The "Risk Problem"
Earlier this week, I playtested a few games made by my fellow students in the Game Design class at Carnegie Mellon's Entertainment Technology Center. There were a variety of tabletop and digital games alike, and it was interesting to see what made them succeed or not work out.
One friend's game, in particular, was conceptually a fun game but it had a balance problem. As I was giving my feedback to him after the playtest, it occurred to me that there are popular games out there that suffer from the same problem. In the end, what I ended up saying to him exactly was "Your game has 'the Risk problem'". I amused myself quite a bit with this phrase, and figured it would be a great starting point for this blog post.
What Is the Risk Problem?
Here is a question to everyone who played Risk - have you ever played the game until a proper end state? The answer will probably be no for most of you. Roughly the last one-fourth to one-third of a typical Risk game is incredibly boring, or even frustrating, for a game this popular.
The reason is that whoever has gained a noticeable edge over their opponent(s) throughout the course of the game is surely going to win, and each turn becomes a formality to establish this victory officially. The losing side does have, perhaps, a one in a million chance to turn the game around, but the unlikelihood of such a reward kills any incentive for both the winning and the losing sides to spend another hour or two playing the game to reach the end state.
Risk is not the only game that suffers from this problem - Monopoly is another well-known example. It is a similarly long and slow-paced board game, and the winner of the game becomes evident pretty much at the halfway point. In the space of digital games, old school real time strategy games such as the Age of Empires or Stronghold series are notable for having the same problem.
What Causes the Problem?
Capitalism.
In a typical capitalist economy, the amount you earn per timeframe has a positive correlation with the amount you already own. This means that the rich get richer at a faster rate than the poor can recover. The same goes for Risk and Monopoly as well - the reason why anyone who gets a noticeable edge is surely going to win is because they are very likely to earn more resources per turn than the other players, which in turn they can spend to crush their opponents further. In Risk, this resource is in the form of troops on the map, whereas it is the game money in Monopoly.
The statement "the rich get richer at a faster rate than the poor," of course, does not take into account one's ability to manage their capital and costs of living, and therefore does not hold true for all cases in real life. It does not exactly parallel the economy in "capitalist games" like Risk and Monopoly either. If it did, every game would end in a tie given that players start with equal resources, since players would earn the same amount of reward every turn.
Games, at least most of them, have an element of randomness in their reward scheme. In Monopoly, the dice rolls determine which square a player's token lands on, and depending on the square, the player may end up paying rent, going to jail, or having the option to buy the square, among many other outcomes. Risk, on the other hand, has a number of random factors in rewarding the player. First of all, the player start states are randomized since different regions on the map are of differing value. During each turn, dice rolls determine how many troops a player loses and whether they succeed in their invasion(s). And then, at the end of each turn, if the player succeeded in at least one invasion, they pick a card at random, and turning in three cards whose troop symbols form a pattern in the beginning of another turn gives the player a troop boost depending on the pattern.
Given these elements of randomness in both games, you may be puzzled at why the "Risk problem" still exists. I will attempt to explain this with a bit of math below, and I promise to try my best in keeping the math as simple as possible for those who are not a fan of math.
Let's say in-game capital advantage, meaning how good your state compared to other players is, is denoted by the value V. For Risk, this value would depend on the number of troops you have on the map, the countries you own, and the troop symbols on your collected cards. For Monopoly, it would depend on the monetary advantage or disadvantage you have and the expected amount of rent you would collect from your properties in the future compared to what other players would be expected to collect. In your nth turn in the game, this value would be represented by Vₙ. The formula for Vₙ, given that we know the state for the previous turn (that is, Vₙ₋₁), would look something like this:
Vₙ = Vₙ₋₁ + Xₙ + Rₙ
Xₙ is the expected reward the player gains in that round. Rₙ, on the other hand, is a random value indicating whether the player performed worse or better in their dice rolls than the expected value. They could be condensed into a single random variable with an expected value Xₙ as well, though it will be easier to keep them separate for comparison purposes later on.
In Risk, Xₙ is driven by the troop supplement in the beginning of the round, and this troop supplement, apart from occasions where the player uses their collected cards, has a roughly linear relationship with Vₙ - the base number is simply the number of countries owned divided by three, and the bonuses added on top come from continents held in whole, with the amount of the bonus being approximately at a fixed ratio to the number of countries in that continent. Meanwhile, towards the middle of the game and onwards, the variance of R stays rather constant across turns - since all players have concentrated their armies on a couple of choke points at this point as part of the dominant strategy, it can be safely assumed that almost all attacks will involve the attacker attacking with three dice and the defender defending with two, making the variance only dependent on the number of attacks the player chooses to make. The number of attacks a player makes is largely independent from Vₙ for Vₙ values above a certain threshold - players with many troops will simply attack until they can move up their frontier to a secure point where they can reconcentrate their troops, or until they disrupt another player's continental troop bonus. If they don't think they have a chance of accomplishing either of these, most players will only attempt one attack to get another troop symbol card - again, independent from Vₙ.
Due to the positive linear correlation between Xₙ and Vₙ, a player who gains an edge over other players could be expected to quickly increase the gap. The lack of such a correlation between Rₙ and Vₙ, on the other hand, means that the greater the value of Vₙ, the more overruled the randomness element will be as the dispersion (variance-to-mean ratio) of the combined reward (Xₙ + Rₙ) gets lower and lower. This translates to lower chances of leading players falling back. This is exactly how winners emerge early in the game - their rewards are almost guaranteed to be higher than the rest, and they are already doing better.
It should be said that the lack of correlation between Vₙ and Rₙ also stops leading players from increasing the gap even further, however, this does not carry much meaning given the already quick increase in Xₙ in relation to Vₙ.
How Can This Problem Be Prevented?
A simple way to get rid of the Risk problem is to keep Xₙ independent from Vₙ. This means that there is no correlation between the reward a player gains from an action and the player's in-game capital at the time the action is taken. For a game in which rewards carry intrinsic value such as in-game currency, however, this seemingly brings another problem. The game now appears faced with the dilemma of losing its flow and sense of progression by keeping all gameplay at a constant difficulty in order to keep the reward at a constant expected value, or having an unsatisfying reward scheme by rewarding the player equally on average after accomplishing hard and easy tasks alike. This, however, will not pose a problem if the game can lead the player into thinking that they are still progressing.
Many games fool players into thinking that they are earning more rewards for actions they take when they have high in-game capital. This usually comes in an unbothersome form of currency devaluation. Take any traditional MMO war game, for instance, in which a Level 3 troop may cost twice the amount of a Level 2 troop while it only has four-thirds of the ability, and your army is capped at a certain number of troops - you will be upgrading your troops regardless, effectively devaluing your currency. Another example that is common in a lot of resource-based games is that despite the usually polynomial improvement pattern of in-game assets, asset upgrades will cost exponentially higher at each level, meaning that the player will have to wait longer to amass enough resources for each upgrade even if they kept upgrading their resource generators in parallel. From a purchasing power stand point, this translates to the player working longer to earn the same amount, no matter how high the numbers in their resource reserves may appear. Clash of Clans is perhaps one of the most popular examples from recent years that uses both of these methods.
Once the game designers are able to fool the player in the described sense, they can pull off even more tricks with resources to prevent leading players from taking off. Specifically, they can make the game's Xₙ decrease with an increasing Vₙ. In fact, the asset upgrade example in the previous paragraph describes exactly that.
