Pennock 2004 / Hanson 2007 / Chen-Pennock 2010 / Tech Buzz Game 実証 / LS-LMSR / Manifold mechanism 移行履歴 まで原典PDFを精読。
UGCにDPMを採用する場合の学術的根拠と限界を、二次解説に頼らず一次情報で固めた版。
"To my knowledge, a DPM is the only known mechanism for hedging and speculating that exhibits all three of the following properties: (1) guaranteed liquidity, (2) no risk for the market institution, and (3) continuous incorporation of information. A standard pari-mutuel fails (3). A CDA fails (1). A CDAwMM, the bookmaker mechanism, and an MSR all fail (2)." — Pennock 2004, §3.2
| 機構 | (1) 流動性 | (2) 運営risk無 | (3) 情報連続反映 |
|---|---|---|---|
| Pari-mutuel | ✓ | ✓ | ✗ |
| CDA | ✗ | ✓ | ✓ |
| CDAwMM / Bookmaker | ✓ | ✗ | ✓ |
| MSR (LMSR) | ✓ | △ bounded | ✓ |
| DPM | ✓ | ✓ | ✓ |
| 種別 | 再分配ルール | 払戻 P_i | Market probability |
|---|---|---|---|
| DPM I | 勝者の元金は返金、負け金のみ再分配 | P₁ = M₂ / N₁ | MPr(A) = M₁N₁ / (M₁N₁ + M₂N₂) |
| DPM II | 全額（勝者元金含む）再分配 | P₁ = T / N₁ | MPr(A) = √(M₁N₁) / (√(M₁N₁) + √(M₂N₂)) |
M₁=YESに賭けられた金額、N₁=YES発行株数、T=M₁+M₂
"We make a critical assumption in order to greatly simplify the analysis; we assume that E[P₁|A] = P₁. ... I conjecture that there are reasonable market efficiency conditions under which assumption (3) is true, though I have not been able to prove that it arises naturally from rational trading." — Pennock 2004, §4.1
つまりPennock自身が「真の確率を表出する根拠を証明できていない」と認めている。これがDPMが LMSR に学術的に劣る根本原因。
"Because initial prices paid are not refunded for winning bets, there is a chance that, if prices swing wildly enough, a wager on the correct outcome might actually lose money. Traders must be aware that if they buy in at an excessively high price that later tumbles allowing many others to get in at a much lower price, they may lose money in the end regardless of the outcome." — Pennock 2004, §5.3
→ これがManifoldが2022年3月にDPMを廃止した直接の原因。
"While there is always a market maker willing to accept buy orders, there is not a market maker accepting sell orders, and thus no guaranteed liquidity for selling: instead, selling is accomplished via a standard CDA mechanism." — Pennock 2004, §7
s_i(r) = a_i + b · log(r_i)m_i(x) = exp((-a_i - x_i) / b) / Σ_j exp((-a_j - x_j) / b)C(q) = b · log(Σ_i exp(q_i / b))"For the logarithmic scoring rule, this maximum expected payment is the entropy, −b·Σ_i π_i log(π_i), of the initial distribution π." — Hanson 2007, "Costs of Market Scoring Rules"
最大損失 = b · log(I)（I = outcome数、初期 uniform の場合）
"For I ≥ 3, if y_i = 0 for i ∉ {j, k} implies q_i = 0 for i ∉ {j, k}, the rule is logarithmic." — Hanson 2007, Theorem 2
→ 対数版だけが「条件付確率の不変性」を満たす唯一の rule。「AをBの条件下で賭ける時、p(B)が変化しないでほしい」という性質を保証するのは log のみ。これがLMSRが「事実上の標準」になった数学的根拠。
"Simple scoring rules do not induce different individuals to form common estimates, while simple betting markets cannot create price estimates unless several people coordinate to bet on the same event, and it typically seems irrational to participate. Market scoring rules, in contrast, act like simple scoring rules when one person estimates an event once, yet can also act like a subsidized betting market with which many people can and rationally should repeatedly interact to produce a common estimate." — Hanson 2007, Introduction
