Skip to main content
Log in

A logarithmic market scoring rule agent-based model to evaluate prediction markets

Journal of Evolutionary Economics Aims and scope Submit manuscript

Cite this article


Prediction Markets (PMs) are markets in which agents trade event contingent assets. Enterprises use PMs to forecast revenues and project deadlines. This paper presents an Agent-based model, called Logarithmic Market Scoring Rule-Automated Market Maker (LMSR-ASM), to evaluate Prediction Markets. Our model is capable of testing different types of Automated Market Makers (AMMs), which are mathematical functions or computational mechanisms needed to provide liquidity in Prediction Markets. The model offers insights into how to set parameters in a PM and how profits react to contrasting settings and AMMs. In addition, we simulate different probability processes, distinct AMMs, and agent behaviors. This paper also utilizes the LMSR-ASM to evaluate the impact of choosing initial prices in profits and revenue opportunities regarding AMM computational implementation. We show that we can use the LMSR-ASM to find optimal parameters for maximizing profits in PMs and how different AMMs affect market results under a variety of settings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Availability of data and materials

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request

Code Availability

the source code is at





  4. A Decentralized Finance (DeFi) Application is defined by Wang (2020) as a smart contract stored in a public distributed ledger (such as a blockchain). It is possible to automate the execution of financial instruments and digital assets using these smart contracts.



  7. The source code for the model is available at The Appendix provide the user guide and offers class descriptions.

  8. Knowledge aggregation refers to the use of a price system as a mechanism for communicating dispersed information, as shown in Hayek (1945).

  9. Different from us, Klingert and Meyer (2018) use a family of zero-intelligence agents. Zero-intelligence unconstraint traders (ZIU) randomly decide to buy or sell and at what price. Zero-intelligence constraint traders (ZIC) are restricted in their trading by their constraint to not sell (or buy) below (or above) a given value. Zero-intelligence plus traders (ZIP) adapt a profit margin based on the success of the last trades. Near-zero-intelligence traders (N-ZI) use the expected value and the last market price. Slamka et al. (2013) use risk-neutral traders that maximize a utility function that depends on the share’s expected value and total costs. Brahma et al. (2012) use two types of fundamental traders and two types of technical traders. The fundamental traders maximize linear utility functions based on different types of pieces of information. While one of the technical traders make choices based on two moving averages, the other uses a range of the price history.

  10. Note that this is true for both LMSR and LS-LMSR. The fact that prices in LS-LMSR do not sum to unity does not affect agents’ portfolio decisions, only the price of the stock they buy does.

  11. Since prices are equivalent to probabilities, the term represents the agent’s probability forecast and price expectations for the next period

  12. Brahma et al. (2012) compare the LMSR-AMM with the Bayesian AMM. They choose the parameter b and a parameter related to the Bayesian AMM to make the average spread of both models approximately the same in a setup where the agents receive Gaussian signals. In their case, they choose \(b=125\). Slamka et al. (2013) choose the b parameter to minimize the mean absolute error between the prediction about the true value of the shares and their actual values. They find three different values for the b parameter: \(b=16\) for the case of no volatility in the true value of the share, \(b=280\) for the case of low volatility, and \(b=49\) for the case of high volatility. Klingert and Meyer (2018), to compare the CDA (Continuous Double Action) with the LMSR, choose the parameter b in a way that the maximal loss of the LMSR market maker is comparable with the costs of the CDA market maker. They choose \(b=17.31\).


  • Agrawal S, Delage E, Peters M, Wang Z, Ye Y (2011) A unified framework for dynamic prediction market design. Operations Research 59(3):550–568

    Article  Google Scholar 

  • Anderson P, Anderson PW, Pines D (eds) (1988) The Economy as an Evolving Complex System. Westview Press

  • Arrow KJ, Forsythe R, Gorham M, Hahn R, Hanson R, Ledyard JO, Levmore S, Litan R, Milgrom P, Nelson FD, Neumann GR, Ottaviani M, Schelling TC, Shiller RJ, Smith VL, Snowberg E, Sunstein CR, Tetlock PC, Tetlock PE, Varian HR, Wolfers J, Zitzewitz E (2008) The promise of prediction markets. Science 320(5878):877–878

