Monty Hall (note the lack of a "U") was the host on the game show Lets Make A Deal, which was cancelled before some of you were born. The show could give away massive amounts of prizes to the lucky (or cunning or destined-to-win or however they pick winners on game shows).
A Monty Haul campaign (with a "U") was the generic label for a Game Master (and his/her campaign) who would run adventures that were like game show giveaways, except the questions weren't as hard. Players would end up staggering under the loads of gold and gems (except the encumbrance rules often were ignored as well) and cherry-picking which magic items they wanted to keep because they had so many to choose from. Think of Conan the Barbarian with a Star Destroyer.
Also, in the first and second editions of Dungeons & Dragons, Player Characters got Experience Points based on how much money they looted, one to one. So the Monty Haul characters would also end up at stratospheric Character Levels, which would lead to situations in which they were assassinating gods to gain their nifty weapons.
This is a situation most gamers greatly deplore and is sternly discouraged in the Game Master's section of all later games, but it is assumed that everyone went through this stage at some early point in their gaming "careers."
In some other cases though, the Monty Haul game master may be a case of Suspicious Game Master Generosity since sometimes they may take it to bring upon an already dangerous foe and give him a slew of new tricks or has their stats tweaked for whatever to get ready for a tougher battle.
Naturally, no troper here ever descended so low.
Many video games seem to either encourage or require the sort of player (or the sort of game play) who would take a broom to a dungeon, to make sure he swept up all the coins.
The opposite to this is the Killer Game Master, who delivers death and horror to the PCs in place of treasure and godlike power.
This doesn't have much to do with the Monty Hall Problem, a probability puzzle whose name also comes from the game show host.
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In Knights of the Dinner Table, the seasoned, regular GMs generally either run balanced campaigns or go the opposite direction of this trope. However, when Bob or Dave step behind the screen to run a game, it goes this route.
This trope plays an unusually huge role in the book, Slathbog's Gold.
Each edition is its own game, balanced around its own set of expectations.
This trope cannot be described in absolute terms, but is entirely relative to the expectations of the game.
While each advancing edition has objectively made magical items more and more common, they have also become increasingly required in order to function at the expected level. Any magical items at all were only a bonus in early (A)D&D, while third edition at the latest made a steady treadmill of ever-more-powerful weapons, armors, and other gear a mandatory part of not dying horribly (especially for non-spellcasters trying to keep up). Trying to apply e.g. a fourth edition approach to AD&D would likely indeed result in Monty Haul gameplay, but someone taking the AD&D approach to fourth edition would instead be a Killer GM.
As a side note, the Advanced Dungeons & Dragons, 2nd Edition Dungeon Master's Guide, in its discussion of where campaigns can go wrong and how to fix them, uses the term "Monty Haul campaign." This likely helped codify the trope name.
The Legend System's magic item rules are specifically designed to prevent this - the progression is the same regardless of the DM's inclinations.
Referenced briefly but incorrectly in Goblins. Forgoth mentions that Minmax cannot simply walk into a blacksmith shop and buy a magic sword, as that would be too Monty Haul, but that's exactly what PCs are expected to do with their treasure. (If you look in the DM's guide, it clearly says a PC can generally buy any magic item on the list, though they need to look for a seller.) Monty Haul only comes in when PCs have too much money for their level.
However, at the same time, the campaign itself definitely is Monty Haul. All the heroes are low-level, but have improbably powerful magic items that even the playing field. The paladin has an axe that acts as the can for a world-destroying demon, the cleric has a spear that multiplies when thrown and then returns to his hand, and the minmaxer recently got a sword made out of oblivion. Which is perhaps fair enough, considering everything else.
Jerry Holkins (Tycho) accuses Mike Krahulik (Gabe) of this in Penny Arcade, then decides to balance it — Tycho being a Killer GM.
Dee Dee from Dexter's Laboratory, after taking over DM-ship of Dexter's game. Considering that Dexter is a Killer GM, the players are much happier with the change. Even funnier is the treasures she hands out: the Archer gets a stronghold (basically the most awesome tree fort ever), the Knight gets a "noble steed", and the Magician gets a Lovely Assistant and a lifetime of sold-out shows in Las Vegas. Dexter's character asks for the chalice they were questing to obtain, and is upset that all it does is "you can drink from it and it'll never spill."
The Diablo series and the majority of its clones tend to be like this in the end game. Bosses and major loot caches will often release a screen-filling fountain of gold and enchanted gear- from which players will pick the one or two very best pieces and leave the rest lying on the floor. At early levels, however, the player will want to keep anything that's better than the standard vendor gear. For a game where the whole point is to constantly upgrade your equipment, the progression is fairly even.
High Rate Private Servers of any given MMORPG are more often then not Monty Hauls meant to fuel a player's armory with the weapon needed to fight big bosses and most of all each other.
A lot of user-created modules or modifications for The Elder Scrolls and Neverwinter Nights are simple dispensers of XP and loot. Many of them who have actual plot can also fall into this due to being made by non-professionals.
Many badly made designs for Unlimited Adventures shower the characters with money. One design in particular (From Beggars to Heroes) was content to throw thousands of money at you for the meekest reason. (For example, as poor beggars in the starting town, you can walk around and meet an unlimited number of rich people every few steps who will shower you with riches every time.) Oh, and you get experience for these, too.
S.T.A.L.K.E.R. also had its share of mods which purported to 'improve' the gameplay but in the process dumped some of the game's best weapons and gear into the player's arms near the very beginning. Kind of ruins the 'scavenge and survive' experience when your stuff is already better than everyone else's...
Dungeon Siege encouraged this by adding pack mules and Traggs, creatures with above normal carrying capacity, but minimal fighting capabilty. Naturally it's a trade off between combat power and hauling size.
