Friday, August 13, 2010


1. Introduction

Time and again here on BGG I have encountered people smugly sure they know all about Risk and how to play. The most obvious dead-giveaways that these people have no idea what they're talking about are the beliefs that the game is dominated by luck and that it takes some horrendously long time to play. The fact is that Risk combines three elements of play with a natural victory condition which is well suited to those elements. The reason so many people have a bad experience playing Risk is that they have bought into the conventional wisdom of how to play, and that conventional wisdom is frankly wrong. This strategy article seeks to explain how the game should be played if one wishes to consistently win the game (and have fun)-- albeit no strategy will eliminate losses completely. The discussion is about standard rules world conquest Risk; missions and other variants are ignored.

The three elements of play in Risk are armies obtained for territories (including continents), the combat mechanism and the cards. The topology of the board plays into these. Of the elements listed, the number of armies received for territories is exactly predictable. The card combinations are governed by strict probabilities and are relatively easy to anticipate. The combat mechanism, i.e., the dice rolls, are not predictable for any given individual roll but are very strictly governed by a Gaussian probability distribution (due to the comparison of dice rolled) and therefore predictable in the aggregate. In other words, in spite of the random elements of the game, strategy can and will dominate play if players know what they're doing.

Some will admit the above, but then object that Risk takes too long for what it is as a game. I've been playing for about thirty years and have certainly played hundreds of games if not into the thousands. My consistent experience time and again is that with capable players, a 6 player game will typically take 5 or 6 rounds. Games with fewer players will take correspondingly more rounds.

2. The basic ideas

Risk is about ruthlessly crushing one's opponents. If one wants a "family game" perhaps one might try The Settlers of Catan. If playing with a child (or other person) who will cry if eliminated, Risk is definitely not the game to play.

Generally speaking, attackers have the advantage. True, on equal numbers of dice, the advantage favors the defender, but of course the attacker gets to choose when, where and whence the attack occurs and so again the advantage will generally go to the attacker.

Hoarding armies for a future attack will often work with inexperienced players but will get one crushed by experienced players. This point is one of the biggest faults of the conventional wisdom. People think taking one territory per turn to get a Risk card but otherwise piling armies onto one's territories is a sure-fire winning strategy. On the contrary, against competent opponents, it's a sure-fire losing strategy. I'll explain how and why below.

At the heart of Risk is an arms race. In the end, the person who consistently gets the most armies and uses them most effectively will win. Therefore the person to attack (all else being equal) is whichever player is in the lead, unless attacking someone else will get one more armies. These additional armies may be from taking an entire continent or from eliminating the player entirely and thereby taking his Risk cards. If no player is significantly more in the lead than any other (at least among one's opponents) and no specific attacks will garner one more armies than any other attack, then one should go for whoever is most vulnerable. If one can take a bite out of another player's continent, one should generally do it even if one has no intention of holding the territory involved.

3. Start of the game

When I was a kid, we did what was then the standard out of the box rules and took territories by placing one army on a territory in turns until all territories were selected. The principles I'll talk about in how to play will naturally extend into the choice of territories if one plays this way.

Later versions of Risk used as standard what some of the earlier versions of Risk included as an optional "quick set-up". Namely, one temporarily removes the wilds and then shuffles and shares out the territory cards as evenly as possible among all players so that all territories on the board are randomly distributed among the players. This means of set-up makes for a better game in many ways and has become standard. In this article it is assumed that this type of set-up is in use hereafter.

When distributing one's armies, one needs bear in mind two opposing factors. Apart from complete continents, all territories are of equal value when collecting armies. On the other hand, territories which are adjacent or very nearly adjacent (e.g., separated by one territory) can form the nucleus of a strategically powerful base from which to build. One also generally knows the order of play before armies are placed. If one is fortunate enough to have a starting position where a territory cannot immediately be attacked (as for example Madagascar if one also controls South Africa and East Africa) one should not waste the armies on that country but use them to fortify the buffer territories (South Africa and East Africa in the example). Armies should then be divided as evenly as possible among all territories that can immediately be attacked. If one then has armies left over so that all these territories cannot get an equal amount, the priority should go to those territories which are mutually adjacent or else nearly so.

