Risk

Risk is a strategy board game produced by Parker Brothers (now a division of Hasbro). Winning Moves also makes a classic 1959 version. It was invented by French film director Albert Lamorisse and originally released in 1957 as La Conquête du Monde ("The Conquest of the World") in France. It was later bought by Parker Brothers and released in 1959 with some modifications to the rules as Risk: The Continental Game, then as Risk: The Game of Global Domination.

Risk is a turn-based game for two to six players. The standard version is played on a board depicting a political map of the Earth, divided into forty-two territories, which are grouped into six continents. The object of the game is to occupy every territory on the board and in doing so, eliminate the other players. Players control armies with which they attempt to capture territories from other players, with results determined by dice rolls.

Gameplay
There are three main phases to a player's turn: getting and placing new armies, attacking, and fortifying.

Getting and placing new armies
Players draft new armies and then distribute these pieces to any of their territories at the beginning of their turn. The number of armies a player may draft hinges upon three factors: number of territories owned; continent bonuses; and redeeming Risk cards. To calculate the number of armies drafted for number of territories owned, players divide their total number of territories by three and round down to the nearest integer. If this result is less than three, round up to three armies. Players also receive bonus armies for occupying an entire continent (see table above). Lastly, players may receive armies for turning in a set of three Risk cards. A set may consist of the three different army units (soldier, cavalry, artillery) or be three of a kind (e.g. all three cards have cavalry pictures). If the player has five cards, the player must trade in a set. The first set to be turned is worth 4 reinforcements; the second is worth 6; third, 8; fourth, 10; fifth, 12; sixth, 15 and for every additional set thereafter 5 more armies than the previous set turned in; the number of reinforcements received is shown by a Golden Cavalry which moves along a grid every time a set is traded in. The probability of having a tradeable set of cards when holding three cards is 33.3(repeating)%, 74% holding four cards, and 100% holding 5 cards.

The player places these armies on any of their territories. If a player owns one or more of the territories depicted on the set of turned in cards, the player may choose one of these territories to be awarded two additional armies that must be placed in that territory.

Attacking
When it is a player's turn to attack, the player can only attack territories that are adjacent to or connected by a sea-lane to a territory already held. A battle's outcome is decided by rolling dice. The attacking player attacks with an army, rolling up to three dice. At least one unit must remain behind in the attacking territory not involved in the attack, as a territory may never be left unoccupied. Before the attacker rolls, the defender must resist the attack with either one or two armies (using at most the number of armies currently occupying the defended territory) by rolling one or two dice. Each player's highest rolling die is compared, as is their second-highest (if both players roll more than one). In each comparison, the highest number wins. The defender wins in the event of a tie. With each dice comparison, the loser removes one army from their territory from the game board. Any extra dice are disregarded and do not affect the results.

If an attack eliminates the final defending army within a territory, the attacker then must occupy the newly conquered territory with at least the number of attacking armies used in the last round of attack. There is no limit to the total number of additional armies that may be sent in to occupy, providing at least one army remains behind in the original attacking territory. Players may attack any number of territories any number of times before yielding the turn to the next player. Attacking is optional; a player may decline to attack at all during their turn. If an attacker occupies a defender's last territory, the defender is eliminated from the game and the attacker acquires all of the defender's Risk cards. If the conquering player then holds five or more cards, the player must trade in sets until the player has fewer than five. The gained armies are placed immediately.

If, at the end of attacking, at least one territory was conquered that turn, the player draws a Risk card from the deck.

Fortifying
When finished attacking and before passing the turn over to the next player, a player has the option to maneuver any number of armies from a single territory occupied by the player into an adjacent territory occupied by the same player. This is sometimes referred to as a "free move". Under an alternate rule, the maneuvering armies may travel through as many territories to their final destination as desired, providing that all involved pass-through territories are connected and occupied by that same player. As always, at least one army must be left in the originating territory. However, the player can only distribute between two territories. Play then proceeds clockwise to the next player.