    Article  Google Scholar 

  • Arthur WB, Durlauf SN, Lane DA (eds) (1997) The economy as an evolving complex system II. Addison-Wesley

  • Berg H, Proebsting TA (2009) Hanson’s automated market maker. Journal of Prediction Markets 3(1):45–59

    Article  Google Scholar 

  • Beygelzimer A, Langford J, Pennock D (2012) Learning performance of prediction markets with elly bettors.arXiv preprint arXiv:1201.6655

  • Blume L, Easley D (2009) The market organism: long-run survival in markets with heterogeneous traders. Journal of Economic Dynamics and Control 33(5):1023–1035

    Article  Google Scholar 

  • Blume LE, Durlauf SN (eds) (2005) The economy as an evolving complex system, III: current perspectives and future directions. Oxford University Press

  • Bonabeau E (2002) Agent-based modeling: Methods and techniques for simulating human systems. Proceedings of the National Academy of Sciences 99(suppl 3):7280–7287

    Article  Google Scholar 

  • Bottazzi G, Giachini D (2017) Wealth and price distribution by diffusive approximation in a repeated prediction market. Physica A: Statistical Mechanics and its Applications 471:473–479

    Article  Google Scholar 

  • Brahma A, Chakraborty M, Das S, Lavoie A, Magdon-Ismail M (2012) A bayesian market maker. In Proceedings of the 13th ACM Conference on Electronic Commerce- EC ’12, pp. 215. ACM Press

  • Chen Y, Pennock DM (2007) A utility framework for bounded-loss market makers. In Proceedings of the Twenty-Third Conference on Uncertainty in Artificial Intelligence, UAI’07, Arlington, Virginia, USA, pp. 49–56 AUAI Press

  • Chen Y, Pennock DM (2010) Dec. designing markets for prediction. AI Magazine 31(4):42–52

    Article  Google Scholar 

  • Cowgill B, Zitzewitz E (2015) Corporate prediction markets: evidence from google, ford, and firm x. The Review of Economic Studies 82(4):1309–1341

    Article  Google Scholar 

  • Das R, Hanson JE, Kephart JO, Tesauro G (2001) Agent-human interactions in the continuous double auction. In Proceedings of the 17th International Joint Conference on Artificial Intelligence - Volume 2, IJCAI’01, San Francisco, CA, USA, pp. 1169-1176. Morgan Kaufmann Publishers Inc

  • Das S (2005) A learning market-maker in the Glosten-Milgrom model. Quantitative Finance 5(2):169–180

    Article  Google Scholar 

  • Dindo P, Massari F (2020) The wisdom of the crowd in dynamic economies. Theoretical Economics 15(4):1627–1668

    Article  Google Scholar 

  • Ehrentreich N (2006) Technical trading in the santa fe institute artificial stock market revisited. Journal of Economic Behavior & Organization 61(4):599–616

    Article  Google Scholar 

  • Ehrentreich N, (2008) Agent-based modeling: the santa Fe Institute artificial stock market model revisited. Number 602 in Lecture notes in economics and mathematical systems. Berlin ; New York: Springer

  • Farmer J, Foley D (2009) The economy needs agent-based modelling. Nature 460:685–686

    Article  Google Scholar 

  • Giachini D (2021) Rationality and asset prices under belief heterogeneity. Journal of Evolutionary Economics 31(1):207–233

    Article  Google Scholar 

  • Gneiting T, Raftery AE (2007) Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association 102(477):359–378

    Article  Google Scholar 

  • Grimm V, Berger U, Bastiansen F, Eliassen S, Ginot V, Giske J, Goss-Custard J, Grand T, Heinz SK, Huse G et al (2006) A standard protocol for describing individual-based and agent-based models. Ecological modelling 198(1–2):115–126

    Article  Google Scholar 

  • Grimm V, Berger U, DeAngelis DL, Polhill JG, Giske J, Railsback SF (2010) The odd protocol: a review and first update. Ecological modelling 221(23):2760–2768

