Strategic dominance is a state in game theory that occurs when a strategy that a player can use leads to better outcomes for them than alternative strategies.
Accordingly, a strategy is dominant if it leads a player to better outcomes than alternative strategies (i.e., it dominates the alternative strategies). Conversely, a strategy is dominated if it leads a player to worse outcomes than alternative strategies (i.e., it is dominated by the alternative strategies).
The concept of strategic dominance has important implications in various domains, so it’s beneficial to understand it. As such, in the following article you will learn more about strategic dominance, and see how accounting for it can help you make better decisions.
Dominant strategies
What is a dominant strategy
A dominant strategy is a strategy that leads to better outcomes for a player than other available strategies (while taking into account the strategies that other players can use).
Dominant strategies can be strictly dominant or weakly dominant:
- A strategy is strictly (or strongly) dominant if it leads to better outcomes than alternative strategies.
- A strategy is weakly dominant if it leads to equal or better outcomes than alternative strategies.
If a strategy is strictly dominant, then it is also weakly dominant, since if it leads to better outcomes than alternative strategies, then it’s possible to say that it leads to equal or better outcomes than alternative strategies. However, if a strategy is weakly dominant, then it may also be strictly dominant, but not necessarily.
In general, a strategy that is both strictly and weakly dominant is referred to as a “strictly dominant strategy”, whereas a strategy that is only weakly dominant is referred to as a “weakly dominant strategy”.
In addition, it’s possible to say that one strategy dominates certain other strategies in particular. For example, if strategy A leads to equal or better outcomes than strategy B, then strategy A weakly dominates strategy B. Similarly, if strategy A leads to better outcomes than strategy B, then strategy A strictly dominates strategy B.
Finally, note that a strategy may dominate another strategy, but still not be the most dominant strategy in the game as a whole. For example, strategy A may strictly dominate strategy B, but it may be the case that strategy C strictly dominates both.
Examples of dominant strategies
An example of a dominant strategy appears in a situation where you can either get $10 now, or you can flip a coin, and if it lands on heads then you get $10, but if it lands on tails then you get nothing. Here, the dominant strategy is to take the money upfront, since this will lead to an outcome that is as good as or better than flipping the coin, because you will either make as much money or make $10 more.
Note that, in this example, the dominant strategy (taking the money upfront) is only weakly dominant, because it sometimes leads to an outcome that is equal to the outcome of the other strategy.
Another example of a dominant strategy appears in a situation where a company needs to choose between two strategies—online and offline advertising—that are equal in all aspects except for their expected payoff. If online advertising will lead to a payoff of $20,000, whereas offline advertising will lead to a payoff of either $15,000 or $10,000, depending on where their competitors advertise, then online advertising is the dominant strategy, because it will lead to a higher payoff.
Note that, in this example, the dominant strategy (online advertising) is strictly dominant, because it always leads to an outcome that is better than the outcome of the other strategy (offline advertising).
Dominated strategies
What is a dominated strategy
A dominated strategy is a strategy that leads to worse outcomes for a player than other available strategies (while taking into account the strategies that other players can use).
Dominated strategies can be strictly dominated or weakly dominated:
- A strategy is strictly (or strongly) dominated if it leads to worse outcomes than alternative strategies.
- A strategy is weakly dominated if it leads to equal or worse outcomes than alternative strategies.
If a strategy is strictly dominated, then it is also weakly dominated, since if it leads to worse outcomes than alternative strategies, then it’s possible to say that it leads to equal or worse outcomes than alternative strategies. However, if a strategy is weakly dominated, then it may also be strictly dominated, but not necessarily.
In general, a strategy that is both strictly and weakly dominated is referred to as a “strictly dominated strategy”, whereas a strategy that is only weakly dominant is referred to as a “weakly dominated strategy”.
Finally, it’s possible to say that one strategy is dominated by certain other strategies in particular. For example, if strategy A leads to equal or worse outcomes than strategy B, then strategy A is weakly dominated by strategy B. Similarly, if strategy A leads to worse outcomes than strategy B, then strategy A strictly dominated by strategy B.
Examples of dominated strategies
An example of a dominated strategy appears in a situation where you can either get $10 now, or you can flip a coin, and if it lands on heads then you get $10, but if it lands on tails then you get nothing. Here, the dominated strategy is to flip the coin, since this will lead to an outcome that is as good as or worse than taking the money upfront, because you will either make as much money or make $10 less.
