Supply-Chain Chained Auctions

• Supply-chain chained auctions are distributed market protocols that coordinate resource allocation across sequential, transformation-linked markets.
• They utilize symmetric and pivot protocols with local double auction rules, ensuring material balance, incentive compatibility, and near-optimal global welfare.
• Randomized α-variants and alternative decentralized methods enhance efficiency, revenue, and robustness against agent heterogeneity and chain asymmetry.

Supply-chain chained auctions constitute a class of distributed market protocols that orchestrate coordinated resource allocation across sequential markets linked by transformation or conversion activities. These mechanisms are foundational for multi-agent supply networks, ensuring material balance, incentive compatibility, and near-optimal global welfare using only local information propagation and minimal communication.

1. Formal Structure of Supply-Chain Chained Auctions

The canonical model comprises a linear supply chain of length $t$, with $t+1$ sequential markets: $M_1$ (initial suppliers), $M_2,\dots,M_t$ (converters), and $M_{t+1}$ (final consumers) (Babaioff et al., 2011). Each market serves agents trading a single unit of a good, with allocations $(x_i\in \{0,1\})$ and payments $(p_i\in\mathbb{R})$; utility is quasi-linear and single-minded:

• For buyers: $u_i = v_i x_i - p_i$
• For sellers: $u_i = -c_i x_i + p_i$

The allocation is materially balanced: every market trades the same quantity $q$, ensuring conservation of goods along the chain. Bids are either true value (buyers) or cost (suppliers/converters); truthful bidding is strictly optimal under incentive-compatible (IC) mechanisms.

Aggregated supply $S = (S_1,\dots,S_n)$ (nondecreasing) and demand $D = (D_1,\dots,D_n)$ (nonincreasing) curves are constructed locally, with synthetic curves embedding upstream cost and downstream demand information via componentwise operations.

2. Distributed Chaining Protocols

Two principal protocols enable decentralized operation without global coordination:

2.1 Symmetric Protocol

Markets propagate supply curves down-chain and demand curves up-chain:

• Down-chain: $M_1$ sends $S^{(1)}$ to $M_2$, then, for $r=2..t$, $M_r$ sends $S^{(r)} = S^{(r-1)} + S^{(r\to r+1)}$
• Up-chain: $M_{t+1}$ sends $D^{(t+1)}$ to $M_t$, then, for $r=t..2$, $M_r$ sends $D^{(r)} = D^{(r+1)} - S^{(r-1\to r)}$

Each $M_r$ runs a local double auction (DA) rule $R$ on $(S^{(r)},D^{(r)})$, choosing trade quantity $q$ and payments using only locally received information. Consistency of $R$ (i.e., matching $q$ across all markets determined by a function of the optimal trade size $\ell$) yields chain-wide material balance.

2.2 Pivot Protocol

Only the consumer market $M_{t+1}$ executes the DA, selecting $(q, P^S, P^B)$ over the aggregated chain. The selected price and quantity $(V=P^S, q)$ are propagated upstream, with each converter updating winner payments: $$p_i = \min(V_{r+1}-S^{(r-1\to r)}, S^{(r-1\to r)}_{q+1})$$ at each stage. Each market thus trades exactly $q$ units at a "critical-value" price determined recursively.

Both protocols require only $O(n)$ or $O(\log n)$ communication per market and render the chain entirely distributed.

3. Incentive-Compatible Double Auction Rules

The DA rules used in the chaining protocol must satisfy IC, individual rationality (IR), and, optionally, budget-balance (BB):

• VCG Double Auction: Trades optimal $\ell$ units; pays $p^B = \max(S_{\ell+1}, D_{\ell+1})$, $p^S = \min(S_{\ell+1}, D_{\ell+1})$. Yields full efficiency and IR/IC, but can run deficits as $p^B \leq p^S$.
• Trade Reduction (TR): Trades $\ell-1$ units, with buyer price $p^B = D_\ell$, seller price $p^S = S_\ell$. Surplus assured and IR/IC, but efficiency is reduced by at most $1/\ell$.
• Randomized α-Reduction: Trades $\ell$ units (VCG prices) with probability $1-\alpha$, and $\ell-1$ units (TR prices) otherwise. Universally IC, expected BB when $\alpha$ chosen appropriately. Expected efficiency interpolates smoothly between VCG and TR. The α-Payment variant matches these properties with reduced payment variance and IC in expectation.

