Tighter Bounds for the Random-Offerer Mechanism
in Bilateral Trade
July 15, 2026
The random-offerer mechanism for bilateral trade selects the seller or the buyer uniformly and lets the selected agent make a profit-maximizing take-it-or-leave-it offer. Let \(\rho_{\rm RO}\) be the infimum, over independent value distributions, of the mechanism’s gains from trade divided by first-best gains from trade. We prove \[\frac{1}{\pi}\le \rho_{\rm RO}<0.460242308085529.\] For the lower bound, we improve the previous guarantee from approximately \(0.317844\) to \(1/\pi\approx 0.318310\). The proof uses a parameterized Lagrangian bound for pointwise-monotone allocations. At multiplier one, this bound has coefficient \(2/\pi\), and the Lagrangian separates into two terms controlled by the optimal seller-offering and buyer-offering profits. For the upper bound, we construct an explicit family consisting of a truncated equal-revenue buyer and a seller distribution with a tilted power-law lower tail and a constant-virtual-cost segment. The family satisfies \(\operatorname{FB}/\operatorname{RO}>2.17276852308451\), improving the previous explicit ratio \(2.0749\); rigorous interval arithmetic certifies the numerical inequality.
Bilateral trade is the basic setting in which private information can prevent all gains from trade from being realized. A seller owns one indivisible item and values it at \(S\), while a buyer values it at \(B\). The values are drawn independently from commonly known distributions, and each agent observes only its own realization. The first-best allocation trades whenever \(B>S\) and generates expected gains from trade \[\operatorname{FB}:=\mathbb{E}[(B-S)_+].\] Myerson and Satterthwaite [1] showed that no mechanism can attain this benchmark for all priors while satisfying Bayesian incentive compatibility, individual rationality, and budget balance.
The random-offerer mechanism is a simple way to conduct trade under these constraints. It selects the seller or the buyer with equal probability and lets the selected agent make a take-it-or-leave-it offer after observing its value. When the seller is selected, a seller of type \(s\) chooses a price \(p\) maximizing \((p-s)\mathbb{P}[B\ge p]\); when the buyer is selected, a buyer of type \(b\) chooses a price \(p\) maximizing \((b-p)\mathbb{P}[S\le p]\). Let \(\operatorname{SO}\) and \(\operatorname{BO}\) denote the gains from trade in the two cases and set \[\operatorname{RO}:=\frac{\operatorname{SO}+\operatorname{BO}}{2}.\] We study the worst-case fraction of first-best gains from trade, \[\rho_{\rm RO}:= \inf_{F_S,F_B:\,\operatorname{FB}>0}\frac{\operatorname{RO}}{\operatorname{FB}}.\]
Deng, Mao, Sivan, and Wang [2] proved the first constant lower bound on \(\rho_{\rm RO}\). Fei [3] improved the guarantee to \(1/T=0.317844432899372\ldots\), where \(T>1\) is the larger solution of \(T=2+\log T\); Hartline and Wang [4] later gave a geometric proof of the same bound. In the other direction, Babaioff, Dobzinski, and Kupfer [5] constructed instances on which random offerer obtains less than one half of first best. Cai, Gupta, Li, and Mehta [6] recently obtained an explicit instance with \(\operatorname{FB}/\operatorname{RO}\ge 2.0749\). Thus the previously known range was \[0.317844432899372\ldots \le \rho_{\rm RO} \le \frac{1}{2.0749} =0.481950937394573\ldots.\]
We tighten both sides of this range: \[\label{eq:main-interval} \frac{1}{\pi}\le \rho_{\rm RO}<0.460242308085529.\tag{1}\] The lower bound holds for every pair of independent priors on a common compact interval.
Theorem 1. For every pair of independent value distributions on a common compact interval, \[\operatorname{RO}\ge \frac{1}{\pi}\,\operatorname{FB}.\]
The upper bound is witnessed by an explicit family of distributions.
Theorem 2. There exist independent value distributions supported on \([0,1]\), with a continuous seller distribution and a buyer distribution having a continuous part and one atom, such that \[\frac{\operatorname{FB}}{\operatorname{RO}}>2.17276852308451.\] Consequently, \(\rho_{\rm RO}<0.460242308085529\).
The exact value of \(\rho_{\rm RO}\) remains open. The lower-bound proof yields a family of inequalities indexed by the Lagrange multiplier, and the choice \(\alpha=1\) is the point at which the Lagrangian matches the two offerer profit problems. The upper-bound construction gives an explicit analytic family whose performance can be evaluated rigorously. Together, the two results substantially narrow the known range of \(\rho_{\rm RO}\).
We next describe the two arguments. Recent work of Liu, Qin, Ren, and Wang [7] proves that second-best bilateral trade obtains at least one half of first-best gains from trade. Their proof uses a Lagrangian reduction and a greedy construction of a pointwise-monotone allocation on a lattice. We keep the Lagrange multiplier \(\alpha\) as a free parameter. For smooth priors, let \(\Lambda(x)\) be the budget-balance functional and define \[V_\alpha:=\sup_x\{\operatorname{GFT}(x)+\alpha\Lambda(x)\},\] where the supremum is over pointwise-monotone allocations. We prove, for every \(\alpha>0\), \[\label{eq:C-beta-intro} V_\alpha\ge \frac{1}{C(\alpha)}\,\operatorname{FB}, \qquad C(\alpha):= \frac{1}{1+\alpha} \frac{\Gamma\!\left(\frac{1}{1+\alpha}\right)^2}{\Gamma\!\left(\frac{2}{1+\alpha}\right)}.\tag{2}\] At \(\alpha=1\), \[C(1)=\frac{1}{2}\mathrm B\!\left(\frac{1}{2},\frac{1}{2}\right)=\frac{\pi}{2}.\] The same choice of multiplier yields \[\operatorname{GFT}(x)+\Lambda(x) =\mathbb{E}[(\phi_B(B)-S)x(S,B)] +\mathbb{E}[(B-\psi_S(S))x(S,B)],\] where \(\phi_B\) is the buyer’s virtual value and \(\psi_S\) is the seller’s virtual cost. A nondecreasing row of a pointwise-monotone allocation is a mixture of seller posted-price rules, and a nonincreasing column is a mixture of buyer posted-price rules. The two terms are therefore bounded by the optimal seller-offering and buyer-offering profits, \(\Pi_S\) and \(\Pi_B\). Since gains from trade dominate proposer profit, we obtain \[\frac{2}{\pi}\operatorname{FB} \le V_1 \le \Pi_S+\Pi_B \le \operatorname{SO}+\operatorname{BO} =2\operatorname{RO}.\]
The proof of 2 begins with the consecutive-lattice analysis of Liu et al. On the lattice, we prove the greedy inequality for every \(\beta\) satisfying \(\beta C(\alpha)\le1\). We then use quantile ironing to remove the seller regularity assumption without decreasing the optimized Lagrangian value. A regularity-preserving lattice approximation transfers the discrete bound to smooth distributions, and a final approximation argument gives the stated result for arbitrary Borel priors on a compact interval.
For the upper bound, the buyer has a truncated equal-revenue distribution. This choice makes every positive seller type optimally post the price \(1\). The seller CDF follows a tilted power law on a lower interval, has constant virtual cost on a middle interval, and equals one above a cutoff. The lower interval makes the buyer’s optimal offer follow a prescribed increasing map. The middle interval makes all larger buyer types offer the same boundary price; seller types in that interval contribute to first best and to the seller-offering mechanism but do not trade when the buyer makes the offer. After sending the buyer truncation and seller perturbation to zero in sequence, the ratio \(\operatorname{FB}/\operatorname{RO}\) reduces to three one-dimensional integrals. An explicit choice of five rational parameters gives the limiting value \[2.172768523084520237\ldots.\] The incentive properties of the construction are proved analytically, and interval arithmetic supplies rigorous enclosures for the three integrals.
The seller’s value \(S\) and buyer’s value \(B\) are independent Borel random variables. Their distributions are common knowledge. We normalize their compact support to \([0,1]\); a common positive affine change preserves every gains-from-trade ratio in the paper. Write \(F_S\) and \(F_B\) for their CDFs. Whenever a density or point mass is present, write it as \(f_S\) or \(f_B\), with the argument indicating which interpretation applies.
Definition 1 (Gains from trade and first best). An allocation rule \(x:[0,1]^2\to[0,1]\) gives the probability of trade at each reported pair \((s,b)\). Its expected gains from trade are \[\operatorname{GFT}(x):=\mathbb{E}[(B-S)x(S,B)].\] The first-best rule is \(x^{\rm FB}(s,b)=\mathbf{1}\{b>s\}\), and \[\operatorname{FB}:=\mathbb{E}[(B-S)_+].\]
We next define the seller- and buyer-offering mechanisms. A proposer may post a price in \([0,1]\) or choose no trade. The responder accepts at equality. On the relevant range (\(p\ge s\) for the seller and \(p\le b\) for the buyer), each profit objective is upper semicontinuous and therefore attains its maximum. When several prices are optimal, choose the smallest one. The lower-bound argument is independent of this tie-breaking rule.
Definition 2 (Seller-offering mechanism). After observing \(s\), the seller chooses a price \(p\in[0,1]\) maximizing \[(p-s)\mathbb{P}[B\ge p].\] The buyer accepts when \(B\ge p\). The mechanism’s gains from trade are denoted \(\operatorname{SO}\). The seller’s ex-ante profit is \[\Pi_S:=\mathbb{E}_S\!\left[\max\!\left\{0,\sup_{p\in[0,1]} (p-S)\mathbb{P}[B\ge p]\right\}\right],\] where a price with zero trade is available.
Definition 3 (Buyer-offering mechanism). After observing \(b\), the buyer chooses a price \(p\in[0,1]\) maximizing \[(b-p)\mathbb{P}[S\le p].\] The seller accepts when \(S\le p\). The mechanism’s gains from trade are denoted \(\operatorname{BO}\). The buyer’s ex-ante profit is \[\Pi_B:=\mathbb{E}_B\!\left[\max\!\left\{0,\sup_{p\in[0,1]} (B-p)\mathbb{P}[S\le p]\right\}\right].\]
Definition 4 (Random-offerer mechanism and worst-case fraction). The random-offerer mechanism selects each offering mechanism with probability one half: \[\operatorname{RO}:=\frac{\operatorname{SO}+\operatorname{BO}}{2}.\] Its worst-case first-best fraction is \[\rho_{\rm RO}:=\inf_{F_S,F_B:\,\operatorname{FB}>0}\frac{\operatorname{RO}}{\operatorname{FB}}.\]
The responder receives nonnegative utility whenever trade occurs. This gives a basic comparison used later.