    Article  Google Scholar 

  • Grimm V, Railsback SF, Vincenot CE, Berger U, Gallagher C, DeAngelis DL, Edmonds B, Ge J, Giske J, Groeneveld J, et al. (2020) The odd protocol for describing agent-based and other simulation models: A second update to improve clarity, replication, and structural realism. Journal of Artificial Societies and Social Simulation 23(2)

  • Hanson R (2003) Combinatorial information market design. Information Systems Frontiers 5(1):107–119

  • Hanson R (2007) Logarithmic market scoring rules for modular combinatorial information aggregation. The Journal of Prediction Markets 1(1)

  • Hanson R, Oprea R (2004) Manipulators increase information market accuracy. George Mason University

  • Hayek FA (1945) The use of knowledge in society. Technical report, Social Science Research Network

    Google Scholar 

  • He XZ, Treich N (2017) Prediction market prices under risk aversion and heterogeneous beliefs. Journal of Mathematical Economics 70:105–114

    Article  Google Scholar 

  • Holland JH, Miller JH (1991) Artificial adaptive agents in economic theory. American Economic Review 81(2):365–71

    Google Scholar 

  • Huang YC, Tsao CY (2018) Evolutionary frequency and forecasting accuracy: Simulations based on an agent-based artificial stock market. Computational Economics 52(1):79–104

    Article  Google Scholar 

  • Johnson PE (2002) Agent-based modeling: What i learned from the artificial stock market. Social Science Computer Review 20(2):174–186

    Article  Google Scholar 

  • Joshi S, Parker J, Bedau MA, et al. (1998) Technical trading creates a prisoner’s dilema: Results from an agent-based model. Santa Fe Institute

  • Kets W, Pennock DM, Sethi R, Shah N (2014) Betting strategies, market selection, and the wisdom of crowds. In Twenty-Eighth AAAI Conference on Artificial Intelligence

  • Klingert FM, Meyer M (2012) Effectively combining experimental economics and multi-agent simulation: suggestions for a procedural integration with an example from prediction markets research. Computational and Mathematical Organization Theory 18:63–90

    Article  Google Scholar 

  • Klingert FMA, Meyer M (2018) Comparing prediction market mechanisms: an experiment-based and micro validated multi-agent simulation. Journal of Artificial Societies and Social Simulation 21(1):1–7

    Article  Google Scholar 

  • Law AM, Kelton WD, Kelton WD. (2007) Simulation modeling and analysis, vol. 3, Mcgraw-hill New York

  • Lebaron B (2002) Building the santa fe artificial stock market. Working paper, graduate. In School of International Economics and Finance, Brandeis, pp. 1117–1147

  • Linn SC, Tay NS (2007) Complexity and the character of stock returns: Empirical evidence and a model of asset prices based on complex investor learning. Management Science 53(7):1165–1180

    Article  Google Scholar 

  • Lorscheid I, Heine BO, Meyer M (2012) Opening the ‘black box’of simulations: increased transparency and effective communication through the systematic design of experiments. Computational and Mathematical Organization Theory 18:22–62

    Article  Google Scholar 

  • Manski CF (2006) Interpreting the predictions of prediction markets. Economics letters 91(3):425–429

    Article  Google Scholar 

  • Montgomery DC (2017) Design and analysis of experiments. John wiley & sons

  • Müller B, Bohn F, Dreßler G, Groeneveld J, Klassert C, Martin R, Schlüter M, Schulze J, Weise H, Schwarz N (2013) Describing human decisions in agent-based models-odd+ d, an extension of the odd protocol. Environmental Modelling & Software 48:37–48

    Article  Google Scholar 

  • Othman A, Pennock DM, Reeves DM, Sandholm T (2013) A practical liquidity-sensitive automated market maker. ACM Transactions on Economics and Computation 1(3):1–25

    Article  Google Scholar 

  • Ottaviani M, Sørensen PN (2007) Outcome manipulation in corporate prediction markets. Journal of the European Economic Association 5(2–3):554–563

