Dominated Strategy in Game Theory Explained

A dominated strategy in game theory occurs when one player has a more dominant strategy over another player.

As we’ve seen, the equilibrium dominated strategies solution concept can be a useful tool. In the Prisoner’s Dilemma, once Player 1 realizes he has a dominant strategy — he doesn’t have to think about what Player 2 will do. Player 1 knows he can just play his dominant strategy and be better off than playing anything else. Games in which all players have dominant strategies are still strategic in the sense that payoff depends on what other players do, but best response does not.

But what if not all players have dominant strategies? What if none of the players do?

What Is Dominant Strategy?

A dominant strategy refers to a strategy that gives a player a higher payoff than anything else they could do, no matter what the other players are doing. If a player has a dominant strategy, expect them to use it. It then follows that a dominated strategy is a strategy that is always inferior to other strategies, regardless of what the other players are doing.

According to game theory, each player will develop their own strategy that they believe gives them the best chance of winning, with their decisions and actions leading to different outcomes. When a player has a dominant strategy, they gain an advantage over their opponents and are in prime position to secure the best possible outcome for themselves.   

How to Identify a Dominated Strategy in Game Theory

There are two types of dominated strategies to know: 

• Strictly dominated strategy: A strategy that always delivers a worse outcome than an alternative strategy, regardless of what strategy the opponent chooses. Players will anticipate a player with a strictly dominated strategy will never play it and choose their actions accordingly. 
• Weakly dominated strategy: A strategy that delivers an equal or worse outcome than an alternative strategy.

A strictly dominant strategy will then always deliver a better outcome than any other strategy, no matter what a player’s opponent does. But a weakly dominant strategy doesn’t always deliver a better outcome, producing an equal outcome against certain strategies.

The iterated deletion of dominated strategies is one common, but tedious, technique for solving games that do not have a strictly dominant strategy. In the first step of this process, at most one dominated strategy is removed from the strategy space of each of the players, since no rational player would ever play these strategies. This results in a new, smaller game.

Some strategies that weren’t dominated before may be dominated in the smaller game. The first step is repeated, creating a new, even smaller game, and so on. The process stops when no dominated strategy is found for any player.

This process is valid since it’s assumed that rationality among players is common knowledge. That is, each player knows that the rest of the players are rational, and each player knows that the rest of the players know that he knows that the rest of the players are rational, and so on.

How Nash Equilibrium Relates to Dominant and Dominated Strategies

Nash equilibrium is when a game reaches a state in which every player has determined a dominant strategy that leads to an optical outcome for each of them. They have no incentive to change their strategy, as long as the other players maintain their strategies. Nash equilibrium is then considered to be a dominant strategy equilibrium. 

Now, let’s return to the process of iteratively deleting dominated strategies discussed above. There are two versions of this process. The first (and preferred) version involves only eliminating strictly dominated strategies. If there is only one strategy for each player remaining after completing this process, that strategy set is the unique Nash equilibrium.

The second version involves eliminating both strictly and weakly dominated strategies. If there is a single strategy for each player at the end of the process, this strategy set is also a Nash equilibrium. Unlike the first process, eliminating weakly dominated strategies may eliminate some Nash equilibria. As a result, the Nash equilibrium found by eliminating weakly dominated strategies may not be the only Nash equilibrium. In some games, if we remove weakly dominated strategies in a different order, we may end up with a different Nash equilibrium.

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Dominated Strategies Example

Consider the following strategic situation, which we want to represent as a game.

Two bars, Bar A and Bar B, are located near each other in the city center. Each bar seeks to maximize revenue and chooses which price to set for a beer: $2, $4 or $5. Each bar has 60 potential customers, of which 20 are locals and 40 are tourists. Locals will buy from the bar setting the lowest price (and will choose randomly if the two bars set the same price). Tourists will choose a bar randomly in any case. 

