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# Random parameters analysis of competitive agents in an imperfect information environment

thesis
posted on 2022-03-28, 13:34 authored by Alex Edward Moore
The focus in this thesis is on the mathematics underpinning the system rather than details such as how to decode on-line sites. What now follows is a summary of the thesis in more detail: Much of the research into poker has built on the concepts developed by Von Neumann [27] and Borel [4]. Game Theory concepts have been developed and used in conjunction with cutting edge techniques such as Neural Networks to develop playing algorithms in a similar way to which games such as Chess or Checkers have been solved. These techniques are not necessarily gambling related, even though poker is usually played as a gambling game. The problem with the Game Theory approach is that analysis of full scale poker rapidly becomes intractable. Prominent researchers in the Game Theory approach ([3], [18], [21],[20],[2], [1]) have recognised this and much of the subsequent research has centred around reducing the size of the problem to be solved. Derivation of a Game Tree for a reduced problem was also the approach adopted by von Neumann and Borel. This approach has been successful and can be summarised as finding an exact solution to an approximating problem. Methods seeking approximations to the full problem that retain the properties of the real problem have been developed and are ingenious. In this thesis a different approach has been used. Instead of seeking approximating problems which may be solved exactly, approximate solutions to the full problem are sought. Regression techniques have been used to find approximate solutions to the full problem. Rather than try to tune a strategy engine using Neural Networks to solve the problem in a human-like way, the population variance in preferences (as measured by coefficients) has been measured and past data for given contexts is used to estimate where on the population distribution of preferences a particular context resides. The resulting system still "learns", but in a different way to a Neural Network. The play has been reduced to a sequence of actions which are chosen to maximise profit expectation. This has much in common with financial market trading where a sequence of trades is likewise chosen to maximise profit. The Game Theory approach also maximises profit expectancy, but because that approach necessitates analysis of a computationally intractable Game Tree, the expectation actually maximised is that of an approximating problem. In a sense the approach adopted here is the opposite of the approach adopted by the authors mentioned above in that they seek exact solutions to an approximating problem whereas this research seeks approximate solutions to the exact problem.

## Table of Contents

1. Trading in uncertainty -- 2. Literature review -- 3. Generic opponents model -- 4. Random parameters model -- 5. Adaptive model -- 6. Tree based contextual coefficients -- 7. Deployment -- 8. Results -- 9. Conclusions -- Appendices

## Notes

This thesis is presented for the degree of Doctor of Philosophy" "September 2011 (original), September 2012 (with corrections) Bibliography: leaves 153-156

## Awarding Institution

Macquarie University

Thesis PhD

## Degree

PhD, Macquarie University, Faculty of Science, Department of Statistics

## Department, Centre or School

Department of Statistics

2012

Peter Petocz

David Bulger

## Rights

Copyright disclaimer: http://www.copyright.mq.edu.au Copyright Alex Moore 2012.

English

## Extent

1 online resource (xiii, 156 leaves) illustrations

## Former Identifiers

mq:28099 http://hdl.handle.net/1959.14/266883 2073512

## Exports

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