In many strategic situations, one person knows something the other doesn't. A job applicant knows their true skill level, but the employer can only see their resume. A used car seller knows the car's real condition, but the buyer only knows what they're told. This is called asymmetric information, and it's the core of signaling games. It creates a game of hide and seek, where one player tries to reveal information and the other tries to interpret it.

To formalize this, we introduce a new player: Nature. Nature makes the first move, but it's not a strategic player. Its job is to randomly assign a private 'type' to the first player, called the Sender. This type is the piece of private information. The Sender observes their own type, but the second player, the Receiver, does not. For instance, Nature could assign a job applicant the type 'high-skill' or 'low-skill'.

Since the Receiver doesn't know the Sender's type, they must form a belief about it. Before the game starts, the Receiver has an initial belief, often called a 'prior'. This is the probability they assign to the Sender being a certain type before seeing any action. Let's say $\theta$ represents the set of possible types, like {High-Skill, Low-Skill}. The Receiver's prior belief is a probability distribution $p$ over these types. For example, they might believe there's a 50% chance the applicant is high-skill, so $p(\text{High-Skill}) = 0.5$.

The Sender, knowing their own type, chooses an action, $a$, from a set of possible actions, $A$. A Sender's strategy is a plan that specifies which action to take for every possible type they could be. For example, a high-skill worker might choose to get a college degree, while a low-skill worker might not. This is a potential separating strategy because different types choose different actions, revealing their private information.

Alternatively, all types might choose the same action. For example, both high-skill and low-skill applicants might decide to wear a formal suit to an interview. This is a pooling strategy, because all types 'pool' into one action. In this case, the action provides no new information, and the Receiver's belief about the Sender's type doesn't change.

The most interesting part of the game happens after the Sender acts. The Receiver sees the action $a$ and must update their belief about the Sender's type. This is where Bayesian updating comes in. The Receiver uses the Sender's action to refine their initial probability assessment. Their initial belief, $p(\theta)$, becomes a new, updated belief, $\mu(\theta|a)$, which is the probability of the Sender having type $\theta$ given that they took action $a$.

Based on this updated belief, the Receiver chooses their own action, $r$, from a set of responses, $R$. The Receiver's strategy specifies which response to choose for every possible action they might observe from the Sender. Both players then receive payoffs, $u_S(\theta, a, r)$ for the Sender and $u_R(\theta, a, r)$ for the Receiver. Notice that the payoffs depend on the Sender's true type, not just the actions. A solution to a signaling game, an equilibrium, is a set of strategies and beliefs where no player has an incentive to change their plan, and all beliefs are consistent with the strategies being played.