    Article  Google Scholar 

  • Palmer RG, Brian Arthur W, Holland JH, LeBaron B, Tayler P (1994) Artificial economic life: a simple model of a stockmarket. Physica D: Nonlinear Phenomena 75(1):264–274

    Article  Google Scholar 

  • Pennock DM (2004). A dynamic pari-mutuel market for hedging, wagering, and information aggregation. In Proceedings of the 5th ACM conference on Electronic commerce, EC ’04, New York, NY, USA, pp. 170–179. Association for Computing Machinery

  • Pennock DM, Sami R (2007) Computational aspects of prediction markets. In: Tardos E, Nisan N, Roughgarden T, Vazirani VV (eds) Algorithmic Game Theory. Cambridge University Press, pp 651–676

    Chapter  Google Scholar 

  • Pierce CA, Block RA, Aguinis H (2004) Cautionary note on reporting eta-squared values from multifactor anova designs. Educational and psychological measurement 64(6):916–924

    Article  Google Scholar 

  • Plott CR, Chen KY (2002) Information aggregation mechanisms: Concept, design and implementation for a sales forecasting problem

  • Restocchi V, McGroarty F, Gerding E (2019) Statistical properties of volume and calendar effects in prediction markets. Physica A: Statistical Mechanics and its Applications 523:1150–1160

    Article  Google Scholar 

  • Restocchi V, McGroarty F, Gerding E (2019) The stylized facts of prediction markets: Analysis of price changes. Physica A: Statistical Mechanics and its Applications 515:159–170

  • Roşu I (2009) A dynamic model of the limit order book. The Review of Financial Studies 22(11):4601–4641

    Article  Google Scholar 

  • Simon HA (1990) Utility and probability: bounded rationality. Springer

    Google Scholar 

  • Slamka C, Skiera B, Spann M (2013) Prediction market performance and market liquidity: a comparison of automated market makers. IEEE Transactions on Engineering Management 60(1):169–185

    Article  Google Scholar 

  • Smith MA, Paton D, Williams LV (2006) Market efficiency in person-to-person betting. Economica 73(292):673–689

    Article  Google Scholar 

  • Springer N (2018) The wisdom of crowds: why the many are smarter than the few and how collective wisdom shapes business, economics, societies, and nations. Macat Library

  • Steinbacher M, Raddant M, Karimi F, Camacho Cuena E, Alfarano S, Iori G, Lux T (2021) Advances in the agent-based modeling of economic and social behavior. SN Business & Economics 1(99):1–24

    Google Scholar 

  • Tay NS, Linn SC (2001) Fuzzy inductive reasoning, expectation formation and the behavior of security prices. Journal of Economic Dynamics and Control 25(3–4):321–361

    Article  Google Scholar 

  • Tesfatsion L (2006) Chapter 16 agent-based computational economics: A constructive approach to economic theory. In: Tesfatsion L, Judd K,(eds). Handbook of Computational Economics, vol. 2 of Handbook of Computational Economics, 831–880. Elsevier

  • Tsao CY, Huang YC (2018) Revisiting the issue of survivability and market efficiency with the santa fe artificial stock market. Journal of Economic Interaction and Coordination 13(3):537–560

    Article  Google Scholar 

  • Wang Y (2020) Automated market makers for decentralized finance (DeFi).arXiv:2009.01676

  • Wolfers J, Zitzewitz E (2004) Prediction markets. Journal of economic perspectives 18(2):107–126

    Article  Google Scholar 

  • Wolfers J, Zitzewitz (2006) Five open questions about prediction markets

  • Wolfers J, Zitzewitz E (2016) Prediction Markets, pp. 1–9. London: Palgrave Macmillan UK

  • Yang H, Chen S (2018) A heterogeneous artificial stock market model can benefit people against another financial crisis. PloS one 13(6):e0197935

  • Yang H, Wang HJ, Sun GP, Wang L (2015) A comparison of us and chinese financial market microstructure: heterogeneous agent-based multi-asset artificial stock markets approach. Journal of Evolutionary Economics 25(5):901–924

    Article  Google Scholar 

  • Zhang Y, Zhang W (2007) Can irrational investors survive? a social-computing perspective. IEEE Intelligent Systems 22(5):58–64

Download references


RAE and DOC thank to cnpq for financial support

Author information

Authors and Affiliations



the authors contributed equally to this work

Corresponding author

Correspondence to Daniel O. Cajueiro.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest

Ethics approval

The manuscript has not been simultaneously submitted or published in any other journal

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Athos V.C. Carvalho, Douglas Silveira and Regis A. Ely contributed equally to this work.


