NTU IO (I) : Auction Theory and Mechanism Design II Groves Mechanism and AGV Mechansim. u i (x, t i, θ i ) = V i (x, θ i ) + t i,

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1 Meng-Yu Liang NTU O : Auction Theory and Mechanism Design Groves Mechanism and AGV Mechansim + 1 players. Types are drawn from independent distribution P i on [θ i, θ i ] with strictly positive and differentiable densities p i. The distributions are common knowledge. private values Agen s preferences: u i x,, θ i Preferences are quasi-linear: For any i 1,..., }, and either selft-interested principal or u i x,, θ i V i x, θ i +, u 0 x, t, θ V 0 x, θ u 0 x, t, θ i1 V x, θ benevolent principal, where V 0 x, θ B 0 x, θ C 0 x, C 0 x is the principal s monetary cost from decision x and B 0 x, θ is nonmonetary benefit. An allocation y is ex post effi cienf x θ K for each θ and i0 x θ maximizes V i x, θ over K, for all θ i0 Budget Balance: θ C 0 x θ for all θ i1 Dominant Strategy vs. Bayesian Mechanisms: Choose agen s transfer so that agen s payoff is the same as the total surplus of all parties up to a constant. Dominant-strategy mechanism: Each agent s optimal announcemens independent of the announcements of the other agents, i.e., C is for each agen 1,..., and for each θ i, ˆθ i and θ i, Bayesian mechanism: C is u i y θ i, θ i, θ i u i y ˆθi, θ i, θ i. E θ i u i y θ i, θ i, θ i E θ i u i y ˆθi, θ i, θ i 1

2 Dominant strategy mechanism is not sensitive to beliefs that players have about each other and it does not require players to compute Bayesian equilibrium strategies. However, focusing on dominant-strategy mechanism restricts the set of mechansims considerably. Thus, implementation in dominant strategies is a nice property to have if feasible, bus not clear how much utility loss a principal should be willing to tolerate in order to have dominant strategies for the agents. Mookherjee and Reichelstein 1989 identify a class of models in which dominant strategy implementation involves no welfare loss relative to Bayesian implementation. Effi ciency Theorems The Groves Mechanism 1973 Dominant Strategy implementation: Choose agen s transfer so that agen s payoff is the same as the total surplus of all parties up to a constant. Let x θ maximizing i0 V i x, θ i denote an effi cient solution for the type profile θ. Define ˆθ j 0,...,} V j x ˆθi, ˆθ j + τ i ˆθ i, where τ i ˆθ i is an arbitrary function of ˆθ i. ˆθi θ Show thas optimal for agen to announce his true type the other agents announcements. This implies that x ˆθ regardless of, t ˆθ is a dominant strategy mechanism which yields effi cient allocation. The proof is simple: Suppose that agen strictly prefers announcing ˆθ i to announcing θ i for some types ˆθ i of the other agents. Then V i x ˆθi, ˆθ i, θ i + V j x ˆθi, ˆθ j > V i x θ i, ˆθ i, θ i + V j x θ i, ˆθ j. But this contradicts the fact that x θ i, ˆθ i is effi cient for type profile θ i, ˆθ i. Budget balance may not be satisfied. Example: Should a bridge be built? u i θ i x +, where x is equal to 0 or 1 and θ i is agen s valuation or willingness to pay for the public good. With c > 0 denoting the cost of supplying the public good, the effi cient rule is x θ 1 if i1 θ i c One Groves mechanism for this example takes the following form: ˆθ ˆθ j c if ˆθ j1 j c and ˆθ x 1 if i1 ˆθ i c 2

