Game theory solution concepts help economists predict how rational players choose strategies when outcomes depend on everyone’s choices. This cheat sheet covers the main tools used to analyze simultaneous and sequential games. Students need these concepts to solve payoff matrices, strategic form games, and extensive form games.
The focus is on identifying stable outcomes and understanding why players may or may not cooperate.
Key Facts
- A strategy si strictly dominates strategy si' for player i if ui(si, s-i) > ui(si', s-i) for every possible strategy profile s-i of the other players.
- A strategy si weakly dominates strategy si' if ui(si, s-i) >= ui(si', s-i) for every s-i and ui(si, s-i) > ui(si', s-i) for at least one s-i.
- A Nash equilibrium is a strategy profile s* where ui(si*, s-i*) >= ui(si, s-i*) for every player i and every alternative strategy si.
- A best response BRi(s-i) is any strategy si that maximizes player i’s payoff given the other players’ strategies s-i.
- In a two-strategy mixed equilibrium, each player chooses probabilities that make the other player indifferent between their pure strategies.
- A subgame perfect Nash equilibrium is a Nash equilibrium that is also a Nash equilibrium in every subgame of an extensive form game.
- Backward induction solves finite sequential games with perfect information by starting at the final decision nodes and working backward.
- A Pareto efficient outcome is one where no player can be made better off without making at least one other player worse off.
Vocabulary
- Strategy
- A complete plan of action that tells a player what to do in every situation where that player may move.
- Payoff
- The numerical value representing a player’s outcome or utility from a particular combination of strategies.
- Dominant Strategy
- A strategy that gives a player a higher payoff than another strategy no matter what the other players do.
- Nash Equilibrium
- A strategy profile where no player can improve their payoff by changing only their own strategy.
- Mixed Strategy
- A probability distribution over pure strategies that allows a player to randomize among actions.
- Subgame
- A part of an extensive form game that begins at a single decision node and includes all later decisions from that node.
Common Mistakes to Avoid
- Calling an outcome a Nash equilibrium without checking every player’s best response is wrong because one profitable unilateral deviation eliminates equilibrium.
- Confusing dominant strategy equilibrium with Nash equilibrium is wrong because Nash equilibrium only requires best responses to the chosen strategies, not to every possible opponent strategy.
- Ignoring weak dominance ties is wrong because weakly dominated strategies can give equal payoffs in some cases and strict elimination rules may not apply.
- Solving sequential games from the beginning is wrong because backward induction requires analyzing later decision nodes first.
- Assuming Pareto efficiency means stability is wrong because an efficient outcome may still give some player an incentive to deviate.
Practice Questions
- 1 In a prisoner’s dilemma payoff matrix, if Confess gives player A payoffs of 5 when B confesses and 10 when B stays silent, while Stay Silent gives A payoffs of 2 when B confesses and 8 when B stays silent, does A have a dominant strategy?
- 2 Find the Nash equilibrium in this simultaneous game: if both choose High, payoffs are (3, 3); if A chooses High and B Low, payoffs are (1, 4); if A Low and B High, payoffs are (4, 1); if both Low, payoffs are (2, 2).
- 3 In a matching pennies game, Player 1 wins 1 if the choices match and loses 1 if they differ. What mixed strategy probability should each player use on Heads in equilibrium?
- 4 Explain why a Nash equilibrium in a sequential game may fail to be subgame perfect if it relies on a threat that would not be optimal when the threatened decision point is reached.