4. Weighing the odds

A number of otherwise well educated people dismiss Risk as being "all luck" because the core combat mechanism relies on dice rolls. Since no given roll of the dice can be predicted, they argue, one has no control over the outcomes in the game and hence the game is all luck supposedly. The flaw in the logic here is that while each individual die roll is random with even probability (i.e., in a fair die no one number is no likely than any other) Risk uses a comparison of dice. Any comparison or combination of dice is governed by a strict Gaussian distribution. Rolling equal numbers of dice, the defender has the advantage because ties go to the defender, but the attacker can roll more dice. The last point gives a very slight advantage to the attacker. The shift is small but enough to have a marked effect on the overall statistics. Roughly what this means is that if one could roll an infinite number of times for an attacker using three dice versus a defender using two dice (or respectively two and one), a good approximation is that the attacker would win slightly more than half the time and the defender slightly less than half. The variance in this result decreases as one rolls more times according to thee square root of the number of rolls. Admittedly, this is not a fine detailed analysis of the probabilities nor is this discussion meant to be. Rather this gives one a good enough approximation to inform strategic play of the game.

What all this means in practice is that as a rule of thumb, if one expects to conquer a territory, one should if possible begin with at least twice the number of armies being attacked (preferably more) if the number of armies involved is large. When the outcome is decided by only a few rolls of the dice, the statistics don't mean much. Flukes do happen, but the more rolls involved, the less influence fluke rolls will have over the outcome.

A corollary of the statistics is that the more often one attacks and the more armies are involved in the process, the more control one has over the game. If every player just distributes his armies over the territories he has and attacks only one or two territories a turn throughout the game, then Risk will become largely luck-driven. Conventional wisdom claims this is a strategy for winning the game. On the contrary, what I've described is a recipe for abrogating one's control of the outcome in the game; experienced players who know how to use the key mechanic of the game will time and again crush players who only attack a territory or two a turn throughout the game. Often the winning approach to Risk is derided as merely aggressive play and assumed to lose more often than it wins. Here one needs to bear in mind that the strategy also involves when and where to attack. Aggressive play does not mean one ignores defensive play; used properly, aggressive play is defensive play.

5. Where and when to attack

Especially in the early part of the game, armies are in limited supply. So one wants to use what armies one has as effectively as possible. The key element here is minimizing one's borders. Building up armies on territories that cannot be attacked (and by the same token cannot therefore attack either) is largely a waste. The benefits are that one delays an advance but one cannot stop an advance which takes the number of armies into account. Yet all armies in territories that cannot attack are just that fewer armies one has to pursue the objective of the game-- conquering the board. In the same way, the fewer attackable territories one has, the more powerful those territories can be with the same number of armies.

This aspect of the game lies at the core of the legend of the supposedly unbeatable tactic of taking control of Australia and using it as a base to build out from. Only Indonesia can be attacked from outside Australia, and one can only make such an attack from Siam. So one could pile all of one's available armies onto Indonesia or better yet Siam, with Australia controlled behind it by one army a territory. Such a position is certainly powerful, but invincible it's not.

The Australia legend however raises two important points germane to choosing a territory to attack. Namely, continental bonuses should not be ignored. The amount of armies added may be small in absolute terms, but relative to the number of armies one gets on a turn, the number is usually significant. That means one should always take a continent if one can, but likewise one should always deny a continent to another player if one can. That's the second point raised. A player gets armies for territories controlled at the start of his turn. If someone takes Ukraine from a player who had control of Europe, that latter player won't get the bonus armies for Europe that turn-- even if he takes Ukraine back immediately. If an opponent is building up a strong position, one absolutely should attack it-- the sooner the better. If one cannot take the territory involved, one should attack anyway because the attacker has the advantage; one will destroy more armies that will otherwise be used against one as the attacker in a losing contest than as the defender on the opponent's turn. That is defensive play. Ideally one takes the territory one would be attacked from, but if not then one either eliminates the ability to attack from the given territory or at least substantially weakens it. Opponents should not be allowed to build up powerful positions, even one has to make a drive specially to do so and even if one does not keep all the territories one takes. That brings up another point as well, one should not be afraid to lose territories. So long as one conquers more territories than one loses, one is advancing toward victory.

What this all means is that when choosing one's target for attack, one should give priority to reducing opponents' ability to attack one's own territory, then to taking continents or creating a buffer zone of territories about a continent one has and finally to minimizing one's borders. These considerations need not be mutually exclusive. Also one should not think in terms of a single territory to attack but a series of them whenever possible.

