Adverse Selection Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Adverse Selection Game

• Players: Two players, Buyer and Seller.
• Strategies: Buyer chooses a price $p$; Seller observes the car quality $q\in\{0,10\}$ and accepts or rejects.

Rules

• Start with privately known seller quality and buyer uncertainty; Players choose whether to offer, accept, or withdraw at the market price.
• The player who acts optimally given adverse selection maximises expected surplus.
• The seller knows the quality of the used car.
• The buyer only knows the prior distribution; The seller accepts an offer if and only if the price is at least the seller’s value.
• The buyer values the car at $1.5q$.

Payoff Matrix

• Seller payoff from trade at price $p$:

• Buyer payoff from trade at price $p$:

• If trade does not occur, both players receive $0$.

Derivation (Best Response Analysis)

If $p=5$:

• Conditions:
  • the low-quality seller with $q=0$ accepts,
  • the high-quality seller with $q=10$ rejects.
• Acceptance at $p=5$ reveals that the car is a lemon with $q=0$.
• Conditional on acceptance, the buyer then expects $q=0$, so expected buyer payoff is

• If $p\geq 10$, both types accept.
• Conditional on acceptance, expected quality is

so the buyer’s valuation is

• The buyer would then get

for every $p\geq 10$.

Derivation (Nash Equilibrium)

• No positive price can give the buyer non-negative expected payoff.
• Therefore, profitable trade cannot be sustained under asymmetric information.

Nash Equilibrium

Result:

There is no positive-trade equilibrium price, the only non-loss-making action is no trade.

Social Optimum

• Under full information, a high-quality car with $q=10$ could trade at any price in the interval $[10,15]$, making both sides weakly better off.
• Adverse selection blocks this efficient trade.

Insights

Insight:

• Acceptance decisions reveal hidden information.
• Hidden quality pushes good types out of the market and destroys gains from trade.