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Does monopoly has a set of notations like Chess for game tracking?
There's a tool-assisted speedrun of Monopoly for the NES that takes a different approach than the one described in this page:

http://tasvideos.org/520M.html

Oddly enough, I came to post exactly that. To spare anyone having to hunt up the obsoleted movie to find the strategy (which hasn't changed in the newest movie, which merely improves luck manipulation), here's a link to their strategy which tells you all the rolls they used: http://tasvideos.org/988S.html

Actually, there were two routes and the one that would, in theory, be faster ended up being slower because they would have been forced to wait too long for the dice to hit the right numbers. Lest you think that's easy, I suggest looking up the TAS of the Shining Force games. The videos are incredibly boring. The interesting part are the complex process they've gone through to disassemble the random number routines and manipulate them to their benefit. They actually have complex programs in Lua, Matlab or possibly a few other things to optimize the randomness for them, which are pretty crazy.

One such discussion of the complex end of luck manipulation can be found here: http://tasvideos.org/2258S.html

And the scripts it mentioned are in the forums proper. There may be more (I haven't read the whole thread), but a lot of them can be found here if anyone is interested:

http://tasvideos.org/forum/viewtopic.php?t=1759&postdays...

It's a LOT more involved than you might expect.

"Whenever you land on an unowned property you may buy that property from the Bank at its printed price. You receive the Title Deed card showing ownership. Place the title deed card face up in front of you. If you do not wish to buy the property, the Bank sells it at through an auction to the highest bidder. The high bidder pays the Bank the amount of the bid in cash and receives the Title Deed card for that property." - http://richard_wilding.tripod.com/monorules.htm#buyingproper...

Player 1 lands on two properties, but this auction does not occur.

This can easily be solved by simply saying "'no auction' house rule". Few people actually honor this rule anyhow. I don't object, I don't think it's a very good rule either. (Then again, I don't think it's a very good game, so my opinion may be suspect.) The biggest problem is the word "auction" is underspecified; it could mean many things and you'll spend more time arguing over it than playing if you try to take this seriously.

I have found it remarkable how few people play by the actual rules, and how few people realize they are not playing by the actual rules. Including some people who claim to hate the game because they are not following the rules. (Often this revolves around not using timers which the rules will often explicitly mention; my version of Scrabble does, for instance, though the online rules I find do not.)

Easily fixed: the auction does occur, but no one bids on the properties.
I have seen it argued that sells at auction does not admit the possibility of not selling it. I do not remember if that was tournament rules or not. You can then buy it for a dollar, if you like, it won't change the results.

If you're going to rules lawyer, you gotta do it all proper like.

I'm not arguing this will necessarily make the game any longer, I'm just saying this isn't a by-the-actual-rules game. It shares this in common with probably 99%+ of all games of Monopoly that are played.

It is not uncommon for games to fail to cover cases in the rules, such as nobody even bidding a dollar for property in Monopoly. Jenga sort of crashes if you get the game into a state where there are no more levels with removable pieces, or at least strictly read mandates that the next person to move is a loser, which is an undesirable outcome.

Using an auction maybe you could make the game shorter by getting player 2 to buy properties, thus loosing money, so when he lands on something player 1 buys she has none left.
Or the auction does occur, someone bids all of their money, and the next turn that person goes bankrupt (making the game take, what, 5 seconds?)
Yeah, I noticed the lack of auction, too, since I always play with auction rules. (As well as the non-official (?) house rule of paying taxes/chance/community chest amounts to the "pot" in the middle, with whomever landing on Free Parking taking the amount.) While auction rules aren't that useful with 2 people, I think it's fun to have them with 3 or more.

Since you don't like Monopoly, I assume you wouldn't like Monopoly-Risk? I've yet to get someone to agree to play that with me, though I'm still not sure I want to. It's just like regular Risk, but battles are decided by games of Monopoly instead of dice rolls.

Have you ever played a game of Monopoly-Risk? It sounds like it would take days of real-time play.
Couldn't this game theoretically be over much faster?