Lemma 1 (Gains from trade dominate proposer profit). The offering mechanisms satisfy \[\operatorname{SO}\ge\Pi_S, \qquad \operatorname{BO}\ge\Pi_B.\]
Proof. At a seller price \(p\), every accepted trade satisfies \(B-S=(p-S)+(B-p)\ge p-S\). At a buyer offer \(p\), every accepted trade satisfies \(B-S=(B-p)+(p-S)\ge B-p\). Taking expectations proves both inequalities. ◻
Definition 5 (Pointwise monotonicity). Let \(\mathcal{R}_2\) be the set of measurable allocations \(x:\mathbb{R}^2\to[0,1]\) such that \(x(s,b)\) is nondecreasing in \(b\) for every \(s\) and nonincreasing in \(s\) for every \(b\).
For the fixed-multiplier argument, first suppose that both distributions have strictly positive continuously differentiable densities on their support intervals. Their ordinary virtual value and virtual cost are \[\label{eq:ordinary-virtual} \phi_B(b):=b-\frac{1-F_B(b)}{f_B(b)}, \qquad \psi_S(s):=s+\frac{F_S(s)}{f_S(s)}.\tag{3}\]
Definition 6 (Budget-balance functional). For \(x\in\mathcal{R}_2\), define \[\label{eq:Lambda} \Lambda(x):=\mathbb{E}[(\phi_B(B)-\psi_S(S))x(S,B)].\tag{4}\]
The envelope formula identifies \(\Lambda(x)\) as the budget-balance functional: \(\Lambda(x)\ge0\) is the expected-payment condition for weak budget balance under envelope-normalized transfers; see Myerson and Satterthwaite [1]. For \(\alpha\ge0\), define \[\begin{align} \phi_B^\alpha(b)&:=(1+\alpha)b -\alpha\frac{1-F_B(b)}{f_B(b)},\notag\\ \psi_S^\alpha(s)&:=(1+\alpha)s +\alpha\frac{F_S(s)}{f_S(s)}. \label{eq:alpha-virtual} \end{align}\tag{5}\] Then \[\label{eq:Lalpha-virtual} \operatorname{GFT}(x)+\alpha\Lambda(x) =\mathbb{E}[(\phi_B^\alpha(B)-\psi_S^\alpha(S))x(S,B)].\tag{6}\]
Definition 7 (Fixed-multiplier value). For smooth priors and \(\alpha\ge0\), set \[V_\alpha:=\sup_{x\in\mathcal{R}_2} \{\operatorname{GFT}(x)+\alpha\Lambda(x)\}.\]
Theorem 3 (Fixed-multiplier bound). For every \(\alpha>0\) and every pair of independent value distributions with strictly positive continuously differentiable densities on a common compact interval, \[V_\alpha\ge\beta(\alpha)\,\operatorname{FB}.\] In particular, \(C(1)=\pi/2\) and \(V_1\ge(2/\pi)\operatorname{FB}\).
For \(\alpha>0\), set \[\label{eq:aq} q:=\frac{1}{1+\alpha}, \qquad a:=\frac{\alpha}{1+\alpha}=1-q.\tag{7}\] The constant in 2 is \[\label{eq:C-beta} C(\alpha)=q\mathrm B(q,q) =q\frac{\Gamma(q)^2}{\Gamma(2q)}, \qquad \beta(\alpha)=\frac{1}{q\mathrm B(q,q)}.\tag{8}\]
We first work on the unit lattice. Let the seller have positive mass \(f_S(s)\) at every \(s\in\{0,1,\ldots,W\}\) and put \(F_S(-1)=0\). Its lattice \(\alpha\)-virtual cost is \[\label{eq:lattice-virtual} \psi_S^\alpha(s):=(1+\alpha)s +\alpha\frac{F_S(s-1)}{f_S(s)}.\tag{9}\] The seller is \(\alpha\)-weakly regular when these costs are nondecreasing. Buyer values range over the nonnegative integers; the local formulation below allows zero buyer masses and therefore covers every finite buyer distribution.
On the unit lattice, write the fixed-multiplier objective directly in terms of probability masses: \[\begin{align} L_\alpha(x;F_S,F_B) &:=(1+\alpha)\sum_{s,b}f_S(s)f_B(b)(b-s)x(s,b)\notag\\ &\quad-\alpha\sum_{s,b}f_S(s)(1-F_B(b))x(s,b) -\alpha\sum_{s,b}f_B(b)F_S(s-1)x(s,b). \label{eq:lattice-L} \end{align}\tag{10}\] It equals \(\operatorname{GFT}+\alpha\Lambda\) when all masses are positive and is also defined when some buyer masses vanish.
As in Liu, Qin, Ren, and Wang [7], an allocation is defined on the whole lattice and is pointwise monotone. Entries with \(b\le s\) can be set to zero without lowering 10 . Indeed, the coefficient of such an entry is \[f_S(s)\bigl((1+\alpha)f_B(b)(b-s) -\alpha(1-F_B(b))\bigr) -\alpha f_B(b)F_S(s-1)\le0.\] Fix a finite buyer truncation \(\{0,\ldots,M\}\). Let \(\Delta_M\) be the buyer probability simplex and let \(\mathcal{R}_2^{W,M}\) be the finite pointwise-monotone allocation polytope. For \[g(x,F_B):=L_\alpha(x;F_S,F_B)-\beta\operatorname{FB}(F_S,F_B),\] bilinearity and compactness allow Sion’s minimax theorem [8] to be applied in the explicit form \[\label{eq:sion-explicit} \inf_{F_B\in\Delta_M}\sup_{x\in\mathcal{R}_2^{W,M}}g(x,F_B) =\sup_{x\in\mathcal{R}_2^{W,M}}\inf_{F_B\in\Delta_M}g(x,F_B).\tag{11}\] Thus it suffices to construct one allocation satisfying \[\label{eq:fixed-maximin} L_\alpha(x;F_S,F_B)-\beta\operatorname{FB}\ge0 \quad\text{for every buyer distribution on \{0,\ldots,M\}}.\tag{12}\] After setting \(x(s,b)=0\) for \(b\le s\), the objective difference is \(g(x,F_B)=\sum_{b=0}^M f_B(b)H(b;x)\), where \[\label{eq:H-local} H(b;x):= \sum_{0\le s<b} f_S(s) \left[ (b-s)((1+\alpha)x(s,b)-\beta) -\alpha\!\sum_{s<s'<b}x(s',b) -\alpha\!\sum_{s<b'<b}x(s,b') \right].\tag{13}\] The sums over intermediate lattice points are the discrete envelope rents. Because a linear function on \(\Delta_M\) attains its minimum at a vertex, the inner infimum in 11 equals \(\min_{0\le b\le M}H(b;x)\). Thus 12 is equivalent to \(H(b;x)\ge0\) for every \(b\le M\). The construction and its no-failure proof are uniform in \(M\), so the same inequalities hold for every finite lattice buyer distribution.
The greedy construction of Liu et al. [7] processes buyer values increasingly, copies the preceding column, and raises the current column from low seller types upward until \(H(b;x)=0\). Their maximin–greedy equivalence is stated for arbitrary \(\beta\ge0\).
Proposition 1 (Parameterized greedy bound). Fix \(\alpha>0\) and a full-support \(\alpha\)-weakly regular seller distribution on a consecutive unit lattice. If \(0<\beta C(\alpha)\le1\), then the greedy construction satisfies every inequality 13 .
Proof. 7 verifies the four steps of Liu et al. [7] with symbolic \(\beta\): threshold-one feasibility, threshold-stationary truncation, nonstationary truncation, and the minimal-counterexample argument. ◻
Corollary 1 (Regular lattice bound). For a full-support \(\alpha\)-weakly regular lattice seller and every finite-support lattice buyer distribution, \[\sup_{x\in\mathcal{R}_2}L_\alpha(x;F_S,F_B) \ge\beta(\alpha)\operatorname{FB}.\]
Proof. Apply 1 with \(\beta=1/C(\alpha)\) and use the maximin–greedy equivalence. ◻
Let \(c(u)=F_S^{-1}(u)\) be the seller’s quantile function, \(0<u<1\), and write \[\label{eq:quantile-virtual} \psi_\alpha(u):=(1+\alpha)c(u)+\alpha u c'(u), \qquad \Phi(u):=\int_0^u\psi_\alpha(v)\,dv.\tag{14}\] The identity \[\label{eq:Phi-M} \Phi(u)=M(u)+\alpha u c(u), \qquad M(u):=\int_0^u c(v)\,dv,\tag{15}\] follows by integration by parts.
For a bounded seller coefficient \(\chi:(0,1)\to\mathbb{R}\) and a smooth buyer prior, let \(\mathcal{X}\) be the set of measurable \(X:(0,1)\times\mathbb{R}\to[0,1]\) that are nonincreasing in the first coordinate and nondecreasing in the second. Define the quantile objective and its value by \[\begin{align} \mathcal{L}_\alpha(\chi;X) &:=\int_0^1\mathbb{E}_B\!\left[ \bigl(\phi_B^\alpha(B)-\chi(u)\bigr)X(u,B)\right]du, \tag{16}\\ \mathcal{V}_\alpha(\chi,F_B) &:=\sup_{X\in\mathcal{X}}\mathcal{L}_\alpha(\chi;X). \tag{17} \end{align}\] When \(c\) is strictly increasing and \(\chi=\psi_\alpha\), the change of variables \(u=F_S(s)\) gives \(\mathcal{V}_\alpha(\psi_\alpha,F_B)=V_\alpha\).
