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A Utility Framework for Bounded-Loss Market Makers
Yiling Chen, David M Pennock
TL;DR
Prediction markets need liquidity for price discovery, but market operators also need to cap subsidized losses. The paper develops utility-based market makers that quote risk-neutral prices, establishes bounded-loss conditions and equivalences to scoring-rule and cost-function formulations, and characterizes the liquidity–loss trade-off. Its main comparison shows that fixed-loss market makers trade off liquidity near uniform versus extreme prices, while minimum liquidity imposes a lower bound on worst-case loss.
Problem
Prediction markets need sufficient liquidity for price discovery, while operators require a bound on the market maker’s possible loss.
Method
The paper develops utility-based market makers that maintain expected utility, set prices to risk-neutral probabilities, and translate among utility, scoring-rule, and cost-function formulations.
Results
For a fixed worst-case-loss bound, no market maker has uniformly higher liquidity across all price regimes; for a fixed minimum liquidity, the paper proves a lower bound on worst-case loss.
Takeaways & Limitations
Liquidity and bounded loss cannot be uniformly optimized across price regimes, so market-maker design must account for the desired liquidity region.
Takeaways & Limitations
The liquidity and loss analysis assumes symmetric, second-order differentiable cost functions and stated regularity conditions.
Abstract
from arXiv · showhide
We introduce a class of utility-based market makers that always accept orders at their risk-neutral prices. We derive necessary and sufficient conditions for such market makers to have bounded loss. We prove that hyperbolic absolute risk aversion utility market makers are equivalent to weighted pseudospherical scoring rule market makers. In particular, Hanson's logarithmic scoring rule market maker corresponds to a negative exponential utility market maker in our framework. We describe a third equivalent formulation based on maintaining a cost function that seems most natural for implementation purposes, and we illustrate how to translate among the three equivalent formulations. We examine the tradeoff between the market's liquidity and the market maker's worst-case loss. For a fixed bound on worst-case loss, some market makers exhibit greater liquidity near uniform prices and some exhibit greater liquidity near extreme prices, but no market maker can exhibit uniformly greater liquidity in all regimes. For a fixed minimum liquidity level, we give the lower bound of market maker's worst-case loss under some regularity conditions.
1 Introduction
The paper targets prediction markets where price discovery is the goal but thin trading can prevent useful information aggregation. It develops bounded-loss utility-based market makers and relates them to existing scoring-rule, cost-function, and liquidity formulations.
- Prediction markets use trading to obtain consensus probabilities or forecasts, but insufficient participation can leave them with little or no price discovery.
- Automated market makers improve liquidity by continuously quoting prices, incorporating information from solitary or asynchronously arriving traders.
- Market operators may subsidize losses to improve incentives and price discovery, but require a maximum loss bound.
- Utility-based market makers keep expected utility constant and accept infinitesimal orders at risk-neutral probabilities, which become instantaneous market prices.
- HARA utility market makers are equivalent to weighted pseudospherical scoring-rule market makers, including an equivalence between negative exponential utility and Hanson’s logarithmic scoring rule.
- The framework also translates into cost-function implementations, while the paper studies the trade-off between worst-case loss and instantaneous liquidity.
2 Background
The background describes proper scoring rules and Hanson’s market scoring rule as mechanisms for eliciting probability estimates while bounding the subsidizing patron’s loss.
- A proper scoring rule motivates truthful reporting of a probability estimate for a discrete or discretized random variable.
- Hanson’s market scoring rule market maker uses a patron subsidy to improve liquidity and overcome traders’ reluctance to participate.
- The patron’s loss is guaranteed not to exceed a fixed subsidy regardless of the number of trades or eventual outcome.
- Because traders update probability estimates sequentially, the market maker effectively pays the last trader and receives payment from the first.
3 Utility-Based Market Makers
The paper defines utility-based market makers by setting prices to risk-neutral probabilities while maintaining expected utility, then characterizes when their losses are bounded.
- Utility-based market makers set security prices equal to current risk-neutral probabilities and accept infinitesimal buy or sell orders at those prices.
- Risk-neutral probabilities combine subjective probabilities with marginal utilities, thereby incorporating the market maker’s risk adjustment.
- A trader’s purchase changes the market maker’s wealth differently across the purchased security’s payoff state and the other states.
- The market maker maintains an invariant expected-utility level rather than maximizing expected utility during trading.
- Bounded loss holds exactly when the utility domain is bounded below or its range is bounded above but not below.
- Neither linear utilities nor strictly convex utilities defined on the entire real line guarantee bounded loss.
4 Relationship Between MSR and Utility-Based Market Makers
The paper establishes equivalence between HARA utility-based market makers and weighted pseudospherical market scoring rules, including logarithmic and negative exponential special cases. It also relates these mechanisms to betting interpretations and bounded-loss conditions.
- Market Maker Equivalence: HARA utility-based market makers are behaviorally equivalent to market scoring rule makers using weighted pseudospherical scoring rules.The equivalence applies to non-linear HARA utility functions and their corresponding weighted scoring functions.
- HARA Utility Class: HARA utility includes CARA and CRRA families, with logarithmic and negative exponential utilities arising as limiting cases.The HARA form has linear absolute risk tolerance; γ → 1 gives logarithmic utility, while γ → ±∞ gives negative exponential utility.