Note that, in this example, the dominated strategy (flipping the coin) is only weakly dominated, because it sometimes leads to an outcome that is equal to the outcome of the other strategy.
Another example of a dominated strategy appears in a situation where a company needs to choose between two strategies—online and offline advertising—that are equal in all aspects except for their expected payoff. If online advertising will lead to a payoff of $20,000, whereas offline advertising will lead to a payoff of either $15,000 or $10,000, depending on where their competitors advertise, then offline advertising is the dominated strategy, because it will lead to a lower payoff.
Note that, in this example, the dominated strategy (offline advertising) is strictly dominated, because it always leads to an outcome that is worse than the outcome of the other strategy (online advertising).
Strategic dominance
Example of strategic dominance
Consider a situation where two companies, called Startupo and Megacorp, are competing in a new market.
This market has one product that is sold in two different versions: the consumer version and the professional version. Both versions are equally profitable for the company selling them, and the companies’ only concern is to make more money by selling more units. However, due to practical constraints, a company can only manufacture one type of product
Most people in the market (80%) are interested in the consumer version, and only a few (20%) are interested in the professional version. Each company can decide whether it wants to sell the consumer version or the professional version of the product. If both companies decide to sell the same type of product, then the companies will have to split the market for that product. Otherwise, each company will have the full consumer or professional market for itself.
As such, each company has two possible strategies to choose from, and there are four possible outcomes to the scenario:
- Both companies enter the consumer market. This means that the companies split the consumer market (which accounts for 80% of the total market share), and that each company therefore gets 40% of the total market share.
- Both companies enter the professional market. This means that the companies split the professional market (which accounts for 20% of the total market share), and that each company therefore gets 10% of the total market share.
- Startupo enters the consumer market, and Megacorp enters the professional market. This means that Startupo gets the full consumer market (80% of the total market share), and Megacorp gets the full professional market (20% of the total market share).
- Megacorp enters the consumer market, and Startupo enters the professional market. This means that Megacorp gets the full consumer market (80% of the total market share), and Startupo gets the full professional market (20% of the total market share).
Based on this, if a company chooses to enter the consumer market, then it will get either 40% or 80% of the total market share. Conversely, if a company chooses to enter the professional market, then it will get either 10% or 20% of the total market share.
Accordingly, for both companies, the (strictly) dominant strategy is to enter the consumer market, since they will end up with a bigger market share this way, regardless of which move the other company makes.
Conversely, for both companies, the (strictly) dominated strategy is to enter the professional market, since they will end up with a smaller market share this way, regardless of which move the other company makes.
Note that this scenario can become more complex by adding factors that often appear in real life, such as additional players, additional products, and different profitability margins for different products. However, although these additional factors make it more difficult to analyze the strategic dominance in a situation, the basic idea behind dominant and dominated strategies remains the same regardless of this added complexity.
Strategic dominance in single-player games
In scenarios where there is only one player, there can still be dominant and dominated strategies.
For example, consider a situation where you are walking along a street, and you need to eventually cross the road. Just as you reach the first of two identical crosswalks that you can use, the crosswalk light turns red. You now have two strategies to choose from:
- Wait for the light at this crosswalk to turn green.
- Keep walking until you reach the next crosswalk, and then cross there.
Given that your goal is to minimize the time spent waiting at the crosswalk, the dominant strategy in this case is to keep going until you reach the next sidewalk. This is because, if you decide to cross at the current crosswalk, you’re going to have to wait for the full length of time that it takes the light to turn green. Conversely, if you keep going until you reach the next crosswalk, then once you get there, one of three things will happen:
- You will reach the second crosswalk while the light is green, in which case you won’t have to wait at all, which represents an outcome that is better than the outcome that you would have gotten if you chose to wait at the first crosswalk.
- You will reach the second crosswalk while the light is already red, in which case you will have to wait for less time than you would have had to wait at the first crosswalk, which represents an outcome that is better than the outcome that you would have gotten if you chose to wait at the first crosswalk.
- You will reach the second crosswalk just as the light turns red again, in which case you will have to wait the same length of time that you would have had to wait at the first crosswalk, which represents an outcome that is equal to the outcome that you would have gotten if you chose to wait at the first crosswalk.
Since the strategy of going for the next crosswalk leads to an outcome that is equal to or better than the outcome of waiting at the current crosswalk, it’s the (weakly) dominant strategy in this case.