When one should attack is generally whenever possible, but one should try to arrange a string of attacks which allows one (after fortification) to leave one's attackable territories with a number of armies comparable to those armies from which each territory can be attacked. When more than one territory can attack, leaving the territory with a number comparable to the total of all territories it could be attacked from is generally a waste if that means one chooses not to attack one of those regions. The more territory one takes, the one armies one gets and the less one's opponents will correspondingly get as well.

6. Distributing armies

When placing armies, one should place armies for the attacks one wishes to make during the turn, whether the armies are from territories held and continental bonuses or from a set of Risk cards. Whenever possible one should before attacking figure out a continuous non-branching path of territories to attack one after another. For example, if one controls Brazil and wants to take all of South America, one should attack Argentina, then Peru and then Venezuela with armies starting in Brazil rather than attacking Peru first and then dividing the armies between an attack into Argentina and one into Venezuela. After all, one must move as many armies as one rolled dice in the last attack when taking a territory. Attacking Argentina first, in the example, is generally superior to attacking Venezuela first because Argentina dead-ends and so stops any advance. (Forcing an opponent to split up armies in this manner is a great defensive tactic because it forces the opponent to effectively waste armies.)

When figuring out if one has enough armies for a series of attacks, a minimum is roughly twice as many armies as the armies occupying the territories to be attacked plus an additional army for each territory one will have to occupy in the process. More armies is better.

One should remember as well that the number of Risk cards a player has is open information,although what cards they have specifically is not. If at any point, one eliminates a player and thereby acquires a total of six or more Risk cards, one must turn in a set immediately-- even in the middle of a turn. So eliminating a player and using his Risk cards to fuel an additional series of attacks is a great tactic whenever possible. So if one has a choice between eliminating various opponents on a turn, if possible pick the opponent that will give one a total of six Risk cards or if that's not possible whichever opponent will get one the most Risk cards.

7. Risk cards

A number of people complain because Risk cards increase in value as the game progresses, but the increasing value of sets of Risk cards is an essential factor of the game. It works both as a leveling mechanism to mitigate the advantage the leader in the game has and to force players to play for the win. The cards fuel an escalation of an arms race which should result a situation where each player must either eliminate other players or face being eliminated himself.

So a natural question arises whether one should hold onto Risk cards, even if one has a set, so that when one turns cards in the set will be worth more. The answer is a resounding no because armies are the limiting factor in how much one conquers on a turn. The more one conquers on a given turn, the more armies one gets and the correspondingly less opponents will get. Moreover holding onto Risk cards makes one a valuable target.

One should also not forget that if one controls the territory on a Risk card in the set one hands in, one will get two additional armies on that territory-- albeit one territory per set. So where one has a choice one should choose the cards in a set to give one additional armies on a territory, again if possible. If one has more than one territory in the set, then one should choose the territory most useful for attack actually or at least potentially as the one on which to place the extra armies. Of course if one can do so when one controls more than one territory shown on one's Risk cards, one should hold onto one of the cards with a controlled territory for the next set.

8. Conclusion

In general, the more armies one gets now the better. The more attacks one makes on a turn, the more control one has over what happens in the game. One should not interpret this to mean that one should make pointless or wasteful attacks that serve only to weaken one's position. If the attacks have a likelihood or success or are worth the Risk of failure, then one should make the attacks-- otherwise not. Preemptive attacks, even when one will not take the territory being attacked, are the heart of defense in this game. Attacks with equal numbers of dice should be avoided. Whether as attacker or defender, one should always roll the most dice possible.

When taking continents, having a buffer zone about that continent is strongly advisable if one can do it. The rule to remember is that one should attack with at least twice as many units as the defending armies, plus one army for each territory to be taken, as a minimum. If the case is borderline, one should usually Risk the attack but stop when the number of armies one has becomes comparable to the number of defending armies.

Players get armies for territories held at the start of a turn and so taking territories one cannot hold but which deny an opponent armies is always advisable. If in the end one does in fact manage to hold the territory, so much the better. To this end, wherever possible attackable borders should be minimized. Armies on those borders should be maximized. When at the end of a turn moving reinforcements, one should place them on the territory from which one most wishes to attack next turn.

If an opponent has done so with a large amount of armies, one should attack the territory, especially if one is massively outnumbered. If such cases, one stops when one can no longer roll three dice unless one has reduced the opponent to a single dice or taken the territory.

If all players play in the manner described, then borders will ebb and flow like tides. All one has to do to win is have one's losses on the whole outweighed by one's gains.