Strategic dominance

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.