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Progress and Problems in Misere Play Rebecca Milley General misere play Progress and Problems Restricted misere play Definitions Invertibility in Restricted Misere Play Comparison Reductions The conjugate property Future directions

  1. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  2. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  3. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  4. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property • Less simplification (domination, reversibility). Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  5. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property • Less simplification (domination, reversibility). Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  6. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property • Less simplification (domination, reversibility). Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  7. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property • Less simplification (domination, reversibility). Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  8. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property • Less simplification (domination, reversibility). Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  9. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property • Less simplification (domination, reversibility). Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  10. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property • Less simplification (domination, reversibility). Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  11. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property • Less simplification (domination, reversibility). Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  12. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property • Less simplification (domination, reversibility). Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  13. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property • Less simplification (domination, reversibility). Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  14. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property • Less simplification (domination, reversibility). Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  15. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property • Less simplification (domination, reversibility). Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  16. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property • Less simplification (domination, reversibility). Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  17. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property • Less simplification (domination, reversibility). Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  18. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property • Less simplification (domination, reversibility). Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  19. Progress and Problems General misere play in Misere Play Rebecca Milley • Zero is trivial. General misere play • Only game equal to zero is {·|·} . Proof... Restricted misere play • G − G � = 0 for all G � = 0; i.e., no inverses. Definitions Invertibility • Equality / inequality is rare and difficult to prove. Comparison Reductions • Can’t use “ G = H ⇔ G − H ∈ P ′′ as normal play. The conjugate property • Less simplification (domination, reversibility). Future directions e.g., 0 and 1 = { 0 |·} are incomparable. Rebecca Milley Progress and Problems in Misere Play October 24, 2017 4 / 17

  20. Progress and Problems Restricted misere theory (Plambeck, Seigel) in Misere Play Rebecca Milley General misere play Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 5 / 17

  21. Progress and Problems Restricted misere theory (Plambeck, Seigel) in Misere Play Rebecca Milley • Equality: G = H means General misere play G + X and H + X have same outcome ∀ games X . Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 5 / 17

  22. Progress and Problems Restricted misere theory (Plambeck, Seigel) in Misere Play Rebecca Milley • Equality: G = H means General misere play G + X and H + X have same outcome ∀ games X . Restricted misere play • A universe is a set of games closed under addition, Definitions Invertibility Comparison negation, and options. Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 5 / 17

  23. Progress and Problems Restricted misere theory (Plambeck, Seigel) in Misere Play Rebecca Milley • Equality: G = H means General misere play G + X and H + X have same outcome ∀ games X . Restricted misere play • A universe is a set of games closed under addition, Definitions Invertibility Comparison negation, and options. Let U be a universe. Then: Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 5 / 17

  24. Progress and Problems Restricted misere theory (Plambeck, Seigel) in Misere Play Rebecca Milley • Equality: G = H means General misere play G + X and H + X have same outcome ∀ games X . Restricted misere play • A universe is a set of games closed under addition, Definitions Invertibility Comparison negation, and options. Let U be a universe. Then: Reductions The conjugate property • Equivalence mod U : G ≡ U H means Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 5 / 17

  25. Progress and Problems Restricted misere theory (Plambeck, Seigel) in Misere Play Rebecca Milley • Equality: G = H means General misere play G + X and H + X have same outcome ∀ games X . Restricted misere play • A universe is a set of games closed under addition, Definitions Invertibility Comparison negation, and options. Let U be a universe. Then: Reductions The conjugate property • Equivalence mod U : G ≡ U H means G + X and H + X have same outcome ∀ X ∈ U . Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 5 / 17

  26. Progress and Problems Restricted misere theory (Plambeck, Seigel) in Misere Play Rebecca Milley • Equality: G = H means General misere play G + X and H + X have same outcome ∀ games X . Restricted misere play • A universe is a set of games closed under addition, Definitions Invertibility Comparison negation, and options. Let U be a universe. Then: Reductions The conjugate property • Equivalence mod U : G ≡ U H means G + X and H + X have same outcome ∀ X ∈ U . Future directions • Inequality modulo U : G ≥ U H means Left wins G + X if she wins H + X , ∀ X ∈ U . Rebecca Milley Progress and Problems in Misere Play October 24, 2017 5 / 17

