--- title: Shifted Myerson Auction Mechanisms url: https://www.emergentmind.com/topics/shifted-myerson-auction type: topic --- # Shifted Myerson Auction Mechanisms Searching arXiv for recent 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 2310.02064", "max_results": 10} {"query":"single-item auctions monopolist intermediary shifted Myerson 2605.21934", "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 [2310.02064]. 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 [2605.21934]. 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 [2310.02064] [2605.21934]. | 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 $x_i(b)\in[0,1]$, and the payment is $p_i(b)\ge 0$. The ex post ROI constraint requires $$ \frac{t_i x_i}{p_i}\ge M_i \quad \Rightarrow \quad p_i\le \frac{t_i}{M_i}x_i. $$ Defining the effective value $v_i:=t_i/M_i$, utility is $$ u_i(v_i,b)= \begin{cases} M_i v_i x_i(b)-p_i(b), & \text{if } p_i(b)\le v_i x_i(b),\\ -\infty, & \text{otherwise.} \end{cases} $$ The design goal is DSIC, IR, and revenue maximization subject to these constraints [2310.02064]. The core characterization states that a single-parameter auction with ROI-constrained bidders is DSIC if and only if each allocation rule $x_i(b_i,b_{-i})$ is non-decreasing in $b_i$ for every fixed $b_{-i}$, and, given such an $x_i$ and $p_i(0,b_{-i})=0$, the unique DSIC payment is $$ p_i(b_i,b_{-i}) = M_i\cdot p_i^M(b_i,b_{-i}) - \max_{0\le z\le b_i}\{M_i\cdot p_i^M(z,b_{-i})-z\cdot x_i(z,b_{-i})\}, $$ where $$ p_i^M(b_i,b_{-i}) = b_i\cdot x_i(b_i,b_{-i})-\int_0^{b_i}x_i(z,b_{-i})\,dz $$ is exactly the classical Myerson payment for bidder $i$ with $x_i$ fixed. Writing $$ r_i(b_i,b_{-i}) := \max_{0\le z\le b_i}\{M_i\cdot p_i^M(z,b_{-i})-z\cdot x_i(z,b_{-i})\}, $$ one obtains $$ p_i(b_i,b_{-i}) = M_i p_i^M(b_i,b_{-i}) - r_i(b_i,b_{-i}). $$ The rebate term is the distinctive shift. It depends on the entire curve $\{x_i(z,b_{-i}),p_i^M(z,b_{-i}):0\le z\le b_i\}$ and hence in particular on $b_{-i}$. This is not a simple rescaling of Myerson payments; it is a path-dependent correction that enforces the ex post inequality $p_i\le v_i x_i$ 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 $i$, every true $v_i$, and every mis-bid $b_i$, $$ u_i(v_i,v_i,b_{-i}) \ge u_i(v_i,b_i,b_{-i}), $$ and $$ u_i(v_i,v_i,b_{-i}) \ge 0. $$ The rebate construction is chosen so that $$ M_i p_i^M(b_i)-r_i(b_i)\le b_i x_i(b_i), $$ which yields nonnegative truthful utility. The proof sketch given in the source emphasizes that monotonicity of $x_i$ preserves the classical Myerson argument, while the rebate handles the additional IR burden created by the ex post ROI feasibility region [2310.02064]. The technical conditions are explicit. The allocation rule $x_i(b_i,b_{-i})$ must be right-differentiable in $b_i$ with only finitely many kinks. The value distributions $F_i$ are continuous on $[0,v_{\max}]$ 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 $v f(v)+F(v)-1$ is non-decreasing in $v$. The single-bidder uniform example illustrates how the shifted mechanism departs from deterministic reserve pricing. With $M=2$ and $v\sim \text{Uniform}[0,1]$, classical Myerson uses $x(v)=1[v\ge \tfrac12]$, charges $p^M(v)=\tfrac12$ for $v\ge \tfrac12$, and has revenue $\tfrac14$. The revenue-maximizing ROI-constrained mechanism instead uses the randomized allocation $$ x(v)= \begin{cases} \frac{4}{3}v, & v\le \frac34,\\ 1, & v>\frac34, \end{cases} $$ for which the rebate is always $r(v)=0$ at $z=0$, so $$ p(v)=2p^M(v)=2v x(v)-2\int_0^v x(z)\,dz. $$ For $v\le \tfrac34$, one checks that $p(v)=v x(v)$, which is first-price style; for $v>\tfrac34$, the payment sticks at $p(v)=\tfrac34$. The resulting revenue is approximately $3/8=0.375>0.25$. 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 $\Pi_S$ whose minimum-payment is $$ p_0=\omega(\Pi_S). $$ Theorem 2.1 in the cited source shows that any deterministic $\Pi_S$ is equivalent to the posted-price mechanism that charges $p_0$ whenever someone wins. The intermediary then faces a standard single-parameter problem in which bidder $i$’s value for the right to buy the item at price $p_0$ is $$ v_i'=\max\{v_i-p_0,0\}. $$ If $\phi_i$ is bidder $i$’s original virtual-value function, the shifted virtual value is $$ \tilde{\phi}_i(v_i):=\phi_i(v_i)-p_0. $$ Equivalently, with truthful reports $t_i=\max\{v_i-p_0,0\}$, the intermediary uses the virtual value $\phi_i(t_i+p_0)-p_0$ [2605.21934]. The allocation rule is direct. The intermediary computes $\phi_i(v_i)$ for each bidder. If $\max_i \phi_i(v_i)