Game theory studies how people, teams, companies, or countries make decisions when each outcome depends on the choices of multiple decision makers. This cheat sheet covers the basic language and tools used to analyze simple strategic games. Students need it to organize payoff matrices, compare strategies, and predict likely outcomes.
It is especially useful for algebra, economics, probability, and decision-making applications.
The main ideas are players, strategies, payoffs, best responses, dominant strategies, and Nash equilibrium. In a payoff matrix, each cell lists the results for both players, often as . A Nash equilibrium occurs when no player can improve by changing only their own strategy.
In mixed strategies, players use probabilities, and expected payoff is found with .
Key Facts
- A strategic game has players, available strategies, and payoffs that depend on the combination of choices made by all players.
- A payoff matrix entry such as means Player 1 receives payoff and Player 2 receives payoff .
- A best response is the strategy that gives a player the highest payoff for a fixed strategy chosen by the other player.
- A dominant strategy gives a player a higher payoff than every other available strategy no matter what the other player does.
- A Nash equilibrium is a strategy combination where each player's strategy is a best response to the other's strategy.
- In a zero-sum game, one player's gain equals the other player's loss, so the total payoff is for every outcome.
- The expected payoff of a random strategy is , where is the probability of outcome and is its payoff.
- For a mixed strategy with two options, if option A is chosen with probability , then option B is chosen with probability .
Vocabulary
- Player
- A player is a decision maker in a game, such as a person, firm, team, or country.
- Strategy
- A strategy is a complete plan for what a player will do in the game.
- Payoff
- A payoff is the numerical value of an outcome for a player, such as points, profit, cost, or utility.
- Best Response
- A best response is the strategy that gives the greatest payoff against a specific strategy chosen by another player.
- Nash Equilibrium
- A Nash equilibrium is an outcome where no player can improve their payoff by changing only their own strategy.
- Mixed Strategy
- A mixed strategy is a strategy where a player chooses among options using probabilities.
Common Mistakes to Avoid
- Reversing the payoff order in a matrix is wrong because usually means Player 1 gets and Player 2 gets .
- Calling any high-payoff outcome a Nash equilibrium is wrong because each player's choice must be a best response to the other player's choice.
- Confusing a dominant strategy with a best response is wrong because a dominant strategy must work best against every possible opponent choice.
- Forgetting that probabilities must add to is wrong because a mixed strategy with two probabilities must satisfy .
- Assuming both players always want the same outcome is wrong because some games are competitive, and in a zero-sum game the payoffs add to .
Practice Questions
- 1 In a payoff matrix, one outcome is listed as . What payoff does Player 1 receive, and what payoff does Player 2 receive?
- 2 A player chooses Strategy A with probability and Strategy B with probability . What is the probability of choosing Strategy B?
- 3 A random payoff is with probability and with probability . Find the expected payoff using .
- 4 Explain why an outcome with the largest total payoff does not always have to be a Nash equilibrium.