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?

How to Identify a Dominated Strategy in Game Theory

The iterated deletion of dominated strategies is one common, but tedious, technique for solving games that do not have a strictly dominant strategy. It involves iteratively removing dominated strategies. There are two types of dominated strategies.

In the first step of the iterative deletion 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 ad infinitum.

There are two versions of this process. The first (and preferred) version involves only eliminating strictly dominated strategies. If, after completing this process, there is only one strategy for each player remaining, that strategy set is the unique Nash equilibrium.

The second version involves eliminating both strictly and weakly dominated strategies. If, at the end of the process, there is a single strategy for each player, this strategy set is also a Nash equilibrium. Unlike the first process, elimination of 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.

I’ve used a lot of terminology, so let’s look at an example to clarify these concepts.

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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.

A 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.” Lets 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 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:

By doing this, 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.) Now let us 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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Dominated Strategies Method Disadvantage

Both methods 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.

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.

A player has a dominant strategy if that strategy gives them 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.

A player has a strictly dominated strategy if that strategy gives them a lower payoff than any other strategy they could use, no matter what the other players are doing. If you have a strictly dominated strategy, expect other players to anticipate you’ll never play it and choose their actions accordingly.

We’ve looked at two methods for finding the “likely” outcome of a game.

1. Look for a Dominant Strategy Equilibrium. This is great if a dominant strategy exists, however, there often isn’t a dominant strategy. This limits the usefulness of this solution concept.
2. Iteratively delete strictly dominated strategies. Iterative deletion is a useful, albeit cumbersome, tool to remove dominated strategies from consideration.

However, neither of these methods is guaranteed to return a tractably small set of expected outcomes. Fortunately, there is a solution concept that does guarantee to return a tractably small set of expected outcomes known as the Nash equilibrium.