In a signaling game, all worker types choose not to pursue an MBA in equilibrium (a pooling equilibrium). A firm claims this is supported by the belief that any applicant who does get an MBA is low-ability. Is this a valid Perfect Bayesian Equilibrium?
ANo — Bayes' rule requires the firm to update toward high-ability upon observing an MBA, making this belief inadmissible
BYes — getting an MBA is an off-path action (probability zero in equilibrium), so Bayes' rule provides no constraint on beliefs there, and pessimistic off-path beliefs can sustain the equilibrium
CNo — in a PBE, all information sets must be reached with positive probability, otherwise the equilibrium is undefined
DYes — but only if the probability of low-ability types matches the unconditional prior distribution of types
PBE requires that beliefs follow Bayes' rule at information sets reached with positive probability on the equilibrium path. In a pooling equilibrium where nobody gets an MBA, the information set reached by observing an MBA has probability zero — it is off-path. At off-path information sets, Bayes' rule cannot be applied (it requires dividing by zero), so PBE leaves beliefs there relatively unconstrained. Pessimistic off-path beliefs (any MBA applicant is low-ability) are thus admissible under PBE, even though they might seem implausible. Further refinements like the Intuitive Criterion specifically address this weakness.
Question 2 Multiple Choice
What is the key difference between a Perfect Bayesian Equilibrium and a Bayesian Nash Equilibrium?
APBE requires mixed strategies; BNE only allows pure strategies in games with complete information
BPBE adds sequential rationality at every information set and requires beliefs to be updated via Bayes' rule on the equilibrium path, ruling out non-credible threats in dynamic games
CPBE applies to static games with private information; BNE applies to sequential games where players move one at a time
DPBE requires complete information about payoffs; BNE is the appropriate concept when payoffs are private
BNE specifies strategies and prior beliefs but does not require that strategies remain optimal at every sequential decision point — it can sustain equilibria where players make non-credible threats that are never tested. PBE adds two requirements: (1) sequential rationality — strategies must be optimal at every information set, given beliefs — and (2) belief consistency via Bayes' rule on the equilibrium path. These requirements eliminate equilibria sustained by implausible off-path threats. PBE is essentially the combination of subgame perfect equilibrium's sequential rationality idea with Bayesian updating in games of incomplete information.
Question 3 True / False
In a Perfect Bayesian Equilibrium, Bayes' rule should be applied to update beliefs at nearly every information set, including those that are seldom reached in equilibrium.
TTrue
FFalse
Answer: False
Bayes' rule can only be applied at information sets reached with positive probability — this is a mathematical requirement, not a design choice. Bayes' rule involves conditioning on an event, and conditioning on a probability-zero event is undefined. PBE therefore only requires Bayes' rule at on-path information sets. At off-path information sets (probability zero in equilibrium), beliefs are required to support sequentially rational play, but PBE leaves the actual belief values relatively unconstrained. This is a genuine limitation of PBE, which is why stronger refinements exist.
Question 4 True / False
A Perfect Bayesian Equilibrium requires players' strategies to be sequentially rational — optimal given their beliefs — at every information set, including those that occur with probability zero in equilibrium.
TTrue
FFalse
Answer: True
Sequential rationality is a requirement at every information set in a PBE, not just those reached in equilibrium. This is what rules out non-credible threats: even if a player's information set is never reached (because the equilibrium path avoids it), the strategy at that information set must still be optimal given whatever beliefs the player holds there. If it weren't, the equilibrium would be sustained by a threat the player would not actually carry out. Sequential rationality at every information set is the key requirement that PBE inherits from the subgame perfect equilibrium concept.
Question 5 Short Answer
Why does defining a Perfect Bayesian Equilibrium require specifying both a strategy profile AND a belief system, rather than just strategies as in a standard Nash equilibrium?
Think about your answer, then reveal below.
Model answer: In dynamic games with incomplete information, players make decisions at information sets where they are uncertain about which node they are at — they don't know which type the other player is or which prior moves were made. Without an explicit belief system, we cannot evaluate whether a strategy is rational: 'maximize expected payoff' requires knowing the probabilities over the nodes in the information set. A strategy that looks rational under one belief might be irrational under another. The belief system specifies exactly what each player believes at each information set, making the rationality criterion well-defined. Bayes' rule then ensures these beliefs are not arbitrary — they must be coherent with the equilibrium strategies wherever possible.
This is why PBE is technically a solution concept for a pair (strategy profile, belief system), not just for a strategy profile. The two components are jointly required to satisfy sequential rationality and belief consistency.