Let \(\overline{\Phi}\) be the greatest convex minorant of \(\Phi\), and let \(\overline{\psi}\) be the right-continuous version of its nondecreasing slope. The zero function is a convex minorant of the nonnegative curve \(\Phi\), so \(0\le\overline{\Phi}\le\Phi\) and \(\overline{\Phi}(0)=\Phi(0)=0\). The minorant also contacts \(\Phi\) at \(1\): otherwise its endpoint value could be raised slightly without violating either convexity or the minorant property. Since \(\Phi(u)=o(u)\) as \(u\downarrow0\), convexity gives \(\overline{\psi}(0+)=0\). Define \[\label{eq:continuous-ironed-c} \overline{c}(u):= \frac{1}{\alpha} u^{-(1+\alpha)/\alpha} \int_0^u v^{1/\alpha}\overline{\psi}(v)\,dv.\tag{18}\] Because a convex nonnegative function starting at zero has nonnegative right derivative, \(\overline{\psi}\ge0\). Then \[\label{eq:continuous-ironed-ode} (1+\alpha)\overline{c}(u)+\alpha u\overline{c}'(u) =\overline{\psi}(u).\tag{19}\] More precisely, \((1+\alpha)\overline{c}(u)\) is a weighted average of \(\overline{\psi}(v)\) over \(0<v<u\). Since \(\overline{\psi}\) is nondecreasing, \((1+\alpha)\overline{c}(u)\le\overline{\psi}(u)\). Equation 19 then gives \(\overline{c}'(u)\ge0\) almost everywhere, so \(\overline{c}\) is a seller quantile function after choosing its left-continuous version.
Lemma 2 (Fixed-multiplier quantile ironing). For every smooth buyer prior, \[\mathcal{V}_\alpha(\psi_\alpha,F_B) =\mathcal{V}_\alpha(\overline{\psi},F_B), \qquad \operatorname{FB}(\overline{c},F_B)\ge\operatorname{FB}(c,F_B).\]
Proof. For \(X\in\mathcal{X}\), let \(y(u)=\mathbb{E}_B[X(u,B)]\), which is nonincreasing. Put \(D=\Phi-\overline{\Phi}\ge0\). Both curves agree at \(0\) and \(1\). The buyer term is unchanged when only the seller coefficient is ironed, and Stieltjes integration by parts gives \[\begin{align} \mathcal{L}_\alpha(\overline{\psi};X) -\mathcal{L}_\alpha(\psi_\alpha;X) &=\int_0^1(\psi_\alpha-\overline{\psi})y\,du\notag\\ &=-\int_0^1D\,dy\ge0. \label{eq:ironing-one-way} \end{align}\tag{20}\] Let \(\mathcal{I}\) be the countable collection of connected components of \(\{u:D(u)>0\}\). On every \(I=(\ell,r)\in\mathcal{I}\), the greatest convex minorant is the chord joining the two contact points, and hence \(\overline{\psi}\) is constant. Define \(AX\) by replacing, for every \(b\), the rows on \(I\) by their Lebesgue average over \(I\), and leave all other rows unchanged. For a nonincreasing function, the average on an interval lies between its one-sided endpoint values. Thus these replacements preserve monotonicity in \(u\); averaging also preserves monotonicity in \(b\). They leave \(\int_0^1X(u,b)\,du\) unchanged for every \(b\). Therefore \[\mathcal{L}_\alpha(\overline{\psi};AX) =\mathcal{L}_\alpha(\overline{\psi};X).\] Moreover, \(AX\) is constant on each \(I\) and \[\int_I(\psi_\alpha-\overline{\psi})\,du =D(r)-D(\ell)=0.\] On the complement of the ironing intervals, \(D=0\) and hence \(D'=\psi_\alpha-\overline{\psi}=0\) almost everywhere. It follows that \(\mathcal{L}_\alpha(\psi_\alpha;AX) =\mathcal{L}_\alpha(\overline{\psi};AX)\). Consequently \(\mathcal{V}_\alpha(\overline{\psi},F_B) \le\mathcal{V}_\alpha(\psi_\alpha,F_B)\), while 20 gives the reverse inequality.
For first best, define \(\overline{M}(u)=\int_0^u\overline{c}(v)\,dv\). The pairs \((M,\Phi)\) and \((\overline{M},\overline{\Phi})\) satisfy \(M+\alpha uM'=\Phi\) and \(\overline{M}+\alpha u\overline{M}'=\overline{\Phi}\). The solution with zero initial value is \[M(u)=\frac{1}{\alpha} u^{-1/\alpha} \int_0^u v^{1/\alpha-1}\Phi(v)\,dv.\] Thus \(\overline{\Phi}\le\Phi\) implies \(\overline{M}\le M\). For every buyer value \(b\), \[\int_0^1(b-c(u))_+\,du =\max_{0\le u\le1}\{bu-M(u)\}.\] Replacing \(M\) by \(\overline{M}\) weakly raises the maximum. Averaging over \(B\) proves the claim. ◻
The next lemma transfers the consecutive-lattice bound to the quantile model while preserving regularity.
Lemma 3 (Regular lattice approximation). Fix \(\alpha>0\). For a bounded nonnegative nondecreasing function \(\psi:(0,1)\to\mathbb{R}\) with \(\psi(0+)=0\), define \[\label{eq:T-alpha} (T_\alpha\psi)(u):= \frac{1}{\alpha} u^{-(1+\alpha)/\alpha} \int_0^u v^{1/\alpha}\psi(v)\,dv.\tag{21}\] Let \(c=T_\alpha\psi\). For every buyer prior having a strictly positive continuously differentiable density on a compact interval beginning at zero, \[\mathcal{V}_\alpha(\psi,F_B) \ge\beta(\alpha)\operatorname{FB}(c,F_B).\]
Proof. We begin with a seller having a positive continuously differentiable density \(f\) on \([0,R]\) and a continuous, strictly increasing value-space \(\alpha\)-virtual cost \[\widehat\psi(s):=(1+\alpha)s+\alpha\frac{F(s)}{f(s)}.\] Write \(c_F:=F^{-1}\) for this seller’s quantile function. For \(W\ge1\), put \(h=R/W\) and \(s_i=ih\). With \(q_i=f(s_i)/F(s_i)\) for \(i\ge1\), define \[\label{eq:lattice-Euler-recursion} G_W=1, \qquad G_{i-1}:=\frac{G_i}{1+hq_i}, \qquad g_i:=G_i-G_{i-1}, \qquad g_0:=G_0.\tag{22}\] Set \(G_{-1}:=0\). The \(g_i\) are positive and sum to one. The original-scale lattice virtual cost at \(s_i\) is \[\widehat\psi_{h,i}:=(1+\alpha)s_i +\alpha h\frac{G_{i-1}}{g_i}.\] For \(i\ge1\), 22 gives \(G_{i-1}/g_i=F(s_i)/(h f(s_i))\), and hence \[\label{eq:exact-lattice-cost} \widehat\psi_{h,i}=\widehat\psi(s_i).\tag{23}\] The equality also holds at \(i=0\), where both sides are zero. Thus the seller has full support and is \(\alpha\)-weakly regular. Dividing all values by \(h\) produces the consecutive unit-lattice seller in 1; both the fixed-multiplier objective and first best scale by \(1/h\).
We next prove the convergence needed for the optimized objective. Iterating 22 gives \[\label{eq:lattice-product} G_i=\prod_{j=i+1}^{W}\left(1+h\frac{f(s_j)}{F(s_j)}\right)^{-1}.\tag{24}\] For every \(\delta>0\), \(f/F\) is bounded and continuously differentiable on \([\delta,R]\). Uniformly over \(s_i\ge\delta\), \[\begin{align} \log G_i &=-\sum_{j=i+1}^{W}\log\left(1+h\frac{f(s_j)}{F(s_j)}\right)\\ &=-\sum_{j=i+1}^{W}h\frac{f(s_j)}{F(s_j)}+O(h) \longrightarrow-\int_{s_i}^{R}\frac{f(t)}{F(t)}\,dt =\log F(s_i). \end{align}\] Hence \(G_i/F(s_i)\to1\) uniformly away from zero. Moreover, \[\label{eq:lattice-density-ratio} \frac{g_i}{h} =\frac{G_i}{F(s_i)} \frac{f(s_i)}{1+h f(s_i)/F(s_i)} \longrightarrow f(s_i)\tag{25}\] uniformly on the same region. The lattice mass below \(\delta\) converges to at most \(F(\delta)\). More precisely, if \(i_h=\max\{i:s_i\le\delta\}\), then \(G_{i_h}\to F(\delta)\) at every continuity point of \(F\); here every point is a continuity point. Thus the mass of the cells meeting \([0,\delta]\) is \(F(\delta)+o(1)\), which vanishes after first taking \(h\downarrow0\) and then \(\delta\downarrow0\).