Note that, in this game, though there is only one player, the concept of “luck”, in the form of whether or not the next light will be green or red, can be viewed as representing a second player, when it comes to assessing the dominance of your strategies.
However, the concept of strategic dominance can occur in even simpler situations, where there is no element of luck. For example, consider a situation where you need to choose between buying one of two identical products, with the only difference between them being that one costs $5 and the other costs $10. Here, if your goal is to minimize the amount of money you spend, then buying the cheaper product is the dominant strategy.
Games with no strategic dominance
There are situations where there is no strategic dominance, meaning that none of the available strategies are dominant or dominated.
For example, in the game Rock, Paper, Scissors, each player can choose one of three possible moves, which lead to a win, a loss, or a draw with equal probability, depending on which move the other player makes:
- Rock wins against scissors, loses against paper, and draws against rock.
- Paper wins against rock, loses against scissors, and draws against paper.
- Scissors win against paper, lose against rock, and draw against scissors.
Accordingly, none of the available strategies dominates the others, because none of the strategies is guaranteed to lead to an outcome that is as good as or better than the other strategies. Rather, there is a cycle-based (non-transitive) relation between the strategies, since choosing rock is better if the other player chooses scissors, and choosing scissors is better if the other player chooses paper, but choosing paper is better if the other person chooses rock.
In addition, note that if multiple strategies always lead to the same outcomes, then they are said to be equivalent. For example, if a company needs to choose between online and offline advertising based on the profit that each option leads to, and both options lead to a profit of $20,000, then these two options are equivalent to one another, at least as long as no other related outcomes are taken into account.
Using strategic dominance to guide your moves
To use strategic dominance to guide your moves, you should first assess the situation that you’re in, by identifying all the possible moves that you and other players can make, as well as the outcomes of those moves, and the favorability of each outcome. Once you have mapped the full game tree, you can determine the dominance of the strategies available to you, and use this information in order to choose the optimal strategy available to you, by preferring dominant strategies or ruling out dominated ones.
For example, let’s say you have three possible strategies, called A, B, and C:
- If strategy A leads to better outcomes than strategies B and C, then strategy A is dominant, and you should use it.
- If strategy A leads to an equal outcome as strategy B, but both lead to better outcomes than strategy C, then strategy C is dominated, and you should avoid it.
In addition, it can sometimes be beneficial to rule out your and your opponents’ strictly dominated strategies, which are always inferior to alternatives strategies, before re-assessing the available moves, in a process called iterated elimination of strictly dominated strategies (or iterative deletion of strictly dominated strategies). It’s possible to also eliminate weakly dominated strategies in a similar manner, but the elimination process can be more complex in that case.
Finally, in situations where multiple available strategies lead to equal outcomes, you can pick one of them at random. In the case of games with multiple players, doing this has the added advantage of helping you make moves that are difficult for other players to predict, which can be beneficial in some situations.
Note: and associated concept is mixed strategies, which involves choosing out of several strategies at random (based on some probability distribution), in order to avoid being predictable.
Using strategic dominance to predict people’s behavior
Understanding the concept of strategic dominance can help you predict other people’s behavior, which can allow you to better prepare for the moves that they will make.
Specifically, there are two key assumptions that you should keep in mind:
- People will generally prefer to use their dominant strategies.
- People will generally prefer to avoid their dominated strategies.
However, it’s also important to keep in mind that in certain situations, people might not pick a strategy that you think is dominant, or might pick a strategy that you think is dominated, for reasons such as:
- They know something that you don’t.
- They value outcomes in a different way than you expect them to.
- They are irrational, so they won’t do what’s best for them from a strategic perspective.
- They lack certain knowledge about the structure of the game (i.e., who are the players, and which strategies and payoffs are available), or about the other players (i.e., about the other players’ rationality and knowledge of the game and the other players).
You can account for such possibilities in various ways. For example, you can incorporate these possibilities into your predictions, in order to make the predictions more accurate, for example by noting that someone tends to choose their strategy based on ego rather than logic. Similarly, you can consider these possibilities when estimating the certainty associated with your predictions, even if you don’t modify the predictions themselves, in order to understand how uncertain your predictions are.
Overall, you can use the concept of strategic dominance to predict people’s behavior, by expecting them to generally use dominant strategies and avoid dominated ones. However, when doing this, it’s important to account for various factors that could interfere with your predictions, such as people’s irrationality, or your incomplete knowledge regarding people’s available moves.
