Dominated Strategy
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Dominated Strategy
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An outcome where one player has superior tactics regardless of the other players
The dominant strategy in game theory refers to a situation where one player has superior tactics regardless of how their opponent may play. Holding all factors constant, that player enjoys an upper hand in the game over the opposition. It means, regardless of the strategies employed by the opponent, the dominant player will always dictate the outcome.
In game theory, players employ different independent strategies to optimize their decision-making with the goal of beating the opponent. Players in an oligopolistic market , military, managers, consumers, or games like the chase, often use game theory as a strategic tool.
In game theory, the outcomes of the actors are different depending on their actions. Some players enjoy an upper hand, while others are less fortunate. The dominant strategy describes a state where one of the players has a superior tactic that always leads to a winning outcome, despite the opponent’s employed choice of strategy.
In game theory , the following are the outcomes players can expect:
In some situations, one player enjoys a strict advantage over their opponent. It means that, no matter how good the losing party’s tactic is, the dominant strategy will always prevail. Here, there is no other possible strategy the opponent can use to alter their odds.
In a weakly dominant outcome, the dominant player dominates the game but against some strategies, only weakly dominates.
In an equivalent outcome, none of the actors benefit or lose against each other. They each choose the one optimal result that is fair for both players. In case one of the players selects the alternative, it would mean an outlandish gain or loss.
In an intransitive outcome, none of the above three outcomes are experienced – no equivalent, strictly, or weak dominant outcome results. The available outcome happens by chance. Either player can win, while the other loses depending on the strategy employed. Therefore, in this outcome, there is no well-defined approach to point to the dominance strategy.
The prisoner’s dilemma is a well-known example used to depict the predicament of two criminals, A and B, when facing persecution – i.e., car theft. During the trials, the prosecutor believes the two suspects might have committed an earlier crime but were not convicted – i.e., burglary. Since there is no hard evidence, the DA employs game theory to force a confession out of the two. They are offered a deal to rat each other out. The following are the terms:
The above information can be plotted in a payoff matrix as below:
The above example represents an equivalent outcome. This is because the dominant strategy for Suspect A and Suspect B will be to confess. Either suspect will always have a dilemma to choose between three years versus seven years and one year versus two years.
In case they committed the burglary, the only rational option available would be to choose the confession strategy. Neither will want to gamble with the loyalty of the other. This is because the alternative is worse – seven years versus a one-year jail term. They will both likely opt for a confession, and this stalemate situation is referred to as the Nash Equilibrium .
The Nash Equilibrium was introduced by American mathematician John Forbes Nash, Jr. in 1950 and was republished in 1952. It refers to an optimal state where the game is considered stable, and no actor has an incentive to change from their chosen strategy, taking all factors constant.
Players in this outcome must consider the affairs of the other. The alternative leaves either of the players in a lesser preferred state. In the prisoner’s dilemma example above, it is the first quadrant where both suspects get an optimal offer of a three-year jail term.
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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.
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 :
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.
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).
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 :
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 .
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).
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:
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.
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:
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:
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.
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:
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.
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 :
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.
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:
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:
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.
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 equilibriu
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