Basic strategy
The official rulebook gives three basic strategy tips for the classic rules: Holding continents is the most common way to increase reinforcements. Players often attempt to gain control of Australia early in the game, since Australia is the only continent that can be successfully defended by heavily fortifying one country (either Siam or Indonesia). Generally, continents with fewer borders are easier to defend as they possess fewer points that can be attacked by other players. South America has 2 access points, North America and Africa each have 3, Europe has 4, and Asia has 5.
 * 1) Players should control entire continents to get the bonus reinforcement armies.
 * 2) Players should watch their borders for buildups of armies that could imply an upcoming attack.
 * 3) Players should build up armies on their own borders for better defense.

Generally, it is thought advisable to hold Risk cards until they can be turned in for maximum reinforcements. This is especially true earlier on in gameplay, because extra armies make a greater difference in the beginning of the game. Eliminating a weak player who holds a large number of Risk cards is also a good strategy, since players who eliminate their opponents get possession of their opponents' Risk cards. In this case, trading in Risk cards earlier may help acquire the necessary troops. If the conquering player has six or more Risk cards after taking the cards of another player, the cards must be immediately turned in for reinforcements until the player has fewer than five cards and then may continue attacking.

"Turtling" is a defensive strategy where a player who feels vulnerable tries to become too expensive to be removed while remaining a threat to harass other players. The objective of this strategy is to make it so that none of the other players want to weaken you, as somebody else may defeat you and take your cards. A player using this strategy might remain in the game all the way to later stages and then mount an attack on the weakest player and start a chain elimination to remove one player after another to win the game. The player who uses this strategy is called a Turtle. The term was popularised in Real-time Strategy games where a player creates a defensive perimeter or a "Turtle Shell" around the base of operations. Solutions to counteract this strategy using cooperation have been proposed by Ehsan Honary.

Alliances
The rules of Risk neither endorse nor prohibit alliances or truces. Thus players often form unofficial treaties for various reasons, such as safeguarding themselves from attacks on one border while they concentrate their forces elsewhere, or eliminating a player who has grown too strong. Alliance making can be one of the most important elements of the game, and it adds human interaction to a decidedly probabilistic game. Some players allow trading of Risk cards, but only during their turn. This optional rule makes alliances more powerful.

Dice probabilities
Defenders always win ties when dice are rolled. This gives the defending player the advantage in "one-on-one" fights, but the attacker's ability to use more dice offsets this advantage, as indicated in the dice probability charts below. Actually capturing a territory depends on the number of attacking and defending armies and the associated probabilities can be expressed analytically using Markov chains, or studied numerically using stochastic simulation.

It is always advantageous to roll the maximum number of dice, unless an attacker wishes to avoid moving men into a 'dead-end' territory, in which case he may choose to roll fewer than three.

Outcome probabilities of one die roll in Risk (various numbers of dice). Thus when rolling three dice against two dice (the most each player can roll), three against one, or two against one, the attacker has a slight advantage, otherwise the defender has an advantage. When large armies face off, a player will tend to gain a greater advantage over his opponent by attacking rather than defending. (Multiple opponents can change the prudence of such a strategy, however.) The number of attacking armies does not include the minimum one army that must be left behind in the territory (e.g. if the attacking territory has 10 armies total, it has maximum 9 attacking armies).

There are online tools available to compute the outcome of whole campaigns (i.e. the attacking of several territories in a row).