Note: the term “mutual knowledge” refers to something that every player in a game knows. The term “common knowledge” refers to something that every player in a game knows, and that every player knows that every player knows, and so on.
Related concepts
There are several concepts that are often discussed in relation to strategic dominance.
One such concept is strategy profile (also known as an action profile or a strategy combination), which is a specific combination of strategies undertaken by each player in a game.
A notable type of strategy profile is the Nash equilibrium, which occurs when no player in a game can gain anything by changing their own strategy unilaterally (i.e., while other players keep their strategies unchanged). There can be any number of Nash equilibria in a game, including none at all in certain games, and a Nash equilibrium is often self-enforcing once it’s reached.
A famous example that illustrates the concept of Nash equilibrium is the prisoner’s dilemma, where two prisoners, who have no way of communicating with each other, have two options: they can either betray the other prisoner, or they can stay silent. This situation can result in the following outcomes:
- If both prisoners betray each other, they each get 2 years in prison.
- If both prisoners stay silent, they each get 1 year in prison.
- If one prisoner betrays the other, and the other stays silent, the one who betrayed gets to go free, while the one who was betrayed gets 3 years in prison.
For both prisoners, the dominant strategy is to betray the other prisoner:
- If the first prisoner stays silent, it’s better for the second prisoner to betray them, since this means that the second prisoner gets to go free instead of spending a year in prison.
- If the first prisoner betrays the second, it’s better for the second prisoner to betray the first one in return, since this means that the second prisoner will get only 2 years in prison, instead of 3.
Here, the Nash equilibrium involves both players selecting their dominant strategy of betraying the other person, which leads them both to spend 2 years in prison, since neither benefits from changing their strategy unless the other prisoner changes their strategy too.
The prisoner’s dilemma illustrates another relevant concept, called the dominant-strategy equilibrium, which occurs when a certain strategy profile involves every player using a dominant strategy. Every dominant-strategy equilibrium is a also Nash equilibrium, but not every Nash equilibrium is a dominant-strategy equilibrium.
Finally, the prisoner’s dilemma also illustrates another relevant concept, called Pareto optimality (or Pareto efficiency), which occurs when a strategy profile cannot be changed in a way that makes all players better off (i.e., when it’s impossible to improve the outcome for any one player without making the outcome worse for at least one other player).
As shown in the example of the prisoner’s dilemma, a Nash equilibrium is not necessarily Pareto optimal, and vice versa. Specifically, in this case, the Nash equilibrium involves both players betraying each other, since this represents their dominant strategy, and neither player benefits from staying silent unless the other player changes their strategy to stay silent too. However, this strategy profile is not Pareto optimal, since the outcome would be improved for both players if they both stay silent (as they would spend only 1 year in prison instead of 2). Furthermore, the Pareto-optimal strategy profile here involves both players staying silent, but this profile is not a Nash equilibrium, since each player can improve their own situation by betraying the other, assuming that the other person stays silent.
Note: various distinctions can be applied to the concepts that are mentioned here. For example, when it comes to the dominant-strategy equilibrium, it’s possible to draw a distinction between a strict dominant-strategy equilibrium and a weak dominant-strategy equilibrium.
Summary and conclusions
- Strategic dominance is a state in game theory that occurs when a strategy that a player can use leads to better outcomes for them than alternative strategies.
- A strategy is dominant if it leads to better outcomes than alternative strategies, and dominated if it leads to worse outcomes than alternative strategies.
- When deciding how to act, you should assess your available strategies, and then prefer to use dominant strategies and avoid dominated ones; you can potentially also insert some randomness into your decisions, if you will benefit from making it harder for other players to predict your moves.
- You can predict people’s behavior by expecting them to prefer to use dominant strategies and avoid dominated ones, but it’s important to account for factors that could interfere with this prediction, such as irrationality, intentional randomness, or your incomplete knowledge of people’s motives and available strategies.
- Not every scenario has strategic dominance; for example, strategies may sometimes lead to equivalent outcomes, or their outcome might depend entirely on which strategies other players choose.
If you found this article interesting and you want to learn more about game theory, take a look at “The Art of Strategy: A Game Theorist’s Guide to Success in Business and Life“. It’s a good read for someone looking to understand basic game theory and how it applies to real-life situations.