Socially efficient mechanism on the minimum budget

TL;DR

This paper tackles designing mechanisms that are socially efficient (SE), dominant-strategy incentive compatible (DSIC), and individually rational (IR) while minimizing the broker’s budget. It introduces a novel budget-optimal mechanism that leverages discrete, known type domains to convert DSIC/IR constraints into a shortest-path problem on per-agent graphs $G_i(v)$, with payments derived from the shortest distances. The authors prove correctness (DSIC/IR) and optimality (minimum budget among justified mechanisms for a fixed SE-maximizing option), and show computational feasibility through graph contractions that yield polynomial-time computation. Numerical experiments demonstrate that the proposed mechanism achieves strictly lower budgets than any VCG-based mechanism for a large majority of instances, highlighting its practical value for trading networks, cloud markets, and related settings where social welfare and budget considerations must be balanced.

Abstract

In social decision-making among strategic agents, a universal focus lies on the balance between social and individual interests. Socially efficient mechanisms are thus desirably designed to not only maximize the social welfare but also incentivize the agents for their own profit. Under a generalized model that includes applications such as double auctions and trading networks, this study establishes a socially efficient (SE), dominant-strategy incentive compatible (DSIC), and individually rational (IR) mechanism with the minimum total budget expensed to the agents. The present method exploits discrete and known type domains to reduce a set of constraints into the shortest path problem in a weighted graph. In addition to theoretical derivation, we substantiate the optimality of the proposed mechanism through numerical experiments, where it certifies strictly lower budget than Vickery-Clarke-Groves (VCG) mechanisms for a wide class of instances.

Figures (6)

• Figure 1: An improper environment parameterized with $\alpha > 0$. Agent $A$ can have one of two types $\mathcal{V}_A \coloneqq \{v_{A}^{(1)}, v_{A}^{(2)}\}$, while agent $B$ has only one possible type $\mathcal{V}_B \coloneqq \{v_{B}\}$.
• Figure 2: The weighted graph $G_A(v)$ and $G_B(v)$ for the type $v = (v_{A}^{(1)},v_B)$, combined at the auxiliary vertex $\star$, for the two possibilities of (\ref{['sub@fig:ex1:case1']}) and (\ref{['sub@fig:ex1:case2']}).
• Figure 3: Difference in budget required by the proposed mechanism relative to VCG-budget, shown as a histogram in (\ref{['sub@fig:exp:hist']}) as well as the average against the number of agents $|\mathcal{N}|$ in (\ref{['sub@fig:exp:plot_vs_n']}), the number of options $|\mathcal{X}|$ in (\ref{['sub@fig:exp:plot_vs_m']}), and the size of type domains $|\mathcal{V}_i|$ in (\ref{['sub@fig:exp:plot_vs_d']}), where $|\mathcal{N}|=16$ is fixed except in (\ref{['sub@fig:exp:plot_vs_n']}).
• Figure 4: An example of the weighted graph $G_{i_1}(v)$ in \ref{['ex:vickrey']} to compute the payment from the winner $i_1 = 1$, where there are $d=5$ types (biddable prices) and the agent $i_2$ makes the second-highest bid of $p_3$ i.e., $k_2 = 3$. While not all edges are shown, each solid arrow represents an edge with a positive weight, each dashed arrow represents an edge with a negative weight, and each dotted arrow represents an edge with a zero weight. Notice that vertices inside a closed dotted line are mutually connected by zero-weight edges, hence they have an equal shortest distance from/to any other vertex. The weight of any positive edge coming into $v_1^{(k)}$ is $p_k$, and the weight of any negative edge going out of $v_1^{(k)}$ is $-p_k$. In the case (\ref{['sub@fig:second-price-ge']}) $i_1 < i_2$, one of the shortest paths from $\star$ to $v_1^{(1)}$ could be $\star\to v_1^{(3)} \to v_1^{(1)}$, along which the distance is $p_3$. In the other case (\ref{['sub@fig:second-price-gg']}) $i_1 > i_2$, one of the shortest paths from $\star$ to $v_1^{(1)}$ could be $\star\to v_1^{(2)} \to v_1^{(1)}$, along which the distance is $p_2$.
• Figure 5: Difference in budget required by the proposed mechanism relative to VCG-budget, shown as a histogram in (\ref{['sub@fig:expn:hist']}) as well as the average against the number of agents $|\mathcal{N}|$ in (\ref{['sub@fig:expn:plot_vs_n']}), the number of options $|\mathcal{X}|$ in (\ref{['sub@fig:expn:plot_vs_m']}), and the size of type domain $|\mathcal{V}_i|$ in (\ref{['sub@fig:expn:plot_vs_d']}), where $|\mathcal{N}|$ is fixed at 8, 16, or 32, as indicated in each row, except in (\ref{['sub@fig:expn:plot_vs_n']}).

Theorems & Definitions (47)

• Definition 1: Environment
• Definition 2: Mechanism
• Definition 3: SE
• Definition 4: DSIC, IR
• Definition 5: Justified mechanism
• Definition 6: VCG mechanism
• Theorem 1: Theorem 1.17 in Nisan2007
• Definition 7: VCG-Clarke
• Lemma 1: Lemma 1.20 in Nisan2007
• Definition 8: VCG-budget