grad-auction-theory
比較四種拍賣形式，評估收益等價和最佳底價
中文翻譯
--- name: "grad-auction-theory" description: "運用拍賣理論比較四種經典拍賣形式，並評估收益等價。當使用者需要選擇拍賣形式、評估競標策略，或判斷收益等價在風險趨避、不對稱或相關價值之下何時失效時使用。" metadata: category: "WP-27 賽局與制度經濟" tags: ["auction-theory", "revenue-equivalence", "Vickrey-auction", "English-auction", "Dutch-auction", "first-price-sealed-bid", "winners-curse"] --- # 拍賣理論：四種經典形式與收益等價 ## 概述 拍賣理論分析在不同銷售機制下的策略性競標行為。四種經典形式（英式（遞增）、荷式（遞減）、第一價格密封競標，以及第二價格密封競標（Vickrey））在標準假設下會產生相同的預期收益。收益等價定理（Revenue Equivalence Theorem，RET）是核心的基準，一切實務上的拍賣設計決策，都來自於偏離它的假設。 ## 何時使用 - 為銷售商品、頻譜、採購或廣告版位，在各種拍賣形式之間做選擇 - 分析特定形式下的競標者策略（出價保留、狙擊出價、跳價） - 評估一個擬議的拍賣是否達到最佳收益或效率 ## 何時不要用 - 定價銷售或協商銷售，沒有競爭性的競標 - 有複雜互補性的多單位拍賣（要用組合拍賣的框架） - 賣方沒有承諾能力去落實拍賣規則 ## 假設 ``` IRON LAW: Revenue equivalence holds ONLY with risk-neutral bidders, independent private values (IPV), and symmetric bidders — violate ANY assumption and auction format matters. ``` - 只賣單一、不可分割的標的 - 競標者是風險中立的預期效用極大化者 - 價值是獨立且同分配的（IPV） - 競標者是對稱的（價值的分配相同） - 支付只是出價的函數（沒有外部性） ## 方法論 **步驟 1：找出價值模型** 分類情境：獨立私有價值（IPV）、共同價值，或相關聯價值。IPV 是指每位競標者確切知道自己的價值；共同價值是指標的物只有一個所有人都不知道的真實價值；相關聯價值則是兩者的推廣。 **步驟 2：推導均衡競標策略** 對每一種拍賣形式，求解貝氏納許均衡。在 IPV 之下的第二價格 / 英式拍賣中，出真實價值是優勢策略。在第一價格 / 荷式拍賣中，競標者會出低於真實價值的價格：b(v) = E[Y1 | Y1 < v]，其中 Y1 是最高的競爭價值。 **步驟 3：套用收益等價，或找出違反之處** 在 RET 的假設下，四種形式產生相同的預期收益。檢查是否有違反：(a) 風險趨避會讓第一價格優於第二價格；(b) 不對稱的競標者打破等價；(c) 共同價值引入贏者的詛咒，並有利於英式拍賣（Milgrom-Weber 連結原則）；(d) 預算限制、進入成本或底價會造成依形式而異的效果。 **步驟 4：建議形式與參數** 根據找出的偏離之處，建議拍賣形式、最佳底價（在 IPV 的情況下 r* 滿足 r* = v0 + [1 - F(r*)] / f(r*)），以及任何額外的設計功能（例如進場費、資訊揭露政策）。 ## 輸出格式 ```markdown ## Auction Analysis: [Context] ### Value Model - **Type**: IPV / Common / Affiliated - **Distribution**: [bidder value distribution] - **Number of bidders**: [N] - **Risk attitude**: risk-neutral / risk-averse ### Format Comparison | Format | Equilibrium Strategy | Expected Revenue | Notes | |--------------------|------------------------------|-----------------|--------------| | English (ascending) | | | | | Dutch (descending) | | | | | First-price sealed | | | | | Second-price sealed | | | | ### Revenue Equivalence Assessment - **RET holds?** Yes / No - **Violation source**: [risk aversion / asymmetry / common values / other] - **Ranking**: [which format generates highest revenue and why] ### Optimal Reserve Price - **r*** = [value] - **Derivation**: [brief] ### Recommendation [Chosen format, reserve price, and rationale] ``` ## 常見陷阱 - 贏者的詛咒是共同價值的現象，在純 IPV 的情境中不適用，但競標者常常表現得好像適用 - 收益等價講的是「預期」收益；各形式之間的變異數不同（第一價格的變異數較低） - 最佳底價會排除一些有效率的交易，賣方是犧牲效率來換取收益 - 實務上，英式拍賣會揭露較多資訊，這對相關聯價值有幫助（Milgrom-Weber 連結原則） - 第二價格和英式拍賣比較容易共謀；第一價格對競標者聯盟比較有抵抗力 - 線上拍賣（eBay 式）不是純粹的英式拍賣，固定的截止時間、代理出價和狙擊出價都會改變均衡行為 ## 參考資料 - Vickrey, W. (1961). "Counterspeculation, Auctions, and Competitive Sealed Tenders." *Journal of Finance*. - Riley, J. & Samuelson, W. (1981). "Optimal Auctions." *American Economic Review*. - Milgrom, P. & Weber, R. (1982). "A Theory of Auctions and Competitive Bidding." *Econometrica*. - Krishna, V. (2010). *Auction Theory*, 2nd ed. Academic Press.
