Tutorial 4, June 6

Microeconomics II (PhD)

Tutorial 4, June 6

Part A

You can use the following envelope theorem in the next exercise.

Theorem (Milgrom and Segal, 2002).

Let X be an arbitrary set,T = [t, t],¹ andf :X×T →R. Denote V(t) = sup

x∈X

f(x, t) (1)

X^∗(t) ={x∈X|f(x, t) =V(t)}. (2)

Suppose that f(x,·) is differentiable for all x ∈ X, f_t(x,·) is uniformly bounded and that X^∗(t) 6= ∅ for almost allt. Then for any selectionx^∗(t)∈X^∗(t),

V(t) =V(t) + Z t

t

f_t(x^∗(s), s)ds. (3)

1. Consider the general mechanism design setting from the lecture, where vi(k, θi) denotes the value of allocation k to agent i with type θi. Suppose that Θi = [θ_i, θi] ⊂ R and that vi is differentiable in θi for all k and the derivative is uniformly bounded. Given a direct revelation mechanism (k, t), let Ui(θ) = vi(k(θ), θi) +ti(θ) be the utility of agent i if θ is the profile of types and all agents report truthfully.

(a) Show that if the direct revelation mechanism (k, t) is implementable in dominant strategies, then U_i(θ) =U_i(θ_i, θ_−i) +

Z θi

θ_i

∂v_i(k(s, θ_−i), s)

∂θi

ds. (ICFOC)

Suppose that vi(k, θi) has the single-crossing property: ^∂2_{∂k ∂θ}^vi(k,θi)

i exists and is stricly positive for all k∈K andθi∈Θi.

(b) Show that if the direct revelation mechanism (k, t) is implementable in dominant strategies, then k(θi, θ−i) is weakly increasing inθi for allθ−i.

(c) Show that any monotone mechanism that satisfies (ICFOC) is implementable in dominant strategies.

Solution: Denote the gains from reporting ˆθi instead of the true typeθi by

`i(θi,θˆi, θ−i) :=vi(k(ˆθi, θ−i), θi) +ti(ˆθi, θ−i)−Ui(θi, θ−i).

Then:

∂`_i(θ_i,θˆ_i, θ_−i)

∂θi

=∂v_i(k(ˆθ_i, θ_−i), θ_i)

∂θi

−v_i(k(θ_i, θ_−i), θ_i)

∂θi

=

Z k( ˆθ_i,θ−i) k(θ_i,θ−i)

∂²v_i(k, θ_i)

∂θi ∂k dk,

where the first line uses (ICFOC) and the second line follows from the fundamental theorem of calculus (assuming, for example, that the cross-derivative is a continuous function).

Because the cross-derivative is greater than 0 andk is weakly increasing, the integral is≥0 if and only if ˆθ_i≥θ_i. Hence,`<sub>i</sub>(θ<sub>i</sub>,θˆ<sub>i</sub>, θ<sub>−i</sub>) is increasing inθ<sub>i</sub>forθ<sub>i</sub>≤θˆ<sub>i</sub>and decreasing forθ<sub>i</sub>≥θˆ<sub>i</sub>. Therefore, for every ˆθi, it is maximized for θi = ˆθi, and the maximum is `i(ˆθi,θˆi, θ_−i) = 0.

Consequently, the gains from lying are weakly negative for all types and the mechanism is DIC.

1This result holds more generally, for example ifT⊂Rⁿis convex.

(d) Discuss the relation of these results to the result that you saw in the lecture.

(e) Show: If a direct revelation mechanism implements the value-maximizing allocation rule in dominant strategies, then it is a VCG mechanism.

Solution: The VCG mechanism withh_i(θ_−i)≡0 implements the value-maximizing allocation rulek^∗ in dominant strategies. Therefore,

t^{V CG}_i (θi, θ−i) =X

j6=i

vj(k^∗(θ), θj) =t^{V CG}_i (θi, θ−i) + Z θ_i

θ_i

∂vi(k^∗(s, θ−i), s)

∂θi

ds.

Moreover, for any mechanism (k^∗, t) that is DIC we have t(θi, θ_−i) =t(θ_i, θ_−i) +

Z θ_i θ_i

∂vi(k^∗(s, θ_−i), s)

∂θi

ds.