- Bounded Loss: All non-linear HARA utility functions satisfy the paper’s bounded-loss condition, whereas linear utility does not guarantee bounded loss.Theorem 2 gives necessary and sufficient utility-domain or utility-range conditions for bounded loss.
- Weighted Pseudospherical Rules: Weighted pseudospherical scoring rules generalize pseudospherical rules through baseline-probability weighting and include logarithmic and spherical special cases.Uniform baseline probabilities yield the unweighted logarithmic rule when all baseline probabilities are equal, while β = 2 yields the spherical rule.
- Scope of Equivalence: Quadratic scoring is outside the weighted pseudospherical class, while weighted pseudospherical scoring rules give bounded MSR loss because their true-outcome payments are finite.The bounded-payment property applies to every weighted pseudospherical scoring rule described here.
- Betting Interpretation: Trades with a utility-based market maker can be viewed as two-person bets, enabling the equivalence proof through a trader’s probability estimate and the maker’s wealth vector.The trader selects a new market-maker wealth vector subject to the maker’s expected-utility constraint.
- Special Cases: A negative exponential utility market maker is equivalent to a weighted logarithmic scoring-rule market maker, and with uniform priors to Hanson’s logarithmic MSR maker.The logarithmic utility case corresponds to another scoring rule specified in the paper.
5 Cost-Function Formulation of Market Makers
The paper presents cost functions as a third formulation connecting utility-based and market-scoring-rule market makers. These functions encode trader spending and share quantities, support price derivation by differentiation, and provide a practical implementation route.
- 5.1 Cost-Function Formulation: A cost function C(q) records total trader spending as a function of outstanding security quantities, and trades update q while charging the change in C.The market starts from an initial quantity vector and charges traders according to the new quantity vector.
- 5.1 Cost-Function Formulation: Arbitrage-free cost-function markets keep security prices in [0, 1] and require their sum to equal 1.These restrictions are stated as the pricing conditions for the cost-function formulation.
- 5.2 Cost Functions and Utility-Based Market Makers: Every utility-based market maker can be translated into a cost-function formulation defined by the paper’s Theorem 6.The cost function reflects money collected from traders and payments owed if each outcome occurs.
- 5.2 Cost Functions and Utility-Based Market Makers: Under continuity, differentiability, and strict monotonicity of utility, the cost function is unique; numerical methods can evaluate it when no explicit form exists.Prices are obtained as partial derivatives of the cost function.
- 5.2 Cost Functions and Utility-Based Market Makers: Logarithmic and negative exponential utilities, both in HARA, yield corresponding cost functions, with an explicit form available for the two-outcome uniform-prior logarithmic case.The negative exponential cost function is also specified in the paper, while price functions follow by differentiation.
- 5.3 Cost Functions and Market Scoring Rules: Weighted pseudospherical scoring-rule makers translate to cost functions through their equivalent utility-based makers, while quadratic scoring requires direct translation.The paper derives the logarithmic MSR cost function by equating trader profits across outcomes and shows its equivalence to the negative exponential formulation under uniform priors.
- 5.3 Cost Functions and Market Scoring Rules: The three formulations provide navigation among utility-based, market-scoring-rule, and cost-function makers, with cost functions presented as the most natural implementation method.MSRs simplify loss analysis, whereas utility-based makers make cost functions easier to obtain for a broad scoring-rule class.
6 Liquidity and Market Maker Loss
The section defines instantaneous liquidity for differentiable market makers and analyzes its trade-off with worst-case loss. Under symmetric, second-order differentiable cost functions, it establishes both an impossibility result for uniformly superior liquidity and a lower bound on loss at a required liquidity level.
- Liquidity: Instantaneous liquidity extends bid-ask and market-depth concepts to continuous, differentiable price functions.It is defined from the slope of the price function, while the slope also approximates the bid-ask spread for small trades.
- Worst-case loss: Under symmetric, second-order differentiable cost functions, worst-case loss can be analyzed through a univariate price function for one outcome.The loss equals the area bounded by the y-axis, p_i = 1, and the price curve in the first quadrant.
- Liquidity-loss trade-off: No market maker can have uniformly greater or equal instantaneous liquidity across all quantity vectors when worst-case loss is fixed.For two-outcome markets, neither compared price function consistently has the smaller slope.
- Lower bound: For N outcomes and minimum instantaneous liquidity ρ, the worst-case loss is bounded below by (N−1)^2ρ under the stated regularity conditions.The bound applies to symmetric, second-order differentiable cost functions and specializes to the two-outcome result associated with Schwarz’s step-wise formulation.
7 Conclusion
The paper introduces utility-based market makers that price orders at risk-neutral probabilities and connects them to scoring-rule and cost-function formulations. It concludes that bounded loss and liquidity involve explicit structural conditions and an unavoidable trade-off across price regimes.
- Utility-based market makers accept orders at risk-neutral prices, with bounded loss characterized by necessary and sufficient conditions.
- HARA utility market makers translate into weighted pseudospherical scoring-rule market makers, while cost functions provide a third equivalent formulation.The paper gives translation procedures and derives cost and price functions for logarithmic and quadratic scoring rules.
- With a fixed worst-case-loss bound, liquidity can be higher near uniform or extreme prices, but no market maker dominates uniformly across regimes.For a fixed minimum liquidity level, the paper also proves a lower bound on worst-case loss.