Creating a Payoff Matrix

Once we’ve identified the players and the strategies, we can begin to create our payoff matrix:

Now, we can fill in the payoffs. We’re told that each bar only cares about maximizing revenue (number of beers sold multiplied by price.) Let’s look at the strategy profile ($2, $5). That is, when Bar A charges $2 and Bar B charges $5. In this case, all the locals will go to bar A, as will half the tourists. This gives Bar A a total of 40 beers sold at the price of $2 each, or $80 in revenue. Bar B only manages to attract half the tourists due to its higher price. This gives Bar B a total of 20 beers sold at a price of $5 each, or $100 in revenue.

We can then fill in the rest of the table, calculating revenues in the same way.

The first thing to note is that neither player has a dominant strategy. For Bar A, there is no price that will give it higher revenues than any other price it could have set, no matter what price Bar B sets. For example, a price of $4 gives Bar A higher payoffs than any other price if Bar B prices at $5. But what if Bar B does not price at $5 and instead prices its beer at $2? In that case, pricing at $4 is no longer Bar A’s best response. Pricing at $5 would be.

Strictly Dominated Strategies Example

The solution concept that we’ve developed so far — equilibrium dominated strategies — is not useful here.

The logic of equilibrium in dominant strategies is that if a player has a strategy that is always best, we would expect him to play it. But what if a player has a strategy that is always worse than some other strategy? It’s reasonable to expect him to never play a strategy that is always worse than another.

Player i’s strategy S is strictly dominated by another strategy S’ if, for every possible combination of strategies by all other players, S’ gives Player i higher payoffs than S.

Does either player have a strictly dominated strategy in the game above? Yes. The strategy “$2” always gives lower payoffs to Bar A than either “$4” or “$5.” Let’s see why the strategy is strictly dominated by the strategy $4 for Bar A:

• If Bar B is expected to play $2, Bar A can get $60 by playing $2 also and can get $80.
• If Bar B is expected to play $4, Bar A can get $80 by playing $2 also and can get $120 by playing $4
• If Bar B is expected to play $5, Bar A can get $80 by playing $2 also and can get $160 by playing $4.

Therefore, Bar A would never play the strategy $2. For any possible strategy by Bar A’s opponent, there is some strategy that gives a higher payoff than the $2 strategy. We can generalize this to say that rational players never play strictly dominated strategies.

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Iterated Deletion of Strictly Dominated Strategies Example

First Round of Deletion

So far, we’ve concluded that Bar A will never play $2, but this is a game of complete information. Bar B knows Bar A’s payoffs. So if we can spot that $2 will never be played because it is a strictly dominated strategy, Bar B can spot this, too. Bar B can thus reasonably expect that Bar A will never play $2.

This is a symmetric game, so the same holds for Bar B. Two dollars is a strictly dominated strategy for Bar B, and Bar A knows this, too. We can delete dominated strategies from the payoff matrix like so:

We’ve lost all cells corresponding to a strategy profile in which a dominated strategy is played. This is exactly our goal, which is to remove outcomes in which dominated strategies are played from the set of outcomes we are considering as feasible.

Second Round of Deletion

We’re now down to four strategy profiles (and four corresponding outcomes.) Let’s put ourselves in the shoes of Bar A again. Bar A knows that it will not play $2, and neither will its opponent. Bar A also knows that Bar B knows this. Now Bar A is comparing the strategies of $4 and $5 and notices that, once the strategy of $2 is taken off the table for both players, the strategy $5 is dominated by the strategy $4. That is:

• If B prices its beer at $4, matching that nets $120, and pricing at $5 nets $100.
• If B prices as $5, pricing at $4 gives $160 while matching at $5 gives $150

Pricing at $5 would only be a best response to $2, but $2 will never be played, so pricing at $5 is never a best response to any strategy a rational player would play. So, we can delete it from the matrix. The game is symmetric so the same reasoning holds for Bar B.

We are now down to exactly one strategy profile — both bars price their beers at $4. We used the iterated deletion of dominated strategies to arrive at this strategy profile. 

