Game Theory: Luck Mitigation

It’s the final round. The game hangs in the balance. All that stands between you and the win is one last roll of the dice. Literally anything but a 4 will give you what you need to win. You hold your breath and roll. The last die spins on its corner for what seems like 10 seconds, before picking a side to land on. It’s a 4. Once again, victory has been snatched out of your hands at the last moment. Practically everyone has encountered a situation like this before. There is no doubt that the element of the unknown brought by luck can lead to some tense and thrilling events in a game.

But sometimes, it just doesn’t seem fair. The game defies probability turn after turn, completely putting you out of the running. Even when you make every effort to plan for bad luck, forces beyond your control dictate your success completely. You can spend an entire game planning, but the 1000:1 odds never come out in your favor. You reflect on the game, and realize that nothing you could have done would have changed the outcome. Angered by the arbitrary nature of the game, you vow never to play it again. Another game has been ruined by luck.

Today, I look at ways to avoid having the luck overshadow all of the careful thought and planning that went into the game. And maybe help players have a better time playing.

I’ll start by giving the motivation in more concrete terms. I’m not saying that randomness is bad in a game. Random factors can make a game replayable, since you won’t always have a fixed outcome, and can’t rely on the same strategy every game. Random factors can also be an extra element that the strategy of the game is built on top of, like many gambling games. But luck is something slightly different. Luck is unbalanced randomness perceived by the player. 

Luck can drive players away from a game. A game that is entirely luck based is completely outside of the player’s control, but these games are usually easy to identify. The vast majority of games include some amount of luck, without eliminating player control. The relationship between a game’s apparent level of luck, and the actual level of luck is an important way for players to identify games. 

I will venture that very rarely are there complaints that a game had less luck than assumed (except, of course, in the case of cheating, in which one player experiences less luck than he or she is is supposed to). For the majority of cases, an aspect of the game that appears random but turns out to be under more control than expected is usually seen as a clever mechanic. The danger is the opposite relationship, when the apparent low level of luck is actually highly dependent on luck. This tends to leave a player feeling powerless in the middle of a game that they have become invested in, which can create negative feelings toward the game. There are several things a designer can do to avoid having players encounter that situation. Generally, it depends on giving a player more control.

The most obvious solution is to eliminate random elements completely, and have the game be completely strategy based. This does not exclude hidden information, just elements that are unknown to all players but can help or hurt an individual more. Chess is a great example.

Another approach is to have randomness affect all players equally. Random setup (at the start of a round or start of the game) is a form of this. This could also be an outcome that affects players proportionally or in a balanced way, so those who lose more get more advantage, or all players are brought toward some average state. The important part is that players have (approximately) the same opportunity to take advantage of the luck and random factors.

A standard way of reducing the effect of randomness is to balance each action, in a cost and benefit sense, so that players who are fortunate to get more powerful cards have a harder time paying the cost. In this way, the net benefits of random elements be equally distributed, so that no random value is inherently “better” than any other. San Juan does a great job of this, where the best randomly obtained buildings have the highest costs. A subset of this is to balance between a little now and more later, so that time is the cost being paid. Macao uses this, letting players select random values but linking the cost (time) and benefit (number of goods)

A designer can also balance risk and reward, where either the cost is paid or reward is gained immediately and the other has a chance of happening later. However, this acts more like shifting the randomness to a different part of the game, rather than eliminating it. This method can either decrease or increase luck, depending on whether the luck comes from something the player has more or less control over. Giving the player the option of where and how to shift luck, is itself a way a of mitigating luck by giving the player more control.

So far these options change the amount of actual luck in the game. But one of the most often cited problems in games involving luck is that the events defy probability. A game may be balanced according to a standard distribution of events, but at any given time, a random sequence of events can fall outside that distribution. Settlers of Catan is often the poster child for this, as it is possible to play an entire game without rolling a given number. In a game so dependent on limited production, being unable to get a resource that should be plentiful can be aggravating. There are ways to reduce the effect of a random distribution.

The first approach is to make the game pseudo-random. Replace a purely random roll of two 6-sided dice with 36 cards showing all of the potential outcomes. There are cards for Catan that do just this. When you reach the bottom of the deck you shuffle and start again. If it is important to maintain some uncertainty, reshuffle before reaching the end of the deck. This will give a slightly less than perfect distribution. The smaller the set of possible outcomes is, the easier this is to achieve.

For a more statistical approach, this problem is often caused by small sample size. The law of large numbers states that the larger your sample is, the more likely it is to match the ideal distribution. (Yes, I just described the solution to a probability problem in terms of probability. That’s math for you.) Practically speaking though, the more times a player encounters a random element, the less each random outcome contributes to the game, and the more likely the player is to encounter the expected distribution of outcomes. In even simpler terms, if the player sees more random events, the overall effect on the player will be less random. This is true on a turn-to-turn basis, or by lengthening the game.

Push your luck mechanics rely on the sample size effects. When a player has good luck, he can keep playing, increasing the sample size and the likelihood that a bad roll will be encountered. Conversely, by reducing the number of random events the player experiences, each event must be assessed separately, so the designer must figure out how important each random event should be.

