Strategic dominance

Imagine playing a game. You want to win. You might have a plan. This plan is called a strategy.

Some plans are better than others. One plan might always win. It works no matter what others do. This is a dominant strategy.

Other plans are not as good. They might always lose. You should not pick those. A smart player avoids them.

Sometimes two plans are just okay. One might be better sometimes. One might be better most of the time.

Players use these ideas to win. They look for the best moves. They try to pick the best plan.

Imagine you are playing a game. You want to get the best result. You make a plan for your moves. In game theory, this plan is called a strategy.

Some plans are better than others. A strategy might be "strictly dominant." This means it always gives a better result than another plan. It works no matter what your opponent does. Other plans might be "strictly dominated." This means they always give a worse result. A smart player will not pick those plans.

Sometimes, a plan is only a little better. We call this "weak dominance." This plan is at least as good as another. It might even be better sometimes.

Players often assume everyone is "rational." This means players try to get what they want most. They also assume "common knowledge." This means everyone knows the rules. They also know that everyone else knows the rules. Players can use these ideas to solve games. They can remove bad plans one by one. This helps them find the best way to play.

Imagine you are playing a game and want the best possible result. You need a complete plan for every choice you might face. In game theory, this full plan is called a strategy. Some strategies are much better than others. A strategy is "strictly dominant" if it always gives a better result than another option. This works no matter how your opponent plays the game. A strategy is "strictly dominated" if it always gives a worse result. Smart players will not pick these bad plans.

Sometimes, a choice is not always better, but it is never worse. We call this "weak dominance." A weakly dominant strategy gives at least as good a result as another plan. It might even give a better result in some situations. You can also have plans that do not dominate each other at all. In Rock, Paper, Scissors, no single move is always the best. Choosing scissors is better if the opponent throws rock. However, choosing rock is better if the opponent throws scissors.

To solve these games, we assume players are "rational." This means players act to get what they prefer most. This preference is often called "utility." It can be money, or it could be something else like justice. We also assume "common knowledge." This means everyone knows the rules and the possible results. Everyone also knows that every other player knows the rules too. This shared understanding helps players guess what others might do.

One way to find the best plan is a method called "iterated elimination." This is a way to simplify a game step by step. First, you remove all the strictly dominated strategies. No rational player would ever pick those bad plans. This leaves you with a new, smaller game. You then look for new dominated strategies in that smaller game. You repeat this process over and over until the game is solved. This process works because we assume rationality is common knowledge.

These ideas help us understand special points called "Nash equilibria." A Nash equilibrium is a state where no player wants to change their plan alone. If a player has a strictly dominant strategy, they will use it in these equilibria. If both players have one, it is called a "dominant strategy equilibrium." However, these points are not always "efficient." This means there might be other outcomes that are better for everyone. The famous "Prisoner's Dilemma" is a classic example of this.

In the field of game theory, players use specific plans to guide their actions. A strategy is a complete contingent plan for a player. This means it is a full specification of how a player will behave. It describes every action a player would take at every possible decision point. In a game, these decision points are known as information sets. By understanding dominance, players can determine which strategies are most likely to succeed.

Dominance occurs when one strategy is clearly superior to another. A player compares two strategies, A and B, to see which provides a better outcome. If choosing B always gives a better result than choosing A, regardless of what opponents do, then B strictly dominates A. Conversely, if B always gives a worse outcome than A, then B is strictly dominated by A. A rational player will never choose a strictly dominated strategy. This is because a strictly dominated strategy can never be part of a Nash equilibrium.

There is also a subtler version called weak dominance. Strategy B weakly dominates strategy A if B always gives at least as good an outcome as A. Additionally, there must be at least one situation where B gives a better outcome than A. If B is weakly dominated by A, it means there is at least one set of opponent actions where B performs worse than A. In all other cases, B performs the same as A. Unlike strict dominance, weakly dominated strategies can still be part of a Nash equilibrium.

Some strategies do not dominate each other at all. In these cases, the best choice depends entirely on the opponent's actions. For example, in Rock, Paper, Scissors, no single move is always the best. Choosing scissors is better if the opponent throws rock. However, choosing rock is better if the opponent throws scissors. In such games, players must make value judgments based on what they believe others will do.

To analyze these choices, mathematicians use two key assumptions: rationality and common knowledge. Rationality assumes that players act to achieve what they prefer most. This preference is measured as "utility." Utility can represent monetary gain, but it can also represent minimizing discomfort or promoting justice. Common knowledge means every player knows the rules and the payoffs. Furthermore, every player knows that every other player knows the rules, and so on, indefinitely.

One way to solve complex games is through iterated elimination. This process is known as IESDS when using strictly dominated strategies. First, you remove all strictly dominated strategies from the game. This creates a new, smaller game. You then repeat the process by looking for new dominated strategies in that smaller version. You continue this until the game is simplified. This method works because rationality is assumed to be common knowledge among all players.

There is also a process called the iterated elimination of weakly dominated strategies, or IEWDS. This procedure removes any strategy that never yields a higher payoff and sometimes yields a lower payoff. While this generalizes the strict method, it is more complex. The results of IEWDS can depend on the order in which strategies are removed. It can also exclude some Nash equilibria that were present in the original game.

Dominance is closely linked to Nash equilibria. If a player has a strictly dominant strategy, they will always play it in any Nash equilibrium. If both players have a strictly dominant strategy, the game has one unique "dominant strategy equilibrium." However, these equilibria are not always efficient. An efficient outcome is one that is better for everyone involved. The Prisoner's Dilemma is a famous example where the Nash equilibrium is not the most efficient result for the players.