  27. Progress and Problems Restricted misere theory (Plambeck, Seigel) in Misere Play Rebecca Milley • Equality: G = H means General misere play G + X and H + X have same outcome ∀ games X . Restricted misere play • A universe is a set of games closed under addition, Definitions Invertibility Comparison negation, and options. Let U be a universe. Then: Reductions The conjugate property • Equivalence mod U : G ≡ U H means G + X and H + X have same outcome ∀ X ∈ U . Future directions • Inequality modulo U : G ≥ U H means Left wins G + X if she wins H + X , ∀ X ∈ U . • These definitions are natural and useful. Rebecca Milley Progress and Problems in Misere Play October 24, 2017 5 / 17

  28. Progress and Problems Restricted misere theory (Plambeck, Seigel) in Misere Play Rebecca Milley • Equality: G = H means General misere play G + X and H + X have same outcome ∀ games X . Restricted misere play • A universe is a set of games closed under addition, Definitions Invertibility Comparison negation, and options. Let U be a universe. Then: Reductions The conjugate property • Equivalence mod U : G ≡ U H means G + X and H + X have same outcome ∀ X ∈ U . Future directions • Inequality modulo U : G ≥ U H means Left wins G + X if she wins H + X , ∀ X ∈ U . • These definitions are natural and useful. • If we are analyzing some rule set (e.g., Domineering), all of the positions we compare are in the universe of that rule set. Rebecca Milley Progress and Problems in Misere Play October 24, 2017 5 / 17

  29. Progress and Problems Restricted misere theory (Plambeck, Seigel) in Misere Play Rebecca Milley • Equality: G = H means General misere play G + X and H + X have same outcome ∀ games X . Restricted misere play • A universe is a set of games closed under addition, Definitions Invertibility Comparison negation, and options. Let U be a universe. Then: Reductions The conjugate property • Equivalence mod U : G ≡ U H means G + X and H + X have same outcome ∀ X ∈ U . Future directions • Inequality modulo U : G ≥ U H means Left wins G + X if she wins H + X , ∀ X ∈ U . • These definitions are natural and useful. • If we are analyzing some rule set (e.g., Domineering), all of the positions we compare are in the universe of that rule set. • A position may have algebraic structure modulo U that it doesn’t have in general (e.g., invertibility, simplification). Rebecca Milley Progress and Problems in Misere Play October 24, 2017 5 / 17

  30. Progress and Problems Some definitions and nice universes. in Misere Play Rebecca Milley General misere play Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 6 / 17

  31. Progress and Problems Some definitions and nice universes. in Misere Play Rebecca Milley • Dicot : at every point, either both players can move General misere play or neither can. Let D be the universe of dicot games. Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 6 / 17

  32. Progress and Problems Some definitions and nice universes. in Misere Play Rebecca Milley • Dicot : at every point, either both players can move General misere play or neither can. Let D be the universe of dicot games. Restricted misere play • Left end: Left has no move now. Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 6 / 17

  33. Progress and Problems Some definitions and nice universes. in Misere Play Rebecca Milley • Dicot : at every point, either both players can move General misere play or neither can. Let D be the universe of dicot games. Restricted misere play • Left end: Left has no move now. Definitions Invertibility Comparison Dead left end: Left has no move now or later (no Reductions move in any follower). The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 6 / 17

  34. Progress and Problems Some definitions and nice universes. in Misere Play Rebecca Milley • Dicot : at every point, either both players can move General misere play or neither can. Let D be the universe of dicot games. Restricted misere play • Left end: Left has no move now. Definitions Invertibility Comparison Dead left end: Left has no move now or later (no Reductions move in any follower). The conjugate property Dead-ending game: All the end followers are dead Future directions ends. Let E be the universe of dead-ending games. Rebecca Milley Progress and Problems in Misere Play October 24, 2017 6 / 17

