Static Games of Incomplete Information

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Static Games of Incomplete Information

• .

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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

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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, ?) ?V(q)-T, V(0)0, V/gt0, V//lt0
• ? is private knowledge for buyer
• Seller knows that ? w.p. and ? w.p.
  
• The game
• 1. Seller offers tariff T(q) specifies a price
  for qty q
• 2. Consumer accepts/rejects
• If seller knows ?, she will charge T ?V(q), her
  profit, ?V(q)-cq. This is maximized at some q
  given by ?V/(q)c

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Nonlinear pricing

• Let be bundle for type and
  for type
• Sellers 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

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Nonlinear pricing

• First note IR1 and IC2 imply IR2
• IR2 cant 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-types indifference curve is always
  steeper than the low types for any allocation
• This implies that high type consumes more than
  low type

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Nonlinear pricing

• Eliminating transfers, principals 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!

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Auctions

• Seller has unit of good and there are two bidders
• Each bidder can have types , with lt
  
• Corresponding probabilities are and
• Buyers expected probability of getting the good
  are
• and payments are
• The constraints are
• IR1 IR2
• IC1 IC2
• What is sellers optimal contract?

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Auctions

• Sellers expected profit is
• Again, IR1 and IC2 are binding. The sellers
  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

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Moral Hazard

• Consider a Principal and an agent who can exert
  costly effort, e
• Let e 0, 1, with costs ?(0)0, ?(1) ?
• Agent receives transfer, t, and has utility
• Uu(t)- ?(e), with u/gt0, u//lt0.
• Production is stochastic, and production level,
• ,
• Stochastic influence of effort on production
• ,

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Moral Hazard

• Principal can offer a contract, t( ), that
  depends on observed, random output
• With two possible outcomes, contract is if
  output is and if output is
• Let Principals profit with qty q be S(q)
• His profit when agent expends effort e0 is
• His profit when agent expends effort e1 is

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Incentive Feasible Contracts

• Induce positive effort and ensure participation
• Incentive constraint
• Participation constraint

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Complete Information Benchmark

• Complete info or First-Best Principal observes
  effort
• Principals problem is
• subject to
• Using Lagrangian, µ, and from FOCs we have,
• From the above equations, we have that
• Thus, Agent obtains full insurance!
• The optimal transfer is t u-1(?)h(?), where
  hu-1

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First Best Case

• When there is complete information
• Principals profit from inducing effort e1
• V1
• If agent exerted 0 effort, principal would earn
• V0
• Inducing effort is optimal for principal if
• , where
• Principals First-Best cost of inducing effort
  is h(?)

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Second-Best In terms of transfers

• Agent is risk-averse
• Principals problem, P, is
• (P)
• subject to , and
• First ensure concavity of (P) Let

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Second-Best In terms of utilities

• The Principals program can be rewritten in terms
  of utilities
• (P/)
• Principals objective function is concave in
• because h(.) is convex, and the constraints are
  linear
• The KKT conditions are necessary and sufficient

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Both IR and IC are binding

• Let ? µ be Lagrange multipliers for IC IR
• The FOCs, upon rearranging terms, are
• where, are second-best optimal
  transfers
• From these, , so IR is binding
• Also, , so IC is binding

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Second-Best Solution

• The variables ( , ?, µ ) are solved
  simultaneously from two FOCs, IC and IR
• The second-best optimal transfers are
• contract does not provide full
  insurance
• 2nd Best cost of inducing effort CSB
• Clearly, for the Principal, CSBgt CFB. So
  Principal induces high effort (e1) less often
  than in first-best
• There is under-provision of effort in the
  second-best

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Mechanism design with a single agent

• Agents 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 C1 decision fn x(.) is
  implementable only if
• whenever and x is differentiable at ?

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Mechanism design with a single agent

• Sketch of proof Type ? 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/d?,

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Mechanism design with a single agent

• The sorting/ single crossing/ constant sign (CS)
  condition is
• Note that is agents 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 agents
  indifference
• curve in (x, t) space, , is decreasing
  in ?
• If ?2gt ?1 , y(?1)(x(?1), t(?1)), y(?2)(x(?2),
  t(?2)), then y(?2)gty(?1)
• 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?

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Optimal mechanisms for one agent

• The assumptions
• A1 Reservation utility independent of type
  
• A2 Quasi-linear utilities
• Principal u0(x, t,?) V0(x, ?)-t Agent u1(x,
  t,?) V1(x, ?)t
• A3 n1, i.e., decision is 1-dim and CS holds.
• A4
• A5
• A6

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Optimal mechanisms for one agent

• The problem Principal maximizes his expected
  utility
• subject to (IR) u1(x(?), t(?), ?) 0, for
  all ?
• (IC) u1(x(?), t(?), ?)
• From A1 A4, if IR satisfied at , it is
  satisfied everywhere
• IR binding at . Thus,
• Let
• From Envelope theorem,
• This implies that,

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Optimal mechanisms for one agent

• Further, u0 V0 V1- U1 Social surplus-Agents
  utility
• Principals objective function
• Since monotonicity is necessary and sufficient
  for implementability, Principals optimization
  program becomes
• s.t. x(.) is monotonic

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Optimal mechanisms

• We solve the principals 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 agents info rent (U1)
• When is it legit to focus on relaxed program?
• When solution x(?) to above eq is monotonic.
  Differentiating,
• When Hazard rate is monotone