Screening and Price Discrimination Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Screening and Price Discrimination Game

• Players: Three players, Seller, Type 1 buyers, and Type 2 buyers.
• Strategies: The seller chooses prices for two software versions: Only Answers and Show Steps; Each buyer chooses the option that gives the highest surplus, or does not buy.

Rules

• Start with a seller facing Type 1 and Type 2 buyers.
• Players choose software-version prices and purchase options.
• The player who selects the profit- or surplus-maximising option is optimal.
• There are $45$ buyers of each type.
• Marginal cost is approximately zero, so revenue maximisation is equivalent to profit maximisation up to fixed cost.

Payoff Details

• Buyer valuations are:

• Buyer surplus equals willingness to pay minus price.
• Seller revenue equals price times quantity sold.

Derivation (Revenue and Incentive Compatibility)

Uniform pricing benchmark

• Under uniform pricing with a single product, the best revenue is:
  • Only Answers at $150$: $150 \cdot 90 = 13{,}500$,
  • Only Answers at $200$: $200 \cdot 45 = 9{,}000$,
  • Show Steps at $180$: $180 \cdot 90 = 16{,}200$,
  • Show Steps at $500$: $500 \cdot 45 = 22{,}500$.
• Therefore the best single-product policy is to sell only Show Steps at price $500$, with revenue $22{,}500$.

Screening menu

• Let $A$ denote Only Answers and $S$ denote Show Steps.
• Set prices:

Type 1 self-selection

• Surplus from Only Answers: $150-150=0$.
• Surplus from Show Steps: $180-449=-269$.
• Since $0>-269$, Type 1 chooses Only Answers.

Type 2 self-selection

• Surplus from Only Answers: $200-150=50$.
• Surplus from Show Steps: $500-449=51$.
• Since $51>50$, Type 2 chooses Show Steps.

Revenue comparison

• The menu $(p_A,p_S)=(150,449)$ is incentive compatible because each type chooses the intended version.
• Total revenue is

• Since $26{,}955 > 22{,}500$, screening dominates the best single-product policy.

Nash Equilibrium

Result:

The self-selection outcome is:

• Price Only Answers at $150$.
• Price Show Steps at $449$.
• Type 1 buys Only Answers.
• Type 2 buys Show Steps. This yields revenue $26{,}955$.

Social Optimum

• Because marginal cost is zero and both types value both products above zero, serving all buyers is efficient.
• The screening menu achieves that allocation while extracting more revenue from the high-value type.

Insights

Insight:

• Screening works by making the premium version attractive mainly to high-value buyers.
• The menu is designed around incentive compatibility, not just around willingness to pay in isolation.