Appendix A: Overview, design concepts and details protocol

In this appendix, we describe our model in terms of the ODD protocol (Grimm et al. , 2006; Grimm et al. , 2010; Müller et al. , 2013; Grimm et al. , 2020). The ODD protocol provides a “template” to present the model overview, including the purpose and main processes that take place in the model, the Design concepts of the model, and the Details necessary for the implementation of the model.

We split this appendix into three subsections. In Appendix A.1, we present the overview of the model. In Appendix A.2, we show the design of the model. In Appendix A.3, we detail the model in terms of the ODD protocol.

1.1 A.1 Overview

Purpose and patterns

The Logarithmic Market Scoring Rule Artificial Stock Market (LMSR-ASM) is an agent-based model capable of evaluating different Automated Market Makers used in Prediction Markets, namely the Logarithmic Market Scoring Rule (LMSR) (Hanson , 2007) and the Liquidity Sensitive Logarithmic Market Scoring Rule (LS-LMSR) (Othman et al. , 2013). The model builds upon the Santa-Fe Institute Artificial Stock Market (SFI-ASM) (Palmer et al. , 1994; Lebaron , 2002; Johnson , 2002) to evaluate how the two AMMs analyzed differ in relation to profits and liquidity.

Patterns observed that show the ability of the model are:

  • Profits decrease with the increase of the parameter b, while volume increases, using LMSR.

  • Profits are higher in LS-LMSR than in LMSR, excluding edge cases.

  • Volume decreases with the increase of the parameter \(\alpha \), using LS-LMSR.

Entities, state variables and scales

The LMSR-ASM includes three types of entities: an Automated Market Maker (AMM), Agents, and Stocks. The AMM sets prices and handles transactions. Agents interact with the AMM, buying and selling according to their own portfolio decisions. Stocks update underlying probabilities and determine the payoff at the end of a simulation.

In Table 3, we present the attributes and state variables associated with the AMM.

Table 3 The attributes and state variables associated with the AMM

In Tables 4 and 5, we present the attributes and state variables associated with the agents and stocks, respectively. Each period does not relate directly to the passing of time but can be interpreted as the opening and closing of a trading session.

Table 4 The attributes and state variables associated with the agents
Table 5 The attributes and state variables associated with the stocks

Process overview and scheduling

The LMSR-ASM follows three different sub-processes: setup, step, and end. At step 0, the model follows the setup, creating initial stocks and setting probabilities. After setup, trading starts, in which all agents interact in order with the AMM. Updates in this phase happen asynchronously. In the last period, the AMM realizes payment for those agents holding the winning stock. Figure 10 presents the full process that rules the LMSR-ASM.

Fig. 10
figure 10

The LMSR-ASM full process

1.2 A.2 Design

Basic principles

The LMSR-ASM is based on the version presented by Ehrentreich (2008) of the Santa-Fe Institute Artificial Stock Market (SFI-ASM). We use two types of agents: ideal and noisy. Ideal agents know the real probability of an underlying event. As in the SFI-ASM, noisy agents are myopic, meaning they only consider prices for the next period. Noisy agents are also homogeneous with respect to their utility function, determining their optimal stock holdings at each period by maximizing a Constant Absolute Risk Aversion (CARA) utility in the form:

$$\begin{aligned} U(W_{i,t+1})=-e^{-\lambda W_{i,t+1}}, \end{aligned}$$

where \(\lambda \) is a strictly positive parameter, representing the degree of risk aversion, and \(W_{i,t+1}\) is the expected wealth of agent i for the next period.