"LMSR has become the de facto market maker mechanism for prediction markets. It is used by many companies including Inkling Markets, Consensus Point, Yahoo! and Microsoft." — Chen & Pennock 2010
"Setting the value of b, often called the liquidity parameter, in LMSR is more art than science in practice. ... If b is too small, the price of a contract changes dramatically after a small number of shares is traded. If b is too large, the price of a contract barely moves even with a large volume of trades." — Chen & Pennock 2010
"Market scoring rules and most cost function based market makers are myopically incentive compatible – a risk-neutral agent will report its probability truthfully if it only participates once. But because an agent can potentially influence other agents by its trading action and can trade more than once, a forward-looking agent may lie about its information to mislead other agents ('bluff') with the hope to obtain greater profit by correcting their mistakes later." — Chen & Pennock 2010
"For DPM, the loss of the market maker is bounded by its initial subsidy as the market is parimutuel." — Chen & Pennock 2010
C(q) = √(Σ q_j²)
p_i(q) = q_i / √(Σ q_j²)
o_i(q^f) = C(q^f) / q_i^f (winning payout)→ surveyは L2-norm DPM を「DPMの commonly used 形」として紹介。Tech Buzz Game でも L2-norm DPM が採用された。
"The contest period initially used the DPM money-ratio price function ... However, this price function is not arbitrage-free. The arbitrage opportunities were exploited by market participants in the second week of the contest, causing prices of all stocks in some markets quickly drop toward zero. The contest was paused and reopened on April 1, 2005 with the share-ratio price function defined in (2), which is arbitrage-free." — Chen, Pennock, Kasturi 2008
C(q) = κ · √(Σ q_j²) (cost function)
p_i(q) = κ · q_i / √(Σ q_j²) (price function)
π_i(q) = q_i² / Σ q_j² (market probability)
o_i(q^f) = κ · √(Σ q_j²) / q_i^f (winning payout)"A trader who wagers on the correct outcome is guaranteed non-negative profit in DPM, because p_i is always less than or equal to κ and o_i is always greater than or equal to κ for the true outcome. ... the market maker's loss is at most C(q^0) whichever outcome is realized." — Chen, Pennock, Kasturi 2008
→ ミライマで言うと、初期seed q^0 が小さければ運営損失は実質ゼロにできる。
| 絶対誤差 |b̂_i − b^f_i| | stocks数 |
|---|---|
| [0, 10) | 162（53%） |
| [10, 20) | 53 |
| [20, 30) | 32 |
| [30, 40) | 21 |
| [40, 50) | 16 |
| [50, 100] | 21 |
→ 305 stocks中 162 (53%) が誤差10未満。中程度の予測力。
| カテゴリ | 人数 | 平均純利益 |
|---|---|---|
| Dishonest（複数アカSybil） | 175 | +$68,567 |
| Other（正直な人間） | 4,644 | −$661 |
| Robot（ランダム） | 100 | −$11,615 |
Sybil攻撃の成立メカニズム：「accounts A and B 開設、A→Bへ price推進buy、B→Aへ売却で価格上げsell」を繰り返すと、価格動的調整によって money 移転が成立する。
ミライマ含意：DPM は本質的に bot/Sybil による self-pumping が起きやすい構造。ミライマで bot 検出を強化する必要性が裏付けられた（既に v4.1 ルール導入済み）。
b(x) = α · Σ x_i (liquidity parameter, dynamic)
C(x) = b(x) · log(Σ exp(x_i / b(x)))"Unlike the LMSR, the OPRS is only defined over the non-negative orthant ... Also unlike the LMSR, the sum of prices in the OPRS is always greater than 1." — Othman & Sandholm 2011
価格合計 > 1 = 「vigorish（運営手数料）」が built-in。typical values: 1〜20%
LS-LMSR は bounded loss を維持。ただし「初期 liquidity」のみ bounded、最大 liquidity は unbounded（参加者が拠出するため）。