Rearranging, we get

t(θ_i, θ_−i) =X

j6=i

v_j(k^∗(θ), θ_j) +t(θ_i, θ_−i)−t^{V CG}_i (θ_i, θ_−i)

and hence (k^∗, t) is a VCG mechanism as well.

Part B

2. Suppose there is one agent, three potential types (θ¹, θ², θ³) and three alternatives (a, b, c). The valuation the agent has for an alternative given his type is given by the following matrix:

θ¹ θ² θ³

a 0 -1 x

b 1 0 -1

c -1 1 0

Consider the functionk⁰ such thatk⁰(θ¹) =a,k⁰(θ²) =b, and k⁰(θ³) =c.

(a)

Definition 1. A decision rulek is weakly monotone if for all θⁱ, θ^j, v(k(θⁱ), θⁱ)−v(k(θ^j), θⁱ)≥v(k(θⁱ), θ^j)−v(k(θ^j), θ^j).

Supposex= 1. Isk⁰ weakly monotone? Is it implementable in the sense that there is a payment rule t such that (k⁰, t) is incentive compatible? How does this relate to the result you saw in the lecture?

(b)

Definition 2. A decision rule k is cyclically monotone if for every sequence of types of length l∈N,(θ¹, θ², ..., θ^l), with θ^l=θ¹, we have

l−1

X

κ=1

v(k(θ^κ), θ^κ+1)−v(k(θ^κ), θ^κ)≤0.

Show that every implementable decision rulekis cyclically monotone.

(c) For which values ofxisk⁰ cyclically monotone?

3. There is one seller with two objects, and one buyer. The seller does not value the objects; the buyer values objectk byθ^k (k= 1,2) and getting both objects byθ¹+θ².

(a) Suppose that valuations are independently distributed and, fork= 1,2, θ^k =

(10 with probability ¹₂ 22 with probability ¹₂.

What are the optimal prices and the corresponding revenue if the seller sells the objects separately?

What is the optimal price and the corresponding revenue if the seller only sells the bundle?

(b) Suppose that valuations are independently distributed and, fork= 1,2, θ^k =

(10 with probability ¹₂ 50 with probability ¹₂.

What are the optimal prices and the corresponding revenue if the seller sells the objects separately?

What is the optimal price and the corresponding revenue if the seller only sells the bundle?

(c) Suppose the seller sets a price for each object and a price for the bundle of both objects. Determine the optimal prices if valuations are identically, independently, and uniformly distributed on [0,1].

(d) Suppose valuations are independently distributed and, fork= 1,2,

θ^k=







1 with probability ¹₆ 2 with probability ¹₂ 4 with probability ¹₃ The expected revenue in the optimal deterministic mechanism is ²⁹₉.

Suppose the seller offers the following menu: A lottery which yields with probability ¹₂ object 1 and nothing otherwise, a lottery which yields with probability ¹₂ object 2 and nothing otherwise, and getting the bundle of both objects for sure. Show that the seller can obtain a larger expected revenue offering this menu compared to the optimal deterministic mechanism.

4. Interdependent value auction

Suppose there is one object for sale andN potential buyers. Each agent privately observes a signalXi, which is independently and identically distributed on [0, X] with cdfF and density f. Denote byGthe cdf of the first-order statistic ofN−1 of these random variables.

Buyers have quasi-linear utilities: in case of winning the object, buyerigets utilityv(x_i, x_−i)−p, where pdenotes the payment made, and he gets utility of 0 in case of not winning. Suppose thatv is positive, strictly increasing in all signals, symmetric in the lastN−1 signals, and denote byv(xi, y) the expected valuation of agentigiven he received signalxi and the highest signal among all other signals has value y.

(a) Show: In a second price auction, each agent bidding according to the bid functionβ(xi) =v(xi, xi) is a Bayes-Nash equilibrium.

Is it a dominant strategy to follow this bid function? Is it an ex-post equilibrium?

