Signaling Games and Perfect Bayesian Equilibrium

What is a Signaling Game?

A signaling game is a specific type of dynamic game with incomplete information. It typically involves two players: a "Sender" who has private information and a "Receiver" who does not. The Sender moves first, choosing an action (a "signal") that the Receiver can observe. The Receiver then updates their beliefs about the Sender's private information and chooses a response.

A simple signaling game unfolds in these steps:

1. Nature's Move: Nature determines the Sender's "type," which is their private information. For example, a worker might be of a "high-ability" or "low-ability" type. The probabilities of these types are common knowledge.
2. The Sender's Move: The Sender, knowing their own type, chooses an action, or signal. For instance, the worker chooses a level of education to attain.
3. The Receiver's Move: The Receiver observes the Sender's signal (e.g., the education level) but not their type. Based on the signal, the Receiver updates their belief about the Sender's type and chooses an action. For example, a firm offers a wage.
4. Payoffs: The payoffs for both players depend on the Sender's true type, the signal chosen, and the Receiver's response.

To see this structure in action, please watch the first couple of minutes of the following video.

As the video illustrates, the crucial element in the game tree is Player 2's information set. This is represented by the dotted line connecting the nodes where Player 2 makes a decision.

This information set captures the essence of the Receiver's problem: they know what action the Sender took, but they don't know from which "type" that action originated. How, then, can the Receiver make an optimal choice? They must form a belief about the probability of being at each node within the information set. This brings us to the need for a new equilibrium concept.

Why We Need a New Solution Concept: Perfect Bayesian Equilibrium

Our existing tools are not quite sufficient for solving these games:

• Subgame Perfect Nash Equilibrium (SPNE), which we used for dynamic games of complete information, relies on breaking the game into subgames. However, an information set that contains more than one node prevents the game tree from being cleanly divided into proper subgames.
• Bayesian Nash Equilibrium (BNE), which we used for static games of incomplete information, requires players to have beliefs and maximize expected utility. However, it's designed for simultaneous-move games and doesn't explicitly handle how beliefs should be updated as the game unfolds.

Perfect Bayesian Equilibrium (PBE) was developed to bridge this gap. It combines the sequential rationality of SPNE with the belief-based reasoning of BNE.

The Two Pillars of Perfect Bayesian Equilibrium

PBE provides rules for both behavior (strategies) and cognition (beliefs). A PBE is an assessment, which is a pair consisting of a strategy profile and a belief system, that satisfies two key requirements.

Let's formalize these two pillars. For additional depth and mathematical precision, you can consult the handout from Northwestern University.

1. Sequential Rationality

A strategy profile is sequentially rational if, at every information set, the player who is to move chooses an action that maximizes their expected payoff, given their beliefs and the subsequent strategies of the other players.

• This extends the logic of backward induction from SPNE.
• The crucial difference is that the optimality of an action is conditional on the player's beliefs. At an information set, a player calculates the expected payoff of each available action by averaging the payoffs over the different nodes in the set, weighted by their belief probabilities.

For example, in the game tree image above, Player II's optimal choice between L' and R' depends entirely on what they believe the probability is that Player I chose L versus M.

2. Belief Consistency (via Bayes' Rule)

A belief system is consistent if the beliefs are derived from the players' strategies using Bayes' rule wherever possible.

• A belief system $\mu$ assigns a probability to each node within an information set. For an information set $I$, $\mu (v)$ is the probability that player $i$ assigns to being at node $v\in I$, given that play has reached the set $I$. Naturally, ${\sum }_{v\in I}\mu (v)=1$.

• On the Equilibrium Path: For any information set that is reached with positive probability under the equilibrium strategies, the beliefs must be calculated using Bayes' rule. If $\text{Pr}(v|\sigma )$ is the probability of reaching node $v$ given strategy profile $\sigma$, then the belief at $v$ is:

  $$\mu (v)=\frac{\text{Pr}(v|\sigma )}{{\sum }_{v^{\prime }\in I}\text{Pr}(v^{\prime }|\sigma )}$$

• Off the Equilibrium Path: The phrase "wherever possible" addresses situations where an information set is reached with zero probability (i.e., someone deviates from the equilibrium path). In this case, Bayes' rule results in a $0/0$ indeterminate form. PBE allows for any beliefs to be assigned at off-path information sets, as long as the strategies are still sequentially rational given those beliefs. These off-path beliefs are critical for determining whether threats are credible.

Formal Definition of Perfect Bayesian Equilibrium

We can now combine these elements into a complete definition.

A Perfect Bayesian Equilibrium (PBE) is an assessment $({\sigma }^{\ast },{\mu }^{\ast })$ — a pair consisting of a strategy profile ${\sigma }^{\ast }$ and a belief system ${\mu }^{\ast }$ — that satisfies two conditions:

1. Sequential Rationality: The strategy profile ${\sigma }^{\ast }$ is sequentially rational given the belief system ${\mu }^{\ast }$. That is, at each information set, the moving player's strategy maximizes their expected payoff, calculated using the beliefs ${\mu }^{\ast }$ for that information set.

2. Belief Consistency: The belief system ${\mu }^{\ast }$ is consistent with the strategy profile ${\sigma }^{\ast }$. That is, for any information set reached with positive probability, beliefs are computed using Bayes' rule.

This equilibrium concept ensures that players act rationally given their beliefs, and their beliefs are rational given the strategies being played. It forces us to explicitly state our assumptions about how players interpret the actions of others, which is the heart of analyzing signaling games.

Conclusion

In this lesson, we have laid the groundwork for analyzing dynamic games of incomplete information.

Key Takeaways:

• Signaling games are dynamic games where an informed Sender sends a signal to an uninformed Receiver, who then updates their beliefs and responds.
• The solution concept for these games is Perfect Bayesian Equilibrium (PBE), which is an assessment consisting of both strategies and beliefs.
• A PBE must satisfy two conditions:
  1. Sequential Rationality: Players' strategies must be optimal at every point in the game, given their beliefs.
  2. Belief Consistency: Beliefs must be formed logically using Bayes' rule based on the observed strategies, at least on the equilibrium path.

We have defined the tools. In our next lesson, we will immediately put them to use by analyzing one of the most famous signaling models in economics: George Akerlof's "Market for Lemons." We will see how the PBE concept allows us to formally model how asymmetric information can lead to severe market failures.