  35. Progress and Problems Some definitions and nice universes. in Misere Play Rebecca Milley • Dicot : at every point, either both players can move General misere play or neither can. Let D be the universe of dicot games. Restricted misere play • Left end: Left has no move now. Definitions Invertibility Comparison Dead left end: Left has no move now or later (no Reductions move in any follower). The conjugate property Dead-ending game: All the end followers are dead Future directions ends. Let E be the universe of dead-ending games. Rebecca Milley Progress and Problems in Misere Play October 24, 2017 6 / 17

  36. Progress and Problems Some definitions and nice universes. in Misere Play Rebecca Milley • Dicot : at every point, either both players can move General misere play or neither can. Let D be the universe of dicot games. Restricted misere play • Left end: Left has no move now. Definitions Invertibility Comparison Dead left end: Left has no move now or later (no Reductions move in any follower). The conjugate property Dead-ending game: All the end followers are dead Future directions ends. Let E be the universe of dead-ending games. Rebecca Milley Progress and Problems in Misere Play October 24, 2017 6 / 17

  37. Progress and Problems Some definitions and nice universes. in Misere Play Rebecca Milley • Dicot : at every point, either both players can move General misere play or neither can. Let D be the universe of dicot games. Restricted misere play • Left end: Left has no move now. Definitions Invertibility Comparison Dead left end: Left has no move now or later (no Reductions move in any follower). The conjugate property Dead-ending game: All the end followers are dead Future directions ends. Let E be the universe of dead-ending games. → Domineering, Hackenbush, NoGo, Snort ⊂ E , D ⊂ E . ֒ Rebecca Milley Progress and Problems in Misere Play October 24, 2017 6 / 17

  38. Progress and Problems Invertibility results in Misere Play Rebecca Milley Write G instead of − G , call it the conjugate . General misere play Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 7 / 17

  39. Progress and Problems Invertibility results in Misere Play Rebecca Milley Write G instead of − G , call it the conjugate . General misere play Theorem (Allen 2009) Restricted misere play Definitions ∗ + ∗ ≡ D 0 . Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 7 / 17

  40. Progress and Problems Invertibility results in Misere Play Rebecca Milley Write G instead of − G , call it the conjugate . General misere play Theorem (Allen 2009) Restricted misere play Definitions ∗ + ∗ ≡ D 0 . Invertibility Comparison Reductions Theorem (McKay, Milley, Nowakowski 2012) The conjugate property Future directions If G + G ∈ N , and H + H ∈ N for all followers H of G, then G + G ≡ D 0 . Rebecca Milley Progress and Problems in Misere Play October 24, 2017 7 / 17

  41. Progress and Problems Invertibility results in Misere Play Rebecca Milley Write G instead of − G , call it the conjugate . General misere play Theorem (Allen 2009) Restricted misere play Definitions ∗ + ∗ ≡ D 0 . Invertibility Comparison Reductions Theorem (McKay, Milley, Nowakowski 2012) The conjugate property Future directions If G + G ∈ N , and H + H ∈ N for all followers H of G, then G + G ≡ D 0 . Theorem (Milley and Renault, 2013) If G is an end then G + G ≡ E 0 Rebecca Milley Progress and Problems in Misere Play October 24, 2017 7 / 17

  42. Progress and Problems Invertibility results in Misere Play Rebecca Milley Write G instead of − G , call it the conjugate . General misere play Theorem (Allen 2009) Restricted misere play Definitions ∗ + ∗ ≡ D 0 . Invertibility Comparison Reductions Theorem (McKay, Milley, Nowakowski 2012) The conjugate property Future directions If G + G ∈ N , and H + H ∈ N for all followers H of G, then G + G ≡ D 0 . Theorem (Milley and Renault, 2013) If G is an end then G + G ≡ E 0 Theorem (Milley 2013) Let U be any universe and let S ⊂ U be closed under followers. If Left wins playing first on G + G + Y for all G ∈ S and left ends Y ∈ U , then G + G ≡ U 0 . Rebecca Milley Progress and Problems in Misere Play October 24, 2017 7 / 17

