Shifted Myerson Auction Mechanisms

• Shifted Myerson auction is a family of auction mechanisms that modify the classical Myerson framework by applying a deterministic shift to account for additional constraints.
• It employs two constructions—rebate-adjusted payments for ex post ROI constraints and virtual value subtraction in intermediary settings—to preserve DSIC and optimize revenue.
• The mechanism retains the threshold-based logic of Myerson’s design while ensuring feasibility and improved extraction of surplus under different economic scenarios.

Searching arXiv for papers on “shifted Myerson auction” and related auction-design contexts. {"query":"\"shifted Myerson\" auction arXiv", "max_results": 10} {"query":"auction design ROI constraints Myerson rebate arXiv (Lv et al., 2023)", "max_results": 10} {"query":"single-item auctions monopolist intermediary shifted Myerson (Liu et al., 21 May 2026)", "max_results": 10} A shifted Myerson auction is a Myerson-derived mechanism in which the classical allocation or payment primitives are modified by an explicit shift induced by an additional constraint or market layer. In current arXiv usage, the term refers to two distinct constructions. In auctions with ex post ROI-constrained bidders, the shift appears in the payment rule: the classical Myerson payment is multiplied by the ROI factor and then reduced by a rebate chosen to satisfy the ROI inequality pointwise (Lv et al., 2023). In single-item auctions with a monopolist intermediary, the shift appears in virtual values: the intermediary subtracts the seller’s minimum price $p_0$ from every bidder’s virtual value and then runs Myerson on those shifted virtuals (Liu et al., 21 May 2026). The common structure is that classical single-parameter optimal auction theory is preserved formally, but the operative object is translated by the relevant economic constraint.

1. Terminological scope and canonical settings

The phrase “shifted Myerson auction” does not denote a single universal mechanism. In the ex post ROI literature, it is a payment-transformed DSIC mechanism for single-parameter bidders. In the intermediary literature, it is a virtual-value-shifted best response to a seller mechanism that collapses to a posted price. The two constructions are related by method rather than by identical formula: each starts from the classical Myerson template and inserts a deterministic shift dictated by feasibility or institutional structure (Lv et al., 2023, Liu et al., 21 May 2026).

Context	Quantity shifted	Resulting mechanism
Ex post ROI constraints	Payment relative to $M_i p_i^M$ via rebate $r_i$	DSIC payment $p_i = M_i p_i^M - r_i$
Monopolist intermediary	Virtual value via $p_0$	Run Myerson on $\tilde{\phi}_i(v_i)=\phi_i(v_i)-p_0$

In the first setting, there are $n$ risk-neutral bidders competing for a single item, or more generally any single-parameter environment. In the second, a seller and a monopolist intermediary interact in a single-item auction with independent private values, and the intermediary controls which bidder messages are forwarded to the seller. A plausible implication is that “shifted Myerson” is best understood as a family of Myerson-compatible reductions rather than a single named auction format.

2. Shifted Myerson under ex post ROI constraints

For bidder $i$, the private value is $t_i>0$, the public ROI-ratio is $M_i>1$, the allocation is $M_i p_i^M$0, and the payment is $M_i p_i^M$1. The ex post ROI constraint requires

$M_i p_i^M$2

Defining the effective value $M_i p_i^M$3, utility is

$M_i p_i^M$4

The design goal is DSIC, IR, and revenue maximization subject to these constraints (Lv et al., 2023).

The core characterization states that a single-parameter auction with ROI-constrained bidders is DSIC if and only if each allocation rule $M_i p_i^M$5 is non-decreasing in $M_i p_i^M$6 for every fixed $M_i p_i^M$7, and, given such an $M_i p_i^M$8 and $M_i p_i^M$9, the unique DSIC payment is

$r_i$0

where

$r_i$1

is exactly the classical Myerson payment for bidder $r_i$2 with $r_i$3 fixed. Writing

$r_i$4

one obtains

$r_i$5

The rebate term is the distinctive shift. It depends on the entire curve $r_i$6 and hence in particular on $r_i$7. This is not a simple rescaling of Myerson payments; it is a path-dependent correction that enforces the ex post inequality $r_i$8 while preserving dominant-strategy truthfulness.

