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May 12, 2023 445 likes •813 views

Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Distributed Prediction Markets modeled by Janyl Jumadinova Raj Dasgupta Weighted Bayesian Graphical Game Outline Introduction Research Problem Janyl Jumadinova

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game

Janyl Jumadinova Raj Dasgupta

C-MANTIC Research Group Computer Science Department University of Nebraska at Omaha

AMEC 2013

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Outline

Problem: Distributed information aggregation - the interaction among multiple prediction markets. Solution: A software agent-based distributed prediction market model where prediction markets running similar events can influence each other. Experimental validation: Comparison with other models and trading approaches.

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation 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. Prediction markets operate similarly to financial markets.

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Event: 2012 Presidential Election

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Event: 2012 Presidential Election

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Event: 2012 Presidential Election

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Event: 2012 Presidential Election

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Event: 2012 Presidential Election

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Event: 2012 Presidential Election

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Event: 2012 Presidential Election

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Do Prediction Markets Work?

Yes, evidence from real markets, laboratory experiments, and theory

I.E.M. beat political polls 451/596 [Forsythe 1999, Berg 2001, Pennock 2002] HP market beat sales forecast 6/8 [Plott 2000] Sports betting markets provide accurate forecasts of game outcomes [Debnath 2003, Schmidt 2002] Market games work [Pennock 2001] Laboratory experiments confirm information aggregation [Forsythe 1990, Plott 1997, Chen 2001] Theory of Rational Expectations [Lucas 1972, Grossman 1981] and more...

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Event: 2012 Presidential Election

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Event: 2012 Presidential Election

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Event: 2012 Presidential Election

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Event: 2012 Presidential Election

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Related Work

The trading protocols used in the market:

Continuous double auction based protocol [Pennock and Sami,

2007; Wolfers and Zitzewitz, 2004]

Dynamic Parimutuel Markets [Pennock 2004]
Automated Market Makers and Scoring Rules [Hanson 2003,

2007; Chen and Pennock, 2007]

Trader behavior and trading strategies:

Play money? [Servan-Schreiber et al. 2004] Incentive Compatibility and Bluffing [Chen et al., 2009; Jian and

Sami, 2012; Conitzer, 2009]

Risk behavior [Dimitrov et al., 2009; Iyer et al., 2010]

Decision making using prediction markets

Traders can take actions to influence the outcome of the decision/event [Chen and Kash, 2011; Shi et al., 2009; Boutilier 2012]

Applications of prediction markets

University/classroom use [Ellis and Sami, 2012; Othman and Sandholm,

2010]

Biomedical research [Pfeiffer and Almenberg; 2010]

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Research Problem

Distributed prediction markets

Multiple prediction markets running simultaneously have

similar events.

The expected outcomes of an event in one prediction

market will influence the outcome of a similar event in a different prediction market. Inter-market effects: evidence from financial markets. Inter-market relationship has not been studied in prediction markets.

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution

Graphical Games Our Model Our Algorithm

Simulation Results Conclusion

Our Solution

1 A software agent-based

distributed prediction market model:

comprises of multiple, parallel running prediction

markets,

uses a graphical structure between the market makers of

the different markets to represent inter-market influence.

2 A graphical game-based algorithm that determines the

best action for the participants in the prediction market using our proposed model.

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution

Graphical Games Our Model Our Algorithm

Simulation Results Conclusion

Graphical Games

Graph representing the local interaction Provide compact representations Each node represents a player Ni is the neighborhood of player i There is an edge (i, j) ∀j ∈ Ni |Ni| is the degree of local interaction for node i The maximum k over the graph k = maxi|Ni| defines the complexity of the representation

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution

Graphical Games Our Model Our Algorithm

Simulation Results Conclusion

Graphical Games

Intuitive graph interpretation: A player’s payoff is only a function of its neighborhood

Implies conditional independence payoff assumption

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution

Graphical Games Our Model Our Algorithm

Simulation Results Conclusion

Model of the Distributed Prediction Market

Weighted Bayesian Graphical Games

Weighted: use weights to model the influence of one market maker on others. Bayesian: used to model the uncertainty of one market maker about the other market makers and incorporate different types of market makers. Graphical Games: allows to capture the interaction between multiple market makers.

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution

Graphical Games Our Model Our Algorithm

Simulation Results Conclusion

Bayes-Nash Equilibrium

Propose an algorithm to calculate the equilibrium of the game efficiently. Determines the best action for the market makers in each prediction market using our proposed model. The best action gives the market price that returns the maximum utility to each market maker.