4. Efficiency, Revenue, and Budget Analysis

The global welfare and budget properties are tightly characterized as follows:

• Welfare Guarantee: Any IR/IC DA rule with local efficiency factor $\epsilon$ yields chain welfare at least $\epsilon \cdot \mathrm{OPT}$. VCG achieves $\epsilon = 1$; TR and α-reduction yield $\epsilon \geq (\ell-1)/\ell$.
• Revenue Bound: Chain revenue $R_\mathrm{chain} = \sum p^B - \sum p^S$ is $\leq 0$ for VCG (deficit), $\geq (\ell-1)L$ (surplus) for TR, and tunable for α-reduction as $$E[R] = (1-\alpha)R_\mathrm{chain}^{VCG} + \alpha R_\mathrm{chain}^{TR}$$.
• Budget Conditions: In the pivot protocol, ex-post BB is achieved if $p^B$ in the consumer DA always exceeds the $(q+1)$th supplier bid. TR rules satisfy this, VCG does not.
• Trade-off Curve: Varying $\alpha \in [0,1]$ in randomized rules traces a revenue–efficiency frontier.

5. Experimental Results and Empirical Performance

Simulations (iid $U[0,1]$, $n_b=n_s$, $100$ runs per scenario) demonstrate:

• VCG-chain: $\approx$100% efficiency, deficit $\approx -0.5n$
• TR-chain: $\approx$99.8% efficiency, surplus $\approx +0.5n$
• α-variant: For $\alpha^* \approx 0.7$, expected revenue $\approx 0$ and efficiency $\approx$99.9%
• In asymmetric allocation ($n_b \gg n_s$), TR and McAfee chains lose $\approx$5% welfare, whereas α-chains recover $>$98% efficiency with zero expected deficit

α-payment substantially reduces payment variance for intermediaries. This suggests the randomized mechanisms are robust to agent heterogeneity and chain asymmetry.

6. Limitations and Generalizations

Current supply-chain chaining protocols impose several structural constraints:

• Linear chain topology; single-minded, unit-demand agent model
• Not designed for combinatorial conversion recipes, multi-unit bids, cycles, probabilistic yields, or private inter-market edges
• Pending extensions to trees, directed acyclic supply graphs (DAGs), stochastic capacities, multi-unit demand, and richer combinatorial exchange constructs

Open directions include algorithmic characterization of efficiency–budget boundaries for α-variants under various distributions, and the integration of chaining protocols with combinatorial resource allocation frameworks.

7. Alternative Decentralized Auction Mechanisms

Decentralized supply-chain formation as formalized by Walsh & Wellman employs simultaneous ascending $(M+1)$st-price auctions (SAMP-SB), mapped on a directed acyclic bipartite graph of goods ($G$), consumers ($C$), and producers ($\Pi$) (Walsh et al., 2011).

• Agents bid myopically in asynchronous auction cycles, chaining input–output dependencies through local price quotes and bid adjustments.
• SAMP-SB converges to exact competitive equilibrium in polytree and no-input-complementarity networks; otherwise, it achieves a $2\varepsilon$–competitive equilibrium within bounded surplus loss.
• Dead ends—where producers hold costly input contracts without output sales—are eliminated in a post-quiescence decommitment phase (SAMP-SB-D), recovering 80–95% of lost surplus, ensuring no agent finishes with negative surplus.
• Empirical results show SAMP-SB-D attains $>$98% efficiency when equilibrium exists, with superior performance over greedy top-down methods in high contention scenarios.

A plausible implication is that dead-end decommitment is essential for practical surplus recovery in protocols where resource contention and multi-level complementarity preclude global equilibria.

Summary

Supply-chain chained auctions realize distributed, incentive-compatible, and nearly efficient resource allocation for sequential production networks. Both symmetric and pivot protocols achieve material balance and adapt to revenue/budget constraints through flexible DA rule selection, including VCG, TR, and randomized α-variants. Extensions to more complex network structures and rich agent types remain open research areas, with decentralized ascending price mechanisms and decommitment procedures critical for practical implementation and equilibrium attainment (Babaioff et al., 2011, Walsh et al., 2011).