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


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.

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  1. In some special cases like the sequencing problem it might be possible to reconcile the three properties e.g. Hain and Mitra (2004). Also, Kosenok and Severinov (2008) show that there exists an ex-post efficient mechanism that is interim incentive-compatible, interim individually-rational and ex-post budget balanced when the joint distribution of the values is known and which satisfies Cremer and McLean (1988) conditions and the identifiability condition.

  2. Long et al. (2017) show that this combinatorial condition is a necessary and sufficient condition for a ranking mechanism to be budget-balanced and strategy-proof in private-value setting. For the exact condition, see Theorem 2 in Sect. 4.

  3. We suppress the dependence of L on s for notational convenience.

  4. The valuation functions \(v_i:[0,1]^n \rightarrow \mathbb {R_+}\), \(i \in N\) satisfy single-crossing if for every \(i, j \in N\), every \(s_{-i} \in S_{-i}\) and every \(s_i > s'_i\),

    $$\begin{aligned} v_i (s_i, s_{-i}) - v_i (s'_i , s_{-i}) > v_j (s_i, s_{-i}) - v_j (s'_i , s_{-i}) \end{aligned}$$
  5. The valuation functions satisfy the SAS condition. Notice that for any two agents i and j, if \(s_i > s_j\) then \(v_i (s) > v_j(s)\) and vice versa. Also, \(s_i = s_j\) implies \(v_i (s) = v_j (s)\) and vice versa.

  6. See Krishna (2009) where it is shown that single-crossing condition is sufficient for ex-post incentive compatibility of generalized Vickrey auction.

  7. The literature of mechanism design in interdependent-value setting emphasizes the importance of single-crossing for efficient mechanisms to be EPIC (see d’Aspremont and Gerard-Varet (1982) and Dasgupta and Maskin (2000) for example).


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Correspondence to Aditya Vikram.

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This paper is based on one of the papers in my PhD thesis written while at Indian Statistical Institute, Delhi. I would like to thank my supervisor Arunava Sen for his valuable guidance. I also thank Debasis Mishra for his comments.



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.


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:

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

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,

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

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. \(\square\)

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Vikram, A. Worst-case efficient and budget-balanced mechanism for single-object allocation with interdependent values. Soc Choice Welf (2023).

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