  43. Progress and Problems Absolute Game Theory in Misere Play Rebecca Milley General misere play Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 8 / 17

  44. Progress and Problems Absolute Game Theory in Misere Play Rebecca Milley • The results so far have been summarized in a survey General misere play paper for Games of No Chance 6 (Milley, Renault). Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 8 / 17

  45. Progress and Problems Absolute Game Theory in Misere Play Rebecca Milley • The results so far have been summarized in a survey General misere play paper for Games of No Chance 6 (Milley, Renault). Restricted misere play • New results for D were published in 2015 (Dorbec, Definitions Invertibility Comparison Renault, Siegel, Sopena). Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 8 / 17

  46. Progress and Problems Absolute Game Theory in Misere Play Rebecca Milley • The results so far have been summarized in a survey General misere play paper for Games of No Chance 6 (Milley, Renault). Restricted misere play • New results for D were published in 2015 (Dorbec, Definitions Invertibility Comparison Renault, Siegel, Sopena). Reductions • Since then, new results have been developed by The conjugate property Future directions applying Absolute CGT (Larsson, Nowakowski, Santos) to misere play, specifically to the dicotic and dead-ending universes. Rebecca Milley Progress and Problems in Misere Play October 24, 2017 8 / 17

  47. Progress and Problems Absolute Game Theory in Misere Play Rebecca Milley • The results so far have been summarized in a survey General misere play paper for Games of No Chance 6 (Milley, Renault). Restricted misere play • New results for D were published in 2015 (Dorbec, Definitions Invertibility Comparison Renault, Siegel, Sopena). Reductions • Since then, new results have been developed by The conjugate property Future directions applying Absolute CGT (Larsson, Nowakowski, Santos) to misere play, specifically to the dicotic and dead-ending universes. • These results are in a paper soon to be submitted, by the union of the above authors. Rebecca Milley Progress and Problems in Misere Play October 24, 2017 8 / 17

  48. Progress and Problems Absolute Game Theory in Misere Play Rebecca Milley • The results so far have been summarized in a survey General misere play paper for Games of No Chance 6 (Milley, Renault). Restricted misere play • New results for D were published in 2015 (Dorbec, Definitions Invertibility Comparison Renault, Siegel, Sopena). Reductions • Since then, new results have been developed by The conjugate property Future directions applying Absolute CGT (Larsson, Nowakowski, Santos) to misere play, specifically to the dicotic and dead-ending universes. • These results are in a paper soon to be submitted, by the union of the above authors. Recall: In general, to show G ≥ U H , we must show o ( G + X ) ≥ o ( H + X ) ∀ X ∈ U . Rebecca Milley Progress and Problems in Misere Play October 24, 2017 8 / 17

  49. Progress and Problems Absolute Game Theory in Misere Play Rebecca Milley • The results so far have been summarized in a survey General misere play paper for Games of No Chance 6 (Milley, Renault). Restricted misere play • New results for D were published in 2015 (Dorbec, Definitions Invertibility Comparison Renault, Siegel, Sopena). Reductions • Since then, new results have been developed by The conjugate property Future directions applying Absolute CGT (Larsson, Nowakowski, Santos) to misere play, specifically to the dicotic and dead-ending universes. • These results are in a paper soon to be submitted, by the union of the above authors. Recall: In general, to show G ≥ U H , we must show o ( G + X ) ≥ o ( H + X ) ∀ X ∈ U . Rebecca Milley Progress and Problems in Misere Play October 24, 2017 8 / 17

  50. Progress and Problems Subordinate comparison (L,M,N,R,S) in Misere Play Rebecca Milley General misere play Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 9 / 17

