The core tools are payoff matrices, best responses, dominant strategies, Nash equilibria, and expected payoff. A dominant strategy is best no matter what the other player chooses, while a Nash equilibrium occurs when no player wants to change strategies alone. Mixed strategies use probabilities when players randomize their choices.
Expected payoff is calculated with weighted averages, using probability times payoff for each possible outcome.
Key Facts
- A payoff matrix lists each player's possible strategies and the payoff each player receives for every strategy combination.
- A dominant strategy is a strategy that gives a player a higher payoff than all other strategies no matter what the opponent does.
- A dominated strategy is a strategy that always gives a lower payoff than another available strategy, so rational players usually avoid it.
- A Nash equilibrium occurs when each player's strategy is a best response to the other player's strategy.
- To find a pure strategy Nash equilibrium, mark each player's best response in every row or column and look for cells where both players are choosing best responses.
- Expected payoff is found using expected payoff = probability of outcome 1 × payoff 1 + probability of outcome 2 × payoff 2 + ...
- In a zero-sum game, one player's gain equals the other player's loss, so Player A payoff + Player B payoff = 0.
- A mixed strategy assigns probabilities to different strategies, such as play Strategy A with probability p and Strategy B with probability 1 - p.
Vocabulary
- Payoff
- A payoff is the numerical value a player receives from an outcome, such as profit, points, utility, or cost savings.
- Strategy
- A strategy is a complete plan of action a player can choose in a game.
- Payoff Matrix
- A payoff matrix is a table that shows the payoff for each player for every possible combination of strategies.
- Best Response
- A best response is the strategy that gives a player the highest payoff given the other player's chosen strategy.
- 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 randomly chooses among actions using assigned probabilities.
Common Mistakes to Avoid
- Confusing a Nash equilibrium with the highest total payoff is wrong because Nash equilibrium is about individual incentives, not the best combined outcome.
- Choosing the largest number in the whole matrix is wrong because each player must compare only their own payoffs for a fixed opponent choice.
- Ignoring dominated strategies is a mistake because eliminating a strategy that is always worse can make the game much easier to analyze.
- Mixing up rows and columns is wrong because one player controls rows and the other player controls columns, so each payoff must be matched to the correct player.
- Forgetting probabilities in expected payoff is wrong because expected payoff must weight each payoff by how likely that outcome is.
Practice Questions
- 1 Player A chooses Top or Bottom, and Player B chooses Left or Right. Payoffs are (A,B): Top-Left (3,2), Top-Right (1,4), Bottom-Left (2,1), Bottom-Right (4,3). Find any pure strategy Nash equilibrium.
- 2 A player has a 0.6 probability of earning 10 points and a 0.4 probability of earning 2 points. What is the expected payoff?
- 3 In a zero-sum game, Player A receives payoffs 5, -2, and 0 in three possible outcomes. What are Player B's payoffs for those same outcomes?
- 4 Explain why a strategy that gives a player the highest payoff in one situation is not always a dominant strategy.
Understanding Game Theory Basics
A game theory model starts by defining the players, their choices, and what each number means. The numbers do not have to be money. They can represent points, time saved, market share, safety, or personal satisfaction.
What matters is the ranking of outcomes for each player. A choice that gives one player a larger number may hurt the other player, benefit them, or leave them unchanged. This is why game theory differs from ordinary optimization.
A player cannot choose the outcome alone. Their result depends on another decision maker.
Before doing any calculations, check whose payoff appears first in each cell and whose appears second. Mixing up the order is one of the most common errors.
The important idea is incentives. People may prefer an outcome that is worse for the group if it is safer or more rewarding for them personally. This can explain price competition between firms, overuse of shared resources, and disagreements about environmental rules.
In the classic prisoner situation, each person has a reason to protect themselves, even though both would be better off with cooperation. A stable result does not always mean a fair or efficient result.
It only means that, given the current choices of others, no single player gains by switching alone. If two players could make a binding agreement, communicate honestly, or face repeated interactions, the result may change.
Random choices become useful when being predictable creates a weakness. Think of a football player choosing where to kick a penalty, or a cybersecurity team deciding when to inspect a network. If one action is used every time, an opponent can prepare for it.
The right probabilities are chosen to make the opponent indifferent between their own main options. For example, if one response becomes much more profitable than another, the probabilities need adjustment until neither response has an advantage. This is not the same as picking randomly without thought.
The probabilities come from the payoffs. Over many rounds, the expected result describes the average outcome, not a guarantee for any one round. A player can still lose several times in a row while using the best long run plan.
Real situations rarely match a small table perfectly. People may lack information, make mistakes, care about reputation, or value fairness more than a numerical reward. A company might accept a lower short term profit to build customer trust.
A student in a group project may contribute effort because future cooperation matters. When solving school problems, state the assumptions clearly. Identify whether choices happen once or repeatedly, whether players know the payoffs, and whether they can communicate.
Watch for strictly worse choices first, since removing them can simplify a large game. Then check each remaining outcome from both players' viewpoints. Good game theory work is careful about what the model leaves out, not just quick at finding a cell in a table.