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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, A Multi-Agent Prediction Market based on Dora Matache*, Raj Dasgupta Boolean Network Evolution Outline Introduction Research Problem Janyl Jumadinova,

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

A Multi-Agent Prediction Market based on Boolean Network Evolution

Janyl Jumadinova, Dora Matache*, Raj Dasgupta

C-MANTIC Research Group Department of Computer Science *Department of Mathematics University of Nebraska at Omaha, USA

IAT 2011

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Outline

Problem: How does a prediction market evolve with respect to different market parameters? Solution: A multi-agent prediction market that uses Boolean network-based rules to capture the evolution of beliefs of the traders as well as to aggregate the market price Experimental validation: Comparison with existing aggregation technique Evaluation of our prediction market with respect to different market parameters

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Prediction Market

A Prediction market is

a market-based mechanism used to

  • combine the opinions on a future event from different

people and

  • forecast the possible outcome of the event based on the

aggregated opinion

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Prediction Market

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Prediction Market

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Prediction Market

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Prediction Market

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Prediction Market

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Prediction Market

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Prediction Market

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Prediction Market

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Prediction Market

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Prediction Market

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Prediction Market

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Prediction Market

Main Features

A prediction market is run for a real-life unknown event Each event has a finite duration Each event’s outcome has a security associated with it Traders buy and sell the securities based on their beliefs about the outcome of the event Traders’ beliefs are expressed as probabilities Market maker aggregates the probabilities from all the traders into a single probability, market price Market price of a security represents the probability of the outcome of an event associated with that security happening Traders get paid according to their reported beliefs

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Boolean Networks (BNs)

The state of the node is either ON (1) or OFF (0) The state of a node is updated according to a Boolean rule The Boolean rule determines state transitions of the nodes Can use BNs to explore the dynamics of the network or just some relevant nodes BNs have been used to model various real networks: genetic regulatory networks, strongly disordered systems common in physics and biology

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Boolean Networks (BNs)

BNs are useful for prediction markets since they reduce the complexity of analyzing a large network Correspondence between Boolean values output by the BN’s rules and the binary outcome of events in a prediction market Can retain the essential aspects of a prediction market, but easy to understand and manipulate

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Research Problem Addressed

Develop a Boolean network-based prediction market using simple Boolean rules for

  • updating the beliefs for each of the market’s

participants,

  • aggregating the participants’ belief information into a

single market price

The Main Research Question:

Under what conditions does a prediction market perform the best?

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Our Solution

Boolean Network-based Prediction Market

A multi-agent prediction market based on Boolean Network evolution Boolean rule for traders’ belief updates Boolean rule for calculating the market price Mathematical model of the states of the traders and the market maker

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Boolean Network-based Prediction Market

  • Our BN-based prediction market consists of:

trading agents, market maker agent, and information sources external to the market

  • Each trading agent has a state representing its belief
  • State can take two values: 1 (believe event will happen)
  • r 0 (believe event will not happen)

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Boolean Network-based Prediction Market

The diagram shows actions of each entity in one time step

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Trading Agents’ Boolean Belief Update

pr(t) - the aggregated market price at trading period t rn(t) - the state of the n-th trading agent at trading period t wn

i - the trust that the n-th trading agent holds for the

accuracy of the posted market price, its own past belief, and the new information signal it obtains, respectively, following Golub [2010] These trusts are represented as weights wn

i ∈ [0, 1],

such that wn

1 + wn 2 + wn 3 = 1

Bn(t) - the information signal received by the n-th trading agent at trading period t Information signal is the value of Bernoulli random variable with probability qn of obtaining 1 (positive information)

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Trading Agents’ Boolean Belief Update

rn(t + 1) =    1 , if wn

1 · pr(t) + wn 2 · rn(t) + wn 3 · Bn(t) > z,

wn

1 + wn 2 + wn 3 = 1, wn i ∈ [0, 1], ∀i = 1, 2, 3;

, otherwise. (1)

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Mean-field Analysis for Calculating the Aggregated Market Price

Market maker agent calculates the aggregated market price, pr(t + 1) We call the current aggregated market price, the density

  • f ones

The density of ones represents the fraction of trading agents that are in state 1 Generate recursive mathematical model for the density

  • f ones using a mean-field approach

The mathematical model for the density of ones can also be used to analyze the dynamics of the prediction market

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Learning the trust values by trading agents

Use neural network representation to find the correct combination of weight parameters Construct a neural network with one hidden layer Inputs: market price, state of the trading agent at time t, information parameter Output: new state of the trading agent at time t + 1

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Patterns of the market price generated by our mathematical model

Generate pattern formation plots Black dot - state of the trading agent is 1 Neutral dot - state of the trading agent is 0 z - threshold parameter for belief update

Equation

q - the probability that the information signal (represented as a Bernoulli random variable) is 1

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Patterns of the market price generated by our mathematical model