3. Incentive structure, regularity, and the single-bidder optimum

Under ex post ROI, DSIC and IR require that for every bidder $r_i$9, every true $p_i = M_i p_i^M - r_i$0, and every mis-bid $p_i = M_i p_i^M - r_i$1,

$p_i = M_i p_i^M - r_i$2

and

$p_i = M_i p_i^M - r_i$3

The rebate construction is chosen so that

$p_i = M_i p_i^M - r_i$4

which yields nonnegative truthful utility. The proof sketch given in the source emphasizes that monotonicity of $p_i = M_i p_i^M - r_i$5 preserves the classical Myerson argument, while the rebate handles the additional IR burden created by the ex post ROI feasibility region (Lv et al., 2023).

The technical conditions are explicit. The allocation rule $p_i = M_i p_i^M - r_i$6 must be right-differentiable in $p_i = M_i p_i^M - r_i$7 with only finitely many kinks. The value distributions $p_i = M_i p_i^M - r_i$8 are continuous on $p_i = M_i p_i^M - r_i$9 so that the usual envelope arguments apply. For revenue maximization with a single bidder, a closed-form optimal mechanism is pinned down under the decreasing marginal revenue condition, namely that $p_0$0 is non-decreasing in $p_0$1.

The single-bidder uniform example illustrates how the shifted mechanism departs from deterministic reserve pricing. With $p_0$2 and $p_0$3, classical Myerson uses $p_0$4, charges $p_0$5 for $p_0$6, and has revenue $p_0$7. The revenue-maximizing ROI-constrained mechanism instead uses the randomized allocation

$p_0$8

for which the rebate is always $p_0$9 at $\tilde{\phi}_i(v_i)=\phi_i(v_i)-p_0$0, so

$\tilde{\phi}_i(v_i)=\phi_i(v_i)-p_0$1

For $\tilde{\phi}_i(v_i)=\phi_i(v_i)-p_0$2, one checks that $\tilde{\phi}_i(v_i)=\phi_i(v_i)-p_0$3, which is first-price style; for $\tilde{\phi}_i(v_i)=\phi_i(v_i)-p_0$4, the payment sticks at $\tilde{\phi}_i(v_i)=\phi_i(v_i)-p_0$5. The resulting revenue is approximately $\tilde{\phi}_i(v_i)=\phi_i(v_i)-p_0$6. The same source notes that bidder welfare under shifted Myerson is generally smaller, since payments are typically higher for low valuations in the first-price portion, although IR is maintained ex post.

4. Shifted Myerson as the intermediary’s best response

In the seller-first Stackelberg model with a monopolist intermediary, the seller commits to a deterministic mechanism $\tilde{\phi}_i(v_i)=\phi_i(v_i)-p_0$7 whose minimum-payment is

$\tilde{\phi}_i(v_i)=\phi_i(v_i)-p_0$8

Theorem 2.1 in the cited source shows that any deterministic $\tilde{\phi}_i(v_i)=\phi_i(v_i)-p_0$9 is equivalent to the posted-price mechanism that charges $n$0 whenever someone wins. The intermediary then faces a standard single-parameter problem in which bidder $n$1’s value for the right to buy the item at price $n$2 is

$n$3

If $n$4 is bidder $n$5’s original virtual-value function, the shifted virtual value is

$n$6

Equivalently, with truthful reports $n$7, the intermediary uses the virtual value $n$8 (Liu et al., 21 May 2026).

The allocation rule is direct. The intermediary computes $n$9 for each bidder. If $i$0, no one wins. Otherwise, letting $i$1, it awards the right to dictate bids to $i$2. For payment, define

$i$3

and the threshold

$i$4

The winner pays the intermediary

$i$5

and then forwards to the seller the bid-vector that achieves the seller’s minimum price $i$6. Hence the total payment is

$i$7

which exactly recovers the standard Myerson threshold corresponding to the original $i$8.

