Strategic Dominance in Single Player Games

In scenarios where there is only one player, there can still be dominant and dominated strategies, and is true in games even as simple as Tic-Tac-Toe.

For example, consider a situation where you are walking along a street, and you know that you need to cross the road. As you reach the first of two possible (and identical) crosswalks that you can use, the light turns red. You now have two strategies to choose from:

  • Wait for the light at this crosswalk to turn green.
  • Keep going until you reach the next crosswalk, and then cross there.

Given that your goal is to minimize the amount of 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 just as the light turns red, 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.
  • 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 while the light is green, in which case you won’t have to wait at all, which also represents an outcome that is better than 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. However, the concept of strategic dominance can occur in even simpler conditions, 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 the products being that one costs $5, and the other costs $10. If your goal is to minimize the amount of money you spend, then clearly buying the cheaper product is the dominant strategy in this case.

Not every game has strategic dominance

There are situations where none of the possible strategies are dominant or dominated.

A strategy that is neither dominant nor dominated is referred to as an intransitive strategy, and a game with no strategic dominance is referred to as a non-transitive game. This represents the fact that there is no transitivity, meaning that just because strategy A is better than strategy B, and strategy B is better than strategy C, that doesn’t mean that strategy A is better than strategy C.

For example, in the case of the game Rock, Paper, Scissors, each player can choose one of three possible moves (i.e. rock, paper, or scissors). Each of these moves can lead to a win, a loss, or a draw, with equal probability, depending on what move that 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 wins against paper, loses against rock, and draws against scissors.

In this case, there is a preference loop with regards to which strategy leads to which outcome, since each strategy is preferable, to an equal degree, in different situations. Accordingly, none of the strategies is dominant over the other strategies, and the game is said to have no strategic dominance.

Using strategic dominance to pick your moves

Accounting for strategic dominance can help you make better decisions.

In order to take strategic dominance into account, you should first assess the situation that you’re in, and take into account all the possible moves that you and your opponents can make, as well as the outcomes of those moves, and the favorability of each outcome.

Once you have mapped the full game tree, and ranked your outcomes in order of favorability, you can move on to identify your dominant and dominated strategies, using the criteria that we saw above. Then, if you have a dominant strategy, use it. Otherwise, try to identify any dominated strategies that you have, and rule them out.

For example, let’s say you have three possible strategies, called AB, and C:

  • If strategy A is better than both strategy B and strategy C, that means that strategy A is a dominant strategy, and that you should therefore use it.
  • If strategy A is equal to strategy B, but both are better than strategy C, that means that strategy C is a dominated strategy, and that you should therefore rule it out.

In more complex situations, you might choose to repeatedly rule out your dominated strategies, in a process called iterated elimination of dominated strategies.

Eventually, doing this will leave you with either a single dominant strategy to use, or with a number of equally-viable strategies that you can choose from.

If all these strategies lead to outcomes that are perfectly equal to one another, one way to choose between them is to add a secondary variable, which will allow you to enhance the way you rank the favorability of the different outcomes.

For example, let’s say you’re trying to decide which laptop to buy, out of five possible options. At first, you might use price and reviews as the two most important factors to consider, which will allow you to filter out most of the options, until you end up with two possible alternatives to choose from, that are ranked equally based on these criteria.

At this stage, you can add an additional variable, such as warranty, that wasn’t important enough to take into consideration during the initial stages of your decision, but which you can now take into account in order to choose between these otherwise equal options.

In the rare situation where all strategies lead to outcomes that are truly equal, and there is absolutely no way for you to discern which one is better, you can simply pick one 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 your opponents to predict.

Using strategic dominance to predict your opponents’ behavior

Understanding the concept of strategic dominance can help you predict your opponents’ behavior, which can allow you to better prepare for the moves that they will make. Specifically, you should make two assumptions regarding how your opponents will act:

  • You should assume that your opponents will use their dominant strategies.
  • You should assume that your opponents will avoid their dominated strategies.

However, there are a few possible exceptions to this. For example, there are situations where other players might not pick a strategy that you think is dominant, or might pick a strategy that you think is dominated, for any of the following reasons:

  • They know something that you don’t.
  • They have different goals than you expect them to.
  • They’re acting irrationally, and not doing what’s best for them from a strategic perspective.

As such, the better you can estimate your opponents’ knowledge and preferences, and the more rational your opponents are, the more likely they are to act in a way that you expect them to, when it comes to strategic dominance.

Furthermore, the more strongly a certain strategy is dominant over others, the more confident you can be that an opponent will use it, and the more strongly a certain strategy is dominated by others, the more confident you can be that an opponent will avoid it.

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