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In [[combinatorial game theory]], the '''Sprague–Grundy theorem''' states that every [[impartial game]] under the [[normal play convention]] is equivalent to a [[nimber]]. The '''Grundy value''' or '''nim-value''' of an impartial game is then defined as the unique nimber that the game is equivalent to. In the case of a game whose positions (or summands of positions) are indexed by the natural numbers (for example the possible heap sizes in nim-like games), the sequence of nimbers for successive heap sizes is called the '''nim-sequence''' of the game. | |||
The theorem was discovered independently by [[Roland Sprague|R. P. Sprague]] (1935) and [[Patrick Grundy|P. M. Grundy]] (1939). | |||
The | ==Definitions== | ||
For the purposes of the Sprague–Grundy theorem, a ''game'' is a two-player game of [[perfect information]] satisfying the ''ending condition'' (all games come to an end: there are no infinite lines of play) and the ''normal play condition'' (a player who cannot move loses). | |||
An ''[[impartial game]]'' is one such as [[nim]], in which each player has exactly the same available moves as the other player in any position. Note that games such as [[tic-tac-toe]], [[checkers]], and [[chess]] are ''not'' impartial games. In the case of checkers and chess, for example, players can only move their own pieces, not their opponent's pieces. And in tic-tac-toe, one player puts down X's, while the other puts down O's. Impartial games fall into two ''outcome classes'': either the next player wins (an ''N-position'') or the previous player wins (a ''P-position''). | |||
An impartial game can be identified with the set of positions that can be reached in one move (these are called the ''options'' of the game). Thus the game with options ''A'', ''B'', or ''C'' is the set {''A'', ''B'', ''C''}. | |||
The normal play convention is where the last player to move wins. Alternatively, the player who first does not have any valid move loses. The opposite - the [[misère]] convention is where the last person to have a valid move or makes the last move loses. | |||
A ''[[nimber]]'' is a special game denoted *''n'' for some ordinal ''n''. We define *0 = {} (the empty set), then *1 = {*0}, *2 = {*0, *1}, and *(''n''+1) = *''n'' ∪ {*''n''}. When ''n'' is an integer, the nimber *''n'' = {*0, *1, ..., *(''n''−1)}. This corresponds to a heap of ''n'' counters in the game of [[nim]], hence the name. | |||
Two games ''G'' and ''H'' can be ''added'' to make a new game ''G''+''H'' in which a player can choose either to move in ''G'' or in ''H''. In set notation, ''G''+''H'' means {''G''+''h'' for ''h'' in ''H''} ∪ {''g''+''H'' for ''g'' in ''G''}, and thus game addition is commutative and associative. | |||
Two games ''G'' and ''G''' are ''equivalent'' if for every game ''H'', the game ''G''+''H'' is in the same outcome class as ''G'''+''H''. We write ''G'' ≈ ''G'''. | |||
A game can refer to two things. It can define a set of possible positions and their moves through its rules, for example, chess, or nim. It can also refer to a certain position, for example, the game *5. Generally, the meaning to be taken is clear from the context. | |||
==Lemma== | |||
For impartial games, ''G'' ≈ ''G''' if and only if ''G''+''G''' is a ''P-position''. | |||
First, we note that ≈ is an [[equivalence relation]] since equality of outcome classes is an equivalence relation. | |||
We now show that for every game G, and ''P-position'' game ''A'', ''A''+''G'' ≈ ''G''. By the definition of ≈, we need to show that ''G''+''H'' is in the same outcome-class as ''A''+''G''+''H'' for all games ''H''. | |||
If ''G''+''H'' is ''P-position'', then the previous player has a winning strategy in ''A''+''G''+''H'': to every move in ''G''+''H'' he responds according to his winning strategy in ''G''+''H'', and to every move in ''A'' he responds with his winning strategy there. If ''G''+''H'' is ''N-position'', then the next player in ''A''+''G''+''H'' makes a winning move in ''G''+''H'', and then reverts to responding to his opponent in the manner described above. | |||
Also, ''G''+''G'' is ''P-position'' for any game ''G''. For every move made in one copy of ''G'', the previous player can respond with the same move in the other copy, which means he always makes the last move. | |||
Now, we can prove the lemma. | |||
If ''G'' ≈ ''G''', then ''G''+''G''' is of the same outcome-class as ''G''+''G'', which is ''P-position''. | |||
On the other hand, if ''G''+''G''' is ''P-position'', then since ''G''+''G'' is also ''P-position'', ''G'' ≈ ''G''+(''G''+''G''') ≈ (''G''+''G'')+''G''' ≈ ''G''', thus ''G'' ≈ ''G'''. | |||
==Proof== | |||