Let \(I_i=[s_i,s_i+h)\), extend continuous seller densities by zero beyond \(R\), and spread each lattice weight uniformly over its cell: \[\label{eq:spread-seller} f_{S,h}(s):=\sum_{i=0}^{W}\frac{g_i}{h}\mathbf{1}_{I_i}(s), \qquad a_{S,h}(s):=\sum_{i=0}^{W} \frac{g_i\widehat\psi_{h,i}}{h}\mathbf{1}_{I_i}(s).\tag{26}\] Put \[a_S(s):=f(s)\widehat\psi(s) =(1+\alpha)s f(s)+\alpha F(s)\] on \([0,R]\), with zero extension outside. The uniform convergence in 25 , the vanishing initial mass, and the \(O(h)\) mass in the cell extending beyond \(R\) imply \[\label{eq:seller-L1} \|f_{S,h}-f\|_1\longrightarrow0, \qquad \|a_{S,h}-a_S\|_1\longrightarrow0.\tag{27}\]
Discretize the buyer upward on the same grid: \(B_h=h\lceil B/h\rceil\) for \(B>0\), and set the zero column to zero. Write \(b_j=jh\), \(p_{j,h}=\mathbb{P}[B_h=b_j]\), and \(\tau_j=\mathbb{P}[B_h>b_j]=\mathbb{P}[B>b_j]\). Zero masses are allowed. The signed buyer virtual coefficient at report \(b_j\) is \[\label{eq:buyer-signed-coefficient} A^B_{j,h}:=(1+\alpha)b_jp_{j,h}-\alpha h\tau_j.\tag{28}\] For \(J_j=((j-1)h,jh]\), spread these weights as \[\label{eq:spread-buyer} f_{B,h}(b):=\sum_{j\ge1}\frac{p_{j,h}}{h}\mathbf{1}_{J_j}(b), \qquad a_{B,h}(b):=\sum_{j\ge1}\frac{A^B_{j,h}}{h}\mathbf{1}_{J_j}(b).\tag{29}\] Only finitely many summands are nonzero. If \(\tau(b)=\mathbb{P}[B>b]\) and \[a_B(b):=(1+\alpha)b f_B(b)-\alpha\tau(b),\] then the first function in 29 is the usual density histogram, while on \(J_j\) the second is \((1+\alpha)b_jp_{j,h}/h-\alpha\tau(b_j)\). Smoothness of the density and Lipschitz continuity of the tail therefore give \[\label{eq:buyer-L1} \|f_{B,h}-f_B\|_1\longrightarrow0, \qquad \|a_{B,h}-a_B\|_1\longrightarrow0.\tag{30}\]
Extend all four densities by zero to a fixed compact rectangle and define \[w_h(s,b):=f_{S,h}(s)a_{B,h}(b) -a_{S,h}(s)f_{B,h}(b), \qquad w(s,b):=f(s)a_B(b)-a_S(s)f_B(b).\] The product structure and 27 30 imply \[\label{eq:objective-density-L1} \|w_h-w\|_1\longrightarrow0.\tag{31}\] For example, \[\|f_{S,h}a_{B,h}-fa_B\|_{L^1(ds\,db)} \le \|f_{S,h}\|_1\|a_{B,h}-a_B\|_1 +\|f_{S,h}-f\|_1\|a_B\|_1.\] The second product is bounded in the same way. This \(L^1\) argument allows the density to be unbounded near the left endpoint. Let \(L_{\alpha,h}\) denote the original-scale version of 10 for these grids, and set \(V_{\alpha,h}:=\sup_{x\in\mathcal{R}_2}L_{\alpha,h}(x)\). If \(x_{ij}\) is a monotone lattice allocation and \(x_h\) is its constant extension to \(I_i\times J_j\), discrete summation by parts gives \[\begin{align} L_{\alpha,h}(x) &=\sum_{i,j}\left[ g_iA^B_{j,h} -p_{j,h}\bigl((1+\alpha)s_i g_i+\alpha hG_{i-1}\bigr) \right]x_{ij}\notag\\ &=\iint w_h(s,b)x_h(s,b)\,ds\,db. \label{eq:spread-objective} \end{align}\tag{32}\] Every such extension is pointwise monotone. Because \(0\le x_h\le1\), 31 yields \[\limsup_{h\downarrow0}V_{\alpha,h} \le \mathcal{V}_\alpha(\widehat\psi\circ c_F,F_B).\] Here the right side is written in quantile coordinates; the change of variables \(u=F(s)\) identifies it with the supremum of \(\iint w(s,b)x(s,b)\,ds\,db\) over value-coordinate pointwise-monotone allocations. Conversely, average any such allocation on each \(I_i\times J_j\). The cell averages form a monotone lattice array, and their step extensions converge at almost every Lebesgue point. Dominated convergence against \(|w|\), together with 31 , gives the reverse inequality. Hence \[\label{eq:V-lattice-limit} V_{\alpha,h}\longrightarrow \mathcal{V}_\alpha(\widehat\psi\circ c_F,F_B).\tag{33}\]
Let \(c_h\) be the lattice seller’s quantile function and set \(\operatorname{FB}_h:=\mathbb{E}[(B_h-S_h)_+]\). The lattice seller CDFs converge to \(F\) by 24 : convergence holds uniformly away from zero, and the mass near zero is controlled by the preceding \(F(\delta)+o(1)\) bound. Since \(F\) is continuous and all distributions have common compact support, their quantile functions converge in \(L^1(0,1)\). Coupling by quantiles and using \(|B_h-B|\le h\) gives \[\label{eq:FB-lattice-limit} |\operatorname{FB}_h-\operatorname{FB}| \le h+\int_0^1|c_h(u)-c_F(u)|\,du\longrightarrow0.\tag{34}\] Applying 1 before taking the limit proves the claim for smooth strictly regular sellers.
For the general \(\psi\) in the statement, extend it monotonically to the real line by setting it equal to zero to the left of \(0\) and equal to its terminal value to the right of \(1\). Convolve with a nonnegative smooth mollifier of radius \(\varepsilon_k\downarrow0\), subtract the value at zero, and add \(\varepsilon_k u\). The resulting bounded functions \(\psi_k\) are smooth and nonnegative on \([0,1]\), satisfy \[\psi_k(0)=0, \qquad \psi_k'(u)\ge\varepsilon_k>0, \qquad \|\psi_k-\psi\|_1\longrightarrow0.\] Put \(c_k=T_\alpha\psi_k\). Differentiating 21 gives \[(1+\alpha)c_k(u)+\alpha u c_k'(u)=\psi_k(u).\] The term \((1+\alpha)c_k(u)\) is a weighted average of the past values of \(\psi_k\). Writing \(r=(1+\alpha)/\alpha\), the derivative lower bound gives \[\psi_k(u)-(1+\alpha)c_k(u) \ge \varepsilon_k\! \left(u-r u^{-r}\int_0^u v^r\,dv\right) =\frac{\alpha\varepsilon_k}{1+2\alpha}\,u.\] Consequently \[\label{eq:ck-derivative-lower} c_k'(u)\ge\frac{\varepsilon_k}{1+2\alpha}>0.\tag{35}\] Smoothness of \(\psi_k\) and \(\psi_k(0)=0\) also make \(c_k\) smooth at zero. Thus \(c_k\) is the quantile function of a seller with a strictly positive smooth density, and its value-space \(\alpha\)-virtual cost is strictly increasing.
The operator \(T_\alpha\) is an \(L^1\) contraction. For \(\delta_k=\psi_k-\psi\), Fubini’s theorem gives \[\begin{align} \|c_k-c\|_1 &\le\frac{1}{\alpha}\int_0^1v^{1/\alpha}|\delta_k(v)| \int_v^1u^{-1-1/\alpha}\,du\,dv\notag\\ &=\int_0^1(1-v^{1/\alpha})|\delta_k(v)|\,dv \le\|\delta_k\|_1. \label{eq:T-contraction} \end{align}\tag{36}\] Also, uniformly over \(X\in\mathcal{X}\), \[\label{eq:V-L1-continuity} |\mathcal{L}_\alpha(\psi_k;X)-\mathcal{L}_\alpha(\psi;X)| \le\|\psi_k-\psi\|_1.\tag{37}\] Thus the corresponding quantile values converge. The positive-part map is one-Lipschitz, so \[|\operatorname{FB}(c_k,F_B)-\operatorname{FB}(c,F_B)|\le\|c_k-c\|_1\longrightarrow0.\]
If \(c\) is constant on a nondegenerate quantile interval \(I\), its defining ODE gives \(\psi=(1+\alpha)c\) almost everywhere on \(I\). Averaging the allocation rows over \(I\) preserves monotonicity and both terms in 16 . Hence the quantile formulation has the same value when all rows corresponding to one seller atom are identified. Apply the smooth strictly regular case to \(c_k\) and pass to the limit. ◻
Proof of 3. Apply 2 to the seller quantile \(c\) and then 3 to \(\overline{\psi}\). Since the original seller quantile is strictly increasing, \[V_\alpha =\mathcal{V}_\alpha(\psi_\alpha,F_B) =\mathcal{V}_\alpha(\overline{\psi},F_B) \ge\beta(\alpha)\operatorname{FB}(\overline{c},F_B) \ge\beta(\alpha)\operatorname{FB}(c,F_B).\] At \(\alpha=1\), \(q=1/2\) and \[C(1)=\frac{1}{2}\mathrm B\!\left(\frac{1}{2},\frac{1}{2}\right)=\frac{\pi}{2}.\] ◻
Lemma 4 (Continuous virtual threshold identities). For smooth priors and every price \(p\), \[\begin{align} \int_p^1(\phi_B(b)-s)f_B(b)\,db &=(p-s)\mathbb{P}[B\ge p], \tag{38}\\ \int_0^p(b-\psi_S(s))f_S(s)\,ds &=(b-p)\mathbb{P}[S\le p]. \tag{39} \end{align}\]
Proof. Substitute 3 and integrate the tail term in the first identity and the CDF term in the second. ◻
Lemma 5 (Lagrangian value is bounded by proposer profits). For smooth positive-density priors, \[V_1\le\Pi_S+\Pi_B.\]
Proof. For \(x\in\mathcal{R}_2\), multiplier one gives \[\begin{align} \operatorname{GFT}(x)+\Lambda(x) &=\mathbb{E}[(\phi_B(B)-S)x(S,B)] +\mathbb{E}[(B-\psi_S(S))x(S,B)]. \label{eq:V-decomp} \end{align}\tag{40}\] Fix \(s\) and choose a right-continuous version of the nondecreasing row \(b\mapsto x(s,b)\). Its Stieltjes measure \(\mu_s\) has total mass at most one and satisfies, almost everywhere, \[x(s,b)=\int_{[0,1]}\mathbf{1}\{b\ge p\}\,d\mu_s(p);\] the missing mass is the no-trade rule. Fubini’s theorem and 38 express the row contribution to the first term of 40 as a mixture of seller posted-price profits. It is at most the seller’s optimal profit at \(s\). Averaging over \(S\) bounds the first term by \(\Pi_S\).
For fixed \(b\), the same construction applied to the nonincreasing column gives a subprobability mixture of rules \(\mathbf{1}\{s\le p\}\). By 39 , its contribution is at most the buyer’s optimal profit at \(b\). Averaging over \(B\) and taking the supremum over \(x\) proves the lemma. ◻
Lemma 6 (Smooth approximation of Borel priors). For any independent Borel priors on \([0,1]\), there are independent priors with strictly positive smooth densities such that \[\operatorname{FB}_k\to\operatorname{FB}, \qquad \Pi_{S,k}\to\Pi_S, \qquad \Pi_{B,k}\to\Pi_B.\]
Proof. We first establish stability under an \(L^\infty\) coupling. Suppose \(|S-S'|\le\varepsilon\) and \(|B-B'|\le\varepsilon\) almost surely. Since the positive-part map is one-Lipschitz in each argument, \[|\mathbb{E}(B-S)_+-\mathbb{E}(B'-S')_+|\le2\varepsilon.\]
Consider seller profit. Fix paired seller types \(s,s'\) and a price \(p\) for the original instance. If \(p-s\le2\varepsilon\), its profit is at most \(2\varepsilon\), so the no-trade option in the perturbed instance gives the required comparison. If \(p-s>2\varepsilon\), post \(p^-=(p-\varepsilon)\vee0\). Then \[\{B\ge p\}\subseteq\{B'\ge p^-\}, \qquad p^- -s'\ge p-s-2\varepsilon>0.\] The profit from \(p^-\) in the perturbed instance is therefore at least the profit from \(p\) minus \(2\varepsilon\). Taking the supremum over \(p\) and then interchanging the primed and unprimed instances shows that the conditional, and hence ex-ante, seller-profit values differ by at most \(2\varepsilon\). For buyer profit, the symmetric transfer is \(p^+=(p+\varepsilon)\wedge1\); the inclusion \(\{S\le p\}\subseteq\{S'\le p^+\}\) gives the same bound.