Solution: Fix a bidder i, suppose this bidder bids b and the highest signal among all other bidders isy. Bidderiwill win the auction if the higest bid among all others is belowb,β(y)≤b, which is equivalent to y ≤β⁻¹(b). Hence, the expected payoff of bidder i with signal xwho bidsb, given the others use the proposed bidding strategies, is

Π(b, x) :=

Z β⁻¹(b) 0

¯

v(x, y)−¯v(y, y)dG(y). (4)

Since ¯vis strictly increasing, the integrand is>0 for ally < xand<0 for ally > x. Therefore, Π is maximized for a bid bsuch that

¯

v(x, β⁻¹(b))−v(β¯ ⁻¹(b), β⁻¹(b)) = 0, which yieldsb=β(x).

Ifn= 2, ¯v and the argument above is completely independent of the distributionGand hence the equilibrium is actually an ex-post equilibrium. For n > 2, ¯v depends onG and one can easily construct an example where an agent regrets his bid ex-post (see your notes from the tutorial for an example).

Clearly, the equilibrium is not in dominant strategies (even isn= 2).

(b) Consider an open English auction. A symmetric strategy in an English auction is a collection β = (β^N, β^N−1, ..., β²) of N −1 functions β^k : [0, X]×R^N−k+ →R+. The interpretation is that β^k(x, pk+1, ..., pN) is the price at which bidder 1 will drop out of the auction if the number of bidders who are still active isk, his own signal isx, and the prices at which the otherN−kbidders dropped out werepk+1≥pk+2≥...≥pN.

Describe a symmetric Bayes-Nash equilibrium of the open English auction and show that this strategy profile constitutes indeed an equilibrium.

Is it an equilibrium in dominant strategies? Is it an ex-post equilibrium?

Solution: Let p^N denote the price at which the first bidder drops out, and define x^N = β^N−1(p^N).

β^N(xi) =v(xi, xi, ...) (5)

β^N−1(xi, p^N) =v(xi, xi, ..., xN) β^k(x_i, p^k+1, ..., p^N) =v(x_i, ..., x_i

| {z }

ktimes

, x^k+1, ..., x^N), (6)

wherex^k+1is defined implicitly byp^k+1=β^k+1(x^k+1, p^k+2, ..., p^N).

Claim: The bidding strategies defined above form an ex-post equilibrium in the open English auction.

Proof. Fix an arbitrary signal realization, suppose all others follow this strategy and focus on bidder 1.

Case (i): Bidder 1 gets the object when following this strategy.

Payoff: v(x1, y1, ..., yN−1)−v(y1, y1, ..., yN−1), whereyk denotes thek−largest of{x2, ..., xN}.

Since strategies are symmetric,x1≥y1. Hence, payoff is weakly positive. There is no profitable deviation: bidding lower does not change the payoff or leads to payoff 0. Bidding higher gives the same payoff.

Case (ii): Bidder 1 does not get the object when following this strategy.

Note thatxi≤y1. Any deviation that leads bidder 1 to win will give him a payoffv(x1, y1, ..., yN−1)−

v(y1, y1, ..., yN−1)≤0, and all other deviations give him payoff 0. By using the proposed strat- egy, he gets payoff 0, hence there is no profitable deviation.

Clearly, the strategies are not dominant.

(c) Show that the symmetric bidding strategies β(x) = _G(x)¹ Rx

0 v(y, y)dG(y) form a Bayes-Nash equi- librium of the first-price auction.

Solution: Note that β is strictly increasing and every bidder uses the same strategy. The expected payoff to a bidder with signalxwho bidsβ(z) is therefore

Π(x, z) :=

Z z 0

¯

v(x, y)−β(z)dG(y) = Z z

0

¯

v(x, y)−¯v(y, y)dG(y).

Hence, Π(x, x)−Π(x, z) = Rx

z v(x, y)¯ −v(y, y)dG(y)¯ ≥ 0 for all z. Therefore there is no profitable deviation in the range ofβ. Clearly, no bid outside this range is profitable, and hence the strategies form a Bayesian equilibrium.

(d) SupposeN = 2, bidderi’s valuation isvi(xi, xj) =ηxi+ (1−η)xj. For whichη is the outcome of the second-price auction efficient?

Solution: Because the bidding strategies are strictly increasing, the bidder with the higher signal receives the object.

Efficiency therefore requires that ¯v(x, y)≥v(y, x) for all¯ x > y. Rearranging, this is equivalent toη ≥¹₂.