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Applied Math
Grade 10-12
Game Theory Basics Cheat Sheet
A printable reference covering payoff matrices, dominant strategies, Nash equilibrium, mixed strategies, and expected payoff for grades 10-12.
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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.