英文原文
--- name: "grad-auction-theory" description: "Apply auction theory to compare the four canonical auction formats and assess revenue equivalence. Use this skill when the user needs to choose an auction format, evaluate bidding strategies, or determine when revenue equivalence breaks down due to risk aversion, asymmetry, or correlated values." metadata: category: "WP-27 賽局與制度經濟" tags: ["auction-theory", "revenue-equivalence", "Vickrey-auction", "English-auction", "Dutch-auction", "first-price-sealed-bid", "winners-curse"] --- # Auction Theory: Four Canonical Formats and Revenue Equivalence ## Overview Auction theory analyzes strategic bidding behavior across different selling mechanisms. The four canonical formats — English (ascending), Dutch (descending), first-price sealed-bid, and second-price sealed-bid (Vickrey) — generate identical expected revenue under standard assumptions. The Revenue Equivalence Theorem (RET) is the central benchmark; deviations from its assumptions drive all practical auction design decisions. ## When to Use - Choosing among auction formats for selling goods, spectrum, procurement, or ad slots - Analyzing bidder strategy (bid shading, sniping, jump bidding) under a specific format - Evaluating whether a proposed auction achieves optimal revenue or efficiency ## When NOT to Use - Posted-price or negotiated sales where no competitive bidding occurs - Multi-unit auctions with complex complementarities (use combinatorial auction frameworks) - The seller has no commitment power to enforce auction rules ## Assumptions ``` IRON LAW: Revenue equivalence holds ONLY with risk-neutral bidders, independent private values (IPV), and symmetric bidders — violate ANY assumption and auction format matters. ``` - Single indivisible object for sale - Bidders are risk-neutral expected-utility maximizers - Values are independently and identically distributed (IPV) - Bidders are symmetric (same distribution of values) - Payment is a function of bids only (no externalities) ## Methodology **Step 1 — Identify the Value Model** Classify the setting: independent private values (IPV), common values, or affiliated values. IPV means each bidder knows their own value with certainty; common values mean the object has one true value unknown to all; affiliated values generalize both. **Step 2 — Derive Equilibrium Bidding Strategies** For each auction format, solve for the Bayesian Nash equilibrium. In second-price / English auctions under IPV, bidding true value is dominant. In first-price / Dutch auctions, bidders shade below true value: b(v) = E[Y1 | Y1 < v] where Y1 is the highest competing value. **Step 3 — Apply Revenue Equivalence or Identify Violations** Under RET assumptions, all four formats yield the same expected revenue. Check for violations: (a) risk aversion favors first-price over second-price; (b) asymmetric bidders break equivalence; (c) common values introduce the winner's curse and favor English auctions (Milgrom-Weber linkage principle); (d) budget constraints, entry costs, or reserve prices create format-dependent effects. **Step 4 — Recommend Format and Parameters** Given the identified deviations, recommend the auction format, optimal reserve price (r* where r* = v0 + [1 - F(r*)] / f(r*) in the IPV case), and any additional design features (e.g., entry fees, information disclosure policy). ## Output Format ```markdown ## Auction Analysis: [Context] ### Value Model - **Type**: IPV / Common / Affiliated - **Distribution**: [bidder value distribution] - **Number of bidders**: [N] - **Risk attitude**: risk-neutral / risk-averse ### Format Comparison | Format | Equilibrium Strategy | Expected Revenue | Notes | |--------------------|------------------------------|-----------------|--------------| | English (ascending) | | | | | Dutch (descending) | | | | | First-price sealed | | | | | Second-price sealed | | | | ### Revenue Equivalence Assessment - **RET holds?** Yes / No - **Violation source**: [risk aversion / asymmetry / common values / other] - **Ranking**: [which format generates highest revenue and why] ### Optimal Reserve Price - **r*** = [value] - **Derivation**: [brief] ### Recommendation [Chosen format, reserve price, and rationale] ``` ## Gotchas - The winner's curse is a common-value phenomenon — it does not apply in pure IPV settings, yet bidders often behave as if it does - Revenue equivalence is about *expected* revenue; variance differs across formats (first-price has lower variance) - Optimal reserve price excludes some efficient trades — the seller sacrifices efficiency for revenue - In practice, English auctions reveal more information, which helps with affiliated values (Milgrom-Weber linkage principle) - Collusion is easier in second-price and English auctions; first-price is more robust to bidder rings - Online auctions (eBay-style) are not pure English auctions — hard close times, proxy bidding, and sniping change equilibrium behavior ## References - Vickrey, W. (1961). "Counterspeculation, Auctions, and Competitive Sealed Tenders." *Journal of Finance*. - Riley, J. & Samuelson, W. (1981). "Optimal Auctions." *American Economic Review*. - Milgrom, P. & Weber, R. (1982). "A Theory of Auctions and Competitive Bidding." *Econometrica*. - Krishna, V. (2010). *Auction Theory*, 2nd ed. Academic Press.