Market Scoring Rules Act As Opinion Pools For Risk-Averse Agents
A market scoring rule (MSR) -a popular tool for designing algorithmic prediction markets -is an incentive-compatible mechanism for the aggregation of probabilistic beliefs from myopic risk-neutral agents. In this paper, we add to a growing body of research aimed at understanding the precise manner in which the price process induced by a MSR incorporates private information from agents who deviate from the assumption of risk-neutrality. We first establish that, for a myopic trading agent with a risk-averse utility function, a MSR satisfying mild regularity conditions elicits the agent's risk-neutral probability conditional on the latest market state rather than her true subjective probability. Hence, we show that a MSR under these conditions effectively behaves like a more traditional method of belief aggregation, namely an opinion pool, for agents' true probabilities. In particular, the logarithmic market scoring rule acts as a logarithmic pool for constant absolute risk aversion utility agents, and as a linear pool for an atypical budgetconstrained agent utility with decreasing absolute risk aversion. We also point out the interpretation of a market maker under these conditions as a Bayesian learner even when agent beliefs are static.
Summary of the paper:The paper aims to address the question of how to combine beliefs to a central judgement in the case that the beliefs held by agents have an information asymmetry in each agents perspective on the future outcomes of particular stochastic events.In particular the paper particularly focuses on the concept of Hanson 2003 known as the prediction market based on a logarithmic market scoring rule. The market scoring rule framework is sometimes also known as a market studied in the context of cost function based market makers. The motivation provided for the MSR setting involves financial markets in which market makers or dedicated specialists are provided incentives, typically in the form of rebates to take trades on both sides of the book ie. the place limit and market orders in such a way that they create liquidity in the market to facilitate buying and selling of an asset. In this context it is argued that a MSR effectively acts as a cost function-based market maker always willing to take the other side of a trade with any willing buyer or seller, and re-adjusting its quoted price after every transaction.The key results of this paper is to develop a new unified understanding of the information aggregation characteristics of a market with risk-averse agents mediated by a MSR, with no regard to how the agents' beliefs are formed. Furthermore, the authors are able to demonstrate that the MSR adjusted market is equivalent to an opinion pool.The main result from a technical perspective are provided in Theorems 2 and 3. Theorem 2 applies to the setting of myopic agents with utility functions of a particular form given by the CARA constant absolute risk aversion form that produce under certain restrictions the equivalence between the markets updated price and the result formed by a logorithmic opinion pool between the current price and the agents subjective beliefs. Note - no budget constraints were utilised - please comment on the implications of this. Theorem 3 applies to the case of decreasing absolute risk aversion for the myopic agents utility.In both cases Bayesian formulations are provided which produces a nice link with the Machine learning literature. Quality:The paper is a high quality paper with an excellent exposition of existing results and newly developed results. It attacks an existing class of problems in an interesting and non-trivial manner, whilst managing to present the results clearly and succinctly.Clarity:The paper is well written and clear. The literature review and contextual background to the work is well developed. All results are clearly defined and explained in detail.Originality:Distinct from other frameworks and existing results in behavioral finance the results developed in this paper do not presuppose any kind of generative model for agent signals, and also do not involve an equilibrium analysis. Consequently, this means that they may be used as tools to analyze the convergence characteristics of the market price in non-equilibrium situations with potentially fixed-belief or irrational agentsIn particular the work takes the approach that they consider the characteristics of the evolution of the price process instead of the properties of prices in equilibrium. In particular, examining the way in which the microstructure noise induces aggregation even when the agents are not considered in a Bayesian manner.By construction an MSR will elicit the subjective probabilities of a risk-neutral agent in an incentive compatible manner but in this context the author show that generally MSR's will elicit risk-neutral probabilities when they interact with risk-averse agents.Significance:The paper studies a well established trading environment microstructure framework in order to establish relationships between traditional belief aggregation methods such as opinion pools and Bayesian inference in two cases that are important from the perspective of the agents utility functional forms being parameterically known, ie. the widely studied CARA class and the decresing absolute risk aversion case.