A designer can also directly let the player change the distribution. Let the players “stack the deck”, so to speak, or make it change through events in the game. New Bedford has a random selection of whales each round, but as players take whales, the distribution changes, and players weigh their choices against how favorable the distribution is. Card counting gives players insight into the changing distribution. It might be bad for casinos, but it can be a good way for players to adjust the luck in their favor in a game.

Besides addressing the total amount of randomness, and changing the random events themselves, the final category is approaches to mitigate luck through external mechanics.

From a purely design side, extra factors can be included to balance the game. This is sometimes considered “rubber-banding”. Ideally, unlucky players are helped more, and lucky players are helped less, or even hurt. This is sometimes criticized as a lazy design approach to artificially balance the game. But when viewed as a form of luck mitigation, not as a purely balance issue, it opens up possibilities to the designer that normal approaches to balance may not address. The designer also needs to be careful or this system can be “gamed” to a skilled player’s advantage. However, that can also be a design goal, to add another level of strategy.

Interaction between players is a powerful way to offset luck. Let players gang up on lucky ones, or help out unlucky ones. By adding interaction between players, often through trade or negotiation, it also changes the valuation of the lucky events. A lucky player who gets a lot of resources will value them less than an unlucky player. Allowing them to trade naturally mitigates that difference in luck. Again, the designer must consider what other strategies this adds to the game, since it can also lead to things like king-making and leader-bashing. Interaction’s main drawback is that players often stop trading and negotiating toward the end, which makes the last few rounds more luck-driven–just when you need to remove luck the most. This can be a strategic element, too, where a player in the lead may convince others to stop trading in order to secure victory.

I have a final note about the difference between perceived luck and actual luck. Especially in games with luck mitigating mechanics, a player who has not yet mastered them will perceive the game to be more luck-based than a more skilled player. In this situation, it may be useful to explain in the rules, how the luck is mitigated, so that new players have a more equal footing.

The opposite of luck is choice. Too much luck can overwhelm choice. And certainly luck can be removed, leaving only choice. Consider how players will experience the luck in the game, and remember that not all players like the same amount of luck in a game. The methods for controlling luck that I discussed above can be a way to bring balance into the game or bring entirely new avenues of strategy to a design. If your game is honest about the role of luck, players will appreciate it for it. And that’s lucky for you as a designer.

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  1. #1 by Isaac Shalev on January 20, 2014 - 9:58 am

    One issue you didn’t address regarding luck mitigation and games is the idea of leverage. Sure, you can try and smooth out outliers by using a deck of cards instead of dice or whatever. But that won’t really solve your problem, because each time you roll the dice, there’s different amount of value at stake. A random fight in Yaktusk in Risk has low leverage, but the same fight in Greenland to break the North American continent has high leverage – the expected value of the outcomes is much different. A game could exhibit perfectly fair and even distribution, but the 3-4 high leverage moments could be the ones where the least-probable results occurred, or where one player got all the breaks. If you’re trying to reduce the impacts of luck on a game, while still keeping randomness as a spice, I think you’re better off creating rubber band mechanisms that give unlucky players compensation some compensation.

    @KindFortress

    • #2 by Oakleaf Games on January 20, 2014 - 10:42 am

      Good point!
      If you address a purely random distribution with a pseudo-random or balanced distribution, it can introduce outcome bias. Card counting takes advantage of this in poker because knowing the previous cards gives you information about the current outcome, while the assumed distribution of outcomes is unbiased. In your example, you might wait to attack until the number of favorable outcomes favors you. Similar to how increasing the sample size approaches the theoretical distribution for achieved outcomes, increasing the sampling pool reduces bias to approach the theoretical distribution of possible outcomes. One way to achieve this is to reset the distribution, for example by shuffling the deck before every card is used.
      This also provides another luck mitigation tool, of being able to actively bias outcomes. Dominion uses this as a mechanism by literally changing the deck you draw from. (Dominion is, in fact, a great case study to demonstrate a lot of these tools, so I might write an entire article about that)
      The bottom line is not every luck mitigation tool is equally effective or balanced, depending on how randomness is used in the game, and some tools may even produce a negative effect on the game. Thanks for making the point!

      • #3 by Isaac Shalev on January 20, 2014 - 10:52 am

        Card-counting as a strategy for dealing with leverage is a good point. In some games, like blackjack, it can work well, b/c you have perfect control over the leverage of a hand, based on how you’re betting it. Not all games feature that level of control – in Settlers, if you need to get the stone roll now to build your city, you need it now. You typically can’t afford to wait another round b/c of hand management concerns. I think leverage is a key point missing from many conversations about probability, and in fact, leverage may be a great concept to build a game around!

  2. #4 by Russel Fleming on July 21, 2017 - 2:09 pm

    Thanks for this article. Between this and a recent D6 Generation podcast, I moved away from using dice for combat to a card system that felt so rewarding during play testing this morning vs the roll dice and hope mechanism. My 10 year old son (co-designer) and I were high-fiving each other as we battle it out, whereas 3 previous play tests with dice ended up with 2 blowouts and less decision making.

    When I explained the new system to my wife, a moderate gamer, she was exited by the idea as she hates it when the dice “roll bad” in games like Rum and Bones. I have a luck mitigation system that may be like Maulifaux’s cheat system (it is inspired by it although I am not sure what th details of the system are…will research later).

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