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution

Graphical Games Our Model Our Algorithm

Simulation Results Conclusion

Bayes-Nash Equilibrium

N−i: the number of market makers market maker i interacts with (its neighborhood). θi: type of market maker i.

Definition

For agent i a strategy si is said to be best response (BR) in a WBGG for type θi to θNi, if ∀s′, EU(si, sN−i, θi) ≥ EU(s′, sN−i, θi) (1)

Definition

A strategy vector s is a Bayes-Nash Equilibrium (BNE) in a WBGG if and only if every agent i is playing a best response to the others.

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution

Graphical Games Our Model Our Algorithm

Simulation Results Conclusion

Bayes-Nash Equilibrium

Message-Passing Algorithm

We extend NashProp algorithm [Ortiz and Kearns, 2003] to incomplete games with an arbitrary graphical structure Table Passing Phase (local optima) Assignment Phase (global optima)

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution

Graphical Games Our Model Our Algorithm

Simulation Results Conclusion

Bayes-Nash Equilibrium

T(w, v) is a binary table indexed by all possible strategies of agent W and V sent by V to W T(w, v) = 1 iff there exists BNE in which agent V plays v when its neighboring agent W plays w

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution

Graphical Games Our Model Our Algorithm

Simulation Results Conclusion

Bayes-Nash Equilibrium

Message Passing Algorithm

Table Passing Phase (local optima)

Local optimal response is found for each agent. Each agent calculates the optimal strategy given its neighbors’ strategies. Each agent then sends the optimal strategy to its neighbors. T(w, v) represents V ’s belief in a BNE in which V plays v and W is “clamped” to w. Table exchanging is iterated until the result is converged.

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution

Graphical Games Our Model Our Algorithm

Simulation Results Conclusion

Bayes-Nash Equilibrium

Message Passing Algorithm

Assignment Passing Phase (global optima)

Global solution is constructed by eliminating inconsistent local optimal response. BNEs are preserved but search still needed. Use heuristic: backtracking local search. Agents incrementally construct BNE obeying constraints imposed.

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution

Graphical Games Our Model Our Algorithm

Simulation Results Conclusion

Approximate Bayes-Nash Equilibrium

Discretization of the action space Player i can now only play action ∈ {0, τ, 2τ, ..., 1} Algorithm takes an extra input parameter ǫ At each node the ǫ-best response is computed

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution

Graphical Games Our Model Our Algorithm

Simulation Results Conclusion

Approximate Bayes-Nash Equilibrium

Theorem

For arbitrary structured graphical games with discrete types stage 1 (Table Passing) converges in at most

nk τ 4(l−1) rounds.

Proposition

Our algorithm applied to a distributed prediction market problem encourages truthful revelation.

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Simulation Results

Comparison

For comparison we use two well-known techniques for trading Greedy strategy

Maximizes immediate utility. Does not consider the types of the market makers.

Influence-less market

Conventional single, isolated markets:

the market price is determined by the market maker

based on that market’s traders’ decisions only.

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Comparison to other strategies

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Comparison to other strategies

Market makers using our Algorithm obtain 56% more utility than the market makers following the next best greedy strategy. Interacting market makers in a distributed prediction market are able to improve their utilities and predict prices with less fluctuations.

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Neighborhood

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Neighborhood

Market makers with a small number of neighbors get less utility than when the number of neighbors is larger, But this relationship is not linear.

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Conclusion

Our novel framework for distributed prediction markets leads to several challenging and important directions that can help to gain a better understanding of the distributed information aggregation problem Results show that our algorithm results in higher utilities and more accurate prices in comparison to a greedy strategy or a disjoint prediction markets

Future Work:

Extend our model to include direct influences between traders across multiple prediction markets Extend our algorithm to study stochastic graphical games to model uncertainty in repeated games Make our algorithm strategy-proof for graphical games in general

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Distributed Prediction Markets modeled by Weighted Bayesian Graphical Game Janyl Jumadinova Raj Dasgupta Outline Introduction Research Problem Our Solution Simulation Results Conclusion

Thank You! Questions?

jjumadinova@unomaha.edu http://myweb.unomaha.edu/∼ jjumadinova C-MANTIC Research Group http://cmantic.unomaha.edu This research has been sponsored as part of the COMRADES project funded by the Office of Naval Research.

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