Individual decision-making

Ideal agents buy stocks until the price matches the real probability of an event. Noisy agents maximize their utility by buying stocks according to their optimal portfolio, given by

$$\begin{aligned} \widehat{x_{i,t}}=\frac{E_{i,t}[p_{t+1}]-p_t}{\lambda \sigma ^2_{t,p}}, \end{aligned}$$

with \(x_{i,t}\) being the amount of stock held by the agent i at period t; \(\sigma ^2_{t,p}\) being the variance of the stock, which follows a Bernoulli distribution; \(p_t\) is the price at time t and \(E_{i,t}[p_{t+1}]\) being agent i’s forecast.

At each period, agents interact in sequence with the market maker, deciding to buy or sell stocks, according to their own expected value calculations of the transaction.

Individual sensing

While ideal agents know exactly the underlying probability of an event, noisy agents make forecasts according to the underlying probability, adjusted by a random factor with normal distribution \(\mathcal {N}(0, 0.025)\). Both agents know market prices at each moment of interaction with the market maker.

Individual prediction

Agents predict according to the probability process. Each period, agents update beliefs, which are equal to underlying probabilities (in the case of ideal agents) or the probability plus a noise factor given by a normal distribution \(\mathcal {N}(0, 0.025)\) with zero mean and 0.025 variance (in the case of noisy agents).


Each agent interacts only with the Automated Market Maker by buying and selling stocks. Trading happens asynchronously at each period.


Agents do not learn but update beliefs each period according to the underlying probability process.


At each step, agents update their optimal holdings considering the changes in their forecast of the underlying event. If the agent’s forecast of the probability of the underlying event is higher than the current stock price, and it does not hold No stocks, it buys Yes stocks according to its optimal demand. If the agent’s forecast of the probability of the underlying event is lower than the current stock price, and it does not hold Yes stocks, it buys No stocks according to its optimal demand. When the agent’s prediction does not match its stock holdings, the agent changes its current holds according to its optimal portfolio. Table 6 summarizes the agents’ adaptation process.

Table 6 Summary of the agents’ adaptation process


The model presents no collective behavior.


Heterogeneity in the model is given by the individual forecasts of each agent.


Stochasticity in the model is given by the probability process of the event and agent behavior. In random walk settings, probabilities are updated at every step, changing in relation to arbitrary shocks that follow a normal distribution \(\mathcal {N}(0, 0.025)\).

Noisy agents also similarly update their beliefs, changing beliefs every period (in the case of noisy agents), with their prediction being the underlying probability plus a noise factor given by a normal distribution \(\mathcal {N}(0, 0.025)\) with zero mean and 0.025 variance.


Emergence in the LMSR-ASM is primarily reflected by market volume and AMM profit. Those emergent variables are the result of trading between agents and the AMM, according to each agent’s portfolio decision and utility-maximization problem.

Non-emergent variables are probabilities and stock prices. Probabilities can be fixed, as set arbitrarily at the start of the simulation, or updated at each period, following a random walk with normal distribution \(\mathcal {N}(0, 0.0025)\). Prices closely follow those set probabilities.


We collect data on trade volume and market profit at the end of the simulation.

Table 7 Initial settings that do not vary among simulations

1.3 A.3 Details

Implementation Details

LMSR-ASM adapts the version of the SFI-ASM due to Ehrentreich (2008) to PM simulations. Ehrentreich (2008) developed the SFI-ASM in Java using the Repast library.


Table 7 presents the initial settings of the model that do not vary among simulations. Other initial parameters vary according to the purpose of the analysis done. We use the LMSR-ASM to evaluate optimal parameters; the difference in profits between LMSR and LS-LMSR; the effects of rounding methods on profits; the effects of setting initial prices. Table 8 presents the initial parameters used to find the optimal parameters of the market. Table 9 presents the initial parameters used to find the difference in profits between LMSR and LS-LMSR. Table 10 shows the initial parameters used to find the effects of rounding methods on profits. Table 11 presents the initial parameters used to find the effects of initial prices.

Table 8 Initial parameters used to find the optimal parameters of the market
Table 9 Initial parameters used to find the difference in profits between LMSR and LS-LMSR
Table 10 Initial parameters used to find the effects of rounding methods on profits
Table 11 Initial parameters used to find the effects of initial prices

Input Data

The model does not utilize external data.


