# Difference between Dominant strategy and Winning Strategy

I am a little bit confused by the definition of dominant strategy and winning strategy, what's the difference between them.

Following are the definitions taken from wikipedia, below every definition I added some thoughts. If you fell that my thoughts are not correct or ambigues please let me know.

Dominant strategy.In game theory, dominant strategy (commonly called simply dominance) occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance.

The first interesting note is that the term dominant strategy is applied only to two different strategies, and more precisely it's should be defined as "A is dominant strategy over B", actually the notion "dominant strategy" might be relevant to the game itself in this case the dominant strategy A dominates over all other strategies of the player.

Winning strategy.A strategy is winning if the player following it must necessarily win, no matter what his opponent plays.

This notion sounds like stronger notion than dominant strategy, because having winning strategy its guaranteed that the player wins if he follows the strategy no matter what the opponent does, on the other hand dominant strategy does not guaranteed the win, it only says that dominant strategy guarantees the best possible outcome for the player over all possible player's strategies and no matter what the opponent does.

I would to emphasize one common property for the dominant strategy and the winning strategy they both determined by the outcome of the game and don't depend on the strategy of the opponent.

In my opinion, winning strategy is always a dominant strategy, but vice verse is not always true.

Do you agree with the intuitive understand and thought?

Your intuition seems mostly correct. However don't forget that not all games are winner vs. loser, see e.g. Coordination games. The meaning of a winning strategy is somewhat meaningless there, unless you define the winner to be the one with the greater outcome (or allow multiple winners which gain more than a certain threshold). Dominance is still an important tool for those strategies.

There is another problem, because your definition of domination includes "better" in any case. This is a strict dominance. A weak dominance means "at least as good as" in all cases and "better" in at least one case, where case means an opponents strategy (or a set of strategies for multiple opponents).

1. Possible outcomes are only $\{\text{win},\text{lose}\}$: A winning strategy for you results in $\text{win}$ for any strategy of the opponent and is thus weakly dominant compared to any non-winning strategy, if there is one. But in many games an opponent can play so badly, that some non-winning strategies lead to a win nevertheless and so they are not strictly dominated by a winning strategy. But more important than the dominance of winning strategies is the dominance between non-winning strategies of your opponent. If s/he has a weakly dominant strategy relative to all other strategies and you can win against this single strategy, you have a winning strategy.

2. Possible outcomes are some set linear ordered set $A$ and you win, if your profit is greater than that of the opponent (this is not a usual definition, one would use other tools like equilibria in this case). For example in the following table each player has two strategies and yours correspond to the rows (each cell is: your outcome, others outcome): $$\matrix{10,12&\mid &10,12\\\hline\\7,4 &\mid&7,4}$$ If you choose the second strategy you'll always win and if you choose the first one, you'll lose, but your outcome is higher, so choosing row $1$ dominates a winning strategy.

Resumé: Dominance is a much broader concept and for many but not all games winning strategies are weakly dominant to at least one other strategy.

• It's a great answer, thank you very much! First point - "that some non-winning strategies lead to a win nevertheless and so they are not strictly dominated by a winning strategy", however it might be strictly dominated by winning strategy, right? Second point - "If s/he has a weakly dominant strategy relative to all other strategies and you can win against this single strategy, you have a winning strategy" it sounds intuitive, but actually this is the case because weakly dominant strategy is the solution for the opponent?
– com
Apr 10 '13 at 16:50
• 1) Yes, this is possible. Take the example for the second part of the answer and change the outcome : 10,12 to lose and (7,4) to win. Then the second one is winning strategy that strictly dominates the other strategy. 2) I don't really understand your question, but weakly dominance is enough, since you only need to be sure s/he can't do better, you don't need that s/he always will do worse using other strategies. Apr 10 '13 at 17:00