In a 2-player game, which of the following correctly describes a Nash equilibrium?
ABoth players are using dominant strategies
BNo player can increase their payoff by unilaterally changing their strategy
CThe sum of all players' payoffs is maximized
DBoth players achieve their single best possible payoff simultaneously
A Nash equilibrium requires only that no player has a profitable unilateral deviation — each player's strategy is a best response to the other's. Option A is too strong (dominant strategy equilibrium implies Nash, but not vice versa). Option C is the definition of Pareto efficiency, not Nash equilibrium. Option D is also too strong — Nash equilibria can occur at payoff combinations far from each player's maximum.
Question 2 True / False
Most Nash equilibrium is also a dominant strategy equilibrium.
TTrue
FFalse
Answer: False
Dominant strategy equilibrium is a strictly stronger concept. A strategy dominates when it is best regardless of what others do. A Nash strategy is only required to be best given what others actually do. Many games (like Coordination and Battle of the Sexes) have Nash equilibria but no dominant strategies at all.
Question 3 Short Answer
Why might a finite game have no pure strategy Nash equilibrium, and what does Nash's theorem say about this?
Think about your answer, then reveal below.
Model answer: In some games (like Matching Pennies), for every pure strategy profile some player prefers to deviate, so best responses cycle with no stable fixed point. Nash's theorem guarantees that every finite game has at least one Nash equilibrium when players are allowed to use mixed strategies — probability distributions over their pure strategies.
The core insight is that mixed strategies expand the strategy space. By randomizing, a player can make opponents indifferent between their choices, breaking the cycle of deviation. The proof uses Kakutani's fixed-point theorem applied to the best-response correspondence over the compact, convex set of mixed strategy profiles.