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GPT 6 Astra Ultra attempt on Econ-35

Exploration · LLM-generated · gpt 6 astra ultra ·

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Contents
  1. Finite posted-price guarantees with correlation, risk, and liquidity
  2. Target and restricted domain
  3. A guarantee against every sequential equilibrium
  4. A matching upper bound for a one-buyer ratio
  5. References
  6. Completion Estimate

Finite posted-price guarantees with correlation, risk, and liquidity

Target and restricted domain

The catalogue target is a broad research family: maximize worst-case revenue across environments and sequential equilibria when correlation, interdependence, risk, budgets, and entry may coexist. We study a precise subfamily with a sharp finite-support benchmark. There is one object, zero seller cost, no entry or information-acquisition cost, and $n$ bidders with finite private types. A type specifies a monetary private value $v_i\geq0$, a budget $b_i\geq0$, and a strictly increasing utility function $U_i$ of monetary surplus. Values are private, but their joint law, budgets, and risk preferences can be arbitrarily correlated. Define willingness $w_i=\min\{v_i,b_i\}$ and $W=\max_iw_i$.

Fix a known finite grid $0<z_1<\cdots<z_m$ containing every positive willingness in every environment; $w_i=0$ is allowed. Set $z_0=0$ and \[ H=\sum_{k=1}^m\frac{z_k-z_{k-1}}{z_k},\qquad \alpha_k=\frac{z_k-z_{k-1}}{H z_k}. \tag{1} \] Thus $(\alpha_k)$ is a probability vector. Benchmark mechanisms are finite extensive forms with commitment, realized nonnegative payments bounded by budgets, and ex-post individual rationality in every realized allocation/payment outcome. An allocated bidder has monetary surplus $v_i-t_i$, a nonrecipient has $-t_i$, and ex-post participation means utility at least $U_i(0)$. Random allocations are permitted but participation applies outcome by outcome. These are stronger restrictions than the target's possible interim participation convention; both the proposed auction and the benchmark obey the same restrictions.

Myerson's characterization concerns a different specified informational domain [Myerson]. Prior-free randomized pricing and competitive-ratio analysis have established antecedents [BS,EM]. We do not assert novelty for the harmonic-support calculation. Its role here is to give complete finite proofs, retain adverse equilibrium selection, and specify exactly which extensions are covered.

A guarantee against every sequential equilibrium

Fix an order of bidders. Draw $k$ publicly with probabilities $\alpha_k$, announce price $p_k=(1-\eta)z_k$ for $0<\eta<1$, and offer the object once to each bidder in that order until someone accepts. A bidder never receives another offer after rejecting. Refusal costs zero; acceptance is feasible only within the bidder's budget. This is a finite extensive-form mechanism.
Theorem (Pointwise revenue guarantee): For every environment in the restricted domain, every type profile in its support, and every sequential equilibrium of the posted-price auction, expected revenue conditional on that profile satisfies \[ {\mathbb{E}}[\mathrm{Rev}\mid(v_i,b_i,U_i)_i]\geq\frac{1-\eta}{H}W. \tag{2} \] Every admissible benchmark mechanism, under every equilibrium satisfying the specified ex-post participation requirement, has realized revenue at most $W$. Consequently, if $V^*$ denotes the catalogue's robust maximin revenue over this same benchmark class, the posted-price auction earns worst-case revenue at least $(1-\eta)V^*/H$. It also attains this factor relative to the optimal revenue for each fixed environment. The approximation ratio is asserted only when the comparison benchmark is positive.
Proof: An offered bidder with $p<w_i$ can afford the price and obtains strictly positive monetary surplus from acceptance. Strict increase of $U_i$ makes acceptance strictly preferable to rejection, whose payoff is zero because there is no later offer. This comparison is independent of all beliefs about other bidders and of the shape of $U_i$. Sequential rationality therefore forces acceptance at every reached information set with $p<w_i$.

If $p<W$, take the first bidder with $w_i>p$. Either a preceding bidder already accepted, producing revenue $p$, or this bidder receives the offer and accepts. In either case the seller receives $p$ with probability one. If $W=z_j$, every $k\leq j$ has $p_k<z_k\leq W$. Revenue is nonnegative at other prices, so its conditional expectation is at least \[ \sum_{k\leq j}\alpha_k(1-\eta)z_k =\frac{1-\eta}{H}\sum_{k\leq j}(z_k-z_{k-1}) =\frac{1-\eta}{H}z_j. \] If $W=0$, nonnegative revenue gives (2) directly. Indifference at other prices cannot reduce this lower bound.

For the upper bound, an unallocated bidder's ex-post participation and strict increase of $U_i$ require $-t_i\geq0$. Together with nonnegative payments this gives $t_i=0$. The sole allocated bidder, if any, must have $t_i\leq v_i$ and $t_i\leq b_i$. Thus total revenue is at most $W$ outcome by outcome, even for randomized mechanisms.