Official licensed Risk games
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 * Castle Risk (1986) – A version focusing only on Europe in which each player has a castle, and the player's goal is to protect the castle from attack. Castle Risk was the first version of Risk released after 27 years of production to depart from standard play. Although it was unsuccessful, it introduced many concepts integrated into later versions of Risk.
 * Risk: Édition Napoléon (1999) – Adds generals, fortresses, and naval units.
 * Risk: Édition Napoléon: Extension Empire Ottoman (2000) – Adds a sixth player to Risk: Édition Napoléon.
 * Risk: 2210 A.D. (2001) – An award winning futuristic version, produced by Avalon Hill, another division of Hasbro. The game features moon territories, ocean territories and commander units and offers a number of official and unofficial expansions.
 * Risk: the Lord of the Rings (2002) – 2–4 player version based on northern Middle-earth.
 * Risk: the Lord of the Rings: Gondor & Mordor Expansion Set (2003) – Extension to Risk: the Lord of the Rings, also includes a 2-player Siege of Minas Tirith mini-game.
 * Risk: the Lord of the Rings: Trilogy Edition (2003) – Combines the first two Lord of the Rings versions, but does not include the Siege of Minas Tirith mini-game.
 * Risk Godstorm (2004) – A version based on the mythological pantheons of various ancient civilizations; produced by Avalon Hill.
 * Risk: Star Wars: Clone Wars Edition (2005) – Set in the Star Wars universe during the Clone Wars. The player can fight on the side of the Separatists or the Republic, using either the classic Risk rules or the Clone Wars variations where altruism pays off.
 * Risk Express (2006) – Designed by Reiner Knizia as part of Hasbro's Express line of games (although not as part of the US-released series). Roll different combinations of infantry, cavalry, artillery & generals to capture the territory cards.
 * Risk: Star Wars Original Trilogy Edition (2006) – Set during the Galactic Civil War, players play as the Galactic Empire, the Rebel Alliance, or the Hutts. This version is unique in that each of the factions has a different set of goals and victory conditions.
 * Risk Junior: Narnia (2006) – Based on The Lion, the Witch and the Wardrobe, players can play as either the forces of Aslan or as the forces of the White Witch.
 * Risk: The Transformers Edition (2007) – Based on the Transformers film, players can either play on the side of the Autobots or the Decepticons on a Cybertron stylised map.
 * Risk: Black OPS (2008) – Limited edition released in early 2008. Print run was limited to a 1000 copies. Most of the copies were given to people in the board game industry to test out new rules for up coming editions.
 * Risk: Balance of Power (2008) – Based on a European map.
 * Risk 1959 (2008) – Winning Moves Games USA released a reproduction of the original game of Risk from 1959. It includes all the original graphics, wood pieces, and individual plastic storage boxes.
 * Risk: Reinvention (2009) – Also called Risk Factor or Risk Revised Edition. This is the commercial released version of Black Ops. It features capitals, cities, missions, and very thin pieces shaped like arrows. It is also available with different components (wooden map, wooden cube pieces, etc.) as Risk Onyx Edition.
 * Risk: Halo Wars Collector's Edition (2009) – Includes UNSC, Covenant, and The Flood. It has 42 territories and 6 sectors.
 * Risk: Factions (2010) – a licensed video game version of the game developed by Electronic Arts, and distributed on Xbox Live Arcade and PlayStation Network. Includes a "classic" mode which allows games played using standard original rules, and a "Factions" variation on the rules.
 * Risk: Metal Gear Solid (2011)
 * Risk Legacy (2011/12) – A game that has permanent changes to the game board and cards each time you play.
 * Risk: Halo Legendary (2012)
 * Risk: Starcraft (2012) – Four play modes with three different races. Each race has two unique hero units.
 * Risk: Star Wars Saga Edition (2013)
 * Risk: Mass Effect Galaxy at War Edition (2013)
 * Risk: The Walking Dead Survival Edition (2013)
 * Risk: Battlefield Rogue (2013)
 * Risk: Plants Vs Zombies (2013)
 * Risk: Doctor Who (2013) – Forbidden Planet Exclusive, UK
 * Risk: Game of Thrones (2015)
 * Risk: Star Wars Edition Game (2015)
 * Risk: Marvel Cinematic Universe Board Game (2015)
 * Risk: Transformers Decepticon Invasion of Earth (2014/15)