"You don't know what your payout is going to be at the time you place your bet. Worse, that payout can be much lower than the odds that you should be getting at the time you place your bet." — Manifold blog "Above the Fold: Market Mechanics"
"Alice expects a 4x-5x return based on moving odds from 20% to 25%, but ultimately receives only 2.1x." — Kevin Zielnicki, 2022/2/17
k = y^p · n^(1-p)（y=YES pool、n=NO pool、p=initial probability）
→ 「Pennock の DPM II が予言した『正解しても損する』問題」が Manifold で実証され、市場 mechanism として捨てられた。
Share prices = ratio of funds invested。具体的な multiplier 曲線は公開ドキュメントには無し（実装は trading_dppm.rs の中）。
→ Tech Buzz Game の L2-norm DPM そのまま + 早期参加ブースト。Manifoldが捨てた DPM を crypto版で復活させた格好。
→ DPM/LMSR どちらでもなく、伝統的 CLOB（NASDAQ/NYSE型）。dApp 化に最適化。
"Augur originally was LMSR or LS-LMSR based but it is no longer using LMSR or LS-LMSR due to gas costs. ... In Augur V2, it was updated to an on-chain orderbook powered by 0x Mesh and in the meantime, it uses CF instead of CAD trading mechanism for price determination." — Decentralized Prediction Markets report (Roughgarden FoB seminar)
教訓: 学術的に最強の LMSR/LS-LMSR でも、実装コスト（gas）に負けて CLOB に逃げた。
→ DPM ですらない、Pure Parimutuel。「価格発見」を完全に放棄して、pool ratio のみ表示。
メカニズム: "Virtual Specialist Technology" は automated market maker系（CDA派生）。LMSRやDPMとは独立に開発された patented system。
→ play-money market が 30年機能しているという生存実例。学術的議論を超えて「実装が続いていれば成立する」証拠。
"For DPM, the loss of the market maker is bounded by its initial subsidy as the market is parimutuel."
"I have not been able to prove that it arises naturally from rational trading."DPMの市場価格は noisy probability estimator にとどまる。LMSRが strict proper scoring rule であるのに対し、DPMはそうではない。
"If prices swing wildly enough, a wager on the correct outcome might actually lose money."Manifold が 2022/3/15 にDPM廃止した直接原因。ミライマの「正解＝確定利益」UX原則と直接矛盾。
C(q) = κ · √(qY² + qN²)
p_i(q) = κ · q_i / √(qY² + qN²)
π_i(q) = q_i² / (qY² + qN²)
payout o_i = κ · √(qY² + qN²) / q_i (winning outcome)| 機構 | 運営loss | UGC適合 | 一次根拠 |
|---|---|---|---|
| LMSR | b·log(n) per market | ✗ | Hanson 2007: 「subsidy が必要」と明記 |
| LS-LMSR | initial b のみ bounded | △ | Othman 2011: 「α tuning is not prescriptive」 |
| DPM (L2) | initial seed のみ | ✓ | Chen-Pennock 2010: 「bounded by initial subsidy」 |
| Pure parimutuel | 0 | ✓ （だが価格発見なし） | Twitch 実装 |
| CLOB | 0 | ✗ （matching必須） | Polymarket 実装、UGCには不向き |
結論: ミライマUGCには L2-norm DPM が学術的・実装的に最適解。LMSRは運営市場（既存b=61）専用継続、UGCには使わない。
"Random walk conjecture. The most important question mark in my mind is whether the random walk assumption (3) can be proven under reasonable market efficiency conditions and, if not, how severely it effects the practicality of the system."
"Incentive analysis. Formally, what are the incentives for traders to act on new information and when?" — Pennock 2004, §9 "Future Work"
→ 22年経過した今も未解決。ミライマでは「実用上機能すれば良い」立場で進めるのが現実的（HSXが30年機能している実績がそれを支える）。
/tmp/dpm_research/
├── pennock_orig.pdf (Pennock 2004 DPM原論文 138KB)
├── hanson_mktscore.pdf (Hanson 2007 LMSR 149KB)
├── hanson_combobet.pdf (Hanson 2003 137KB)
├── pennock_dpm_v2.pdf (Tech Buzz Game empirical 147KB)
├── chen_pennock_survey.pdf (Chen-Pennock 2010 survey 253KB)
├── othman_lslmsr.pdf (Othman-Sandholm 2011 165KB)
├── othman_practical.pdf (Practical LS-LMSR 2.5MB)
└── decentralized_pm.pdf (Roughgarden FoB seminar 1.6MB)各 .txt 変換済み（pdftotext）も同フォルダにあり、grep で再検索可能。