When the most weight is given to the information signal, market price tends to oscillate in a narrow range of values When the most weight is given to the trading agent’s state or the past market price, the market price is stable When the weights are more evenly distributed, the market price reaches stability with time

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Patterns of the market price generated by our mathematical model

When the most weight is given to the information signal, market price tends to oscillate in a narrow range of values When the most weight is given to the trading agent’s state or the past market price, the market price is stable When the weights are more evenly distributed, the market price reaches stability with time

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Patterns of the market price generated by our mathematical model

When the most weight is given to the information signal, market price tends to oscillate in a narrow range of values When the most weight is given to the trading agent’s state or the past market price, the market price is stable When the weights are more evenly distributed, the market price reaches stability with time

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Verify the mathematical model for the aggregated market price

  • - mathematical model for the market price
  • - evolution of the actual Boolean network

Observe that our mathematical model for pr(t) is a good approximation of the fraction of nodes in state 1 from evolving the actual BN

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Verify the mathematical model for the aggregated market price

  • - mathematical model for the market price
  • - evolution of the actual Boolean network

Observe that our mathematical model for pr(t) is a good approximation of the fraction of nodes in state 1 from evolving the actual BN

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Behavior of the aggregated market price

Red line - pr(t) = pr(t + 1)

Behavior with noise 28 / 37

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Behavior of the aggregated market price

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Behavior of the aggregated market price

Observe four main behaviors:

1

Aggregated market price converges to 1

2

Aggregated market price converges to 0

3

Aggregated market price converges to the value of information signal

4

Aggregated market price does not converge (chaos)

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Behavior of the aggregated market price

Observe four main behaviors:

1

Aggregated market price converges to 1

2

Aggregated market price converges to 0

3

Aggregated market price converges to the value of information signal

4

Aggregated market price does not converge (chaos)

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Behavior of the aggregated market price

Observe four main behaviors:

1

Aggregated market price converges to 1

2

Aggregated market price converges to 0

3

Aggregated market price converges to the value of information signal

4

Aggregated market price does not converge (chaos)

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Comparison to Conventional Prediction Markets

Compare with Logarithmic Market Scoring Rule (LMSR) aggregation technique [Hanson 2007, Chen and Pennock 2007] Market price using LMSR - x, using BN-based model - ⋄ Observe similar results but less market price oscillations with BN-based approach

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Comparison to Conventional Prediction Markets

Compare with Logarithmic Market Scoring Rule (LMSR) aggregation technique [Hanson 2007, Chen and Pennock 2007] Market price using LMSR - x, using BN-based model - ⋄ Observe similar results but less market price oscillations with BN-based approach

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Robustness to noise

Prediction markets can be affected by manipulation or misinformation Model disturbance in prediction market by changing the belief of some trading agents Analyze response to disturbances of the prediction market Use a noise procedure called a flip rule

  • At each time period t randomly select j trading agents
  • Flip their state

Modify the mathematical model for the aggregated market price

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Robustness to noise

Stability of the prediction market is either preserved or induced by the introduction of noise

Behavior without noise 32 / 37

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Scalability

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Scalability

The accuracy of our mathematical model for the aggregated market price improves as the number of trading agents increase

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Scalability

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Experimental Results

Scalability

The market price becomes less dynamic as the number of trading agents increases But only up to some number of trading agents

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Conclusion and Future Work

In this work we have:

  • described a Boolean Network based representation of the

prediction market

  • used it to calculate the aggregated market price and analyze

the behavior of trading agents in response to various market parameters

  • show BN approach gives results similar to LMSR with less

fluctuations in market price

  • show how our model can be used to analyze and predict the

dynamics of the prediction market

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Conclusion and Future Work

In this work we have:

  • described a Boolean Network based representation of the

prediction market

  • used it to calculate the aggregated market price and analyze

the behavior of trading agents in response to various market parameters

  • show BN approach gives results similar to LMSR with less

fluctuations in market price

  • show how our model can be used to analyze and predict the

dynamics of the prediction market In the future we plan to: Statistical comparison of market prices generated by our model and real data Relax some of the assumptions Extend the model to more than two trading agent’s states

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

References

1 Y. Chen, D. Pennock. Utility Framework for

Bounded-Loss Market Maker. Proc. of the 23rd Conference on Uncertainty in Artificial Intelligence (UAI 2007), pages 49-56, 2007.

2 R. Hanson. Logarithmic Market scoring rules for

Modular Combinatorial Information Aggregation. Journal of Prediction Markets, 1(1):3-15, 2007.

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A Multi-Agent Prediction Market based on Boolean Network Evolution Janyl Jumadinova, Dora Matache*, Raj Dasgupta Outline Introduction Research Problem BN-based Prediction Market Experimental Results Conclusion

Thank You! Questions?

jjumadinova@unomaha.edu C-MANTIC Research Group http://cmantic.unomaha.edu/

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