Choose \(\varepsilon_k\downarrow0\). Map each original value \(v\in[0,1]\) to \(2\varepsilon_k+(1-4\varepsilon_k)v\) and add independent noise with a smooth density supported on \([-\varepsilon_k,\varepsilon_k]\). The resulting prior has a smooth density supported in \([\varepsilon_k,1-\varepsilon_k]\) and can be coupled to the original value with displacement at most \(3\varepsilon_k\). Mix it with the uniform distribution on \([0,1]\) with probability \(\delta_k\downarrow0\). The mixture has a smooth density that is strictly positive on \([0,1]\).
Mixing changes each prior by total variation at most \(\delta_k\). Conditional proposer profit and the first-best integrand lie in \([0,1]\); responder tails and CDFs change by at most \(\delta_k\). Thus changing both priors by these mixtures changes each proposer-profit value and first best by at most \(2\delta_k\). Combining this estimate with the coupling bounds above proves all three convergences. ◻
Proof of 1. For smooth positive-density priors, 3 at \(\alpha=1\) and 5 1 give \[\frac{2}{\pi}\operatorname{FB}\le V_1\le\Pi_S+\Pi_B\le\operatorname{SO}+\operatorname{BO}=2\operatorname{RO}.\] For arbitrary Borel priors, apply the smooth inequality to the approximating sequence in 6 and pass to the limit: \[\frac{2}{\pi}\operatorname{FB}\le\Pi_S+\Pi_B\le2\operatorname{RO}.\] Dividing by two proves the theorem. ◻
Fei’s constant is \(1/T\), where \(T>1\) is the larger solution of \(T=2+\log T\), and \[\frac{1}{T}=0.317844432899372\ldots <\frac{1}{\pi}=0.318309886183790\ldots.\] The value \(\pi\) is the beta integral at the multiplier where the two proposer-profit objectives enter symmetrically.
We now prove 2. The family depends on two vanishing parameters: the buyer truncation \(\varepsilon>0\) and a seller perturbation \(\eta>0\). We first define the limiting shape.
Fix parameters \(a,k_1,k_2,x,z>0\) with \(0<x<z<1\), and define \[\label{eq:Q-A} Q(y):=y^a\exp\!\bigl(k_1(y-1)+k_2(y^2-1)\bigr), \qquad A(y):=a+k_1y+2k_2y^2, \quad 0\le y\le1.\tag{41}\] Then \(Q(1)=1\) and \(Q'(y)=Q(y)A(y)/y\).
Definition 8 (Truncated equal-revenue buyer). For \(\varepsilon\in(0,1)\), let \(B_\varepsilon\) be the buyer value with survival function \[\mathbb{P}[B_\varepsilon\ge b]= \begin{cases} 1, & 0\le b\le\varepsilon,\\ \varepsilon/b, & \varepsilon<b\le1. \end{cases}\] Equivalently, the distribution has density \(\varepsilon/b^2\) on \([\varepsilon,1)\) and an atom of mass \(\varepsilon\) at \(1\).
Definition 9 (Three-region seller). For \(\eta>0\), set \[q_\eta:=\frac{1+\eta-z}{1+\eta-x}.\] Let \(S_\eta\) be the seller value with CDF \[\label{eq:Geta} G_\eta(s):= \begin{cases} q_\eta Q(s/x), & 0\le s\le x,\\[2mm] \dfrac{1+\eta-z}{1+\eta-s}, & x<s<z,\\[3mm] 1, & z\le s\le1. \end{cases}\tag{42}\]
For \(s=xy\in(0,x]\), define \[\label{eq:psi-lower} \psi(s):=xy\frac{1+A(y)}{A(y)}.\tag{43}\]
Lemma 7 (Validity and optimal offers). Assume \(a>2k_2\) and \(\psi(x)<1\). Then \(G_\eta\) is a continuous CDF. The seller-offering mechanism posts \(1\) almost surely, and the buyer’s unique optimal offer is \[\label{eq:buyer-offer-rule} p(b)= \begin{cases} \psi^{-1}(b), & 0<b\le\psi(x),\\ x, & \psi(x)<b\le1. \end{cases}\tag{44}\]
Proof. The pieces agree at \(x\) and \(z\). Moreover, \(Q(0)=0\), \(Q(1)=1\), and \(Q'(y)=Q(y)A(y)/y>0\), so \(G_\eta\) is a continuous CDF. Let \(g_\eta\) denote its density on \((0,z)\).
For every seller value \(s>0\), a buyer price \(p\in(\varepsilon,1]\) gives seller profit \[(p-s)\frac{\varepsilon}{p} =\varepsilon\left(1-\frac{s}{p}\right),\] which is increasing in \(p\). Prices below \(\varepsilon\) give weakly smaller profit: their best value is attained at \(p=\varepsilon\), and \(\varepsilon(1-s)\ge\varepsilon-s\). Thus the seller posts \(p=1\), uniquely for \(s>0\). Since \(G_\eta\) has zero mass at \(s=0\), \[\label{eq:SO-limit} \frac{\operatorname{SO}_{\varepsilon,\eta}}{\varepsilon} =\mathbb{E}[1-S_\eta] =\int_0^1G_\eta(s)\,ds.\tag{45}\]
For the buyer’s problem, define the seller virtual cost \[\psi_\eta(s):=s+\frac{G_\eta(s)}{g_\eta(s)}.\] On the lower region, with \(y=s/x\), one has \(\psi_\eta(s)=\psi(s)\). Its derivative is strictly positive: \[\label{eq:psi-derivative} \frac{1}{x}\frac{d\psi}{dy} =1+\frac{A(y)-yA'(y)}{A(y)^2} =1+\frac{a-2k_2y^2}{A(y)^2}>0.\tag{46}\] On the middle region, direct differentiation gives \[\label{eq:psi-cap} \psi_\eta(s)=1+\eta.\tag{47}\]
On each region where \(G_\eta\) has a density, \[\frac{d}{dp}\bigl((b-p)G_\eta(p)\bigr) =g_\eta(p)\bigl(b-\psi_\eta(p)\bigr).\] The lower-region virtual cost is continuous and strictly increasing from \(0\) to \(\psi(x)\). Thus, for \(b\le\psi(x)\), profit increases up to \(p=\psi^{-1}(b)\) and decreases afterward. For \(b>\psi(x)\), profit increases throughout the lower region and is maximized there at \(x\). On the middle region, \(\psi_\eta=1+\eta>b\), so profit is strictly decreasing; on \([z,1]\), it equals \(b-p\) and is also decreasing. This proves 44 and uniqueness. ◻
Let \(G\) be 42 at \(\eta=0\), and set \[\label{eq:F-S-B-integrals} \mathcal{F}:=\int_0^1\frac{G(s)}{s}\,ds, \qquad \mathcal{S}:=\int_0^1G(s)\,ds, \qquad \mathcal{B}:=\int_0^x \left(1-\frac{s}{\psi(s)}-\log\psi(s)\right)dG(s).\tag{48}\]
Lemma 8 (Sequential limit). For the distributions in 8 9, \[\lim_{\eta\downarrow0}\lim_{\varepsilon\downarrow0} \frac{\operatorname{FB}_{\varepsilon,\eta}}{\operatorname{RO}_{\varepsilon,\eta}} =\frac{2\mathcal{F}}{\mathcal{S}+\mathcal{B}}.\]
Proof. The first-best identity \[\operatorname{FB}_{\varepsilon,\eta} =\int_0^1G_\eta(s)\mathbb{P}[B_\varepsilon\ge s] \,ds\] gives \[\lim_{\varepsilon\downarrow0} \frac{\operatorname{FB}_{\varepsilon,\eta}}{\varepsilon} =\int_0^1\frac{G_\eta(s)}{s}\,ds.\] Near zero, \(G_\eta(s)\) is a constant multiple of \(s^a\), so \(G_\eta(s)/s\) is integrable. Moreover, \(\mathbb{P}[B_\varepsilon\ge s]/\varepsilon\le1/s\) for \(s>0\). Dominated convergence therefore proves the limit. 45 gives the seller-offering limit.
Under 44 , a seller type \(s\le x\) trades precisely when \(B_\varepsilon\ge\psi(s)\). For every fixed \(v\in(0,1)\), \[\begin{align} \lim_{\varepsilon\downarrow0} \frac{1}{\varepsilon}\mathbb{E}[(B_\varepsilon-s) \mathbf{1}\{B_\varepsilon\ge v\}] &=\int_v^1\frac{b-s}{b^2}\,db+(1-s)\\ &=1-\frac{s}{v}-\log v. \end{align}\] Substitute \(v=\psi(s)\) and integrate over \(dG_\eta(s)\) on \([0,x]\). Seller types above \(x\) never trade under a buyer offer. Since \(v=\psi(s)>s\), the scaled integrand has the common bound \[\frac{1}{\varepsilon}\mathbb{E}[(B_\varepsilon-s) \mathbf{1}\{B_\varepsilon\ge v\}] \le 1+|\log v|.\] Indeed, the displayed limit formula gives the bound when \(\varepsilon\le v\); when \(v<\varepsilon\), the left side equals \(1-\log\varepsilon-s/\varepsilon\le1-\log v\).