  51. Progress and Problems Subordinate comparison (L,M,N,R,S) in Misere Play Rebecca Milley Theorem (Subordinate Comparison in D ) General misere play Let G , H ∈ D . Then G ≥ D H if and only if Restricted misere play Definitions 1 o ( G ) ≥ o ( H ) ; Invertibility Comparison 2 ∀ H L ∈ H L , either ∃ G L ∈ G L : G L ≥ D Reductions H L or ∃ H LR ∈ H L R : G ≥ D H LR ; The conjugate property Future directions 3 ∀ G R ∈ G R , either ∃ H R ∈ H R : G R ≥ D H R or ∃ G RL ∈ G R L : G RL ≥ D H . Rebecca Milley Progress and Problems in Misere Play October 24, 2017 9 / 17

  52. Progress and Problems Subordinate comparison (L,M,N,R,S) in Misere Play Rebecca Milley Theorem (Subordinate Comparison in D ) General misere play Let G , H ∈ D . Then G ≥ D H if and only if Restricted misere play Definitions 1 o ( G ) ≥ o ( H ) ; Invertibility Comparison 2 ∀ H L ∈ H L , either ∃ G L ∈ G L : G L ≥ D Reductions H L or ∃ H LR ∈ H L R : G ≥ D H LR ; The conjugate property Future directions 3 ∀ G R ∈ G R , either ∃ H R ∈ H R : G R ≥ D H R or ∃ G RL ∈ G R L : G RL ≥ D H . Theorem (Subordinate Comparison in E ) Let G , H ∈ E . Then G ≥ E H if and only if o ( G ) ≥ ˆ ˆ o ( H ) ; 1 2 For all H L ∈ H L , either ∃ G L ∈ G L : G L ≥ E H L or ∃ H LR ∈ H L R : G ≥ E H LR ; 3 For all G R ∈ G R , either ∃ H R ∈ H R : G R ≥ E H R or ∃ G RL ∈ G R L : G RL ≥ E H . Rebecca Milley Progress and Problems in Misere Play October 24, 2017 9 / 17

  53. Progress and Problems Strong outcome in Misere Play Rebecca Milley General misere play Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 10 / 17

  54. Progress and Problems Strong outcome in Misere Play Rebecca Milley Definition General misere play Let G ∈ E . The strong left outcome and strong right Restricted misere play outcome of G are defined as Definitions Invertibility Comparison { o L ( G + X ) } ; Reductions o L ( G ) = ˆ min ���� The conjugate property left end X ∈E Future directions o R ( G ) = ˆ max { o R ( G + Y ) } . ���� right end Y ∈E Rebecca Milley Progress and Problems in Misere Play October 24, 2017 10 / 17

  55. Progress and Problems Strong outcome in Misere Play Rebecca Milley Definition General misere play Let G ∈ E . The strong left outcome and strong right Restricted misere play outcome of G are defined as Definitions Invertibility Comparison { o L ( G + X ) } ; Reductions o L ( G ) = ˆ min ���� The conjugate property left end X ∈E Future directions o R ( G ) = ˆ max { o R ( G + Y ) } . ���� right end Y ∈E Definition Let G ∈ E . We define the strong outcome of G as  L , if (ˆ o L ( G ) , ˆ o R ( G )) = ( L , L );    if (ˆ o L ( G ) , ˆ o R ( G )) = ( L , R ); N , o ( G ) = ˆ P , if (ˆ o L ( G ) , ˆ o R ( G )) = ( R , L );    if (ˆ o L ( G ) , ˆ o R ( G )) = ( R , R ) . R , Rebecca Milley Progress and Problems in Misere Play October 24, 2017 10 / 17

  56. Progress and Problems Domination & reversibility in Misere Play Rebecca Milley General misere play Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 11 / 17

  57. Progress and Problems Domination & reversibility in Misere Play Rebecca Milley • Domination works as normal in restricted misere play. General misere play Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 11 / 17

  58. Progress and Problems Domination & reversibility in Misere Play Rebecca Milley • Domination works as normal in restricted misere play. • Reversibility does not! General misere play Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 11 / 17