Stock creation To set initial prices, the AMM creates artificial stocks, which are stocks held by the AMM, and, therefore, do not have to be paid at the last period. During stock creation, the AMM adds new stocks until the price of the marginal stock is greater than the price set. If initial prices are \(\$0.50\), no additional stocks are created.

Internal probability updates Before interacting with the AMM, agents observe the underlying probability of the model at that period. For ideal agents, the internal probability is the underlying probability. For noisy agents, the internal probability is the underlying probability plus a noise factor given by a normal distribution \(\mathcal {N}(0, 0.025)\).

Internal optimal portfolio updates After updating probabilities, noisy agents update their optimal portfolio calculations, given by

$$\begin{aligned} \widehat{x_{i,t}}=\frac{E_{i,t}[p_{t+1}]-p_t}{\lambda \sigma ^2_{t,p}}, \end{aligned}$$

where \(x_{i,t}\) is the amount of stock held by the agent i at period t, \(\sigma ^2_{t,p}\) is the variance of the stock, which follows a Bernoulli distribution, \(p_t\) is the price at time t and \(E_{i,t}[p_{t+1}]\) is agent i’s forecast.

The amount of stocks the agent will buy or sell is given by the difference between their holdings and this portfolio update.

Agent interactions with AMM

Agents send a buy or sell order to the AMM and the AMM provides a price, according to

$$\begin{aligned} P(\overline{q}-q) = C(\overline{q}) - C(q), \end{aligned}$$

where \(P(\overline{q}-q)\) is the price of the order, C(q) is the cost function at the time of the order and \(C(\overline{q})\) is the cost function after the order.

The cost function for the LMSR is given by

$$\begin{aligned} C(q)=b\log \left( e^{\frac{q_\textsc {Y}}{b}} + e^{\frac{q_\textsc {N}}{b}} \right) , \end{aligned}$$

where \(q_\textsc {Y}\) and \(q_\textsc {N}\) are the two types of stocks available, and b a strictly positive parameter that controls liquidity in the market.

The cost function for the LS-LMSR is given by

$$\begin{aligned} C(q)=b(q)\log \left( e^{\frac{q_\textsc {Y}}{b(q)}} + e^{\frac{q_\textsc {N}}{b(q)}} \right) , \end{aligned}$$


$$\begin{aligned} b(q)=\alpha (q_\textsc {Y}+q_\textsc {N}), \end{aligned}$$

with \(\alpha \) being a strictly positive parameter set by the owner of the PM.

Appendix B: Variance and factor analysis

1.1 B.1 Variance analysis

To evaluate how many runs are necessary, we follow the protocol proposed in Lorscheid et al. (2012), presenting a pre-experimental error variance matrix created by analyzing four preliminary design points. Table 12 describes the settings used in the simulations. “N. of Agents” stands for the number of agents. The initial probability and the probability after shock are shown in the columns “Initial Prob.”, and “Prob. after shock”, respectively.

Table 12 Description of the parameters used in the Variance Analysis

We consider Noisy agents, 2-digits rounding methods, and fixed probability process

The parameters represent an assortment of standard and extreme values we use during simulations. These design points aid in the decision of the required number of runs. Table 13 shows the results of these simulations, in an error variance matrix.

Table 13 Error Variance Matrix

Even at a low number of runs, the model presents convergence in the coefficient of variation. However, convergence happens only at a higher number of runs, for some variables. The low number of runs necessary for convergence is intuitive, given stochasticity in the model is derived from a normal distribution with mean 0 and low standard deviation (\(\mathcal {N}(0, 0.025)\)). In Section 4, we decided to run each specification 100 times since our pre-simulations show convergence in the analyzed parameters at that level. This helps maintain a low computational cost, allowing for the analysis of more simulation specifications. To further demonstrate the validity of the chosen number of runs, we present in Table 14 the results of an F-test between all samples and the result of a t-test between the largest sample and the sample with 100 runs. All t-tests and F-tests failed to reject the null hypothesis that the means are equal.

