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The Large Finite Model

In this section, I consider a finite model with N agents and show that an appropriate modification of the optimal welfare/revenue mechanisms found here are approximately welfare/revenue optimal for large N . Notice that in the finite setting, there is a possibility that an agent’s allocations and transfers depend not only upon his own declaration, but upon the entire vector of declarations. Therefore, I use the bold notation (v, w) to denote this vector. The finite optimization problem is: Definition: Welfare-Maximization Problem (Finite) Maximize

"P WN (x, t) := E

N i=1

1

xi (v, w)vi N

# s.t

(1)

N X i=1 N X i=1

ti (v, w) ≥ 0

∀v, w

(BB)

xi (v, w) ≤S N

∀v, w

(LS)

0 ≤ xi (v, w)

∀i, v, w

(NN)

ti (v, w) ≤ wi

∀v, w

(BC)

1{w0 ≤wi } (vi xi (v 0 , v−i , w0 , w−i ) − t(v 0 , v−i , w0 , w−i ))

∀v, w, v 0 , w0

(IC)

∀v, w

(IR)

≤ vi xi (v, w) − ti (v, w) vi xi (v, w) − t(v, w) ≥ 0

The above conditions are respectively: (BB) budget balance, (LS) limited supply, (NN) non-negative consumption, (BC) budget constraints, (IC) incentive compatibility, and (IR) individual rationality. Notice that the supply constraint is written as a per-consumer supply constraint. This is done to aid comparison between the finite and continuum settings. The finite Revenue-Maximization problem is identical to the welfare-maximization problem except the objective 1 is replaced by hP i N ti (v,w) RN (x, t) = E . i=1 N I use the notation WN , RN to denote the optimal welfare/revenue achieved in the N agent setting and the notation W∞ , R∞ to denote the optimal welfare/revenue achieved in the continuum setting. The following proposition states that the welfare/revenue achieved in the optimal mechanisms in the continuum setting yields an upper bound for the per-capita welfare/revenue in the finite setting. Proposition 1 For all N , WN ≤ W∞ and RN ≤ R∞ . Proof: Given a finite mechanism x1 , . . . , xN , t1 , . . . , tN , define a symmetric mechaPN Ev−i ,w−i [xi (v,w):vi =v,wi =w] nism x(v, w) = and i=1 N PN Ev−i ,w−i [ti (v,w):vi =v,wi =w] t(v, w) = . This is an average of what a player with i=1 N type (v, w) receives according to the finite mechanism when his identity is chosen at random and all other player types may be realized. Notice in particular that x, t satisfies all constraints of the continuum mechanism design problem and that WN (x1 , . . . , xN , t1 , . . . , tN ) = W(x, t) ≤ W∞ . Taking the supremum over all possible finite mechanisms yields WN ≤ W∞ (and similarly RN ≤ R∞ ). 2

I now define an approximately welfare-optimal mechanism xW , tW based upon the welfare-maximizing mechanism in the continuum setting. Its approximate optimality is shown in the subsequent theorem. The mechanism is defined according to the following algorithm: • The price pW is defined according to (1 − F (pW ))E[w] = S · pW · F (pW ). • Agents specify their valuations and wealths (v1 , w1 ), . . . , (vN , WN ). • All agents receive S of the good. wi if vi ≥ p W • Agents’ interim demand is defined as xˆ(vi , wi ) = pW . 0 if vi < pW • Agents’ interim supply is defined as yˆ(vi , wi ) =

0

if vi ≥ pW

S

if vi < pW

.

P ˆ(vi , wi ). • Aggregate demand is defined as XN = N i=1 x PN • Aggregate supply is defined as YN = i=1 yˆ(vi , wi ). • If XN ≥ YN , then all suppliers supply S of the good and agents’ demands for the good are proportionally rationed. • If XN < YN , then all demands for the good are met and agents’ supply of the good is proportionally rationed. Notice that incentive compatibility is guaranteed because the above algorithm uses the price derived in the continuum setting. Therefore, agents’ declarations have no price effects but only determine the quantity that they receive or sell. Also, an agent with vi > pW receives S +

wi pW

of the good (which is equal to his continuum

allocation) if his interim demand is satisfied and an intermediate amount if his demand is rationed. The following defines an approximately revenue-optimal finite mechanism xR , tR based upon the revenue-maximizing mechanism in the continuum setting: • The price pR is defined according to S = (1 − F (pR )) E[w] . p R

• Agents’ interim demand is defined as xˆ(vi , wi ) = • Aggregate demand is defined as X = 3

PN

i=1

wi

if vi ≥ pR

0

if vi < pR

pR

xˆ(vi , wi ).

