In a market entry game, an incumbent threatens to start a price war if a challenger enters — a war that would hurt both firms. This threat sustains an equilibrium where the challenger stays out. Which statement correctly describes this situation?
AThis is not a Nash equilibrium because the incumbent's threat is not credible
BThis is a Nash equilibrium, but it fails subgame perfection because the incumbent would not rationally carry out the threat if entry actually occurred
CThis is both a Nash equilibrium and a subgame perfect equilibrium, since no player deviates from their strategy on the equilibrium path
DSubgame perfection does not apply to market entry games — it is only relevant to repeated games
Nash equilibrium only requires that no player wants to deviate *given the current strategies*. Because the challenger stays out, the incumbent's threat is never actually tested — so the threat can sustain the equilibrium even if carrying it out would be irrational. This is the key flaw that refinements address. Subgame perfection applies backward induction: it asks whether the incumbent's strategy is a Nash equilibrium in every subgame, including the subgame that starts *after* entry occurs. Since a price war hurts the incumbent, the credible response to entry is accommodation — so the price-war threat fails subgame perfection, and the only subgame perfect equilibrium is for the challenger to enter and the incumbent to accommodate.
Question 2 Multiple Choice
Perfect Bayesian Equilibrium requires that beliefs be updated using Bayes' rule. Why might beliefs still be underdetermined even when all Bayes' rule conditions are met?
ABayes' rule requires prior probabilities that are never well-defined in game-theoretic settings
BBeliefs at information sets that are never reached on the equilibrium path are not constrained by Bayes' rule, leaving them free to be specified arbitrarily
CPBE only pins down beliefs when there are exactly two player types; more types leave the system underdetermined
DBayes' rule only applies to complete-information games where all types are observable
Bayes' rule updates probabilities based on observed events: P(type | signal) ∝ P(signal | type) × P(type). But Bayes' rule only applies when the conditioning event (the signal or action observed) has positive probability under the equilibrium. If an off-equilibrium action is never taken in equilibrium, Bayes' rule places no constraint on what the receiver should believe upon observing it — the conditioning event has probability zero and the formula is undefined. Different off-equilibrium beliefs can support different equilibria, which is why PBE is often further refined by criteria like the Intuitive Criterion that restrict what off-equilibrium beliefs are 'reasonable.'
Question 3 True / False
Subgame perfection eliminates Nash equilibria that rely on threats that would be irrational to actually carry out if the moment to act arrived.
TTrue
FFalse
Answer: True
This is precisely the point of subgame perfection. It requires that strategies form a Nash equilibrium in every subgame of the original game — including subgames that are never reached in the equilibrium play. A threat that would hurt the threatening party if executed fails this test: in the subgame that begins when the threat must be carried out, following through is not the optimal action, so the strategy cannot be part of a subgame perfect equilibrium. Backward induction systematically removes such threats by determining optimal play at each final decision node and working backwards.
Question 4 True / False
Nearly every subgame perfect equilibrium is also a trembling-hand perfect equilibrium.
TTrue
FFalse
Answer: False
Subgame perfection and trembling-hand perfection impose different requirements and neither is a subset of the other in general. Trembling-hand perfection requires that a strategy remain a best response when opponents occasionally make small random errors ('trembles'). This can eliminate equilibria that survive subgame perfection — for example, equilibria where a player is indifferent between two strategies and the equilibrium relies on choosing the one that would be suboptimal under any perturbation. Conversely, trembling-hand perfection is defined for strategic form games and can select equilibria that would be eliminated by subgame perfection in the extensive form. The two refinements capture different kinds of implausibility.
Question 5 Short Answer
Why does Nash equilibrium alone permit 'non-credible threats' to sustain equilibria, and how does subgame perfection address this problem?
Think about your answer, then reveal below.
Model answer: Nash equilibrium only requires that no player wants to deviate given what the other players are currently doing. A threat can sustain an equilibrium even if executing the threat would be irrational — as long as the threat deters the opponent from acting in a way that would trigger it, the threat is never tested. Because Nash equilibrium only checks optimality along the equilibrium path, it cannot rule out strategies that specify irrational behavior off the path. Subgame perfection fixes this by requiring strategies to be a Nash equilibrium in every subgame, including those reached only if a threat is called. This forces each player's strategy to be optimal at every decision node regardless of history, eliminating any strategy that relies on a commitment that rational players would abandon if tested.
The key distinction is on-path vs. off-path optimality. Nash equilibrium only enforces optimality on the path that is actually played; subgame perfection enforces it everywhere. This is why backward induction is the tool for finding subgame perfect equilibria — it starts from the end of the game tree, where there is no future strategy to hide behind, and works backwards, enforcing rationality at every node.