Bayes‑Nash Equilibrium in Cournot Competition and Auction Theory

Two firms compete in quantities while the market demand intercept, θ, can be either low (θ_L) or high (θ_H) with equal probability. Firm 1 observes the actual state of θ and can condition its output on that information, choosing Q₁(θ_L) and Q₁(θ_H). Firm 2 lacks this private signal and must commit to a single output Q₂. A Bayes‑Nash equilibrium requires that each type of each firm maximizes its expected profit given the other firm’s strategy. This means the optimality condition must hold for Firm 1 when θ = θ_L, when θ = θ_H, and for Firm 2’s unconditional choice. The equilibrium quantities solve the system of first‑order conditions derived from the profit functions under each demand state.

Introduction to Auction Theory

Auctions appear whenever buyers’ valuations are uncertain or when a market needs price discovery. Real‑world examples include Treasury securities, fundraising events, and the massive private auction platform Google Ads, which generates roughly $250 billion in annual revenue. In such settings, the auction format shapes how participants bid and how efficiently the item is allocated. The lecture highlights that “we see auctions generally when firms are uncertain about how much people are willing to pay.”

Auction Mechanisms

Two sealed‑bid formats dominate the discussion. In a first‑price auction, the winner pays the amount it bid; the bid therefore influences both the probability of winning and the profit margin. In a second‑price auction, the winner pays the second‑highest bid, making truthful bidding a dominant strategy because “if you win the item and pay your valuation, it’s just as good as not winning at all.” Classroom experiments using MobLab illustrate how participants tend to overbid in first‑price settings, driven by the “excitement of winning,” while second‑price auctions yield bids close to true valuations.

Solving for First‑Price Auction Equilibrium

Consider two bidders whose private valuations V_i are drawn independently from a uniform distribution on [0, 1]. Each bidder’s payoff is U_i = (V_i − b_i) if b_i > b_j and 0 otherwise. The optimal bidding strategy balances the higher winning probability from a larger bid against the reduced profit margin. Using a “guess and check” approach, the symmetric linear equilibrium is found to be b = V/2. This “bid shading” result means each bidder submits a bid equal to half of its valuation, confirming the theoretical prediction that the first‑price equilibrium strategy for two uniform bidders is b = V/2.

Mechanisms & Explanations

The Bayes‑Nash equilibrium condition demands optimality for every type of every player, ensuring that each mental state—such as Firm 1’s knowledge of θ_L or θ_H—has its own first‑order condition. In first‑price auctions, raising a bid raises the chance of winning but cuts the profit per unit; the equilibrium bid precisely balances these forces. By contrast, the second‑price format removes the incentive to shade because the winner’s own bid does not affect the price paid, making truthful bidding optimal.