2.10 Supplementary Appendix
2.10.4 A sufficient condition for optimality of post ∗
In this section, we will identify some restrictions on the distribution that ensures thatpost∗ is anoptimalmechanism for the private budgets case. We summarize our assumptions below.
Definition 2.10 We say distribution Φ satisfies Assumption A if
Values and budget are distributed independently, i.e., there exists a prior G over V ≡ [0, β]×[0, β] and a prior Π over [0, β] such that Φ(v, B) =G(v)Π(B) for all (v, B).
Marginal G1 satisfies the property that H1(x) := xG1(x) ∀ x is strictly convex.
Finally, define K¯ as before: K¯ := arg maxr∈[0,β]r(1−G1(r)) - this is well defined because H1 is strictly convex. Then, the following must hold:
[1−G( ¯K, β)−G(β,K) +¯ G( ¯K,K)]¯ Z K¯
0
( ¯K−B)dΠ(B)≥ Z K¯
0
B[G1( ¯K)−G1(B)]dΠ(B) If G is the uniform distribution over [0,1]×[0,1] and Π is uniform over [0,1], then the resulting distribution satisfies Assumption A.
Proposition 2.8 If Φsatisfies Assumption A, then a post∗ mechanism is optimal.
Proof: Fix any B in (0, β) and consider the optimal post-1 mechanism in M− derived in Proposition 2.4. We use this mechanism for eachB (using the expression in Proposition 2.2) to define a new mechanism (f0, v0) for the private budget case - for B ∈ {0, β}, we use the limiting mechanisms of the post-1mechanism suggested in Proposition 2.2.
(f0(v), p0(v)) =
(1, B) if v1 > B and B <K¯ (1,K) if¯ v1 >K¯ and B ≥K¯ (0,0) otherwise.
Of course, this mechanism is not incentive compatible in the private budget case - when v1 > B > 0, the (agent, manager) pair has an incentive to report a budget equal to zero get the outcome (1,0). But notice that the expected revenue of the optimal mechanism in the class of incentive compatible and individually rational mechanisms that are not manager non-trivial cannot exceed the expected revenue of (f0, p0).
Now, consider the post∗ mechanism by setting K = ¯K:
(f∗(v), p∗(v)) =
(1,K)¯ if {v1 >K¯ and B ≥K¯} or{v1, v2 >K¯ and B <K¯} (KB¯, B) if v1 >K¯,v2 ≤K, and¯ B <K¯
(0,0) otherwise
The two mechanisms are shown in Figures 2.8 and 2.9 below.
We argue that post∗ generates weakly greater expected revenue that (f0, p0) under As- sumption A. Hence, the optimal mechanism must be a post∗ mechanism by Theorem2.2.
K¯ v1 v2
(0,0,0)
p0(v, B) = ¯K
B p0(v, B) = 0
¯K
p0(v, B) =B
Figure 2.8: Upper bound
K¯ K¯
v1 v2
(0,0,0)
p∗(v, B) = ¯K
B p∗(v, B) = 0
p∗(v, B) =B
¯K
Figure 2.9: Lower bound
Note that (f0, p0) and (f∗, p∗) yield the same revenue for the following types:
(v, B) such that B ≥K¯
(v, B) such that v1 >K,¯ v2 ≤K¯, andB <K¯ (v, B) such that v1 ≤B, andB <K¯
So, we ignore these types and focus on rest of the types.
for any type (v, B) such thatv1, v2 >K¯ andB <K, revenue from (f¯ ∗, p∗) is ¯Kwhereas revenue from (f0, p0) is B; so the difference in revenue is ¯K−B.
for any type (v, B) such thatv1 ∈(B,K] and¯ B <K, revenue from (f¯ ∗, p∗) is 0 whereas revenue from (f0, p0) is B; so the difference in revenue is B.
Then the condition for revenue from (f∗, p∗) to be more than that of (f0, p0) is:
[1−G( ¯K, β)−G(β,K) +¯ G( ¯K,K¯)]
Z K¯ 0
( ¯K−B)dΠ(B)≥ Z K¯
0
B[G1( ¯K)−G1(B)]dΠ(B)
This holds because of Assumption A.
Chapter 3
Selling two complementary goods
3.1 Introduction
An agent consumes a pair of goods only in a specific ratio of quantities. For instance, a firm needs two inputs in a particular ratio to produce a final product; that is, the firm has a Leontief production function. A consumer treats a pair of goods, coffee and sugar, for example, as perfect complements. That is, the consumer has Leontief preferences, and hence discards excess in either of the goods (after consuming them in the ratio of quantities). A seller who owns one unit each of the two divisible goods is selling to such an agent; what is the revenue-maximizing optimal mechanism in this setting?
We set up this as a mechanism design problem. For any allocation of the bundle of goods, the agent evaluates it using the ratio in which she consumes. The agent has quasilinear preferences across such bundles of goods whose quantities are in the desired ratio. The agent’s payoff is determined by a value, the ratio, and quantity of the bundle he consumes.
The value is interpreted as the payoff from consuming one unit of one of the goods combined with the other good in the desired ratio. Both per unit value from the consumption of the bundle of goods and the ratio itself are private information of the agent. The central theme of the paper is in finding the revenue-maximizing mechanism in such an environment.
For each report, a mechanism assigns quantities of both the goods and payment to be made by the agent. Due to the revelation principle, we focus, without loss of generality, on direct mechanisms that are incentive compatible. An agent could potentially misreport both on value and ratio dimensions. Dealing with incentive constraints in multi-dimensional mech- anism design problems is difficult (Manelli and Vincent, 2007; Carroll, 2017). We present this natural two-dimensional mechanism design model and show that a simple class of non- wasteful mechanisms are optimal under some conditions over seller’s beliefs on agent’s type.
Note that we consider adivisiblegoods model, while the optimal mechanism in the indivisible model may involve randomization. (Hart and Reny(2015); Thanassoulis(2004)).
We show that a posted price mechanism or a ratio-dependent posted price mechanism is optimal. The former is a mechanism in which the seller offers one unit of one of the goods and the other good in the desired ratio at some fixed price. In the latter mechanism, each type gets the same bundle as in the former mechanism, but the price depends on the reported ratio. We first show that it is without loss of generality to focus on mechanisms in which allocations to any type are in the desired ratio; that is, the agent, after a truthful report, does not dispose of either of the goods that the mechanism allocates. This result allows us to use Myersonian techniques. We then characterize incentive compatible mechanisms and provide sufficient conditions over the seller’s belief on the type-space for simple non-wasteful mechanisms to be optimal. We fully describe these mechanisms over the parameters of the problem.