  59. Progress and Problems Domination & reversibility in Misere Play Rebecca Milley • Domination works as normal in restricted misere play. • Reversibility does not! General misere play Restricted misere play • The problem is when an otherwise reversible option Definitions would reverse through an end... Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 11 / 17

  60. Progress and Problems Domination & reversibility in Misere Play Rebecca Milley • Domination works as normal in restricted misere play. • Reversibility does not! General misere play Restricted misere play • The problem is when an otherwise reversible option Definitions would reverse through an end... Invertibility Comparison • Other (“open”) reversibility works as normal in any Reductions misere universe (LMNRS). The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 11 / 17

  61. Progress and Problems Domination & reversibility in Misere Play Rebecca Milley • Domination works as normal in restricted misere play. • Reversibility does not! General misere play Restricted misere play • The problem is when an otherwise reversible option Definitions would reverse through an end... Invertibility Comparison • Other (“open”) reversibility works as normal in any Reductions misere universe (LMNRS). The conjugate property • “End” reversibility has been solved for dicotic Future directions (DRSS 2015) and dead-ending (LMNRS) universes. Rebecca Milley Progress and Problems in Misere Play October 24, 2017 11 / 17

  62. Progress and Problems Domination & reversibility in Misere Play Rebecca Milley • Domination works as normal in restricted misere play. • Reversibility does not! General misere play Restricted misere play • The problem is when an otherwise reversible option Definitions would reverse through an end... Invertibility Comparison • Other (“open”) reversibility works as normal in any Reductions misere universe (LMNRS). The conjugate property • “End” reversibility has been solved for dicotic Future directions (DRSS 2015) and dead-ending (LMNRS) universes. Before we get to end-reversible reductions, we need perfect murder games: M ( n ) = {· | 0 , M ( n − 1) } . Rebecca Milley Progress and Problems in Misere Play October 24, 2017 11 / 17

  63. Progress and Problems Domination & reversibility in Misere Play Rebecca Milley • Domination works as normal in restricted misere play. • Reversibility does not! General misere play Restricted misere play • The problem is when an otherwise reversible option Definitions would reverse through an end... Invertibility Comparison • Other (“open”) reversibility works as normal in any Reductions misere universe (LMNRS). The conjugate property • “End” reversibility has been solved for dicotic Future directions (DRSS 2015) and dead-ending (LMNRS) universes. Before we get to end-reversible reductions, we need perfect murder games: M ( n ) = {· | 0 , M ( n − 1) } . M (0) M (1) M (2) M (3) M (4) Rebecca Milley Progress and Problems in Misere Play October 24, 2017 11 / 17

  64. Progress and Problems End-reversible reductions in Misere Play Rebecca Milley General misere play Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 12 / 17

  65. Progress and Problems End-reversible reductions in Misere Play Rebecca Milley In any universe: General misere play Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 12 / 17

  66. Progress and Problems End-reversible reductions in Misere Play Rebecca Milley In any universe: General misere play 1. Remove dominated options. Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 12 / 17

  67. Progress and Problems End-reversible reductions in Misere Play Rebecca Milley In any universe: General misere play 1. Remove dominated options. Restricted misere play 2. Reverse open-reversible options. Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 12 / 17

  68. Progress and Problems End-reversible reductions in Misere Play Rebecca Milley In any universe: General misere play 1. Remove dominated options. Restricted misere play 2. Reverse open-reversible options. Definitions Invertibility In D : Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 12 / 17

  69. Progress and Problems End-reversible reductions in Misere Play Rebecca Milley In any universe: General misere play 1. Remove dominated options. Restricted misere play 2. Reverse open-reversible options. Definitions Invertibility In D : Comparison Reductions 3. Replace end-reversible options by ∗ . The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 12 / 17

  70. Progress and Problems End-reversible reductions in Misere Play Rebecca Milley In any universe: General misere play 1. Remove dominated options. Restricted misere play 2. Reverse open-reversible options. Definitions Invertibility In D : Comparison Reductions 3. Replace end-reversible options by ∗ . The conjugate property Future directions 4. Replace {∗ | ∗} by 0. In E : Rebecca Milley Progress and Problems in Misere Play October 24, 2017 12 / 17