.

• If X ≥ S, then demands are proportionally rationed. • Otherwise, all demands for the good are met. As in the welfare mechanism, incentive compatibility is guaranteed because the above algorithm uses the price derived in the continuum setting. Therefore, agents’ declarations have no price effects but only determine that quantity that they purchase. As these mechanisms are admissible, it is clear that W(xW , tW ) ≤ WN and R(xR , tR ) ≤ RN . The following theorem shows that the average welfare (resp. revenue) generated by these mechanisms heads towards that of the continuum setting which has already been shown to be an upper bound. Therefore, these mechanisms are approximately optimal in the large finite settings. Theorem 2 As N → ∞, W(xW , tW ) → W∞ and RN → R∞ .

Therefore,

WN − W(xW , tW ) → 0 and RN − R(xR , tR ) → 0. Proof of Theorem 2: The proof here proceeds by calculating lower bounds for W(xW , tW ) and R(xR , tR ). Focusing on welfare first, by the strong law of large numbers, ∃n such that ∀N > n, Pr(|XN | < (1 + )S) > 1 − and Pr(|YN | > (1 − )S) > 1 − . Moreover, Pr((|XN | < (1 + )S) ∩ (|YN | > (1 − )S)) ≥ 1 − 2 ¯ − P r(B). ¯ because for any two events (correlated or not) P r(A ∩ B) ≥ 1 − P r(A) Therefore, for any agent i who demands the good, he receives on average at least xˆ(vi , wi ) 1− at least 1 − 2 fraction of the time. Therefore, his finite utility is at least 1+ (1 − 2) 1− of his continuum utility. Thus, an agent’s ex-ante utility (before his type 1+ 1− is realize) is at least (1 − 2) 1+ of his ex-ante utility in the continuum model. A

similar argument applies for suppliers of the good. Therefore, for N large enough 1− WN (xW , tW ) ≥ (1 − 2) 1+ W∞ . As N increases, may be taken ever smaller and

WN (xW , tW ) → W∞ . The revenue argument proceeds even more simply as there are only demanders. While the continuum setting is often taken as an approximation for a large population, there are frequent instances where the behavior in the continuum setting is quite different than that in the finite setting. The above theorem shows that the optimal welfare/revenue achieved in the continuum setting represents an upper bound which is approximately achievable in the finite settings. Furthermore, the solution 4

of the continuum settings is useful for analyzing the finite case because it guides the construction of the approximately optimal finite mechanisms and provides a means to demonstrate the approximate optimality of the constructed mechanisms. Remark: This is not the first paper to find approximately optimal mechanisms in the large finite case which satisfy both ex-post IC and ex-post IR constraints. One such paper is Hashimoto (2013) who uses a random priority mechanism to find approximately optimal large mechanisms which satisfy ex-post IC and ex-post IR constraints. However, that result is not directly applicable here because it is a setting with no restrictions on monetary transfers. Several other papers have considered approximate optimality in other large finite settings (for example, Segal (2003) and Bodoh-Creed (2013)).

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Continuum Implementation

Turning to implementation, one may note that the continuum optimal mechanisms do not specify what happens if a mass of agents misreports their types. A simple solution is to say that in this event the planner chooses to just not give the good to anybody. However, this heavy handed solution introduces many undesirable equilibria. The mechanisms provided in the previous subsection provide a dominant strategy implementation in the case with a finite number of agents. This suggests that a similar approach may work in the continuum setting. To implement the welfare optimal mechanisms, the planner can transfer a uniform amount S of the good to every agent and then allow agents to buy/sell the good among themselves. This is a dominant strategy allocation rule. Note: This implementation closely resembles one of the mechanisms found in Che, Gale and Kim (2013) called Random Allocation with Resale. A key difference here is that they have indivisible goods which are resold, and so agents receive the good with final probability 0, S, or 1 based upon their type. In the mechanism proposed here, intermediate allocation possibilities will be realized as well. With a finite number of agents, optimal auctions with resale are investigated in Zheng (2002). A dominant strategy implementation is to ask each agent to post a demand schedule. The planner then finds the maximal market-clearing price and clears the market