From http://www.hasbro.com/monopoly/en_US/discover/strategy/rules...

  Whenever a player lands on an unowned MONOPOLY property he may buy that property
  from the Bank at its printed price. He receives the Title Deed card showing 
  ownership and places it face-up in front of him.

  If he does not wish to buy the MONOPOLY property it is sold at auction by the 
  Banker to the highest bidder. The buyer pays to the Bank the amount of the bid
  in cash and receives the Title Deed card for that property. Any player,
  including the one who declined the option of buying it at the printed price,
  may bid. Bidding may start at any price.
This article doesn't seem to be following the official rules (at least, according to the multiple "Action: None"s scattered throughout).

Faster variation:

Player 1: Rolls a number causing him to land on ANY property. Declines to purchase, but bids the price up such that Player 2 receives the deed but pays all of his money.

Player 2: Lands on community chest, has to pay some amount of money greater than the mortgaged value of property received on last play, goes bankrupt.

Am I missing something?

This is also brought up in the comments of the article itself. Yes, the game could be over in fewer turns/dice rolls this way. But then, would someone really pay that much for the property at auction? Maybe not.
Agreed, but we are talking about the shortest theoretically possible game anyway, so might as well put all options on the table.

One could argue that rolling doubles 5 times in the first 7 roles is just as unlikely. :)

Ah, right. My thinking was that in the scenario proposed in the article, it doesn't matter what player 2 does (apart from the dice rolls, which is beyond their control anyway) they'll still lose. You only have to control the actions of player 1. While in the auction scenario you need to control both players to make it work. So if you were actually playing as player 1 and all the dice rolls lined up, you would only be guaranteed to pull off the scenario in the article. But I guess it doesn't matter since both scenarios are theoretically possible.
Why buy Park Place?
You have to own all of one color before you can put houses/hotels on them.
Oh yeah! It's been a while since I've played.
The article itself is quite interesting, but I almost skipped the video, which is at least as amusing! It was worth the watch.
Shortest game I ever played was less than 3 minutes. On my first turn I bought St. Charles Place. On the other player's first turn he bought on of the light blue properties. On my second turn I bought Virginia. On the other player's second turn he bought States Ave. I mortgaged my 2 properties and traded him them for his one property and all his cash. On his next turn he landed on Community Chest and couldn't pay the school fees.
While I could see why you'd offer the deal (he wouldn't be able to build on his new monopoly until he passed Go a couple times to unhock), I'm not sure why he'd take it. I would think he'd know that being out of cash would keep him from getting any new properties for a long time...

You did make sure he had the money to pay the 10% interest fee on transfer, right?

Yes he paid the 10%. He likes to play with rules I don't like: Free Parking pot, double money for landing on Go, and no even build rule. So he may have been lucky and I may have been unlucky. He took a chance and it could have paid off for him. Either way it was probably going to be a short game.
If you can enumerate all the rules, properties, cards, etc., it seems to me the shortest game problem can be solved with dynamic programming. :)
another solution (i think)

player 1

  1, 1 --> community chest, collect 200
  4, 4 --> lands on jail, just visiting
  2, 4 --> purchases st. james
player 2

  2, 2 --> -150 from income tax
  3, 7 --> lands on virginia avenue
player 1

  1,1 --> purchase tennesse
  2,2 --> chance (go to new york), purchase new york
  purchase 3, 3, 4 houses (costs 1000)
player 2

  1,1 -- lands on st. james, pay -550
  1,2 -- lands on new york avenue, pay -800

  GAME OVER

shit, this is a hard game...have spent the past half hour going for sub-seven moves...anyway
I found this article completely incomprehensible until I remembered that Monopoly is localised. Park Place? Boardwalk? You mean Park Lane and Mayfair, surely? :)
A shorter way would be for one player to buy a property then have another player trade all his money for it. Then that player would have to mortgage the property. Then they would get a community chest that makes them lose more money than the mortgage value. This would take only 2 turns.