Dominated Strategy

A dominated strategy is a strategy which doesn’t result in the optimal outcome in any case. A strategy is dominated if there always exist a course of action which results in higher payoff no matter what the opponent does.

Identifying strategic dominance in a game is important in identifying its Nash equilibrium, an outcome which no player would want to change. In a two-strategy game, if one strategy is dominant, the other must be a dominated strategy. Since not all games have a dominant strategy, it is not necessary for all games to have dominated strategies. But if there are more than two strategies available, it is possible for a game to have a dominated strategy even if there is no dominant strategy (as illustrated in example 2).

Examples

Example 1

Let’s consider two firms, A and B, each of which must decide whether to hire a lawyer to represent it in an arbitration. The following matrix shows the payoffs:

Payoffs represent is
in USD in millions
Firm B
Hire a Lawyer No Lawyer
Firm A Hire a Lawyer 45,45 75,25
No Lawyer 25,70 50,50

In the example on dominant strategy, we identified that hiring a lawyer is a dominant strategy for both firms. Hence, we can conclude that not hiring a lawyer is the dominated strategy for both firms.

Example 2

Now, let’s see what happens in a game in which there are more than two strategies available to each player. Assume that Firm A and Firm B are firms who must decide about whether to decrease their advertising budget, not change it or increase it. The following payoff matrix shows different scenarios:

Payoffs represent is
in USD in millions
Firm B
Cut Advertising No Change Increase Advertising
Firm A Cut Advertising 80,80 -10,110 -50,150
No Change 110,-10 0,0 30,130
Increase Advertising 150,-50 130,30 -20,-20

In the payoff matrix above, rows show strategies of Firm A and columns show strategies available to Firm B. The numbers in red to the left of the comma in each cell are payoffs of Firm A and those to the right (in blue) are payoffs to Firm B.

In this game there is no dominant strategy for Firm A. It is because when Firm B cuts advertising, Firm A’s maximum payoff occurs when it increases its budget (see Row 3 and Column 1). Similarly, when Firm B doesn’t change its advertising budget, maximum payoff for Firm A occurs when it increases its budget (see Row 3 and Column 2). But if Firm B also increases its advertising, maximum payoff for Firm A exists when it doesn’t change its budget (see Row 2 and Column 3). Since the maximum payoff changes with change in strategy of the opponent, there is no dominant strategy.

But the game has a dominated strategy, a strategy which is no good no matter what the opponent do. In order to identify the dominated strategy, we need to find if a row exists which corresponds to the lowest Firm A payoffs in all columns.

Payoffs represent is
in USD in millions
Firm B
Cut Advertising No Change Increase Advertising
Firm A Cut Advertising 80,80 -10,110 -50,150
No Change 110,-10 0,0 30,130
Increase Advertising 150,-50 130,30 -20,-20

The lowest red values all occur in the highlighted row in the payoff matrix shown above. Hence, cutting advertising is the dominated strategy for Firm A. You can see that cutting advertising is also a dominated strategy for Firm B, but it is just a coincidence. It is not necessary for both firms to have dominated strategies.

Written by Obaidullah Jan, ACA, CFA and last modified on