Worst-case efficient and budget-balanced mechanism for single-object allocation with interdependent values

Abstract

We study a model in which a single object is to be allocated among a set of agents whose valuations are interdependent. We define signal-ranking mechanisms and show that if the signal-ranking allocation rule satisfies a combinatorial condition and the valuation functions are additively separable, there exist budget-balanced and ex-post incentive compatible signal-ranking mechanisms. For a restricted setting, we show that the worst-case efficient mechanism of Long et al. (Games Econ Behav 105:9–39, 2017) continues to be worst-case efficient. We also give an example to show that their mechanism is no longer optimal when restrictions are relaxed.

Appendix

The following lemma gives the sufficient condition for a s-ranking allocation rule to be ex-post implementable.

Lemma 1

An s-ranking mechanism is EPIC if the valuation functions $(v_1 (s), v_2 (s), \ldots , v_n (s) )$ are increasing in their own signal.

Proof

We show that the signal-ranking mechanism is EPIC. Consider an arbitrary signal profile s and without loss of generality let $s_1> s_2> \cdots > s_n$. The payment of agent ranked i is:

$$\begin{aligned} p_i (s)&= v_i (s) \pi _i - v_i (0, s_{-i}) f_i (0,s_{-i}) - {\mathop {\underset{\scriptscriptstyle {0}}{\int }}\limits ^{\scriptscriptstyle {s_i}}} f_i (x, s_{-i}) \frac{\partial v_i(x, s_{-i})}{\partial s_{i}} dx \nonumber \\&= {\mathop {\underset{\scriptscriptstyle {j = 1}}{\sum }}\limits ^{\scriptscriptstyle {n-i}}} v_i (s_{i+j} ,s_{-i}) (\pi _{i+j-1} - \pi _{i+j}) \end{aligned}$$

(7)

If the agent i reports $s_{i'} > s_i$ such that his rank is $i' < i$, the payment of the agent is

$$\begin{aligned} p_{i'} (s_{i'}, s_{-i'})&= v_{i'} (s_{i'}, s_{-i'}) \pi _{i'} - v_{i'} (0, s_{-i'}) \pi _{n} - {\mathop {\underset{\scriptscriptstyle {0}}{\int }}\limits ^{\scriptscriptstyle {s_{i'}}}} f_{i'} (x, s_{-i'}) \frac{\partial v_{i'} (x, s_{-i'})}{\partial s_{i'}} dx \\&= {\mathop {\underset{\scriptscriptstyle {j = 1}}{\sum }}\limits ^{\scriptscriptstyle {i-i'-1}}} v_{i'} (s_{i'+j} ,s_{-i'}) (\pi _{i'+j-1} - \pi _{i'+j}) + v_{i'} (s_{i'-1}, s_{-i'}) (\pi _{i-1} - \pi _{i}) \\&\qquad + {\mathop {\underset{\scriptscriptstyle {j = i+1}}{\sum }}\limits ^{\scriptscriptstyle {n-1}}} v_{i'} (s_{j} ,s_{-i'}) (\pi _{j} - \pi _{j+1}) \end{aligned}$$

Let $\Delta u (s_{i'}, s_i) = v_i (s) \pi _i - p_i (s) - \big ( v_{i'} (s) \pi _{i'} - p_{i'} (s_{i'}, s_{-i'}) \big )$ be the difference between utilities obtained by agent ranked i with signal type $s_i$ when he reports the true signal and when he falsely reports $s_{i'}$. Hence,

$$\begin{aligned} \Delta u (s_{i'}, s_i)&= v_i (s) \pi _i - v_{i'} (s) \pi _{i'} - {\mathop {\underset{\scriptscriptstyle {j = 1}}{\sum }}\limits ^{\scriptscriptstyle {n-i}}} v_i (s_{i+j} ,s_{-i}) (\pi _{i+j-1} - \pi _{i+j}) + {\mathop {\underset{\scriptscriptstyle {j = 1}}{\sum }}\limits ^{\scriptscriptstyle {i-i'-1}}} v_{i'} (s_{i'+j} ,s_{-i'}) (\pi _{i'+j-1} - \pi _{i'+j}) \nonumber \\&\quad + v_{i'} (s_{i'-1}, s_{-i'}) (\pi _{i-1} - \pi _{i}) + {\mathop {\underset{\scriptscriptstyle {j = i+1}}{\sum }}\limits ^{\scriptscriptstyle {n-1}}} v_{i'} (s_{j} ,s_{-i'}) (\pi _{j} - \pi _{j+1}) \nonumber \\&= - v_i (s) (\pi _{i'} - \pi _{i}) + {\mathop {\underset{\scriptscriptstyle {j = 1}}{\sum }}\limits ^{\scriptscriptstyle {i-i'}}} v_{i'} (s_{i'+j-1} ,s_{-i'}) (\pi _{i'+j-1} - \pi _{i'+j})\nonumber \\&= {\mathop {\underset{\scriptscriptstyle {j = 1}}{\sum }}\limits ^{\scriptscriptstyle {i-i'}}} \big ( v_{i'} (s_{i'+j-1} ,s_{-i'}) - v_i (s_i , s_{-i}) \big ) (\pi _{i'+j-1} - \pi _{i'+j}) \end{aligned}$$

