ Normal Play: Players place an X in any open square on a single
   board. Once a board has 3 in row, that board is removed from play.
   The player that gets 3 in a row on the last remaining board wins.
   ∙ Notakto: In the misère version, the player to get three in a row on
   the last remaining board loses.
   Optimal Play
   ∙ For Notakto, a complete analysis involves an 18 element
   commutative monoid. Check out “The Secrets of Notakto: Winning
   at X-only Tic-Tac-Toe” at http://arxiv.org/abs/1301.1672.
   ∙ Also, check out the free iPad app called Notakto.
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10. increasing rigor
    Definition
    An impartial game is a finite set X of positions together with a
    starting position and a collection
    {Opt(Q) ⊆ X | Q ∈ X}
    of possible options. Two players take turns choosing a single
    available option in Opt(Q) of current position Q. Player who
    encounters empty option set cannot move and loses. Note that
    positions are games in there own right.
    Definition
    Given a position, two possible states:
    ∙ P-position: previous player (player that just moved) wins
    ∙ N-position: next player (player that is about to move) wins
    From perspective of the player that is about to move, a P-position is
    a losing position while an N-position is a winning position.
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11. p-position vs n-position
    Examples
    ∙ ∗n is an N-position
    ∙ ∗1 + ∗1 is a P-position
    ∙ ∗1 + ∗2 is an N-position
    ∙ Empty X-Only Tic-Tac-Toe board is an N-position
    ∙ X-Only Tic-Tac-Toe board with X in middle is an P-position
    ∙ Two empty X-Only Tic-Tac-Toe boards is an P-position
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12. game sums
    Definition
    If G and H are games, then G + H is the game where each player
    makes a move in one of the games. Set of options:
    Opt(G + H) := {Q + H | Q ∈ Opt(G)} ∪ {G + S | S ∈ Opt(H)}
    Theorem
    G + G is a P-position.
    Proof
    Copy cat.
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13. game equivalence
    Definition
    G1 = G2
    if G1 + G2
    is a P-position.
    Intuition: something akin to “copy cat” works.
    Examples
    ∙ ∗1 + ∗1 = ∗0 since ∗1 + ∗1 + ∗0 is a P-position.
    ∙ ∗1 + ∗2 = ∗3 since ∗1 + ∗2 + ∗3 is a P-position.
    Theorem
    G1 = G2
    iff G1 + H and G2 + H have the same outcome for all H.
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14. minimum excludant
    Definition
    If A is a set of ordinals, then mex(A) is the smallest ordinal not in A.
    Examples
    ∙ mex({0, 1, 2, 4, 5}) = 3
    ∙ mex({1, 3}) = 0
    ∙ mex({0, 1}) = 2
    ∙ mex(∅) = 0
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15. nim-number of a game
    Definition
    If G is a game, then
    nim(G) := mex({nim(Q) | Q ∈ Opt(G)}).
    This is a recursive definition. We start computing with terminal
    positions (empty option set).
    Examples
    ∙ nim(∗0) = mex(∅) = 0
    ∙ nim(∗1) = mex({nim(∗0)}) = mex({0}) = 1
    ∙ nim(∗2) = mex({nim(∗0), nim(∗1)}) = mex({0, 1}) = 2
    ∙ nim(∗n) = n
    ∙ nim(∗1 + ∗1) = mex({nim(∗1)}) = mex({1}) = 0
    ∙ nim(∗1 + ∗2) = mex({nim(∗2), nim(∗1), nim(∗1 + ∗1)})
    = mex({2, 1, 0}) = 3
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16. sprague–grundy theorem
    Theorem
    Every game is equivalent to a single Nim pile:
    G = ∗ nim(G).
    Examples
    ∙ ∗1 + ∗1 = ∗ nim(∗1 + ∗1) = ∗0
    ∙ ∗1 + ∗2 = ∗ nim(∗1 + ∗2) = ∗3
    Theorem
    β wins G iff G = ∗0.
    Big Picture
    Fundamental problem in the theory of impartial combinatorial
    games is the determination of the nim-number of the game.
    We can think of nim-numbers as “isomorphism” classes of games.
    15
    

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17. achievement & avoidance games on finite groups
    Let G be a finite nontrivial group. Players take turns picking a new
    element gi ∈ G.
    Generate game (achievement)
    GEN(G): First player to build ⟨g1, g2, . . . , gi⟩ = G wins.
    Do Not Generate game (avoidance)
    DNG(G): The position ⟨g1, g2, . . . , gi⟩ = G is not allowed. Terminal
    positions are the maximal subgroups of G.
    Both games introduced by F. Harary in 1987.
    16
    

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18. gen for cyclic groups
    Example
    GEN(Zn) is boring. Let’s look at
    GEN(Z6) with non-optimal play.
    Choice Set Generated
    a4 {a4, a2, e}
    a3 {a4, a2, e, a3, a, a5}
    In this case, β wins. However, α
    could have won immediately by
    choosing a or a5.
    Optimal play for Zn
    α can always win GEN(Zn) in one move by choosing any ak, where n
    and k are relatively prime.
    17
    