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at this price. This is similar to how several sovereign debt auctions operate. Theorem 3 Under Assumptions 1, 2, both the welfare-maximizing and the revenuemaximizing mechanisms are implementable through dominant strategies. Proof of Theorem 3: 1). Allow every agent to make a report of their type θ(v, w). Find the market clearing price p. This price may not be unique, in which case, take p to be the maximal market clearing price. Market clearing works by everyone who declares a value v > p receives x(v, w) = w/p quantity of the good in exchange for a transfer −px(v, w). Agents who declare a value v < p give up their S quantity of the good in exchange for a transfer of pS. Finally, agents who declare v = p of which there may be a non-zero measure randomly receive or sell a proportionate share of the total leftover aggregate supply/demand so that markets clear. If a measure 0 set of agents report v such that v = p, then to tie break, we say that all of these agents fully purchase the good. Now, suppose that an agent made a report θ(v, w) 6= (v, w). If an agent overreports their value, it is either the case that they receive the same allocation as truthful revelation, or they receive the good at a price p > v. In either event, they weakly prefer to report their true type v. The same goes for underreports of values where agents may accidentally sell the good when they do not wish to. As for misreports of wealth, an agent cannot overreport his wealth. He can underreport his wealth, but this strictly hurts him when he is a buyer and offers no benefits when he is a seller. Therefore, it is weakly dominant for every agent to report their wealth truthfully. 2). For the revenue-maximizing mechanism, it is dominant strategy for an agent to demand w/p for all prices p < v and to demand 0 for all prices p > 0. When the price is p itself, an agent’s demands are immaterial, so for tiebreaking, let him demand w/p in this situation as well. The planner looks at all reports, calculates the maximal market-clearing price and clears the market at this price. If agents’ reports are such that any price p0 > p has insufficient demand and price p has excess demand, then agents who demand at the price p are only partially filled.

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An agent has two types of misreports available. He can demand less when p < v or more when p > v. The first type of misreport is undesirable because it means that the agent may purchase less of the good than he wishes. The second type of misreport is undesirable because it means that the agent may purchase the good when he does not wish to. Agents cannot demand more for p < v because that would exceed their budget constraints and cannot demand less when p > v because their demand is already 0 and negative demands are not possible. The dominant strategy implementations of both the welfare- and revenue-maximizing mechanisms have the benefit of being both intuitive and low information in the sense that agents need not know the underlying distribution of other agents’ types. Additionally, the above dominant strategy implementations do not depend upon the assumptions on type distributions. These assumptions are necessary for the optimality arguments, but not for the implementation arguments.

2.1

Production

The fact that the optimal mechanisms found in this paper are well-behaved linear mechanisms makes extensions of the given model fairly tractable. In this subsection, I analyze the situation where the principal is able to produce the good at a linear cost c ∈ [v, v¯) instead of taking supply to be fixed. The setting remains the same except that S is now a choice variable and the budget balance condition (BB) is replaced with:

Z Z t(v, w)f (v)dvg(w)dw − cS ≥ 0 W

(BB’)

V

Notice that since S is now a choice variable, a mechanism is now the tuple (x, t, S). However, to apply the analysis from Section 3, one can think of a mechanism as being chosen from a two-stage process. Specifically, the planner chooses the production level S and then the allocation/transfer rules x, t. From this point of view, which is without loss of generality, one can see that Lemmas 1-3 still apply, assuming that

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the mechanism (x, t, S) is admissible in the first place. This leads to the following theorem. Theorem 4 Under Assumptions 1 & 2 and linear production costs, the following four statements hold: 1. The welfare-optimal mechanism is a linear mechanism with uniform transfers (pW , TW ) and supply SW . The welfare-optimal price, transfers, and supply are such that (1 − F (pW ))E[w] = SW pW , pW = c, and TW = 0. 2. The revenue-optimal mechanism is a linear mechanism with uniform transfers (pR , TR ) and supply SR . The revenue-optimal price, transfers, and supply are such that (1 − F (pR ))E[w] = SR pR ,

1−F (pR ) c f (pR )

= pR (pR − c), and TR = 0.