As \(s\downarrow0\), \(\psi(s)\asymp s\) and \(dG_\eta(s)=O(s^{a-1})\,ds\), uniformly for small \(\eta\). Hence \(s^{a-1}(1+|\log s|)\) is a common integrable majorant, and the buyer-offering integral converges to \(\mathcal{B}\). Also, \(G_\eta\to G\) pointwise, while \(G_\eta(s)/s\) is bounded by a constant multiple of \(s^{a-1}\) on \((0,x]\) and uniformly on \([x,1]\). Dominated convergence gives \[\int_0^1\frac{G_\eta(s)}{s}\,ds\longrightarrow\mathcal{F}, \qquad \int_0^1G_\eta(s)\,ds\longrightarrow\mathcal{S}.\]
Combining the first-best, seller-offering, and buyer-offering limits and using \(\operatorname{RO}=(\operatorname{SO}+\operatorname{BO})/2\) gives the stated ratio. Taking the limits in this order keeps the buyer’s boundary offer unique during the \(\varepsilon\) limit. ◻
The three quantities have convenient unit-interval formulas. Put \[\label{eq:I-defs} I_F:=\int_0^1\frac{Q(y)}{y}\,dy, \qquad I_S:=\int_0^1Q(y)\,dy,\tag{49}\] and \[\label{eq:IB-def} I_B:=\int_0^1 \left[ \frac{1}{1+A(y)} -\log\!\left(xy\frac{1+A(y)}{A(y)}\right) \right]Q'(y)\,dy.\tag{50}\] With \(q_0=(1-z)/(1-x)\), elementary integration on the middle and terminal regions gives \[\begin{align} \mathcal{F} &=q_0I_F +(1-z)\log\!\frac{z(1-x)}{x(1-z)}-\log z, \tag{51}\\ \mathcal{S} &=q_0xI_S +(1-z)\log\!\frac{1-x}{1-z}+(1-z), \tag{52}\\ \mathcal{B}&=q_0I_B. \tag{53} \end{align}\]
Take the following terminating decimals as exact rational parameters: \[\begin{align} a&=0.095412847191313230,\notag\\ k_1&=0.0034331918146865968,\notag\\ k_2&=0.000042063244380725156,\label{eq:parameters}\\ x&=0.027131876557804906,\notag\\ z&=0.84530040073301205.\notag \end{align}\tag{54}\] They satisfy \(a>2k_2\) and \[A(1)=0.098930165494761277112, \qquad \psi(x)=0.3013846934020427791\ldots<1,\] which verifies the conditions used in 46 44 .
Arb interval arithmetic at 100 decimal digits gives the enclosures in 1; every displayed radius has been rounded upward.
| Quantity | Certified enclosure midpoint and radius |
|---|---|
| \(I_F\) | \(10.44755489238983467309805621763\;\pm 10^{-28}\) |
| \(I_S\) | \(0.91137899753252926492220359010\;\pm 10^{-29}\) |
| \(I_B\) | \(12.52948706468993154373219525462\;\pm 10^{-28}\) |
| \(\mathcal F\) | \(2.645835354262006463976354346239\;\pm 10^{-29}\) |
| \(\mathcal S\) | \(0.443087560739101222493372791649\;\pm 10^{-30}\) |
| \(\mathcal B\) | \(1.992363179780563445946824002812\;\pm 10^{-29}\) |
| \(2\mathcal F/(\mathcal S+\mathcal B)\) | \(2.172768523084520237024664850580\;\pm 10^{-29}\) |
In particular, \[\frac{2\mathcal{F}}{\mathcal{S}+\mathcal{B}} >2.17276852308451.\] By 8, sufficiently small positive \(\eta\) followed by sufficiently small positive \(\varepsilon\) yields an actual pair of distributions with the same strict inequality. This proves 2. The interval-arithmetic verification is given in 8.
We proved \[\frac{1}{\pi}\le\rho_{\rm RO}<0.460242308085529.\] The lower bound follows from the fixed-multiplier guarantee at \(\alpha=1\), where the Beta-function constant is \(\pi/2\) and the Lagrangian decomposes into seller and buyer posted-price profit terms. The upper bound follows from an explicit hard family with certified limiting ratio \(\operatorname{FB}/\operatorname{RO}>2.17276852308451\).
Determining the exact value of \(\rho_{\rm RO}\) remains open. The lower bound is governed by a multiplier-dependent monotone-allocation argument, whereas the hard family is governed by the seller’s lower-tail virtual-cost curve. A tight characterization must connect these two structures.
This appendix proves 1 by extending the state argument of Liu et al. [7] from their target \(\beta=1/2\) to a symbolic target. Their maximin–greedy equivalence already holds for arbitrary \(\beta\); the work here is to verify the no-failure lemmas with the target left symbolic. Fix \(\alpha>0\), let \(q=1/(1+\alpha)\) and \(a=1-q\), and assume \[\label{eq:beta-condition-app} 0<\beta C(\alpha)\le1, \qquad C(\alpha)=q\mathrm B(q,q).\tag{55}\] Since \[C(\alpha)=\int_0^1q t^{q-1}(1-t)^{q-1}\,dt>1,\] the condition also gives \(0<\beta<1\), as required by the state recursion.
For a full-support lattice seller, set \[r_i:=\frac{F_S(i-1)}{f_S(i)},\qquad i\ge1,\] and set \(r_0:=r_{\rm base}:=0\). Thus \(\psi_S^\alpha(i)=(1+\alpha)i+\alpha r_i\) for \(i\ge1\). At a successful tight step \(H(b;x)=0\), the current column has threshold form \[x(s,b)=1\;(s<t),\qquad x(t,b)=y\in(0,1],\qquad x(s,b)=0\;(s>t).\] Its clear state is \(\omega=(y;r_t,\ldots,r_{b-1})\). The base clear state is \(\omega_0=(\beta q;r_{\rm base})\). Given a clear state reached at buyer value \(b_0\), the compressed execution evaluates the positive normalization \[\widehat H_{t,b_0}(b;x) :=\frac{H(b;x)-H(b_0;x)}{F_S(t)},\qquad b>b_0.\] The clear-state coordinates and the future ratio tail determine every later greedy update. Write \(\operatorname{ALG}(\omega;\mathbf{r})\) for this compressed execution with future ratios \(\mathbf{r}\).
Two clear states are comparable when one ratio vector is a suffix of the other. Write \(\omega\prec\omega'\) when the ratio vector of \(\omega'\) is a proper suffix of that of \(\omega\), or when the vectors agree and the threshold allocation of \(\omega'\) is larger. Thus \(\omega'\) is the more demanding state. The following lemma records the two state properties used below.
Lemma 9 (Greedy-state toolkit). Fix \(\alpha\ge0\), \(\beta\ge0\), and an \(\alpha\)-weakly regular full-support lattice seller.
For a fixed terminal value \(M\), the maximin inequality 12 holds exactly when the greedy construction succeeds through \(M\). The inequalities for all finite \(M\) are therefore equivalent to never failing.
Clear states are sufficient statistics. If \(\omega\prec\omega'\) and \(\operatorname{ALG}(\omega';\mathbf{r})\) succeeds, then \(\operatorname{ALG}(\omega;\mathbf{r})\) succeeds and its output is dominated by the output from \(\omega'\). Equivalently, failure from \(\omega\) implies failure from \(\omega'\).
Proof. Part (i) is Liu et al.’s Maximin–Greedy Equivalence [7], stated there for arbitrary \(\beta\).
For part (ii), subtract the last clear equality \(H(b_0;x)=0\) from each later constraint and divide by the current threshold CDF. This is the normalized state equation used in their Proposition 5.1. Its coefficients depend only on the clear state and the future ratio tail. In the comparison of two aligned states, the term proportional to \(\beta\) is identical in both normalized equations and cancels. Suppose their threshold order reverses for the first time at buyer value \(b\). Earlier columns are ordered by the induction hypothesis, and their coefficients in the normalized equation are nonpositive; the current-column coefficients have the opposite strict order. The two equations therefore satisfy \[\widehat H(b;x')<\widehat H(b;x)=0\] for the more demanding run, contradicting its clear equality. Successful outputs preserve the dominance order.
For failure propagation, the ordered thresholds and weak regularity imply that a virtual-cost-test failure in the less demanding run also occurs in the more demanding run. The other failure mode is exhaustion of the boundary candidates. A feasible fractional boundary value in the more demanding run, together with the same normalized comparison, would give one in the less demanding run. Thus failure from the less demanding state implies failure from the more demanding state. These are the remaining steps of Liu et al. [7]; after the target term cancels, they do not depend on the value of \(\beta\). ◻
An execution is threshold-stationary through length \(N\) when it clears at threshold zero for every buyer value below \(N\). A proper-suffix output has deleted at least one positive leading ratio. For a ratio sequence, normalize \(F_N(0)=1\) and set \[F_N(m)=\prod_{i=1}^{m}\left(1+\frac{1}{r_i}\right).\] The long execution uses the sequence \((r_{\rm base},r_1,\ldots,r_{k+1})\). The short execution deletes \(r_1\) and uses \((r_{\rm base},r_2,\ldots,r_{k+1})\). The quantities \(\mathcal{D}_\beta\) and \(Q_b\) below are the corresponding normalized gaps between the long state and the artificial short state.
The next three subsections establish the parameterized versions of their threshold-one feasibility, threshold-stationary truncation, and nonstationary truncation results. The final subsection applies their minimal-counterexample argument.
While the threshold remains zero, successive differences of the tight constraints \(H(b;x)=0\) give \[\label{eq:G-recurrence-app} x_b=\beta G_b, \qquad G_b=\left(1-\frac{q}{b}\right)G_{b-1} +\frac{q}{b}\frac{F_S(b-1)}{F_S(0)}, \qquad G_1=q.\tag{56}\] Unrolling the recurrence gives \[G_b=\sum_{t=1}^b\frac{q}{t}\frac{F_S(t-1)}{F_S(0)} \prod_{j=t+1}^b\left(1-\frac{q}{j}\right).\] Since \[\sum_{t=1}^b\frac{q}{t}\prod_{j=t+1}^b\left(1-\frac{q}{j}\right) =1-\prod_{j=1}^b\left(1-\frac{q}{j}\right)<1,\] monotonicity of \(F_S\) gives \(G_b<F_S(b)/F_S(0)\). Subtracting consecutive instances of 56 now yields \[G_{b+1}-G_b=\frac{q}{b+1} \left(\frac{F_S(b)}{F_S(0)}-G_b\right)>0.\]
Suppose the virtual-cost test failed at threshold one for buyer value \(b\). Put \(\kappa=(1+\alpha)/\alpha=1/a\) and normalize \(F_S(0)=1\). Weak regularity would imply, for \(1\le m\le b-1\), \[r_m=\frac{F_S(m-1)}{f_S(m)}\ge\kappa(b-m).\] Consequently, for \(0\le i\le b-1\), \[F_S(i)=\prod_{m=1}^i\left(1+\frac{1}{r_m}\right) \le\prod_{m=1}^i\left(1+\frac{a}{b-m}\right) =:\widetilde{F}^{(b+1)}(i).\] All coefficients in the unrolled formula for \(G_b\) are nonnegative, so \(G_b\) is bounded by the corresponding extremal value \(\widetilde{G}_b^{(b+1)}\). The beta–gamma calculation of Liu et al. [7] is independent of the target and gives \[\label{eq:G-C-app} G_b\le\widetilde{G}_b^{(b+1)}<C(\alpha)=q\mathrm B(q,q).\tag{57}\] The threshold-zero candidate therefore respects the copied-column lower bound, and \(x_b=\beta G_b<\beta C(\alpha)\le1\). It is feasible and is encountered before threshold one. This contradicts the hypothetical failure and proves the parameterized form of Proposition 5.4.