We prove the theorem by [[structural induction]] on the set representing the game. | |||
Consider a game <math>G = \{G_1, G_2, \ldots, G_k\}</math>. By the induction hypothesis, all of the options are equivalent to nimbers, say <math>G_i \approx *n_i</math>. We will show that <math>G \approx *m</math>, where <math>m</math> is the [[mex (mathematics)|mex]] of the numbers <math>n_1, n_2, \ldots, n_k</math>, that is the smallest non-negative integer not equal to some <math>n_i</math>. | |||
Let <math>G'=\{*n_1, *n_2, \ldots, *n_k\}</math>. The first thing we need to note is that <math>G \approx G'</math>. Consider <math>G+G'</math>. If the first player makes a move in <math>G</math>, then the second player can move to the equivalent <math>*n_i</math> in <math>G'</math>, and conversely if the first player makes a move in <math>G'</math>. After this the game is a P-position (by the lemma), since it's the sum of some option of <math>G</math> and a nim pile equivalent to that option. Therefore, <math>G+G'</math> is a P-position, and by another application of our lemma, <math>G \approx G'</math>. | |||
So now, by our lemma, we need to show that <math>G+*m</math> is a P-position. We do so by giving an explicit strategy for the second player in the equivalent <math>G'+*m</math>. | |||
Suppose that the first player moves in the component <math>*m</math> to the option <math>*m'</math> where <math>m'<m</math>. But since <math>m</math> was the ''minimal'' excluded number, the second player can move in <math>G'</math> to <math>*m'</math>. | |||
Suppose instead that the first player moves in the component <math>G'</math> to the option <math>*n_i</math>. If <math>n_i < m</math> then the second player moves in <math>*m</math> to <math>*n_i</math>. If <math>n_i > m</math> then the second player, moves in <math>*n_i</math> to <math>*m</math>. It's not possible that <math>n_i = m</math> because <math>m</math> was defined to be different from all the <math>n_i</math>. | |||
Therefore, <math>G'+*m</math> is a P-position, and hence so is <math>G+*m</math>. By our lemma, <math>G \approx *m</math> as desired. | |||
==Development== | |||
The Sprague–Grundy theorem has been developed into the field of [[combinatorial game theory]], notably by [[E. R. Berlekamp]], [[John Horton Conway]] and others. The field is presented in the books ''[[Winning Ways for your Mathematical Plays]]'' and ''[[On Numbers and Games]]''. | |||
==See also== | |||
*[[Genus theory]] | |||
*[[Indistinguishability quotient]] | |||
==References== | |||
*{{cite journal | |||
| author = Sprague, R. P. | |||
| title = Über mathematische Kampfspiele | |||
| url = http://www.jstage.jst.go.jp/article/tmj1911/41/0/41_0_438/_article | |||
| journal = [[Tohoku Mathematical Journal]] | |||
| year = 1935–36 | |||
| volume = 41 | |||
| pages = 438–444}} | |||
*{{cite journal | |||
| author = Grundy, P. M. | |||
| title = Mathematics and games | |||
| journal = [[Eureka (University of Cambridge magazine)|Eureka]] | |||
| url = http://www.archim.org.uk/eureka/27/games.html | |||
| year = 1939 | |||
| volume = 2 | |||
| pages = 6–8 |archiveurl = http://web.archive.org/web/20070927192024/http://www.archim.org.uk/eureka/27/games.html |archivedate = 2007-09-27}} Reprinted, 1964, '''27''': 9–11. | |||
*{{cite journal | |||
| author = Schleicher, Dierk; Stoll, Michael | |||
| title = An introduction to Conway's games and numbers | |||
| year = 2004 | |||
| arxiv = math.CO/0410026}} | |||
*{{cite journal | |||
| author = Milvang-Jensen, Brit C. A. | |||
| year = 2000 | |||
| url = http://www.itu.dk/people/brit/Brits%20thesis.pdf | |||
| title = Combinatorial Games, Theory and Applications}} | |||
==External links== | |||
* [http://www.cut-the-knot.org/Curriculum/Games/Grundy.shtml Grundy's game] at [[cut-the-knot]] | |||
* [http://www.math.ucla.edu/~tom/Game_Theory/comb.pdf Easily readable, introductory account from the UCLA Math Department] | |||
* [http://sputsoft.com/blog/2009/04/the-game-of-nim.html The Game of Nim] at [http://sputsoft.com sputsoft.com] | |||
{{DEFAULTSORT:Sprague-Grundy theorem}} | |||
[[Category:Combinatorial game theory]] | |||
[[Category:Theorems in discrete mathematics]] | |||
Revision as of 14:32, 21 July 2013
Template:Cleanup In combinatorial game theory, the Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a nimber. The Grundy value or nim-value of an impartial game is then defined as the unique nimber that the game is equivalent to. In the case of a game whose positions (or summands of positions) are indexed by the natural numbers (for example the possible heap sizes in nim-like games), the sequence of nimbers for successive heap sizes is called the nim-sequence of the game.