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19. gen for dihedral groups
    Example
    Let’s look at GEN(D4).
    Choice Set Generated
    r2 {r2, e}
    e {r2, e}
    f {r2, e, f, r2f}
    r D4
    β wins!
    18
    

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20. dng for arbitrary groups
    Theorem (F.W. Barnes, 1988)
    Let G be any finite nontrivial group. Then α wins DNG(G) iff there is a
    g ∈ G such that
    ∙ ⟨g⟩ has an odd number of elements;
    ∙ ⟨g⟩ ̸= G;
    ∙ ⟨g, h⟩ = G for all non-identity elements h ∈ G such that h2 = e.
    Otherwise, β wins.
    Corollary
    ∙ α wins DNG(Zn) iff n is odd or not a multiple of 4. Otherwise, β
    wins.
    ∙ α wins DNG(Dn) iff n odd. Otherwise, β wins.
    19
    

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21. exploring nim-numbers
    ∅
    ∗0
    {0}
    ∗1
    {2}
    ∗1
    {0, 2}
    ∗0
    ∅
    ∗1
    {1}
    ∗0
    {0}
    ∗2
    {2}
    ∗2
    {3}
    ∗0
    {0, 1}
    ∗0
    {0, 2}
    ∗1
    {0, 3}
    ∗0
    {1, 2}
    ∗0
    {2, 3}
    ∗0
    {0, 1, 2}
    ∗0
    {0, 2, 3}
    ∗0
    DNG(Z4) = ∗0 GEN(Z4) = ∗1
    20
    

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22. nim-numbers for cyclic groups
    Theorem (Ernst, Sieben)
    If n ≥ 2, then nim(GEN(Zn)) = nim(DNG(Zn)) + 1.
    Theorem (Ernst, Sieben)
    If n ≥ 2, then
    DNG(Zn) =
    
    
    
    
    
    
    
    
    
    
    
    
    
    ∗1, n = 2
    ∗1, n ≡2
    1
    ∗0, n ≡4
    0
    ∗3, n ≡4
    2
    and
    GEN(Zn) =
    
    
    
    
    
    
    
    
    
    
    
    
    
    ∗2, n = 2
    ∗2, n ≡2
    1
    ∗1, n ≡4
    0
    ∗4, n ≡4
    2
    21
    

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23. nim-numbers for dihedral groups
    Theorem (Ernst, Sieben)
    For n ≥ 3, we have
    DNG(Dn) =
    {
    ∗3, n ≡2
    1
    ∗0, n ≡2
    0
    and
    GEN(Dn) =
    
    
    
    
    
    
    
    ∗3, n ≡2
    1
    ∗0, n ≡4
    0
    ∗1, n ≡4
    2
    22
    

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24. nim-numbers for abelian groups
    Theorem (Ernst, Sieben)
    If G is a finite nontrivial abelian group, then
    DNG(G) =
    
    
    
    
    
    
    
    
    
    
    
    
    
    ∗1, G is nontrivial of odd order
    ∗1, G ∼
    = Z2
    ∗3, G ∼
    = Z2 × Z2k+1
    with k ≥ 1
    ∗0, else
    GEN(G) =
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    ∗2, |G| is odd and d(G) ≤ 2
    ∗1, |G| is odd and d(G) ≥ 3
    ∗2, G ∼
    = Z2
    ∗1, G ∼
    = Z4k
    with k ≥ 1
    ∗4, G ∼
    = Z4k+2
    with k ≥ 1
    ∗1, G ∼
    = Z2 × Z2 × Zm × Zk
    for m, k odd
    ∗0, else
    23
    

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25. general results for dng
    Theorem (Ernst, Sieben)
    ∙ If G is any finite nontrivial group, then DNG(G) is ∗0, ∗1, or ∗3.
    ∙ If |G| is odd and nontrivial, then GEN(G) is ∗1 or ∗2.
    Conjecture
    If |G| is even, then GEN(G) is one of ∗0, ∗1, ∗2, ∗3, ∗4.
    24
    

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26. general results for dng
    Theorem (Benesh, Ernst, Sieben)
    Let G be a finite nontrivial group.
    ∙ If |G| = 2, then DNG(G) = ∗1.
    ∙ If |G| is odd, then DNG(G) = ∗1.
    ∙ If |Φ(G)| is even, then DNG(G) = ∗0.
    ∙ If every maximal subgroup of G is even, then DNG(G) = ∗0.
    ∙ If the even maximal subgroups cover G, then DNG(G) = ∗0.
    ∙ Otherwise, DNG(G) = ∗3.
    Using our “checklist” criteria, we have completely characterized DNG
    for nilpotent, generalized dihedral, generalized quaternion, Coxeter
    (includes symmetric groups), alternating, and some Rubik’s cube
    groups.
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27. future work
    Open questions
    ∙ What is spectrum of GEN(G)?
    ∙ What is GEN(G) for generalized dihedral? (In progress)
    ∙ Are there nice results for products and quotients? (In progress)
    ∙ Is it possible to characterize the nim-numbers of GEN(G) in terms
    of covering conditions by maximal subgroups similar to what we
    did for DNG(G)? (In progress)
    Thanks!
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