Beyond Winner-Take-All Procurement Auctions

Abstract

Blockchain protocols often seek to procure computationally challenging work from a decentralized set of participants. While there are simple procurement auctions that result in the minimal cost of acquisition and maximal efficiency, they also lead to concentration in the provider set due to the winner-take-all market structure. We design and analyze single-good procurement auctions that balance social-cost minimization (at the extreme, a winner-take-all auction) with decentralization (at the extreme, a uniform allocation). We first give a dominant-strategy incentive-compatible (DSIC) mechanism explicitly designed to implement non-winner-take-all allocations. Our allocation rule uniquely solves an optimization with respect to a modified social-cost metric that penalizes large, single-player concentrations and is parameterized with a curvature value, $α$, with $α\rightarrow 0$ implementing the uniform allocation and $α\rightarrow \infty$ implementing the winner-take-all allocation. We further quantify the loss in social cost of this mechanism as a function of $α$. We then propose two alternative mechanisms, each addressing a limitation of the DSIC mechanism, namely a lack of Sybil-resistance and a complex payment rule. First, we examine a variation of Tullock contests to achieve a non-winner-take-all Sybil-proof procurement mechanism. Second, we consider a mechanism with the same allocation rule as the DSIC mechanism but with an alternative payment rule in which producers are simply paid proportionally to their bids. This provides a much simpler payment rule which, while not DSIC, still results in the mechanism being ex-post ``safe'' (where there exists a bidding strategy that is guaranteed to result in non-negative utility) for participating bidders. For both non-DSIC mechanisms, we characterize the equilibrium allocations and prove price of anarchy bounds.

Figures (5)

• Figure 1: Allocations (left) and social cost (right) with $n=2$ and $c_1=1$ under the DSIC mechanism for various values of $\alpha$.
• Figure 2: Equilibrium allocations and the resulting social costs in the Tullock procurement mechanism as a function of $c_2$ with $c_1=1$ and various values of $n$. Note that we only show $x_2$ in the left subplot because $x_2=x_3=\ldots=x_n$.
• Figure 3: Honest and equilibrium-induced bids (for the paid-as-bid payment rule, \ref{['def:payment-proportional']}) and allocations for the two-player game with $\alpha=5$. We see that the equilibrium allocations approach their limits of $1/5$ and $1-1/5$ as $c_2$ increases. Also note that the equilibrium bids for player one increase in $c_2$, while honest bids remain constant at $c_1=1$.
• Figure 4: Equilibrium allocations for the two-player game and corresponding social costs. The left plot shows the $1/\alpha$ and $1-1/\alpha$ bounds for the lower and higher allocations, respectively, as horizontal lines.
• Figure 5: Price of anarchy scaling in $C = c_{\max} /c_{\min}$ for $\alpha=4$ and various values of $n$.

Theorems & Definitions (37)

• definition 1: Player utility function
• definition 2: Dominant-strategy incentive-compatible
• definition 3: PNE
• definition 4: Sybil-proof
• definition 5: Ex-post safe
• definition 6: Social cost
• definition 7: Price of anarchy (PoA)
• definition 8: $\alpha$-PARs
• lemma 1: Optimization problem
• theorem 1