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In [[combinatorial game theory]], '''Maker-Breaker games''' are a subclass of [[positional game]]s.<ref>J. Beck: ''Combinatorial Games: Tic-Tac-Toe Theory'', Cambridge University Press, 2008.</ref> | |||
It is a two-person [[perfect information|game with complete information]] played on a [[hypergraph]] ''(V,H)'' where ''V'' is an arbitrary set (called the board of the game) and ''H'' is a family of subsets of ''V'', called the ''winning sets''. The two players alternately occupy previously unoccupied elements of ''V''. | |||
The first player, ''Maker'', has to occupy a winning set to win; and the second player, ''Breaker'', has to stop Maker from doing so; if Breaker successfully prevents maker from occupying a winning set to the end of the game, then Breaker wins. Thus, in a Maker–Breaker positional game, Maker wins if he occupies all elements of some winning set and Breaker wins if he prevents Maker from doing so. There can be no draw in a Maker-Breaker positional game: one player always wins. | |||
The definition of Maker-Breaker game has a subtlety when <math>|V|=\infty</math> and <math>|H|=\infty</math>. In this case we say that Breaker has a winning strategy if, for all ''j'' > 0, Breaker can prevent Maker from completely occupying a winning set by turn ''j''. | |||
When [[tictactoe]] is played as a Maker–Breaker positional game, Maker has a winning strategy (Maker does not need to block Breaker from obtaining a winning line) | |||
.<ref name="kruczek2010">{{cite journal| last=Kruczek| first=Klay|coauthor=Eric Sundberg|title=Potential-based strategies for tic-tac-toe on the integer latticed with numerous directions| journal=[[The Electronic Journal of Combinatorics]]| year=2010| volume=17| pages=R5}}</ref> | |||
==Maker-Breaker games on graphs== | |||
There has been quite some research done on playing Maker-Breaker games when the board of the game is the edge-set of a [[Graph (mathematics)|graph]] <math>G=(V,E)</math> (usually taken as the complete graph) and the family of winning sets is <math>\mathcal{F}=\{F\subset E\vert G[F]\hbox{ has property }\mathcal{P}\}</math>, where <math>\mathcal{P}</math> is some [[graph property]] (usually taken to be monotone increasing) such as connectivity (see, e.g.,<ref name="ChvatalErdos1978">{{cite journal|last=Chvatal|coauthor=Erdos|title=Biased positional games| journal=[[Annals of discrete mathematics]]| year=1978| volume=2| pages=221–229}}</ref>). | |||
==References== | |||
{{reflist}} | |||
[[Category:Combinatorial game theory]] | |||
Revision as of 00:41, 11 January 2014
In combinatorial game theory, Maker-Breaker games are a subclass of positional games.[1]
It is a two-person game with complete information played on a hypergraph (V,H) where V is an arbitrary set (called the board of the game) and H is a family of subsets of V, called the winning sets. The two players alternately occupy previously unoccupied elements of V.
The first player, Maker, has to occupy a winning set to win; and the second player, Breaker, has to stop Maker from doing so; if Breaker successfully prevents maker from occupying a winning set to the end of the game, then Breaker wins. Thus, in a Maker–Breaker positional game, Maker wins if he occupies all elements of some winning set and Breaker wins if he prevents Maker from doing so. There can be no draw in a Maker-Breaker positional game: one player always wins.
The definition of Maker-Breaker game has a subtlety when and . In this case we say that Breaker has a winning strategy if, for all j > 0, Breaker can prevent Maker from completely occupying a winning set by turn j.
When tictactoe is played as a Maker–Breaker positional game, Maker has a winning strategy (Maker does not need to block Breaker from obtaining a winning line) .[2]
Maker-Breaker games on graphs
There has been quite some research done on playing Maker-Breaker games when the board of the game is the edge-set of a graph (usually taken as the complete graph) and the family of winning sets is , where is some graph property (usually taken to be monotone increasing) such as connectivity (see, e.g.,[3]).
![{\displaystyle {\mathcal {F}}=\{F\subset E\vert G[F]{\hbox{ has property }}{\mathcal {P}}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24842dc64a9ce52d11c05c08d4b34333c93516fd)
References
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- ↑ J. Beck: Combinatorial Games: Tic-Tac-Toe Theory, Cambridge University Press, 2008.
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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