Questions: Strategic Form Games and Nash Equilibrium

Question 1

In the Prisoner's Dilemma, both players confessing is a Nash equilibrium even though both players would receive higher payoffs if they both stayed silent. Why does mutual confession qualify as a Nash equilibrium?

• A. Both players prefer confessing because they anticipate the other will confess, making it a self-fulfilling prophecy
• B. Neither player can improve their individual payoff by unilaterally switching to silence, given that the other player is confessing
• C. The Nash equilibrium always selects the strategy that maximizes the sum of all players' payoffs
• D. Both players are playing dominant strategies, and dominant strategies always produce Nash equilibria

Answer: (B)

A Nash equilibrium requires only that no single player can profitably deviate given the others' strategies — not that the outcome is collectively optimal. If Player 1 is confessing, Player 2's best response is also to confess (silence would give the worst payoff). And if Player 2 is confessing, Player 1's best response is to confess. So neither player wants to deviate unilaterally — mutual confession is stable. Option A hints at a reasoning process, not the formal definition. Option C is wrong — Nash equilibria can be collectively suboptimal (PD is the canonical example). Option D is correct that confessing is a dominant strategy, and dominant strategies are always Nash equilibria — but the reasoning in B is the direct definition.

Question 2

A game has three pure strategies for each of two players and no cell in the payoff matrix has both payoffs underlined (i.e., no pure-strategy Nash equilibrium exists). What does Nash's existence theorem guarantee?

• A. The game has no equilibrium and players should use maximin strategies
• B. The theorem does not apply because not all finite games have equilibria
• C. The game must have at least one Nash equilibrium in mixed strategies
• D. Players must coordinate on a correlated equilibrium instead

Answer: (C)

Nash's existence theorem guarantees that every finite game — any game with a finite number of players and finite strategy sets — has at least one Nash equilibrium, possibly in mixed strategies. When no pure-strategy equilibrium exists, the theorem guarantees a mixed-strategy equilibrium exists: probability distributions over pure strategies such that each player is indifferent among the strategies they mix over (and therefore willing to randomize). The theorem rests on Kakutani's fixed-point theorem applied to the best-response correspondence.

Question 3

A Nash equilibrium is a strategy profile in which every player is playing a best response to the strategies of all other players.

• T. True
• F. False

Answer: True

This is the precise definition of Nash equilibrium: mutual best response. If every player is already responding optimally to what everyone else is doing, no single player has an incentive to unilaterally change their strategy — the profile is stable. The best-response interpretation also suggests the systematic way to find Nash equilibria: for each player, underline their best payoff given each strategy of the opponent; cells where all players have their best payoffs underlined are Nash equilibria.

Question 4

A Nash equilibrium usually produces the outcome that maximizes the total combined payoffs for most players in the game.

• T. True
• F. False

Answer: False

The Prisoner's Dilemma is the classic counterexample. Both players confessing is the unique Nash equilibrium, but mutual silence gives each player a higher payoff — total surplus is maximized at mutual silence, not at the equilibrium. Nash equilibrium is about strategic stability (no individual incentive to deviate), not social optimality. This gap between individual rationality and collective welfare is one of the central lessons of game theory and underlies problems ranging from arms races to climate agreements.

Question 5

Why is a Nash equilibrium described as 'stable' rather than 'optimal,' and what classic game illustrates the difference most clearly?

Model answer: A Nash equilibrium is stable in the sense that no player has a unilateral incentive to deviate — each is already doing the best they can given others' choices. But stability has nothing to do with producing the best collective outcome. The Prisoner's Dilemma illustrates this: the unique Nash equilibrium (both confess) is worse for both players than the alternative (both cooperate/stay silent), which is not an equilibrium because each player individually wants to deviate from it. The equilibrium is 'trapped' at a suboptimal outcome because individual incentives undermine the collectively better choice.

This distinction — between individual strategic rationality and collective optimality — is foundational to applied game theory, mechanism design, and policy analysis. It explains why markets can fail (equilibria exist but are inefficient), why arms races persist (mutual disarmament is not a Nash equilibrium), and why achieving cooperation often requires changing the payoff structure (e.g., through contracts, regulations, or repeated interaction) rather than appealing to rationality alone.