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.
Understanding Game Theory Basics Reference
A useful way to analyze a small game is to work one player at a time. Hold the other player's choice fixed and compare only the payoffs belonging to the player you are studying. Mark that player's highest payoff in each relevant row or column.
Then switch roles and repeat. A cell marked for both players is a stable outcome under the assumption that each person knows the available choices and acts in their own interest.
This procedure prevents a common error. Students sometimes compare the two numbers inside one cell, even though those numbers belong to different players and cannot be compared directly.
Dominant strategies can make a game much easier, but many games do not have one. It is important not to force the idea onto every matrix. A strategy is only dominant when it performs better against every possible opponent choice.
If it is better in one case but worse in another, it is not dominant. Sometimes a strategy is strictly worse than another strategy in every case. Removing that inferior option can simplify the matrix.
This is called eliminating a dominated strategy. Use care when doing this, since a strategy that looks weak may still matter if the comparison is not better in every case.
A Nash equilibrium does not mean the outcome is fair, kind, or best for the group. It only describes a situation where a one-person change does not help the person making that change. Two drivers choosing the same crowded route, two shops setting high prices, or students choosing how much effort to put into a group task can create stable results that leave everyone worse off than they could be through cooperation.
This difference between individual incentives and group welfare is one of the most important lessons in game theory. Real decisions often include trust, communication, repeated interactions, and rules. Those features can change what people choose.
Mixed strategies are needed when a player would become predictable by always choosing one action. In a two-option game, the probabilities are chosen so that the opponent gets the same expected payoff from either response. If one response paid more on average, the opponent would always use it, so the proposed probabilities would not be stable.
To find the needed probability, calculate the opponent's average payoff for each response. Multiply each possible payoff by the probability of the outcome, then add the results. Set the two averages equal and solve for the probability.
Check that the answer lies from zero to one. This method appears in sports play calling, security patrols, pricing decisions, and card games.
Zero-sum games are a special case where one side's loss exactly matches the other side's gain. In those games, players focus on limiting the worst possible loss while seeking the best guaranteed result.