(8)

As the function $v_i (s_i, s_{-i})$ is increasing in $s_i$, the expression in (8) is positive. Similarly if $s_{i'} < s_i$, then also $\Delta u (s_{i'}, s_i) > 0$. Hence, the mechanism is EPIC. $\square$

Proof of Theorem 3:

Let there be an arbitrary s-ranking mechanism $\pi = (\pi _1, \pi _2, \ldots , \pi _n)$. The efficiency ratio of the mechanism at arbitrary signal profile s is:

$$\begin{aligned} \frac{\pi _1 v_{[1]}(s) + \pi _2 v_{[2]}(s) + \cdots + \pi _n v_{[n]}(s)}{v_{[1]}(s)} = \pi _1 + \pi _2 \frac{v_{[2]}(s)}{v_{[1]}(s)} + \cdots + \pi _n \frac{v_{[n]}(s)}{v_{[1]}(s)} \end{aligned}$$

The worst-case efficiency ratio is

$$\begin{aligned} \mu = {\underset{\scriptscriptstyle {s \in S^n}}{\text {inf}}} \bigg ( \pi _1 + \pi _2 \frac{v_{[2]}(s)}{v_{[1]}(s)} + \cdots + \pi _n \frac{v_{[n]}(s)}{v_{[1]}(s)} \bigg ) \end{aligned}$$

If the signal profile is such that $s_1 \ge s_2 \ge \cdots \ge s_n$, then

$$\begin{aligned} \mu&= \pi _1 + \pi _2 \frac{v_2(s)}{v_1(s)} + \cdots + \pi _n \frac{v_n(s)}{v_1(s)} \\&= \pi _1 + \pi _2 \bigg ( \frac{\gamma h (s_2) + \underset{\scriptscriptstyle {j \ne 2}}{\sum } h(s_j) }{\gamma h (s_1) + \underset{\scriptscriptstyle {j \ne 1}}{\sum } h(s_j) } \bigg ) + \cdots + \pi _n \bigg ( \frac{\gamma h (s_n) + \underset{\scriptscriptstyle {j \ne n}}{\sum } h(s_j) }{\gamma h (s_1) + \underset{\scriptscriptstyle {j \ne 1}}{\sum } h(s_j) } \bigg ) \end{aligned}$$

The minimum value of each of the $n-1$ ratios is $\frac{1}{\gamma }$ and the minima occurs at $(s_1,s_2,\ldots ,s_n)=(1,0,0,\ldots , 0)$. Hence,

$$\begin{aligned} \mu = \pi _1 + \frac{1}{\gamma } (\pi _2 + \pi _3 + \cdots + \pi _n) = \Big (1 - \frac{1}{\gamma } \Big ) \pi _1 + \frac{1}{\gamma } \end{aligned}$$

The optimization problem is:

$$\begin{aligned} {\underset{\scriptscriptstyle {(\pi _1,\pi _2,\ldots ,\pi _n)}}{\text {max}}} \text { } \Big (1 - \frac{1}{\gamma } \Big ) \pi _1 + \frac{1}{\gamma } \\ \text {s.t. } \hspace{15mm} \pi _i&\ge 0 \hspace{10mm} \forall i \in \{ 1,2, \ldots , n \} \\ \pi _1 + \pi _2 + \cdots + \pi _n&= 1 \\ {\underset{\scriptscriptstyle {j \in N}}{\sum }} (-1)^j \left( {\begin{array}{c}n-1\\ j-1\end{array}}\right) \pi _j&= 0 \\ \pi _{i+1} - \pi _{i}&\le 0 \hspace{10mm} \forall i \in \{ 1,2,\ldots , n-1 \} \end{aligned}$$

This optimization problem is equivalent to the optimization problem solved by Long et al. (2017) to find the optimal worst-case efficient ranking mechanism in the class of dominant strategy incentive-compatible and BB mechanisms. This is because all the constraints in both the problems are identical and the objective function above is a monotonic transformation of objective function of their optimization problem. Hence, $\pi ^*$ also solves our optimization problem and maximizes the worst-case efficiency ratio. $\square$