10.3 An Example of Postcontractual Hidden Knowledge: The Salesman Game If the customer type is a , ð Pushover the efficient sales effort is low and sales should be moderate . If the customer type is a , ð Bonanza the effort and sales should be higher . The Salesman Game Players ð a manager and a salesman r
The order of play ð 1 The manager offers the salesman a contract of the form [ ( ), ( )], w m q m where w is the wage , is q sales , and is a m message . 2 The salesman decides whether or not to accept the contract. 3 Nature chooses whether the customer type t is a or Bonanza a with probabilities 0.2 and 0.8. Pushover The salesman observes the type, but the manager does not .
4 If the salesman has accepted the contract, he chooses his effort . e œ His sales level is q e , so his sales perfectly reveal his effort. œ 5 The salesman's wage is ( ) if he chooses ( ) w m e q m and zero otherwise.
Payoffs ð The manager is risk-neutral and the salesman is risk-averse . r If the salesman rejects the contract, r _ œ his payoff is 8 and the manager's is zero. U If he accepts the contract, r then 1 œ , and 1 œ ( , , ), q w U e w t manager salesman 2 2 ` Î ` ` Î ` ` Î ` where 0, 0, 0, U e U e U w 2 2 and ` Î ` 0. U w The manager can perfectly deduce effort, even out of equilibrium. ð
The optimal contract The manager's indifference curves are straight lines with slope 1. ð The salesman's indifference curves slope upwards , and are convex . ð The salesman has two sets of indifference curves, r solid for and dashed for . Pushovers Bonanzas
Figure 10.1 ð The optimal truth-telling contract is the pooling contract r that pays the intermediate wage of w 3 for the intermediate quantity of q 3 , and zero for any other quantity, regardless of the message . The pooling contract is a second-best contract, r a compromise between the optimum for Pushovers and the optimum for . Bonanzas The contract must satisfy the participation constraint, r 0.8 ( , , ) 0.2 ( , , ) 8. U q w Pushover U q w Bonanza 3 3 3 3
The nature of the equilibrium depends on the shapes of the indifference ð curves. Figure 10.2 ð The equilibrium is separating , not pooling, and r there does exist a first-best fully revealing , contract. The contract induces the salesman to be truthful , and r the incentive compatibility constraints are satisfied. The idea is to reward salesmen not just for high effort, ð but for appropriate effort.
Another way to look at a separating equilibrium is to think of it as a choice of contracts rather than as one contract with different wages for different outputs . In this interpretation, the manager offers a menu of contracts and ð the salesman selects one of them after learning his type .
The Salesman Game illustrates a number of ideas . It can have either a pooling or a separating equilibrium. ð The revelation principle can be applied to avoid ð having to consider contracts in which the manager must interpret the salesman's lies . It shows how to use diagrams when the algebraic functions are ð intractable or unspecified.
10.4 The Groves Mechanism The principal is an altruistic government ð that cares directly about the utility of the agents. a benevolent government r The mayor is considering installing a streetlight costing $100. ð He will only install it if he decides that the sum of the residents' r valuations for it is greater than or equal to the cost . The mayor's problem is to discover their valuations. r
The Streetlight Game Players ð the mayor and five householders r The order of play ð 0 Nature chooses the value v i that householder places on having a streetlight installed, i using distribution f v ( ). i i Only householder observes . i v i
1 The mayor announces a mechanism M , , which requires a householder who reports to pay ( ) m w m if the streetlight is installed, and installs the streetlight 5 if ( ) ´ 100 0. g m , m , m , m , m m 1 2 3 4 5 j œ j 1 2 Householder reports value i simultaneously i m with all other householders. 3 If ( ) 0, g m , m , m , m , m 1 2 3 4 5 the streetlight is built and householder pays ( ). i w m i
Payoffs ð The mayor tries to maximize social welfare , r including the welfare of taxpayers besides the 5 householders . His payoff is zero if the streetlight is not built. r Otherwise, it is r 5 1 mayor œ 100, v j œ 1 j 5 subject to the constraint that ( ) 100, w m j œ 1 j so he can raise the taxes to pay for the light.