  71. Progress and Problems End-reversible reductions in Misere Play Rebecca Milley In any universe: General misere play 1. Remove dominated options. Restricted misere play 2. Reverse open-reversible options. Definitions Invertibility In D : Comparison Reductions 3. Replace end-reversible options by ∗ . The conjugate property Future directions 4. Replace {∗ | ∗} by 0. In E : 3. Remove non-fundamental end-reversible options, including removal of lone options as long as the result is in E . Rebecca Milley Progress and Problems in Misere Play October 24, 2017 12 / 17

  72. Progress and Problems End-reversible reductions in Misere Play Rebecca Milley In any universe: General misere play 1. Remove dominated options. Restricted misere play 2. Reverse open-reversible options. Definitions Invertibility In D : Comparison Reductions 3. Replace end-reversible options by ∗ . The conjugate property Future directions 4. Replace {∗ | ∗} by 0. In E : 3. Remove non-fundamental end-reversible options, including removal of lone options as long as the result is in E . 4. Simultaneously remove lone left and lone right end-reversible options; i.e., replace { A | C } with 0 if A and C are end-reversible. Rebecca Milley Progress and Problems in Misere Play October 24, 2017 12 / 17

  73. Progress and Problems End-reversible reductions in Misere Play Rebecca Milley In any universe: General misere play 1. Remove dominated options. Restricted misere play 2. Reverse open-reversible options. Definitions Invertibility In D : Comparison Reductions 3. Replace end-reversible options by ∗ . The conjugate property Future directions 4. Replace {∗ | ∗} by 0. In E : 3. Remove non-fundamental end-reversible options, including removal of lone options as long as the result is in E . 4. Simultaneously remove lone left and lone right end-reversible options; i.e., replace { A | C } with 0 if A and C are end-reversible. 5. Replace other end-reversible options by {· | M ( n ) } for left options or {− M ( n ) | ·} for right. Rebecca Milley Progress and Problems in Misere Play October 24, 2017 12 / 17

  74. Progress and Problems Unique canonical form in Misere Play Rebecca Milley General misere play Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 13 / 17

  75. Progress and Problems Unique canonical form in Misere Play Rebecca Milley • If G , H ∈ D are reduced and G ≡ D H then G and H General misere play are identical. (Dorbec et al.) Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 13 / 17

  76. Progress and Problems Unique canonical form in Misere Play Rebecca Milley • If G , H ∈ D are reduced and G ≡ D H then G and H General misere play are identical. (Dorbec et al.) Restricted misere play • If G , H ∈ E are reduced and G ≡ E H then G and H Definitions Invertibility Comparison are identical. (Larsson et al.) Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 13 / 17

  77. Progress and Problems The conjugate property in Misere Play Rebecca Milley General misere play Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 14 / 17

  78. Progress and Problems The conjugate property in Misere Play Rebecca Milley • Recall, in general, we have G + G � = 0. General misere play Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 14 / 17

  79. Progress and Problems The conjugate property in Misere Play Rebecca Milley • Recall, in general, we have G + G � = 0. General misere play • For some U , we might have G + G ≡ U 0... Restricted misere play Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 14 / 17

  80. Progress and Problems The conjugate property in Misere Play Rebecca Milley • Recall, in general, we have G + G � = 0. General misere play • For some U , we might have G + G ≡ U 0... Restricted misere play or even G + H ≡ U 0, with H �≡ U G . Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 14 / 17

  81. Progress and Problems The conjugate property in Misere Play Rebecca Milley • Recall, in general, we have G + G � = 0. General misere play • For some U , we might have G + G ≡ U 0... Restricted misere play or even G + H ≡ U 0, with H �≡ U G . !? Definitions Invertibility Comparison Reductions The conjugate property Future directions Rebecca Milley Progress and Problems in Misere Play October 24, 2017 14 / 17

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