Consider the long and short executions in Proposition 5.6, of lengths \(k+2\) and \(k+1\), and let \(F_{k+2}\) be the normalized CDF generated by the long inverse-hazard-ratio sequence. Put \[\varphi_i:=(i+1)(1+\alpha)+\alpha r_{i+1}, \qquad i=0,\ldots,k.\] Weak regularity is \(\varphi_0\le\cdots\le\varphi_k\). Replace the factors \(1/2\) in the two auxiliary quantities in their Appendix C by \[\begin{align} K_\beta &:={} \sum_{m=0}^{k+1} \frac{\beta F_{k+2}(m)}{(1+\alpha)(m+1)} \prod_{j=m+2}^{k+2}\left(1-\frac{q}{j}\right), \tag{58}\\ J_\beta &:={} \sum_{m=1}^{k+1} \frac{\beta\alpha(k+1-m)F_{k+2}(m)}{(1+\alpha)m(m+1)} \prod_{j=m+2}^{k+1}\left(1-\frac{q}{j}\right). \tag{59} \end{align}\] Thus \[\label{eq:KJ-scaling} K_\beta=2\beta K_{1/2}, \qquad J_\beta=2\beta J_{1/2}.\tag{60}\] Here \(K_\beta\) is the allocation required to clear the threshold-zero candidate at buyer value \(k+2\). If \(K_\beta<1\), that candidate is feasible; if \(K_\beta=1\), it clears with allocation one. In either case the output retains the first positive ratio. Thus a proper-suffix output implies \(K_\beta>1\), and the weaker inequality \(K_\beta\ge1\) will suffice.
For a candidate whose ratio suffix begins at \(t\ge2\), let \(x^{\rm long}\) denote the long candidate and let \(x^{\rm short}\) be the corresponding artificial candidate in the short execution, with the same boundary allocation. The candidate-dependent terms in the two normalized constraints cancel, as in equations (12)–(15) of Liu et al. Define \(\mathcal{D}_\beta\) by \[\label{eq:D-residual-app} \widehat H^{\rm long}(k+2;x^{\rm long}) -\left(1+\frac{1}{r_1}\right) \widehat H^{\rm short}(k+1;x^{\rm short}) =\mathcal{D}_\beta.\tag{61}\] The right side is independent of \(t\) and of the boundary allocation.
The normalized CDFs of the two executions satisfy \(F_{k+2}(m+1)=(1+1/r_1)F_{k+1}(m)\). Substituting the threshold-zero recurrence 56 into the long-minus-short state residual and collecting the terms gives \[\label{eq:D-beta-app} \mathcal{D}_\beta =J_\beta+1-\beta(k+2) \prod_{j=2}^{k+2}\left(1-\frac{q}{j}\right).\tag{62}\] The two terms after \(J_\beta\) are independent of the virtual-cost vector. Thus, on every level set of \(K_\beta\), minimizing \(\mathcal{D}_\beta\) is the same as minimizing \(J_\beta\). Equation 60 already shows that the level sets and every first-order comparison are exactly those in the \(\beta=1/2\) proof, up to the common positive factor \(2\beta\).
The adjacent-pooling calculation is as follows. Let \[d_\ell:=\varphi_\ell-(\ell+1)(1+\alpha)>0.\] For \(m\ge\ell+1\), \[\label{eq:F-derivative-app} \frac{\partial F_{k+2}(m)}{\partial\varphi_\ell} =-\frac{\alpha}{d_\ell(d_\ell+\alpha)}F_{k+2}(m),\tag{63}\] and the derivative is zero for \(m\le\ell\). Hence \(\partial_{\varphi_\ell}K_\beta<0\). Put \[R_\ell:= \frac{\partial_{\varphi_\ell}J_\beta}{\partial_{\varphi_\ell}K_\beta}.\] The common factor \(2\beta\) cancels in \(R_\ell\). After the common negative factor in 63 is removed, \(R_\ell\) is a weighted average over \(m=\ell+1,\ldots,k+1\). For each summation index, the quotient between the numerator and denominator coefficients is \[\frac{\alpha(1+\alpha)(k+2)(k+1-m)}{m((1+\alpha)(k+2)-1)},\] which is strictly decreasing in \(m\). The average defining \(R_{\ell+1}\) is the same weighted tail with its first, largest term removed. Therefore \(R_\ell>R_{\ell+1}\) for \(0\le\ell<k\).
If \(\varphi_i<\varphi_{i+1}\), the implicit-function theorem gives a local level-set direction satisfying \[\frac{d\varphi_{i+1}}{d\varphi_i} =-\frac{\partial_{\varphi_i}K_\beta}{\partial_{\varphi_{i+1}}K_\beta}<0, \qquad \frac{dJ_\beta}{d\varphi_i} =\partial_{\varphi_i}K_\beta(R_i-R_{i+1})<0.\] Hence increasing the smaller adjacent virtual cost and decreasing the larger one preserves \(K_\beta\), preserves weak regularity for a sufficiently small move, and strictly lowers \(J_\beta\). Repeating these pooling moves forces every minimizer in the compactified admissible level set to satisfy \(\varphi_0=\cdots=\varphi_k\). The compactification and continuity argument in Liu et al. [7] applies without change: by 60 , fixing \(K_\beta\) is equivalent to fixing \(K_{1/2}\), and \(J_\beta\) is the same positive multiple of \(J_{1/2}\).
For that constant-virtual-cost case, let \(r>0\) be the terminal positive ratio and write \(\lambda:=ar\). The long and short sequences are, respectively, \[r_i^{(k+2)}=\frac{k+1-i+\lambda}{a}\quad(1\le i\le k+1), \qquad r_i^{(k+1)}=\frac{k-i+\lambda}{a}\quad(1\le i\le k).\] For \(t\ge0\), let \((a)_t=a(a+1)\cdots(a+t-1)\), with \((a)_0=1\). Define \[\label{eq:U-app} U_N(\lambda):= \sum_{t=1}^{N}\frac{q}{t}F_N(t-1) \prod_{j=t+1}^{N}\left(1-\frac{q}{j}\right).\tag{64}\] Rejection of the threshold-zero candidate gives \(\beta U_{k+2}(\lambda)\ge1\). The summands are decreasing in \(\lambda\), and the gamma–beta calculation in Lemma 5.8 gives \[\label{eq:U-cutoff-app} U_{k+2}(q) <\sum_{j=0}^{\infty}\frac{q}{j+q}\frac{(a)_j}{j!} =q\mathrm B(q,q)=C(\alpha)\le\frac{1}{\beta}.\tag{65}\] It follows that \(\lambda<q\).
Set \[L_k(\lambda):= \left(1+\frac{a}{k+\lambda}\right)(k+1)U_{k+1}(\lambda) -(k+2)U_{k+2}(\lambda).\] Substituting the two \(U\)-expressions into 62 and collecting the common factor \(\beta/q\) gives \[\label{eq:D-L-identity} \mathcal{D}_\beta =\frac{\beta}{q}\left(\frac{q}{\beta}+L_k(\lambda)\right).\tag{66}\] To bound \(L_k\), write \(U_{k+1}(\lambda)=\sum_{t=1}^{k+1}A_t(\lambda)\), where \[A_t(\lambda):=\frac{q}{t}F_{k+1}(t-1) \prod_{j=t+1}^{k+1}\left(1-\frac{q}{j}\right).\] The constant-cost ratios give \[\frac{k+\lambda+1-t}{k+\lambda+1-t+a}A_t =\frac{t-1}{t-1+a}A_{t-1}\quad(2\le t\le k+1), \qquad \left(1+\frac{a}{k+\lambda}\right)F_{k+1}(t-1)=F_{k+2}(t).\] Substitution yields \(L_k(\lambda)=\sum_{t=1}^{k+1}w_tA_t(\lambda)\) with \[w_1=\frac{ak(k+\lambda+a)}{(1+a)(k+\lambda)}-(k+1+a), \qquad w_t=\frac{a(k+\lambda+a)(k+1-t)}{(k+\lambda)(t+a)}\quad(t\ge2).\] Because \(\lambda<q=1-a\), one has \(A_1(\lambda)=A_1(q)\), \(A_t(\lambda)\ge A_t(q)\) for \(t\ge2\), and \((k+\lambda+a)/(k+\lambda)\ge(k+1)/(k+1-a)\). Hence \[\begin{align} L_k(\lambda) &\ge \left(\frac{ak(k+1)}{(1+a)(k+1-a)}-(k+1+a)\right)A_1(q)\\ &\quad+ \sum_{t=2}^{k+1} \frac{a(k+1)(k+1-t)}{(k+1-a)(t+a)}A_t(q) =-qS_k, \end{align}\] where the recurrence for \(A_t(q)\) gives the final telescoping identity and \[S_k:=\frac{(a)_{k+1}}{(k+1)!} +\sum_{j=0}^{k}\frac{q}{j+q}\frac{(a)_j}{j!}.\] Put \(c_j=(a)_j/j!\). For \(j\ge k+1\), \[\frac{q}{j+q}c_j>\frac{q}{j+1}c_j=c_j-c_{j+1}.\] Thus the omitted infinite tail is larger than \(c_{k+1}\), the first term in \(S_k\). Integrating the binomial series \((1-t)^{-a}=\sum_{j\ge0}c_jt^j\) against \(qt^{q-1}\) gives \[\label{eq:finite-beta-sum-app} S_k<\sum_{j=0}^{\infty}\frac{q}{j+q}\frac{(a)_j}{j!} =q\mathrm B(q,q)=C(\alpha)\le\frac{1}{\beta}.\tag{67}\] Combining 66 67 gives \[\mathcal{D}_\beta \ge\beta\left(\frac{1}{\beta}-S_k\right)>0.\] If the long candidate clears, 61 and \(\mathcal{D}_\beta>0\) imply that the corresponding short residual is negative. The short execution must therefore fail or increase its boundary allocation, which produces a dominating state. If the long execution fails a virtual-cost test, weak regularity transfers the failure to the corresponding short threshold. If it exhausts all boundary candidates, the same residual comparison is negative for every corresponding short candidate, so the short execution also fails. This proves the parameterized threshold-stationary truncation result.
