Nash equilibrium and dominant strategies help students analyze situations where two or more decision makers affect each other’s outcomes. This cheat sheet covers how to read payoff matrices, compare strategies, and identify stable outcomes. Students need these tools for applied math, economics, political science, biology, and any setting involving strategic choice.
The core idea is that each player chooses a strategy to maximize their own payoff given what the other player might do. A dominant strategy is best no matter what the opponent chooses, while a Nash equilibrium is a pair of strategies where no player can improve by switching alone. Payoff matrices organize the information, and best responses help locate equilibria quickly.
Key Facts
- A payoff matrix lists each player’s payoff for every possible combination of strategies.
- A strategy is dominant if it gives a player a higher payoff than every other strategy no matter what the other player chooses.
- A strategy is strictly dominant if it always gives a greater payoff, and weakly dominant if it gives at least as much payoff and sometimes more.
- A best response is the strategy that gives the highest payoff for a player given the other player’s chosen strategy.
- A Nash equilibrium occurs when each player is choosing a best response to the other player’s strategy.
- In a 2 by 2 game, mark each player’s best response in every row or column, then look for cells where both players’ best responses meet.
- A player should not switch unilaterally from a Nash equilibrium because switching alone would not increase that player’s payoff.
- For a mixed strategy in a 2 by 2 game, the probability often comes from setting the opponent’s expected payoffs equal, such as pA + (1 - p)B = pC + (1 - p)D.
Vocabulary
- Payoff
- A payoff is the numerical value a player receives from a particular outcome in a game.
- Strategy
- A strategy is a complete choice or plan of action available to a player.
- Dominant strategy
- A dominant strategy is a strategy that gives a player the best payoff regardless of what the other player does.
- Best response
- A best response is the strategy that gives the highest payoff against a specific strategy chosen by another player.
- Nash equilibrium
- A Nash equilibrium is an outcome where every player is using a best response, so no one benefits by changing alone.
- Mixed strategy
- A mixed strategy is a plan in which a player chooses among strategies using probabilities.
Common Mistakes to Avoid
- Choosing the largest total payoff instead of checking each player’s incentive is wrong because Nash equilibrium depends on individual best responses, not combined benefit.
- Calling a strategy dominant after checking only one opponent choice is wrong because a dominant strategy must work against every possible opponent choice.
- Ignoring the order of payoff pairs is wrong because the first number usually belongs to the row player and the second number belongs to the column player.
- Assuming every game has a pure strategy Nash equilibrium is wrong because some games only have mixed strategy equilibria.
- Thinking a Nash equilibrium must be fair or socially best is wrong because it only means no single player can improve by switching alone.
Practice Questions
- 1 In a payoff matrix, if Player A gets 4 from Up and 2 from Down when Player B chooses Left, and 3 from Up and 1 from Down when Player B chooses Right, does Player A have a dominant strategy?
- 2 For the payoff cells (Up, Left) = (3, 2), (Up, Right) = (1, 4), (Down, Left) = (2, 1), and (Down, Right) = (0, 3), identify Player A’s best response to each of Player B’s choices.
- 3 If a player earns expected payoff 5p + 1(1 - p) from Strategy X and 2p + 4(1 - p) from Strategy Y, find the value of p that makes the player indifferent.
- 4 Explain why an outcome can be a Nash equilibrium even if both players would receive higher payoffs at a different outcome.