Connections

The catalog reads like a list. It is closer to a few facts of convex analysis wearing different institutional clothes. Four of them tie the repository together. The map shows the edges; this page states the theorems.

1. Convex duality

Fix outcomes $\{1,\dots,n\}$ and a convex function $G$ on the probability simplex. Savage's characterisation says a strictly proper scoring rule is exactly

$S(p,i) = G(p) + \langle\, \nabla G(p),\; e_i - p\,\rangle,$

and the edge a truthful forecaster $q$ holds over any lie $p$ is the Bregman divergence $D_G(q,p)\ge 0$ of that same $G$ (Savage 1971; Banerjee et al. 2005; Gneiting & Raftery 2007). Take the convex conjugate: $C = G^\ast$ is a cost-function market maker with prices $\nabla C$, arbitrage-free because $C$ is convex, and worst-case loss equal to the range of the conjugate. Hanson's LMSR is the entropic case $C(q)=b\log\sum_i e^{q_i/b}$ with loss $b\log n$ (Hanson 2007; Abernethy, Chen & Wortman Vaughan 2013).

The involution $C \leftrightarrow C^\ast$ is the maker-to-CFMM duality: Uniswap and Balancer are the same potential read from the reserve side, not a different mechanism (Frongillo, Papireddygari & Waggoner 2024). And the same duality reaches the order book: a call or batch auction's uniform clearing price is the dual variable of “maximise matched volume,” and a parimutuel is a convex program whose dual yields the implied odds (Agrawal et al. 2011). So scoring rules, market makers, CFMMs, batch auctions, and parimutuels are conjugate views of one convex potential.

2. Infimal convolution is composition

The operation that combines two of these objects is the infimal convolution

$(f \mathbin{\square} g)(x) = \inf_{y}\ \{\, f(y) + g(x - y) \,\},$

whose one essential property is that it is dual to addition: $(f \mathbin{\square} g)^\ast = f^\ast + g^\ast$ (Rockafellar 1970; Moreau 1965). Summing in one space is inf-convolving in the other. That single identity shows up as the composition law in two literatures that never cite each other:

Markets. Merge two cost-function makers and the valid trades are exactly the infimal convolution of their cost functions; dually their conjugate scoring-rule generators add, so liquidity (the Hessian of the conjugate) is additive (Bhaskara, Frongillo & Papireddygari 2023, arXiv:2311.08725).
Risk sharing. The market aggregate of several agents' convex risk measures is their infimal convolution; the minimiser is the Pareto-optimal allocation and the agents' common subgradient is the clearing price (Barrieu & El Karoui 2005; Jouini, Schachermayer & Touzi 2008).

Same operation, three readings: pooling liquidity, sharing risk, combining agents. The hybrid CLOB+AMM and the combinatorial market are both instances of merging makers this way. Inf-convolution with a quadratic is the Moreau envelope, whose non-Euclidean form is the mirror map of the next section.

3. Exponential weights is the engine

Three mechanisms in three different clusters are one update in disguise, mirror descent with the negative-entropy mirror:

LMSR prices by Follow-the-Regularised-Leader with the entropic regulariser; the liquidity $b$ is the learning rate and bounded loss is the regret bound (Abernethy, Chen & Wortman Vaughan 2013).
The logarithmic opinion pool is the normalised geometric mean, the same multiplicative (Bayesian) mixture; the wealth-weighted pool update $w \leftarrow w\cdot p(y)/\pi$ is the Hedge update.
Kelly sizing maximises log-wealth growth $\sum_i p_i\log(p_i/\pi_i) = \mathrm{KL}(p\,\Vert\,\pi)$, which is exactly the regret of the log score, by that same multiplicative step.

The softmax that prices LMSR is $\nabla$ of the log-sum-exp, the gradient of the conjugate of entropy. The market maker, the aggregator, and the sizing rule are reading the same object off different faces.

4. The scoring substrate

Underneath all of it is the dictionary proper scoring rule ↔ convex generator ↔ Bregman divergence (Savage 1971; Banerjee et al. 2005). The three classics are three choices of $G$:

$\text{log} \to \mathrm{KL},\qquad \text{Brier} \to \lVert q-p\rVert_2^2,\qquad \text{spherical} \to \text{angular separation.}$

Each then has a market-maker twin. The log score is LMSR; the Brier (quadratic) score is a quadratic-potential maker, the Euclidean sibling of LMSR's entropic one (the family cmm.py's quadratic_potential belongs to). Further links:

The energy score and CRPS are kernel scoring rules: energy distance equals maximum mean discrepancy with the distance kernel (Sejdinovic et al. 2013), which is why “submit a bag of samples” is well founded, it is an MMD estimator.
Peer prediction is the same machinery with a peer's report standing in for ground truth: the multi-task mechanisms maximise a Bregman / $f$-mutual-information between reports (Kong & Schoenebeck 2019).
The object the Hyvärinen score evaluates is the marginal score $\nabla\log p(y)$, and Tweedie's formula $\;\mathbb{E}[\theta \mid y] = y + \sigma^2\,\nabla\log p(y)\;$ (Efron 2011) is the exact posterior-mean identity for that same gradient. Hansen & Tong (2026) use it to show that the score-driven filters of econometrics (Creal, Koopman & Lucas 2013), which step a latent parameter along the conditional likelihood score, are exact Bayesian updates in expectation space under local precision discounting, and the leading approximation otherwise.

Companion: the relationship map (the same structure as a graph). Worked pipeline and proofs: An Algebra of Prediction-Rewarding Mechanisms. Sources: bibliography.