# Dominant Strategy

In game theory, a dominant strategy is the course of action that results in the highest payoff for a player regardless of what the other player does.

Not all players in all games have dominant strategies; but when they do, they can blindly follow them. It is because a dominant strategy is the optimal strategy unconditionally i.e. there is no dependence on the strategy the other player choses.

## Example

Let’s consider two firms, A and B, who have appointed an arbitrator to resolve a contractual dispute of \$100 million. If neither firm hires a lawyer, the expected payoff is \$50 million each. If Firm A hires a lawyer but Firm B doesn’t, the payoff to A increases to \$70 million and that of B reduces to \$25 million. Similarly, if Firm B hires a lawyer but Firm A doesn’t, Firm B gets \$70 million and Firm A \$25 million. If both firms hire a lawyer, they each get \$45 million.

The following payoff matrix lays out the game:

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

The strategies of Firm A are listed in rows and those of Firm B are listed in columns. The payoffs to the left of the comma (in red color) belong to Firm A and those in blue to the right accrue to Firm B.

Finding a dominant strategy for Firm A involves the following steps:

• Select the first strategy of the opponent i.e. Firm B i.e. hiring a lawyer. Look inside the ‘Hire a Lawyer’ column for Firm A’s payoff and see which row has the higher payoff. In this example, it is 45 (as 45 is greater than 25).
• Place a check mark against the higher payoff or draw an arrow from lower payoff towards higher payoff.
• Select the second strategy of Firm B i.e. not hiring a lawyer. Looking inside the ‘No Lawyer’ column, compare the red payoffs, select the higher payoff i.e. 70 and place a check mark against it or draw an arrow from lower payoff towards higher payoff.
• If both check marks (or arrow heads) appear in the same row, that row represents the dominant strategy of the row player i.e. Firm A. In this case, hiring a lawyer is the dominant strategy.

## Equilibrium in Dominant Strategies

Hiring a lawyer is a dominant strategy for Firm A because if Firm B hires a lawyer, it is better to hire a lawyer and get \$45 million instead of not hiring and getting only \$25 million. If Firm B doesn’t hire a lawyer, it is better for Firm A to hire a lawyer and get \$70 million instead of only \$25 million.

Using the steps above, we can find out that hiring a lawyer is also a dominant strategy for Firm B.

It follows that both firms will follow their dominant strategies and hire a lawyer. Even though they will have to share \$10 million of their payoff with the lawyers, it is the best strategy to adopt and no firm has any incentive to deviate from it. In other words, existence of dominant strategies for all players results in an equilibrium which his called equilibrium in dominant strategies.

by Obaidullah Jan, ACA, CFA and last modified on

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