The shift arises because once the seller’s mechanism has collapsed to a posted price, each bidder’s surplus from winning the intermediary’s auction is $i$9. By Myerson’s lemma, the revenue-optimal single-parameter auction awards to whoever has nonnegative virtual value, and in this transformed environment the virtual value becomes $t_i>0$0. Economically, the intermediary extracts an extra margin of $t_i>0$1 on top of what the seller would get.

5. Theorems, bounds, and impossibility statements in the seller-first model

Under regularity, meaning that each $t_i>0$2 is regular and $t_i>0$3 is nondecreasing, the intermediary’s unique best response is exactly the shifted Myerson auction: personalized reserve

$t_i>0$4

allocation to the bidder with largest $t_i>0$5, and threshold payment $t_i>0$6 as above (Liu et al., 21 May 2026).

The revenue consequences are sharply split by distributional assumptions. For general regular distributions, there exists a single-bidder distribution $t_i>0$7 such that for any seller mechanism $t_i>0$8, once the intermediary best-responds, the seller’s revenue can be made arbitrarily small compared to the no-intermediary optimum. In fact, when $t_i>0$9, $M_i>1$0 even though $M_i>1$1. By contrast, if each $M_i>1$2 is $M_i>1$3-strongly regular for $M_i>1$4, then there is a simple posted price $M_i>1$5 for the seller such that under the shifted-Myerson response the seller obtains a constant $M_i>1$6-dependent fraction of $M_i>1$7. In the identical case,

$M_i>1$8

and in the nonidentical case,

$M_i>1$9

These bounds are stated to be tight up to constants.

The same source states that randomization does not help the seller: even if randomized mechanisms are allowed, the best deterministic posted price, and thus the best $M_i p_i^M$00, is still as good as any lottery. This separates the seller-first shifted-Myerson setting from environments in which randomization can enlarge the seller’s design space.

6. Computational remarks and conceptual comparison

The intermediary version of shifted Myerson is algorithmically light. The intermediary must know each $M_i p_i^M$01, or at least be able to compute $M_i p_i^M$02 and its inverse $M_i p_i^M$03. In standard parametric families such as exponential, uniform, and power-law, these are closed form or available by one-dimensional root-finding. The only extra parameter is $M_i p_i^M$04, announced by the seller up front. Running the shifted auction costs $M_i p_i^M$05 evaluations of $M_i p_i^M$06 and $M_i p_i^M$07 plus a single maximum over bidders. If $M_i p_i^M$08 is nonmonotone, one must iron it by replacing $M_i p_i^M$09 with its smallest nondecreasing majorant; the source states that this can be done in $M_i p_i^M$10 time on an $M_i p_i^M$11-point grid or by the Pool-Adjacent-Violators algorithm in the continuous case (Liu et al., 21 May 2026).

A worked special case is given by the one-bidder family

$M_i p_i^M$12

Here

$M_i p_i^M$13

on $M_i p_i^M$14, so

$M_i p_i^M$15

If the seller posts $M_i p_i^M$16, the intermediary’s personalized reserve is

$M_i p_i^M$17

which is capped at $M_i p_i^M$18, yielding

$M_i p_i^M$19

The source describes this as a disguised single-item posted price, and the trade-off in optimizing $M_i p_i^M$20 yields the inapproximability result.

Comparing the two meanings of shifted Myerson clarifies the term’s current technical usage. In the ROI-constrained model, the operative modification is a rebate-adjusted payment that preserves DSIC and ex post IR while enforcing $M_i p_i^M$21. In the intermediary model, the operative modification is a subtraction of $M_i p_i^M$22 from virtual values, induced by the seller’s minimum price and the intermediary’s margin. This suggests that the unifying idea is not a single formula but a structural recipe: begin with the Myerson characterization in a single-parameter environment, then apply the exact shift needed to encode the additional constraint without abandoning the threshold-based logic of optimal auction design.