We discuss dominant strategies, dominated strategies and the iterative deletion of strictly dominated strategies (IDSDS) to solve or simplify strategic form (simultaneous) games.
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Dominant and dominated strategies
Introductory game theory assumes that players are rational, and that rationality is common knowledge between players. That is, players take actions to maximize their own payoffs, given we expect everybody else to act rationally too. With this in mind, we can begin to formulate beliefs about the behaviors of decision makers in strategic situations. One useful technique is to eliminate inferior, nonsensical and irrational strategies. To do so, we must first introduce the concept of dominance.
Dominance helps us to rank and compare preferences, strategies, and payoffs. In introductory game theory, we like to distinguish between dominant and dominated strategies, and whether dominance is strict or weak. Let’s begin with strict dominance:
Strict dominance
We say that Strategy A strictly dominates Strategy B when the payoff from Strategy A is greater than the payoff from Strategy B, regardless of the strategies that other players are using.
A player’s strategy is a strictly dominant strategy if it strictly dominates every other strategy that the player can use. If a strictly dominant strategy exists, a rational player will always use that strategy.
A strategy is strictly dominated when it always produces an inferior payoff to some other strategy, regardless of what other players are doing. A rational player will never use a strictly dominated strategy.
Example I: Group project at college (strictly dominant strategy)
In the example above, Studious Jane and Lazy Joe are working on a college group project. A higher grade, of course, is associated with the level of combined effort. If both Joe and Jane exhibit low effort, they will earn a poor grade. But if they both put it high effort, they will share in a good grade.
Studious Jane has a strictly dominant strategy to contribute high effort. Her payoff from exhibiting high effort is always greater than the payoff associated with medium or low effort, regardless of what Joe is doing. After all, she is studious and wants to earn the highest possible grade.
Joe, by contrast, is lazy. He earns the greatest payoff from exhibiting low effort, regardless of Jane’s behavior. Indeed, Joe has a strictly dominant strategy to contribute low effort. The solution to this game then is not surprising. Jane works hard, and Joe does not. Jane gets a payoff of three, while Joe enjoys a payoff of five.
Weak dominance
Strategy A weakly dominates Strategy B when the payoffs from strategy A is at least as great as the payoff from strategy B, regardless of the strategies that other players use.
Likewise, a player’s strategy isa weakly dominant strategy if it weakly and/or strictly dominates every other strategy that the player can use. If a weakly dominant strategy exists, a rational player will always use it.
A strategy is weakly dominated when it produces an equal or inferior payoff to some other strategy, regardless of what other players are doing. A rational player will never use a weakly dominated strategy.
Example II: Group project at college (weakly dominant strategy)
In this example, Indifferent Joe has a weakly dominant strategy to contribute low effort to the group project. If Studious Jane gives medium or low effort, Joe’s payoff is 3 no matter what he does. But if Jane gives high effort, then Indifferent Joe gets a higher payoff of 4 by choosing low effort (or 3 otherwise). So, Joe contributes low effort because it gives him at least a good a payoff as medium or high effort, regardless of what Jane does. Once again, the solution to this game is not surprising: Jane works hard, and Joe does not.
Dominance, rationality and solutions
In the above examples, we found a solution to each game because all players had a strictly or weakly dominant strategy. Sometimes, that isn’t enough to solve a game right away. But if a player knows or believes that the other player is rational, then solutions might present themselves.
Example III: Group project at college
In the example above, Average Joe does not have any dominant or dominated strategies. Rationality alone is not enough for Joe to decide on his course of action. If Joe believes, however, that Studious Jane is rational, then Joe can formulate the belief that she will contribute high effort. High effort, after all, is Jane’s strictly dominant strategy. Given this belief, Joe will free ride with low effort to maximize his own payoff.
Iterative deletion of strictly dominated strategies (IDSDS)
The example above was a two-step approach to solving a strategic form game. We can generalize the approach, deleting as many inferior strategies as we can to simplify and solve the game. This is what game theorists call an iterative deletion of strictly dominated strategies (IDSDS). Generally speaking, we can undertake IDSDS in several steps:
- Step 1: Eliminate every strictly dominated strategy from the game. The assumption here is that all players behave rationally.
- Step 2: Eliminate every strictly dominated strategy that emerges from the simplified game after Step 1. The assumption here is that all players believe that other players behave rationally.
- Step 3: Eliminate every strictly dominated strategy that emerges from the resulting game after Step 2. The additional assumption here is that all players believe that all other players believe that all other players are rational.
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. - Step N: Repeat the process until we cannot eliminate any more strictly dominated strategies.
You’ll notice the accumulation of assumptions about common knowledge of rationality with each successive step in the IDSDS. It’s this combination of rationality, dominance and common knowledge that allows us to simplify the game and find potential solutions.
Dominance solvable
A game is dominance solvable if a unique single solution exists at the end of the IDSDS process. The games between Jane and Joe, for example, are dominance solvable. Unfortunately, games like the game of rock-paper-scissors below have no strictly dominated strategies. So, we cannot use IDSDS to simplify these games. Fortunately, IDSDS is only one of many methods. In the next post, we turn to other solution concepts like the Nash equilibrium for guidance.
Example IV: Rock-Paper-Scissors
Further reading
- Strategic interdependence, preferences, and beliefs — The building blocks of game theory
- Extensive and normal form games — The structure of strategic situations
- Common knowledge, rationality, and reasonableness in introductory game theory
- Dominant strategies, dominated strategies and iterative deletion — Simplifying the game
- Nash equilibrium — A fundamental solution concept, and the hallmark of game theory
- Sequential games and subgame perfect Nash equilibrium