Fable, while in itself the game isn't a Monty Haul, if you spend an hour or two doing a job to afford some real estate, the returns on the investment are used to buy more real estate (which pays you every five minutes) eventually become so huge that you have so much money you have no idea what to do with it all.
Fallout 3's Operation Anchorage DLC takes the player through a short, relentlessly linear, and relatively easy (depending on character build) campaign, at the end of which you're awarded with a treasure trove of goodies not available anywhere in the core game.
Ditto Mothership Zeta(powerful alien weapons, weapon repair epoxy, and crystals that can be sold for a motherload of caps), and the steel ingot fetch quest in The Pitt.
Many Fallout players consider Fallout 3 to be a Monty Haul campaign in general, especially compared to previous installments and Fallout: New Vegas.
The Black Jack tables in New Vegas (and more so the slot machines if you have high luck and too much time on your hands) are ridiculously easy ways to clean out every casino that has these minigames and earn up to 100,000 caps.
The treasure at the end of the Dead Money DLC is 37f gold bars worth 10,000 caps each (though each weight 35 lbs and there's a time limit before your collar self-destructs). However, that is a subversion: not only are you expected to leave the vast majority of them there, they are also worth less (per pound) than many other items. This in turn sets up a Double Subversion: while the gold haul is probably a bit less awesome than the player was hoping for, the real prizes are the Sierra Madre Chips that can be used in the Sierra Madre's Matter Replicators to buy various useful and valuable items, among them Stimpaks and Weapon Repair Kits, two items you can never have enough of. After completing the campaign, you get your own vending machine, and a fresh supply of chips every three days.
Dead Money itself however is practically a video game adaptation of Tomb of Horrors, so calling it a Monty Haul campaign doesn't really do it justice...
Dragon Age: Origins has the "Soldier's Peak DLC". Accept a quest, spend 10 minutes mowing down underpowered Mooks, proceed to a vendor offering the best weapons and armor in the game. Granted, you might have to spend another 10 minutes mowing down money spiders and selling off their Vendor Trash... Still, the DLC takes about an hour to get through and is fairly difficult. And by the time you'll be able to afford and use that vendor's high-level equipment, you'll damn well need it.
Dragon Age: Origins – Awakening also throws more money at the player then they may know what to do with. This is in stark contrast to the original campaign, where money is quite tight (barring DLCs and a single developer oversight that requires a lot of loading screens). Players new to the expansion may be shocked to find quest related NPCs requesting of them things on the order of dozens of gold pieces to commission some sort of order—a small fortune in the original campaign, pocket change in Awakening. With so much money available, you wonder why the Warden doesn't just hire a couple hundred mercenary soldiers to build up Amaranthine's otherwise vastly outnumbered forces.
The Architect Entertainment buildings in City of Heroes were created to allow players to write their own story arcs for other players to enjoy. Among most of the community, however, it's better known for its "farm" missions, as they're called. It's gotten to the point where some farms have earned "Hall of Fame" status for having so many favorable ratings among players. The earliest farms would take advantage of exploits to allow characters to go from creation to level cap in a single day of beating up defenseless enemies for disproportionately high XP. Naturally, the devs did not take kindly to this, and closed such loopholes whenever they find them, even banning the most egregious exploiters of them. Newer farms aren't quite as efficient, and are not cracked down upon as forcefully, however.
In the MMO Star Trek Online, it is ridiculously easy to gain the best gear and equipment for little risk via earning special ingame currency, and via crafting said gear using easiy farmed resources called "data samples". Also any dropped gear that's rare but you don't need can be sold on the player exchange for insane amounts of ingame currency. Many players become multi-millionaires this way, making said currency practically worthless - save for some special items like rare bridge officers that boost your in-space abilities (only one type which is so rare that it sells for millions on its own).
Subverted now for crafting when the game went free to play, as crafting high level items now requires Dilithium. Which can only be produced in quantities of 8,000 a day, and most items require around 12,000 Dilithium to craft. Only players willing to spend real money on the game to trade it to others for Dilithium can really craft the high level items.
Firkraag's Dungeon (Chapter 2) in particular had a metric ton of unique loot, including a +3 bow, a Dragon helm/shield/sword/scales (all with bonus resistances), Full Plate Mail +1, the best two-handed sword in the game (Carsomyr), a shield cloak, not to mention a ton of ammunition and various minor weapons. Any party that ventures in will find enough gear to make a killing on the resale market.
To be fair, Carsomyr is only available if you decide to fight the Bonus Boss inside. And unless you cheat, defeating him in chapter 2 is a tenuous suggestion at best.
For a savvy party, going into Watcher's Keep in Amn and stealthing your way past most of the enemies on the first two floors will net you tons of +3 and +4 weapons and ammo to sell, long before you'd get anything else in the main plotline that compares.
The Pokémon games, starting from Black and White and progressing as it goes on. In the first pair of Kalos titles, as a mandatory part of the story, you can choose to take on a Pokemon that will completely wreck the rest of the game singlehandedly, about a quarter of the way in. This is not getting into the numerous useful, initially rare items random NPCs will throw at you for no effort on your part other than talking to them.
Truth in Television for most professional GMs who are promoting a game system for a company. Their real job (other than to run a great session) is to get people to want to buy the source books themselves. Naturally, its a good idea to end the session with the player characters fabulously wealthy and successful, because they leave the table feeling good about themselves (this is also why the company usually supplies their GMs with Feelies to pass out as prizes or participation gifts).
A notable non-Table Top Games example comes from a review of DayZ, whose creator allowed a couple of gaming journalists to spawn the basic pistol and some ammunition by server command where normally they'd be completely unarmed at first. This was the exact same weapon that used to be part of everyone's Starter Equipment until it was removed, somewhat controversially, and the player base did not find this at all amusing.