The nonstationary allocations are affine in \(\beta\). The following calculation exposes the cancellation in the long–short clear equations. We use the local indexing of Liu et al.: in this subsection, \(r_0>0\) denotes the first positive ratio deleted in the short execution. At the first turning point \(j+2\), let \(r_j^*\le r_j\) be the ratio at which the short execution reaches allocation one. Their equations (38) and (40), with symbolic \(\beta\), are \[\begin{align} &(1+\alpha)(j+1)(1-x_{j+1}^{(k+1)}) =\beta\prod_{i=1}^{j-1}\left(1+\frac{1}{r_i}\right) \left(\frac{1}{r_j^*}-\frac{1}{r_j}\right), \tag{68}\\ &\frac{R_{j+2}}{r_0}(1-x_{j+2}^{(k+2)}) >\beta\left(1+\frac{1}{r_0}\right) \prod_{i=1}^{j-1}\left(1+\frac{1}{r_i}\right) \left(\frac{1}{r_j^*}-\frac{1}{r_j}\right), \tag{69} \end{align}\] where \(R_b:=(b-1)(1+\alpha)-\alpha r_0\). The long execution passed the threshold-one virtual-cost test at the turning point, which gives \(R_{j+2}>0\); thereafter \(R_b\) increases by \(1+\alpha\). Subtracting the second multiple of 68 from 69 gives \[Q_{j+2}:= \frac{R_{j+2}}{r_0}(1-x_{j+2}^{(k+2)}) -\left(1+\frac{1}{r_0}\right)(1+\alpha)(j+1) (1-x_{j+1}^{(k+1)})>0\] for every \(\beta>0\).
For propagation, write \(y_b=x_b^{(k+2)}\), \(z_{b-1}=x_{b-1}^{(k+1)}\), and \(P_b=\prod_{\ell=1}^{b-2}(1+1/r_\ell)\). When the long execution raises its allocation and clears the current constraint, the two recurrences are \[\begin{align} &(b-1)(1+\alpha)(1-z_{b-1}) =((b-1)(1+\alpha)-1)(1-z_{b-2})+1-\beta P_b, \tag{70}\\ &0=-\frac{R_b}{r_0}(1-y_b) +\frac{R_b-1}{r_0}(1-y_{b-1}) +\left(1+\frac{1}{r_0}\right)(1-\beta P_b). \tag{71} \end{align}\] The identical affine terms \(1-\beta P_b\) cancel. With \[Q_b:=\frac{R_b}{r_0}(1-y_b) -\left(1+\frac{1}{r_0}\right)(b-1)(1+\alpha)(1-z_{b-1}),\] subtraction yields \[\label{eq:Q-propagation} Q_b=Q_{b-1}+\alpha\left[ \frac{1-y_{b-1}}{r_0} -\left(1+\frac{1}{r_0}\right)(1-z_{b-2}) \right].\tag{72}\] The definition of \(Q_{b-1}\) gives \[R_{b-1}\left[ \frac{1-y_{b-1}}{r_0} -\left(1+\frac{1}{r_0}\right)(1-z_{b-2}) \right] =Q_{b-1}+\alpha(r_0+1)(1-z_{b-2})>0.\] Thus \(Q_b>Q_{b-1}>0\).
For a copied long column, \(y_b=y_{b-1}\). The short threshold-zero allocations are nondecreasing, so \(1-z_{b-1}\le1-z_{b-2}\). Moreover, \[\frac{R_b}{(b-1)(1+\alpha)} =1-\frac{\alpha r_0}{(b-1)(1+\alpha)}\] is increasing in \(b\). The inequality \(Q_{b-1}>0\) therefore gives \(Q_b>0\) directly from the definition of \(Q_b\). At the terminal step, let \(\widehat H_{\rm long}\) and \(\widehat H_{\rm short}\) denote the two constraint residuals after the positive normalization used in the definition of \(Q_b\). The same subtraction gives \[\widehat H_{\rm long} -\left(1+\frac{1}{r_0}\right)\widehat H_{\rm short} =Q_{k+2}>0.\] If the long execution clears at its terminal candidate, the artificial short candidate has negative residual and the short execution must either fail or raise its boundary allocation, producing a dominating state. If the long execution fails, the same strict comparison transfers failure to the short execution. This proves both conclusions of the parameterized nonstationary-truncation proposition.
Assume that the greedy construction fails, and choose a failing global ratio vector \[(r_0,r_1,\ldots,r_m),\qquad r_0=0,\quad r_i>0\;(i\ge1),\] of minimum length. Failure occurs after the full positive tail has been processed; otherwise the prefix ending at the first failure is a shorter counterexample.
First suppose that no successful clear step before failure has positive threshold. The execution is threshold-stationary. Delete \(r_1\) and apply the threshold-stationary truncation proved above. Failure transfers to \((r_0,r_2,\ldots,r_m)\), contradicting minimality.
Now suppose a first positive-threshold clear step exists. If its threshold is at least two, call its buyer value \(k+2\). The long prefix \((r_0,\ldots,r_{k+1})\) is threshold-stationary before this step, and its output has a proper ratio suffix. Threshold-stationary truncation says that the prefix with \(r_1\) deleted either fails or produces a state dominating the long-prefix state. Failure is already a shorter counterexample. In the dominance case, append the common remaining tail \((r_{k+2},\ldots,r_m)\). State sufficiency identifies the continuation from the long-prefix state with the original failing execution, and 9(ii) transfers failure to the dominating truncated state. This again yields a shorter failing vector.
It remains that the first positive threshold equals one. Let \(k+2\) be the first later buyer value at which the execution either fails or clears at a threshold at least two. Such a value exists because the full execution fails. Every earlier threshold in this prefix is at most one. The nonstationary truncation proved above says that deleting \(r_1\) either makes the short prefix fail or makes its output dominate the long-prefix output. The first alternative is a shorter counterexample. In the second, append the common remaining tail and invoke state sufficiency and 9(ii) to transfer the eventual failure to the short execution.
Every possible first departure from threshold zero produces a shorter failing ratio vector. This contradicts minimality. The greedy construction never fails, and 9(i) proves 1.
This appendix gives the interval calculation underlying 1. The decimal parameters in 54 are interpreted as exact rationals, and all transcendental quantities are enclosed with Arb ball arithmetic at 100 decimal digits. Direct interval evaluation establishes the conditions in 7: \(0<a<1\), \(0<x<z<1\), \(a>2k_2\), \(q_0\in(0,1)\), and \(\psi(x)<1\). The remaining calculation consists of the following finite polynomial and interval operations.
Write \[h(y)=k_1y+k_2y^2, \qquad C_0=e^{-(k_1+k_2)}, \qquad Q(y)=C_0y^ae^{h(y)}.\] Then \[\begin{align} I_F&=C_0\int_0^1y^{a-1}e^{h(y)}\,dy,\notag\\ I_S&=C_0\int_0^1y^ae^{h(y)}\,dy. \end{align}\] For \(I_B\), define \[H(u):=\frac{u}{1+u}-u\log x-u\log(1+u)+u\log u.\] Using \(Q'(y)=C_0y^{a-1}e^{h(y)}A(y)\) and integrating the \(\log y\) term in closed form gives \[\label{eq:IB-series-form} I_B=C_0\int_0^1y^{a-1} \left[e^{h(y)}H(A(y))-e^{h(y)}A(y)\log y\right]dy.\tag{73}\]
Set \(L=30\). Let \(E_{L-1}(y)\) be the Taylor polynomial obtained by truncating the expansion of \(e^{h(y)}\) after order \(L-1\) in \(h(y)\). Let \(H_{L-1}(y)\) be the Taylor polynomial of \(H(a+d)\) through order \(L-1\) in \(d\), after substituting \(d=A(y)-a\). Define the finite polynomials \[P(y):=E_{L-1}(y)H_{L-1}(y)=\sum_np_ny^n, \qquad J(y):=E_{L-1}(y)A(y)=\sum_nj_ny^n.\] Their contribution to 73 is evaluated as \[C_0\left(\sum_n\frac{p_n}{a+n} +\sum_n\frac{j_n}{(a+n)^2}\right).\]
Set \(K=k_1+k_2\) and \(D=k_1+2k_2\). On \([0,1]\), the exponential remainder is bounded by \[R_E\le e^K\frac{K^L}{L!}.\] For \(L\ge2\), termwise differentiation of \(H\) gives the Taylor remainder bound \[\begin{align} R_H\le{}& \frac{D(D/a)^{L-1}}{L(L-1)(1-D/a)} +\frac{D(D/(1+a))^{L-1}}{L(L-1)(1-D/(1+a))}\notag\\ &+\frac{(D/(1+a))^L}{L(1-D/(1+a))} +\frac{(D/(1+a))^L}{(1+a)(1-D/(1+a))}. \label{eq:RH} \end{align}\tag{74}\] The exact parameters satisfy \[0<a\le A(y)\le A(1)<0.1, \qquad -\log x<4, \qquad -\log a<3.\] Thus \(|\log A(y)|\le-\log a<3\), and, for \(u=A(y)\), \[|H(u)|<0.1+0.4+0.01+0.3<1.\] Hence the uniform remainders for the two products in 73 are at most \[R_P=R_E+e^K R_H, \qquad R_J=R_E(a+D).\] After integration, the error bounds are \[|\Delta I_F|\le C_0R_E/a, \quad |\Delta I_S|\le C_0R_E/(a+1), \quad |\Delta I_B|\le C_0(R_P/a+R_J/a^2).\] For the exact parameters in 54 , outward-rounded Arb evaluation gives error bounds smaller than \(6.71\times10^{-106}\), \(5.84\times10^{-107}\), and \(1.19\times10^{-46}\), respectively. Evaluating the finite sums and propagating the resulting balls through 51 52 53 gives the enclosures in 1. Directed rounding yields \[\frac{2\mathcal{F}}{\mathcal{S}+\mathcal{B}} >2.17276852308451.\]