VCG Auctions and Perfect Bayesian Equilibrium Explained

A Vickrey‑Clarke‑Groves (VCG) auction allocates items so that the total reported utility is maximized. The rule “a bid is essentially a report or claim about how much you value the items” captures the core idea: each bidder states a valuation, and the mechanism chooses the efficient allocation.

Payments are set by the externality each bidder imposes on the others. To compute a payment, first determine the optimal utility that all other bidders would obtain if the current bidder were absent; then subtract the utility those bidders actually receive when the bidder participates. The difference is the amount the bidder must pay.

Because a bidder’s payment depends only on others’ valuations, reporting the true value maximizes the bidder’s own payoff. Truthful reporting is therefore a dominant strategy, and payments are always non‑negative.

Example
- Bidder 1 values item A at 5, item B at 3, and the bundle at 8.
- Bidder 2 values item A at 2, item B at 6, and the bundle at 7.

The efficient allocation gives A to bidder 1 and B to bidder 2, yielding total utility 11.

• Bidder 1’s payment equals the loss imposed on bidder 2: without bidder 1, bidder 2 would obtain item B for a utility of 6; with bidder 1 present, bidder 2 receives B for utility 3, so the externality is 1.
• Bidder 2’s payment equals the loss imposed on bidder 1: without bidder 2, bidder 1 would obtain A for utility 5; with bidder 2 present, bidder 1’s utility from A drops to 2, so the externality is 3.

Thus bidder 1 pays 1 and bidder 2 pays 3, and both receive their allocated items.

Dynamic Bayesian Games

Dynamic games with incomplete information require solution concepts that handle both strategic timing and private types. Nash equilibrium alone is insufficient because it can sustain non‑credible threats—strategies that a rational player would never actually carry out. Subgame‑perfect equilibrium improves on this by eliminating threats that are not optimal in every subgame, but it still fails when information sets contain multiple nodes that a player cannot distinguish.

Perfect Bayesian Equilibrium (PBE) resolves these issues by combining subgame perfection with Bayesian reasoning. It refines Nash equilibrium by demanding that players’ beliefs be updated consistently with Bayes’ rule wherever possible, and that actions be optimal given those beliefs.

Perfect Bayesian Equilibrium

An “assessment” in PBE consists of a strategy profile together with a belief system—probability distributions over the nodes in each non‑singleton information set. Two conditions must hold:

1. Sequential rationality – each player’s action maximizes expected payoff given the beliefs and the strategies of the others.
2. Belief consistency – beliefs are derived from Bayes’ rule at information sets reached with positive probability. When an information set is off the equilibrium path (reached with zero probability), Bayes’ rule does not apply, allowing flexible belief assignment. As the lecture notes, “Bayes' rule only allows us to define conditional probabilities conditional on events that have positive probability.”

Pooling vs. Separating Equilibria

• Pooling equilibrium: Both types of players choose the same action, so no private information is revealed. “It's called a pooling equilibrium because both types of students… choose the same action.”
• Separating equilibrium: Different types choose different actions, thereby signaling their private information. A classic example is a costly essay: only high‑value types find the essay worthwhile, while low‑value types avoid the cost.

For a costly signal to separate types, the cost (c) must satisfy two inequalities: it must exceed the benefit a low‑value type would obtain from sending the signal (so the low type refrains), yet be lower than the benefit a high‑value type gains (so the high type proceeds). Formally, (c > \text{benefit}{\text{low}}) and (c < \text{benefit}{\text{high}}). This condition ensures that the signal credibly distinguishes the types.

Signaling & Screening

Signaling and screening are mechanisms that use costly actions to extract private information. In the “Application Game” discussed in class, students may submit an optional essay. The essay acts as a signal: if the cost of writing the essay lies between the benefits for low‑ and high‑valuation students, only the high‑valuation students will submit, revealing their type.

The lecture emphasized that such costly signals are essential for achieving separating equilibria in dynamic Bayesian games, allowing the designer to screen participants based on their willingness to incur the cost.