3. SW > SR 4. pW < pR Proof of Theorem 4: From the Main Theorem, it is known that the optimal mechanism after a supply level S is chosen is a linear pricing mechanism with uniform transfers (p, T ). Welfare-Maximization: As in the main theorem, the Budget Balance and Supply conditions can be assumed to hold with equality. Otherwise, either money or good can be uniformly distributed and this weakly improves welfare. Therefore, setting the supply demanded equal to the overall supply and letting transfers be maximal yields:

R (1 − F (p))

W

wg(w)dw + T =S p Sp − Sc = T

(2) (3)

which in turn implies Z (1 − F (p))

wg(w)dw + Sp − Sc

W

and therefore 8

= Sp

R (1 − F (p)) W wg(w)dw =S F (p)p + (1 − F (p))c

(4)

The Welfare Equation is: R v¯ W=S

p

vf (v)dv

1 − F (p)

Substituting into the Welfare Equation yields: R v¯ W=

p

vf (v)dv

F (p)p + (1 − F (p))c

Z wg(w)dw W

Now, the numerator is decreasing in p and the denominator is increasing in p (since its derivative is f (p)(p − c) + F (p)), so overall welfare is decreasing in p. Similarly, if one looks at Equation (4), there too the numerator is decreasing in p and the denominator is increasing in p, so supply is inversely related to price p. Therefore, if welfare is decreasing in p, one wants the smallest price p possible, which in turn implies the largest supply S possible due to their inverse relationship. This occurs when transfers are equal to 0 in Equation (3). So, at the welfare optimum, it is the case that T R (1 − F (p)) W wg(w)dw = Sp.

= 0, p = c and

Revenue-Maximization: As in the main theorem, the Supply condition can be assumed to hold with equality, and transfers can be taken to be zero. This yields:

R (1 − F (p))

W

wg(w)dw =S p 0=T

The Revenue Equation is: R = Sp − Sc Substituting into the Revenue Equation yields:

9

(5) (6)

(p − c)(1 − F (p)) R= p

Z wg(w)dw W

Maximizing with respect to p requires: [(1 − F (p)) − (p − c)f (p)] p + (p − c)(1 − F (p)) = 0 Rearranging gives: p(p − c) =

c(1 − F (p)) f (p)

Restricting attention to p > c, the left hand side (LHS) is increasing in p and the right hand side (RHS) is decreasing in p. Furthermore the LHS > RHS when p = v¯ and the LHS < RHS when p = c. Moreover, one knows that the First Order Condition is sufficient because R = 0 when p = c or v¯ and R > 0 for intermediate p ∈ (c, v¯). There are a few important points about the above theorem that I would like to note. The first is that there are no transfers from the principal to the agents. This is not surprising in the revenue-maximization mechanism, but perhaps is for the welfareoptimal mechanism. This can be taken as a statement that, for linear production costs, it is a better use for the principal to spend any excess money manufacturing the good, that to distribute that money via transfers. This also points to the fact that the planner does not want to produce the good and distribute the good itself to the agents and then allow resale because this also provides a monetary transfer to the agents who sell the good which is not a feature of the optimal mechanism. An intuitive understanding is that additional production has the advantage that of being more targeted than lump-sum transfers. Since the welfare-maximizing principal is no longer making transfers to the agents, the original rationale of why prices are higher in the welfare-maximizing mechanism goes away. In the main theorem, the welfare-maximizing principal made uniform lump-sum transfers and allowed the market to clear. So, the welfare-maximizing linear price was higher than the revenue-maximizing linear price because of these transfers. When there is linear production, the welfare-maximizing principal will manufacture

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more of the good than the revenue-maximizing principal and make no transfers. This leads to the opposite conclusion about the relevant prices, namely that the market price under the welfare-maximizing principal is lower than the market price under the revenue-maximizing principal. Finally note that the same analysis could be carried out for convex production costs. In that setting, transfers will be zero for the revenuemaximizing mechanism, but may be nonzero in the welfare-maximizing mechanism.

References Bodoh-Creed, Aaron, “Efficiency and information aggregation in large uniformprice auctions,” Journal of Economic Theory, 2013, 148 (6), 2436 – 2466. Che, Yeon-Koo, Ian Gale, and Jimwoo Kim, “Assigning Resources to BudgetConstrained Agents,” Review of Economic Studies, 2013, 80 (1), 73–107. Hashimoto, Tadashi, “The Generalized Random Priority Mechanism with Budgets,” Working Paper, 2013. Segal, Ilya, “Optimal Pricing Mechanisms with Unknown Demand,” American Economic Review, 2003, 93 (3), 509–529. Zheng, Charles Z, “Optimal auction with resale,” Econometrica, 2002, 70 (6), 2197–2224.

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