• Iterated deletion of dominated strategies: This is a method that involves first deleting any strictly dominated strategies from the original payoff matrix. Once this first step of deletion is completed, the reduced matrix is then studied and any strategies that are dominated in this new, reduced matrix are deleted. This process continues until no more strategies can be deleted.
• Rationalizable strategies: When iterated deletion of dominated strategies results in just one strategy profile, the game is said to be dominance solvable. More generally, the strategies that remain after a process of iterated deletion of strictly dominated strategies are known as rationalizable strategies
• Equilibrium in strictly dominant strategies: Whereas looking for an equilibrium in strictly dominant strategies involves finding a strategy that is always the best response for each player, looking for an equilibrium via iterated deletion involves iteratively discounting from consideration strategies that are never best responses. Notice that a dominant strategy (when one exists), by definition, strictly dominates all the others.

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Real-Life Applications of Dominant and Dominated Strategies

Classic examples of dominant and dominated strategies have been developed over the centuries, with some rooted in everyday life. Below are some of the most well-known examples that speak to the nature of human decision-making. 

Prisoner’s Dilemma

Two prisoners are interrogated individually, with the option to rat out the other or stay silent. Each prisoner reasons that it’s better to confess either way and hope the other doesn’t confess, leading to the optimal payoff. Because both prisoners reach the same conclusion, they both confess and receive harsher sentences. Both would’ve received better outcomes if they both stayed silent, but individual interest wins out in this scenario. 

Battle of the Sexes

A couple wants to spend time together over the weekend, but each wants to do a different activity. The dominant strategy is for each person to attend their preferred activity, but that defeats the purpose of spending time together. A couple could compromise by attending one person’s preferred activity one weekend and attending the other person’s preferred activity the next. This problem highlights the importance of communicating and sometimes accepting a less-than-optimal outcome to ensure the other person also receives some benefits.  

Stag Hunt 

Two hunters who are strangers to each other have the option of working together to hunt a stag — something neither can hunt on their own — or separately hunting hares, which are a smaller but more sure catch. Hunting a stag together can lead to the highest possible payoff, but hunting hares removes a lot of the risk involved in working with another hunter. This situation emphasizes how crucial trust is in getting people to work together.  

Chicken Game

The chicken game refers to two drivers driving at top speed toward each other. If one driver ‘chickens out’ and swerves first, they lose. If both drivers swerve, neither has more shame than the other, although each driver would rather be the one not to swerve and win. If neither swerves, they risk a dangerous outcome. This more extreme scenario explores the kind of risks people are ready to take and whether they’re willing to cooperate with each other.   

Matching Pennies

Players A and B each secretly place a penny heads- or tails-up on a table, revealing it after they’ve placed it. Player A wins if both pennis are heads or tails, and player B wins if the pennies don’t match. The interests of the players are in conflict since each one desires an outcome that hurts the other. However, there is a Nash equilibrium because the best strategy is to randomly place the penny on the table. There’s always a 50 percent chance of it being heads or tails, so neither player has any reason to deviate from this strategy.

Disadvantages of Dominated Strategies Method

While dominated strategies can be helpful for approaching games and real-life scenarios, there are some flaws to consider.

Ineffective When Equilibrium Isn’t Achieved

Dominated strategies have in common one major shortcoming — they do not always narrow down what may happen in a game to a tractably small number of possibilities. For example, a game has an equilibrium in dominant strategies only if all players have a dominant strategy. If this is not the case, this solution concept is not very useful.

Inapplicable to Games Where Strategies Can’t Be Deleted

Similarly, some games may not have any strategies that can be deleted via iterated deletion. Even among games that do have some dominated strategies, the remaining set of rationalizable strategies may be very large. The predictive power may not be precise enough to be useful.

Incorrect in Some of the Assumptions Made About Players

Dominant and dominated strategies operate under the assumption that all players involved are rational. But this is often not the case. People may ignore personal interests and make decisions based on a sense of fairness or community. People are also prone to making emotional decisions and mistakes, undermining the logic of dominant and dominated strategies.

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