The theorem was discovered independently by R. P. Sprague (1935) and P. M. Grundy (1939).
Definitions
For the purposes of the Sprague–Grundy theorem, a game is a two-player game of perfect information satisfying the ending condition (all games come to an end: there are no infinite lines of play) and the normal play condition (a player who cannot move loses).
An impartial game is one such as nim, in which each player has exactly the same available moves as the other player in any position. Note that games such as tic-tac-toe, checkers, and chess are not impartial games. In the case of checkers and chess, for example, players can only move their own pieces, not their opponent's pieces. And in tic-tac-toe, one player puts down X's, while the other puts down O's. Impartial games fall into two outcome classes: either the next player wins (an N-position) or the previous player wins (a P-position).
An impartial game can be identified with the set of positions that can be reached in one move (these are called the options of the game). Thus the game with options A, B, or C is the set {A, B, C}.
The normal play convention is where the last player to move wins. Alternatively, the player who first does not have any valid move loses. The opposite - the misère convention is where the last person to have a valid move or makes the last move loses.
A nimber is a special game denoted *n for some ordinal n. We define *0 = {} (the empty set), then *1 = {*0}, *2 = {*0, *1}, and *(n+1) = *n ∪ {*n}. When n is an integer, the nimber *n = {*0, *1, ..., *(n−1)}. This corresponds to a heap of n counters in the game of nim, hence the name.
Two games G and H can be added to make a new game G+H in which a player can choose either to move in G or in H. In set notation, G+H means {G+h for h in H} ∪ {g+H for g in G}, and thus game addition is commutative and associative.
Two games G and G' are equivalent if for every game H, the game G+H is in the same outcome class as G'+H. We write G ≈ G'.
A game can refer to two things. It can define a set of possible positions and their moves through its rules, for example, chess, or nim. It can also refer to a certain position, for example, the game *5. Generally, the meaning to be taken is clear from the context.
Lemma
For impartial games, G ≈ G' if and only if G+G' is a P-position.
First, we note that ≈ is an equivalence relation since equality of outcome classes is an equivalence relation.
We now show that for every game G, and P-position game A, A+G ≈ G. By the definition of ≈, we need to show that G+H is in the same outcome-class as A+G+H for all games H. If G+H is P-position, then the previous player has a winning strategy in A+G+H: to every move in G+H he responds according to his winning strategy in G+H, and to every move in A he responds with his winning strategy there. If G+H is N-position, then the next player in A+G+H makes a winning move in G+H, and then reverts to responding to his opponent in the manner described above.
Also, G+G is P-position for any game G. For every move made in one copy of G, the previous player can respond with the same move in the other copy, which means he always makes the last move.
Now, we can prove the lemma.
If G ≈ G', then G+G' is of the same outcome-class as G+G, which is P-position.
On the other hand, if G+G' is P-position, then since G+G is also P-position, G ≈ G+(G+G') ≈ (G+G)+G' ≈ G', thus G ≈ G'.
Proof
We prove the theorem by structural induction on the set representing the game.
Consider a game . By the induction hypothesis, all of the options are equivalent to nimbers, say . We will show that , where is the mex of the numbers , that is the smallest non-negative integer not equal to some .
Let . The first thing we need to note is that . Consider . If the first player makes a move in , then the second player can move to the equivalent in , and conversely if the first player makes a move in . After this the game is a P-position (by the lemma), since it's the sum of some option of and a nim pile equivalent to that option. Therefore, is a P-position, and by another application of our lemma, .
So now, by our lemma, we need to show that is a P-position. We do so by giving an explicit strategy for the second player in the equivalent .
Suppose that the first player moves in the component to the option where . But since was the minimal excluded number, the second player can move in to .
Suppose instead that the first player moves in the component to the option . If then the second player moves in to . If then the second player, moves in to . It's not possible that because was defined to be different from all the .
Therefore, is a P-position, and hence so is . By our lemma, as desired.
Development
The Sprague–Grundy theorem has been developed into the field of combinatorial game theory, notably by E. R. Berlekamp, John Horton Conway and others. The field is presented in the books Winning Ways for your Mathematical Plays and On Numbers and Games.
See also
References
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Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang Reprinted, 1964, 27: 9–11. - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang