# Mathematics, Game Theory and Expert Models

#### Written by Robert Coyle

The life of many mathematicians has been an interesting adventure, a story worth being told in books and movies. An outstanding example of this is the life of John Forbes Nash Jr, narrated in the bestselling Pulitzer prize-nominated book ''*A beautiful mind''* written by Sylvia Nasar, which also inspired the well-known movie of the same title, and winner of four Academy Awards. John Nash was one of the most inspiring and genial mathematicians of the last century, whose career was regrettably cut short by paranoid schizophrenia. His interests varied over a broad range of mathematics where he made significant findings and his contributions were fundamental to the development of the new mathematics field, Game Theory, for which he won the Nobel Prize.

Game Theory is a field of mathematics that attempts to create tools to help people make better decisions. But what does making better decisions mean? In some cases the answer is quite simple. Take for instance the following example; if you’ve earned a sum of money from an investment that was a good decision. However, do you remember how you made that decision? Did you decide to invest your money because you had a feeling it was a good investment or you thought it would be profitable? Most people can relate to a time when they had to make a decision and were uncertain if the decision was the right choice. In most cases, people make decisions based on a set of assumptions about what will happen in the future because the information they have is partial. Therefore there should be a general consensus that the more data is provided, the better people can make decisions.

However, should the quantity of information influence making better decisions? Sometimes data can be quite complex and therefore it is not always easy to understand the correct way to use it. This is why an “Expert Model” is necessary and beneficial when making better decisions. A model for making a better decision must have two features: it must be able to define the best goal a person can achieve and the way to reach this goal. The best goal is usually easily defined, particularly when the decision relates to something that can be quantified, like money. A good model is one that maximises the quantity a person can obtain, however, the way to reach the best goal is not as easy to define. Any decision taken by somebody usually provokes the reaction of somebody else who can affect the chances to reach that goal. Moreover every person has a personal way to react to a situation. Even in games, where every player is forced to follow precise rules, it is difficult to predict the game development: you make a move guessing how the opponent will react. Therefore how is it possible to create an expert model?

Game Theory tries to answer this question by analysing different types of game. Games provided a good starting point: the rules were well defined and all the players have the same aim, which is winning the game or reach the best possible output. Two of the basic principles stated by mathematicians over which building the Game Theory are:

- People behave in a rational way
- People’s aim is to maximise their payoff

After a set of rules is determined and the rules are known by all the players, it is possible to determinate a strategy: a player’s strategy is the combination of successive moves that he makes to reach the maximum pay-off. The combination of each player’s strategy determines how the game ends and the payoff of each player. If the game ends in a finite number of moves, it is theoretically possible to determine the result for each combination of strategies. Therefore each player can know his payoff in advance according to the strategy chosen by all the players. We assume that each player behaves rationally to maximise his payoff, it is then possible to determine which strategy will be chosen: it is therefore said that the game has equilibrium and each player has made the best decision to reach the maximum pay-off.

A lot of games were invented by mathematicians for modelling different situations: there are games where players have to collaborate to maximise their payoffs, others where players have no choice but to oppose each other, etc. Some games involved other area of mathematics, like statistics, to determine the best strategy for the players.

Game Theory is now applied in different fields, not only mathematics, including economics, political science, biology and philosophy.

A good practical example is the application of Game Theory to risk analysis. Several models were developed to assess the risk of a project or of an investment: one of these models is the “principal-agent model of possible shortcuts and induced failure risks”. A “manager” and a “agent” are in charge of a project involving the development and maintenance of a physical system which is compounded by several components. The project is behind schedule and the agent fears he will face penalties but he has also the option to take shortcuts in one or more tasks of the components’ development. However the shortcuts increase the probability of having system failures which incur penalties. The manager instead can choose the level of the penalties on the agent for being in late or for causing a system failure. Otherwise she can monitor the agent’s work but she is not able to judge if a shortcut keeps the failure probability under an acceptable level. Moreover, monitoring activity has its costs. It is evident that the manager’s and the agent’s decisions are deeply related: the manager’s decision affects the agent’s decision and vice-versa. The aim of the model is to find the optimal set of options for manager (monitoring/penalties) and agent (taking shortcuts or not) for meeting the project deadlines and keeping the probability of a system failure under controls. The output of this game depends on different factors: the complexity of the system, the probability of system failures, the size of the penalties, the monitoring costs, etc. The change of one of these factors can significantly change the output of the game, therefore the players’ strategies may vary accordingly to the factors.

Game Theory is about the creation of algorithms to make predictions, and therefore, to help us make better decisions. Here at Creme, we analyse a huge quantity of data using algorithms and statistical models with the aim to provide new, more efficient and reliable tools for making better decisions. Algorithms, such as those used in Game Theory, with a combination of real data and expert models can really help people making better decisions.

Check out www.cremeglobal.com for more information on our capabilities and expert models.

References:

Game example from: Paté-Cornell, Elisabeth; Garber, Russ; Guikema, Seth; and Kucik, Paul, “Games and Risk Analysis” (2009). Published Articles & Papers. Paper 5.