Table 14 Statistics and p-values for equality of means between number of runs

1.2 B. 2 ANOVA

To evaluate the impact of the parameters on profits and volume, we run a factorial ANOVA, varying multiple parameters simultaneously. Table 15 presents all the different initial settings used. Due to the number of simulations needed, a lower number of 50 runs was used. It is important to notice that this value is still consistent with the process presented in 1. There are a total of 6,000 different initial specifications for the LS-LMSR and 4,000 for the LMSR, with the total number of simulations used in the ANOVA being 500,000. We present ANOVA results for four different specifications. Table 16 are results for volume and profit, using LMSR. Table 17 are results for the same variables, using LS-LMSR.

Table 15 Initial parameters used in the factorial ANOVA
Table 16 Analysis of variance for the LMSR
Table 17 Analysis of variance for the LS-LMSR

ANOVA supports our major findings. In particular, when it comes to the optimal values for parameters b and \(\alpha \) we have discussed in Section 4.1, we can infer that they are significant for the LMSR and LS-LMSR markets, respectively. Observe that it holds for volume and market profit. With respect to the analysis based on the variation of the probability after shock and the initial probability provided in Section 4.2, Tables 16 and 17 tell us that – by themselves – the initial probability and probability after shock are not significant. However, these two terms have significant interactions with initial prices, which leads to the interpretation that, in a market where market creators decide initial prices, volumes, and profits are similar even between markets with different overall probabilities, given other characteristics are equal. Also, due to the difference in the effect of these interactions, we can see the final probability has a bigger effect on outcomes, due to it determining payoff. Regarding the rounding methods discussed in Section 4.3, we observe that their effects are not significant for the LMSR but significant for the LS-LMSR. The initial prices discussed in Section 4.4 are significant for the market profit both in the LSMR and LS-LMSR. As prices reflect probabilities, the AMM will have to bear the costs if initial prices are far from their actual value.

1.3 B.3 Effect sizes

We utilize the partial \(\eta ^{2}\) (Pierce et al. , 2004), derived from the analysis of variance presented in Section 2, to measure effect sizes of the parameters on volume and profit. This approach is also used in Klingert and Meyer (2018). The results support our main findings in the paper while also revealing new insights. Table 18 shows the partial \(\eta ^{2}\) of the varying parameters on volume under LMSR. As expected, the main drivers of variance are the number of agents and the value of parameter b. One other interesting effect is that the initial probability of the event does not affect volumes, and surprisingly, neither does the probability after the shock. This shows that volume is similar in a market with relatively homogeneous traders – regardless of how much probabilities vary over time. It is important to notice that only one shock is possible in the model, so we cannot test how various probability shifts affect market volume.

Table 18 Effect sizes on volume using LMSR

Table 19 shows the effects of the factors on profit, for the LMSR. The main effects are related to choosing b and initial prices. This reinforces our findings regarding the importance of setting initial prices according to the market creator’s beliefs of the probability of the outcome. The effect of b is also intuitive, as the parameter determines the worst-case loss for the market maker, which is given by \(b\log n\).

Table 19 Effect sizes on profit using LMSR

The following tables show the results regarding the LS-LMSR. Table 20 presents the effects on volume. Results are similar to the LMSR case, with the number of agents being the main factor. However, in the LS-LMSR, initial prices also affect market volume with a low impact.

Table 20 Effect sizes on volume using LS-LMSR

Results for the effects on profits are also similar to LMSR. However, there is a distinct effect of rounding methods – showing that the way trades are (computationally) handled has a significant impact on LS-LMSR.

Table 21 Effect sizes on profit using LS-LMSR

These measures strengthen the main findings of our paper, such as rounding methods having little to no impact on volume but being an important decision that affects profits under the LS-LMSR. The results also show the importance of defining initial prices, so market makers are advised to study underlying events before creating a market and setting prices to their beliefs.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carvalho, A.V.C., Silveira, D., Ely, R.A. et al. A logarithmic market scoring rule agent-based model to evaluate prediction markets. J Evol Econ 33, 1303–1343 (2023).

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


JEL Classification