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.
Understanding Nash Equilibrium & Dominant Strategies
Start by reading the order of the numbers in every cell. In many two-player tables, the first number belongs to the row player and the second belongs to the column player. Mixing up that order gives the wrong answer even if every comparison is done correctly.
A payoff can represent money, points, time saved, survival chances, or personal satisfaction. It does not have to be a literal cash amount. What matters is that larger numbers mean a player prefers that outcome within the model.
Dominant strategies are useful because they can simplify a game before you search for stable choices. If one option is worse than another in every possible case, a rational player has no reason to use it. You can remove that strictly dominated option, then inspect the smaller table.
This process is called elimination of dominated strategies. It must be used carefully.
Weak dominance can create ties, so removing a weakly dominated strategy may hide an outcome that could still occur. Students should state whether a comparison is strict or weak rather than treating them as identical.
A stable outcome is not automatically a fair or efficient outcome. Each player may be protecting their own result while both end up worse off than they could have been through cooperation. This is the important lesson behind situations like price competition between shops, countries choosing whether to reduce pollution, or students deciding how much effort to put into a shared project.
The model separates individual incentives from what would be best for the group. Real situations often need rules, trust, contracts, or repeated interaction to change those incentives.
Some games have more than one stable outcome. In that case, the matrix alone may not predict which one people choose. Past habits, communication, timing, and expectations can matter.
Other games have no stable cell made of pure choices. Players then may need to randomize. A mixed strategy does not mean acting carelessly.
It means choosing each option with a planned probability so that an opponent cannot gain by predicting a pattern. To find the probability, calculate the opponent’s average payoff from each available choice and make those averages equal. This makes the opponent indifferent between their choices.
When solving problems, show the reasoning in a clear order. Identify who controls rows and columns. Mark each player’s best choices while holding the other player’s action fixed.
Check every cell for mutual best choices. Then look for dominated options only when the comparisons support removal. For mixed strategies, remember that probabilities must be between zero and one and must add up to one.
Finally, interpret the answer in words. Explain what a player would do, why switching alone does not help at a stable outcome, and what the result says about the situation being modeled.