3 The AGV Mechanism: nstead of being paid the surpluses of the other agents on the basis of their reports, each agens paid the expected value of the other agents surpluses conditional on his own report. ˆθ E θ i j 0,...,} V j x ˆθi, θ i, θ j + τ i ˆθ i The function τ i will be determined later to ensure BB. x, s Bayesian incentive compatiable: ˆθi θ i must maximize E θ i V i x ˆθi, θ i, θ i + V j x ˆθi, θ i, θ j. But ˆθ i θ i maximizes the term inside the expectation operator for all θ i and, therefore, maximizes the expectation. Suppose cost C 0 x 0. BB requires i1 ˆθ 0. Let V j x ˆθi, θ i E i ˆθi E θ i, θ j denote the expected externality for agen when he announces ˆθ i. Since E i ˆθi is the first part of the transfer to agen and τ i is supposed not to depend on ˆθ i, E i ˆθi must be paid be other agents, i.e., τ i ˆθ i E j ˆθj 1 1 E θ j V k x ˆθj, θ j, θ k. 1 k j For C 0 x > 0. BB requires i1 ˆθ C 0 x ˆθ. Consider the fictional problem where the agents utility functions are Ṽ i x, θ i V i x, θ i C 0 x and the principal s coss C 0 x 0. We then compute the transfers for this fictional problem, and set t C 0 x ˆθ /. Ex post R may not be satisfied. Need agents to sign a contract before they learn their types privately. Exercise: A seller has a single indivisible unit of a good that he wants to sell to one of N buyers. The seller values the good at zero. The buyer i s utility is described as: u i x i,, θ i θ i x i +, where θ i is a random variable uniformly distributed over [0, 1], x i denotes the probability that buyer i obtains the good, and denotes buyer i s transfer. The seller s utility is u 0 0 x 0 + t 0 t 0. Any feasible vector x must satisfy : x i 0 for all i 0,, N, and N i0 x i 1. 3

4 1. Characterize a Pareto-effi cient allocation x θ. Hint: You may find it useful to define z i θ i max θ j } and work with it. 2. Design a Groves mechanism thamplements x θ. 3. Now, impose the R constraints, u i 0, and BB constraint, N i0 0. Find the Groves mechanism implementing x θ that yields the highest t 0 θ. 4. Find the Bayesian Nash equilibrium for the second-price auction game. i.e. the highest bidder gets the object and pays the second highest bidding price. 5. Give an economically meaningful explanation of the relationship between 3 and 4. 1 Find the first best decision x θ x θ arg max Σ N i0v i x i, θ i s.t. Σ N i0x i 1 arg max Σ N i0θ i x i s.t. Σ N i0x i 1 Σ N i0 θ ix i is maximized when all the x goes to the individual with largest value of θ. Hence, x 1 if θ i z i ˆθ i i θ for i 0, 1,..., n 0 2. Groves Mechanism: By definition Σ v j x ˆθi, ˆθ j + τ i ˆθ i and x i ˆθ x i 1 0 ˆθ θ i + ˆθ if θ i z i ˆθ i i, we have payoff to i when reporting ˆθ i θ i + Σ v j x ˆθi, ˆθ i, ˆθ j + τ i ˆθ i Σ v j x ˆθi, ˆθ i, ˆθ j + τ i ˆθ i θ i + τ i ˆθ i z i ˆθ i + τ i ˆθ i if ˆθ i z i ˆθ i if ˆθ i z i ˆθ i An agent would only prefer ˆθ i z i ˆθ i if θ i z i ˆθ i. An agent would only prefer ˆθ i < z i ˆθ i if θ i < z i ˆθ i. Hence, is the besnterest of the agent to set θ i ˆθ i. All agents tell the truth. 3 Seller : max tθ t 0 θ s.t. Σ N i0 θ 0 and u i x i,, θ i 0. Without loss of generality, suppose θ 1 < θ 2 <... < θ N. Seller s maximization problem becomes: max τ iθ i ΣN 1 i1 z i θ i + τ i θ i τ N θ N s.t.z i θ i + τ i θ i 0 for i 1,..., N 1 θ N + τ N θ N 0 4

5 n order to maximize this expression, make the R constrains for the low valuation types bind. This implies τ i θ i z i θ i i 1,..., N 1. Hence, the question becomes max τ iθ i τ N θ N s.t. θ N + τ N θ N 0 Since τ N θ N can not depend on θ N, the minimum τ N θ N to satisfy R for all possible θ is z N θ N. Hence, t 0 θ τ N θ N z N θ N θ N 1 and τ i θ i z i θ i i 1,..., N. 5 Under the mechanism in 3, all agents bid their valuation. The agent with the highest valuation receives the good and pays a transfer payment equals to the 2nd highest bid. All other agents pay nothing. This is exactly a second price auction. This shows that the second price auction is the most profitable Groves mechanism for the seller which satisfies the R and BB constraints. With uncertainty, the seller has to pay information rent θ N θ N 1 for the buyer with the highest valuation for telling the truth. Hence, the seller can only receive the highest virtual surplus θ N θ N θ N 1 θ N 1. 5