8.3 Perfect Bayesian equilibrium

Perfect Bayesian equilibrium (PBE) combines strategic decision-making with belief updating in games where players don't know each other's types. It's the go-to solution concept for dynamic games with incomplete information, showing up in signaling games, auctions, and bargaining models.

PBE ties two things together: players must play optimally given their beliefs, and those beliefs must be consistent with how players actually behave. This creates a tight logical loop where strategies and beliefs reinforce each other.

Perfect Bayesian Equilibrium

Definition and Requirements

Perfect Bayesian equilibrium extends subgame perfect equilibrium to handle games with incomplete information. A PBE has two components that must work together:

1. Strategies for every player at every information set
2. Beliefs (probability distributions) over other players' types at every information set

For these to form a PBE, two conditions must hold:

• Sequential rationality: Each player's strategy must be optimal at every information set, given their beliefs about other players' types and the strategies those players will follow going forward. This isn't just about the start of the game; it applies at every point where a player might act.
• Belief consistency: Beliefs must be derived from Bayes' rule and the equilibrium strategies wherever possible. On the equilibrium path (where actions actually occur with positive probability), Bayes' rule pins down beliefs exactly. Off the equilibrium path, beliefs can't be pinned down by Bayes' rule, so they're less restricted, though they still need to be "reasonable."

Applications and Examples

PBE is the standard tool for analyzing dynamic games with asymmetric information: signaling games, auctions, bargaining, and reputation models.

Consider Spence's job market signaling game. A job seeker knows their own productivity type (high or low), but employers don't. The job seeker chooses an education level, and then an employer observes that choice and offers a wage.

• The job seeker's strategy specifies how much education to get, depending on their type.
• The employer's strategy specifies what wage to offer based on the observed education level.
• The employer holds beliefs about the job seeker's type after seeing the education choice.

In a PBE, the job seeker's education choice must be optimal given the wage the employer will offer, and the employer's beliefs about the seeker's type must follow from Bayes' rule applied to the equilibrium education strategies. The wage offer must then be optimal given those updated beliefs.

Beliefs and Bayesian Updating

Forming and Updating Beliefs

In these games, players can't observe each other's types directly. Instead, they start with prior beliefs (often determined by nature's move at the start of the game) and update those beliefs as they observe actions.

A belief at an information set is a probability distribution over the nodes in that set. When a player observes an action, they revise their belief using Bayes' rule:

$P(\text{type} \mid \text{action}) = \frac{P(\text{action} \mid \text{type}) \cdot P(\text{type})}{P(\text{action})}$

where $P(\text{action})$ is the total probability of that action across all types:

$P(\text{action}) = \sum_{\text{all types}} P(\text{action} \mid \text{type}) \cdot P(\text{type})$

This is just the standard Bayes' rule calculation. The numerator captures how likely a specific type is to produce the observed action, weighted by the prior. The denominator normalizes everything so the posterior probabilities sum to 1.

Consistency and Off-Path Beliefs

Bayes' rule works cleanly when the observed action has positive probability under the equilibrium strategies. But if an action is never taken in equilibrium (it's "off the equilibrium path"), then $P(\text{action}) = 0$, and the formula breaks down: you'd be dividing by zero.

In this case, Bayes' rule simply can't be applied. PBE allows beliefs at off-path information sets to be any probability distribution over types. This flexibility is both a feature and a limitation of PBE. Stronger refinements (like the Intuitive Criterion in signaling games) place additional restrictions on off-path beliefs.

One natural restriction that PBE does impose: beliefs should assign zero probability to types that could not possibly have taken the observed action, regardless of the equilibrium. For example, if taking a certain action is strictly dominated for low types no matter what beliefs the other player holds, then observing that action should lead to a belief that assigns probability 0 to the low type.

Solving for Perfect Bayesian Equilibria

Step-by-Step Process

Finding a PBE typically involves a mix of forward reasoning (about beliefs) and backward reasoning (about optimal play). Here's the general approach:

1. Set up the game. Identify the types, the prior distribution over types (nature's move), each player's available actions at each information set, and the payoffs.

2. Conjecture a strategy profile. Propose a strategy for each type of each player. Common conjectures in signaling games include separating (different types take different actions), pooling (all types take the same action), and semi-separating (some types mix).

3. Derive beliefs using Bayes' rule. Given the conjectured strategies, apply Bayes' rule to determine what the uninformed player should believe after each observable action. For off-path actions, assign beliefs (you'll need to check that these support the equilibrium).

4. Optimize the responding player's strategy. Given the derived beliefs, find the responding player's best action at each information set by maximizing expected payoff.

5. Check the informed player's incentives. Verify that each type of the informed player is actually willing to follow the conjectured strategy, given how the responding player will react. No type should want to deviate.

6. Verify all conditions. Confirm sequential rationality at every information set and belief consistency via Bayes' rule on the equilibrium path.

If any step fails, revise the conjectured strategy and try again.

Verifying a PBE

Once you have a candidate equilibrium, check these three things:

• Sequential rationality at every information set. Each player's action must maximize their expected payoff given their beliefs and the continuation strategies. This must hold even at information sets that are never reached in equilibrium.
• On-path belief consistency. At every information set reached with positive probability, beliefs must follow exactly from Bayes' rule and the equilibrium strategies.
• Off-path belief reasonableness. At information sets that are never reached, beliefs must support the equilibrium (no player should want to deviate given the off-path beliefs assigned).

Perfect Bayesian Equilibrium vs. Other Concepts

Comparison to Subgame Perfect Equilibrium

Both PBE and subgame perfect equilibrium (SPE) require sequential rationality: strategies must be optimal at every point in the game, not just at the start. The key difference is what they can handle.

SPE works in games of perfect information (or at least games with well-defined proper subgames). It uses backward induction to eliminate non-credible threats. But in games with incomplete information, most information sets don't begin proper subgames, so SPE has very little bite.

PBE fills this gap by introducing beliefs at information sets and requiring those beliefs to be updated via Bayes' rule. Where SPE refines Nash equilibrium in complete-information dynamic games, PBE does the analogous job in incomplete-information dynamic games.

Relationship to Bayesian Nash Equilibrium

Bayesian Nash equilibrium (BNE) is the baseline solution concept for static games with incomplete information. Each player's strategy must be optimal given their prior beliefs about other players' types. But BNE doesn't impose any requirements about what happens during the game: there's no sequential rationality and no belief updating.

PBE is strictly stronger than BNE. Every PBE is a BNE, but not every BNE is a PBE. The additional requirements of PBE eliminate equilibria that rely on non-credible threats or beliefs that contradict observed behavior.

PBE effectively combines the sequential rationality of SPE with the Bayesian reasoning of BNE, making it the right tool for dynamic games where players learn about each other through their actions.