Integrating (2), and taking the infimum over equilibria and environments, gives at least $(1-\eta)\inf_\omega{\mathbb{E}}_\omega W/H$. Every benchmark mechanism has worst-case revenue at most $\inf_\omega{\mathbb{E}}_\omega W$, so this proves the robust maximin comparison. Applying the same upper bound separately to each environment proves the environmentwise comparison. The finite posted-price game has sequential equilibria: backward induction at each offer prescribes acceptance, refusal, or any mixture at a tie, and the finite tree permits consistent beliefs from completely mixed trembles. Thus the equilibrium infima used here are nonempty. ∎

Proposition (Size of the guarantee): For the grid in (1), \[ 1\leq H\leq m,\qquad H\leq1+\log(z_m/z_1). \] The upper logarithmic bound is interpreted as $H=1$ when $m=1$.
Proof: The first summand of $H$ equals one and every subsequent summand lies in $(0,1)$, proving the first inequalities. For $k\geq2$, $(z_k-z_{k-1})/z_k\leq\int_{z_{k-1}}^{z_k}dx/x$ because $1/x\geq1/z_k$ throughout the interval. Sum these inequalities. ∎

A matching upper bound for a one-buyer ratio

The robust maximin revenue and a ratio to each environment's optimum are different objectives. The following sharpness result concerns the latter. Consider one risk-neutral buyer, budget at least $z_m$, and possible values $z_1,\ldots,z_m$. The ambiguity set contains the point-mass environment at each value. In the point-mass environment $j$, the supremum of revenue under adverse equilibrium selection is $z_j$: prices strictly below $z_j$ approach it and participation bounds revenue above by it.
Lemma (A finite menu inequality): Suppose a finite menu of expected allocations $x_j\in[0,1]$ and payments $t_j$ satisfies \[ z_jx_j-t_j\geq0,\qquad z_jx_j-t_j\geq z_jx_k-t_k\quad\text{for all }j,k. \] Then \[ \sum_{j<m}\left(\frac1{z_j}-\frac1{z_{j+1}}\right)t_j +\frac{t_m}{z_m}\leq1, \qquad \min_j\frac{t_j}{z_j}\leq\frac1H. \tag{3} \]
Proof: Participation for type 1 gives $t_1/z_1\leq x_1$. Incentive compatibility for type $j\geq2$ against report $j-1$ gives $(t_j-t_{j-1})/z_j\leq x_j-x_{j-1}$. Adding yields \[ \frac{t_1}{z_1}+\sum_{j=2}^m\frac{t_j-t_{j-1}}{z_j}\leq x_m\leq1. \] Collecting the coefficients of $t_j$ proves the first inequality. They are positive, and their scalar product with $(z_j)_j$ equals $1+\sum_{j=2}^m(z_j-z_{j-1})/z_j=H$. If every $t_j/z_j$ exceeded $1/H$, the first inequality would be violated. The weak conclusion follows. ∎
Theorem (Sharp supremal competitive ratio): Among all finite committed one-buyer mechanisms in this subfamily, the supremum guaranteed ratio of revenue to the corresponding point-mass optimum, with worst-case sequential-equilibrium selection, is exactly $1/H$.
Proof: Fix a mechanism. For each value $z_j$, choose any sequential-equilibrium buyer strategy and denote its expected allocation/payment by $(x_j,t_j)$. Because there is only one strategic buyer and the mechanism is committed, a buyer of value $z_j$ can mimic the entire strategy chosen by value $z_k$. The random mechanism has the same distribution under that mimicry. Root optimality and participation imply exactly the inequalities of Lemma Lemma (A finite menu inequality). Thus for some $j$ this chosen equilibrium has revenue at most $z_j/H$. The worst equilibrium in that environment has no higher revenue, proving the upper bound. Theorem Theorem (Pointwise revenue guarantee), for every $\eta>0$, proves the matching lower bound in the supremum as $\eta\downarrow0$. ∎

For the grid $(1,2)$, $H=3/2$ and (1) assigns probabilities $2/3$ and $1/3$. Prices just below one and two approach a $2/3$ guarantee for each value. Pricing at the grid points without a tie convention need not attain the supremum: the lowest type can reject an offer exactly equal to its value. The strict discount in the finite mechanism addresses this issue explicitly.

Unresolved boundary. The theorem handles arbitrary correlation, private hard budgets, and arbitrary strictly increasing utility, under ex-post participation and free participation. It does not handle interdependent values, costly entry, information acquisition, or the larger interim-participation benchmark. The first missing step is a replacement for the pointwise willingness and payment upper bound in that broader class: cross-state insurance, loser payments, and entry incentives can invalidate the bound. No universal optimum for the catalogue family is claimed. All results above are analytic; no numerical search or empirical verification is asserted.

References

Completion Estimate

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