Static Games of Incomplete Information

Static Games of Incomplete Information .

Mechanism design • Typically a 3-step game of incomplete info Step 1: Principal designs mechanism/contract Step 2: Agents accept/reject the mechanism Step 3: Agents that have accepted, play the game specified by mechanism • Constant theme: Incomplete information and binding individual rationality constraints prevent efficient outcomes

Nonlinear pricing • A monopolist produces good at marginal cost c and sells quantity q • Consumer transfers T to seller and has utility u1(q, T, θ)= θV(q)-T, V(0)=0, V/>0, V//<0 • θ is private knowledge for buyer • Seller knows that θ= w.p. and θ= w.p. • The game: 1. Seller offers tariff T(q): specifies a price for qty q 2. Consumer accepts/rejects • If seller knows θ, she will charge T=θV(q), her profit, θV(q)-cq. This is maximized atsome q given by θV/(q)=c

Nonlinear pricing • Let be bundle for type and for type • Seller’s expected profit: • Seller faces two constraints: 1. Individual Rationality (IR):Consumer should be willing to purchase 2. Incentive Compatibility (IC): Consumer should consume the bundle intended for his type • IR1: ; and IR2: • IC1: ; and IC2: • First step: To show that only IR1 and IC2 are binding

Nonlinear pricing • First note: IR1 and IC1 imply IR2 • IR2 can’t be binding unless =0 • However, IR1 must bind. Else seller can increase by same amount and increase revenue • Also, IC2 must be binding, else seller can increase , satisfy all constraints and increase revenue • The high-type’s indifference curve is always steeper than the low type’s for any allocation • This implies that high type consumes more than low type:

Nonlinear pricing • Eliminating transfers, principal’s objective function is: • FOC wrt • FOC wrt • Check that IC1 is satisfied • Note: Quantity purchased by high-type is optimal Quantity purchased by low-type is sub-optimal • Seller sacrifices efficiency for rent-extraction!

Auctions • Seller has unit of good and there are two bidders • Each bidder can have types , with < • Corresponding probabilities are and • Buyer’s expected probability of getting the good are and payments are • The constraints are: IR1: ; IR2: IC1: ; IC2: • What is seller’s optimal contract?

Auctions • Seller’s expected profit is: • Again, IR1 and IC2 are binding. The seller’s profit: • Also, ex-ante prob of a player getting good, • Moreover, • Case 1: . The seller sets and Optimal mechanism: Not to sell if both announce low-type; sell to high-type if they announce different types; sell wp ½ to each if both announce high type • Case 2: . The seller sets and Optimal mechanism: Sell to high-type if bidders announce different types, and sell wp ½ to each if they both announce high-type or low-type

Mechanism design with a single agent • Agent’s type with distribution/density • Type-contingent allocation is fn. • Defn: A decision function is implementable if there exists a transfer t(.) such that allocation y(.) is incentive-compatible, i.e. • Theorem: A piecewise C1decision fn x(.) is implementable only if whenever and x is differentiable at θ

Mechanism design with a single agent • Sketch of proof: Type θ announces to maximize The FOC and SOC are Totally differentiating the first equation, The (local) SOC becomes or, Rewrite the FOC we get, Eliminating, dt/dθ,

Mechanism design with a single agent • The sorting/ single crossing/ constant sign (CS) condition is: • Note that is agent’s marginal rate of substitution between decision k and transfer t • Consider x to be output supplied by agent, i.e., • Then sorting condition means that the agent’s indifference curve in (x, t) space, , is decreasing in θ • If θ2> θ1 , y(θ1)=(x(θ1), t(θ1)), y(θ2)=(x(θ2), t(θ2)), then y(θ2)>y(θ1) • Theorem: If decision space is 1-dim and CS holds, then a necessary condition for x(.) to be implementable is that it is monotonic. • What about sufficiency?

Optimal mechanisms for one agent • The assumptions: A1: Reservation utility independent of type A2: Quasi-linear utilities: Principal: u0(x, t,θ)= V0(x, θ)-t;Agent: u1(x, t,θ)= V1(x, θ)+t A3: n=1, i.e., decision is 1-dim and CS holds. A4: A5: (satisfied if V0 is independent θ) A6:

Optimal mechanisms for one agent • The problem: Principal maximizes his expected utility subject to: (IR) u1(x(θ), t(θ), θ)≥ =0, for all θ (IC) u1(x(θ), t(θ), θ)≥ • From A1 & A4, if IR satisfied at , it is satisfied everywhere • IR binding at . Thus, • Let • From Envelope theorem, • This implies that,

Optimal mechanisms for one agent • Further, u0= V0+ V1- U1≡ Social surplus-Agent’s utility • Principal’s objective function: • Since monotonicity is necessary and sufficient for implementability, Principal’s optimization program becomes s.t. x(.) is monotonic

Optimal mechanisms • We solve the principal’s program ignoring monotonicity • The solution to the relaxed program is • The principal faces a trade-off between maximizing total surplus (V0+ V1) and appropriating the agent’s info rent (U1) • When is it legit to focus on relaxed program? When solution x*(θ) to above eq is monotonic. Differentiating, WhenHazard rate is monotone:

Static Games of Incomplete Information