The payoff of householder is zero r i if the streetlight is not built. Otherwise, it is r 1 i ( ) œ ( ). m , m , m , m , m v w m 1 2 3 4 5 i i
Mechanisms Mechanism M 1 ð 5 œ ( ) 20, 100 r w m Build iff m i j œ 1 j Talk is cheap, and r the dominant strategy would be to overreport or underreport . a flawed mechanism r
Mechanism M 2 ð 5 œ w m ( ) Max m { , 0}, Build iff m 100 r i i j œ 1 j If all the householders knew each other's values perfectly, r then there would be a continuum of Nash equilibria that attained the efficient result. Each householder would announce up to his valuation r if necessary.
Mechanism M 3 ð 5 œ ( ) 100 , 100 r w m m Build iff m i j j Á œ 1 j i j a Nash equilibrium in which all the players are truthful r a dominant-strategy mechanism r Truthfulness is weakly dominant . ñ The players are strictly better off telling the truth ñ whenever lying would alter the mayor's decision . It is not budget-balancing. r The total tax revenue could easily be negative . r
10.5 Price Discrimination A problem of mechanism design under adverse selection ð Varian's Nonlinear Pricing Game Players ð one seller and one buyer r The order of play ð
0 Nature assigns the buyer a type s , . The buyer is "unenthusiastic" with utility function or u "valuing" with utility function , with equal probability. v The seller does not observe Nature's move, but the buyer does . 1 The seller offers mechanism { , } w q m m under which the buyer can announce his type as and m buy amount for lump sum . q w m m 2 The buyer chooses a message m or rejects the mechanism entirely and does not buy at all.
Payoffs ð The seller has a zero marginal cost, so his payoff is . r w w u v The buyers' payoffs are 1 œ ( ) and 1 œ ( ) r u q w v q w u u u v v v q œ if is positive, and 0 if q 0, w w ww ww with , 0 and , 0. u v u v The marginal willingness to pay is greater for the valuing buyer: r for any , q w w ( ) ( ). (10.27) u q v q
Condition (10.27) is an example of the single-crossing property. ð œ œ Combined with the assumption that (0) (0) 0, r v u it also implies that ( ) ( ) for any value of . u q v q q
Perfect Price Discrimination The game would allow perfect price discrimination ð if the seller did know which buyer had which utility function. The seller's maximization problem ð r Maximize w w u v , , , q q w w u v u v subject to the participation constraints u q ( ) w 0 ñ u u ( ) 0 ñ v q w v v
The constraints will be satisfied as equalities . ð œ ( ) r w u q u u œ ( ) r w v q v v The seller's maximization problem rewritten ð ( ) ( ) r Maximize u q v q u v , q q u v w w * * œ œ ( ) 0 ( ) 0 ð u q v q u v * * * * œ ( ) œ ( ) w u q w v q u u v v The entire consumer surpluses are eaten up. r
Interbuyer Price Discrimination The interbuyer price discrimination problem arises ð when the seller knows which utility functions Smith and Jones have and can sell to them separately . Assume that the seller must charge each buyer a single price per unit ð and let the buyer choose the quantity.
The seller's maximization problem ð r Maximize p q p q u u v v , , , q q p p u v u v subject to the participation constraints ( ) 0 ñ u q p q u u u ( ) 0 ñ v q p q v v v and the incentive compatibility constraints œ [ ( ) ] ñ q argmax u q p q u u u u œ [ ( ) ] ñ q argmax v q p q v v v v
The buyers' quantity choice problems ð w ( ) œ 0 r u q p u u w ( ) œ 0 r v q p v v The seller's maximization problem rewritten ð w w ( ) ( ) r Maximize u q q v q q u u v v , q q u v subject to the participation constraints w ( ) ( ) 0 ñ u q u q q u u u w ( ) ( ) 0 ñ v q v q q v v v
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