# Complexity Explorer Santa Few Institute

## Competition, Cooperation and Game Theory

16 Mar 2017

In advance of his upcoming two-part tutorial, we asked instructor and Santa Fe Institute postdoctoral researcher Justin Grana about the subject matter, applications, and excitement of Game Theory.

What is game theory and what are you covering in this tutorial?

Game theory is the standard quantitative tool for analyzing multiple, interacting decision makers.  This is distinct from a decision maker in isolation because each decision maker must anticipate what the other decision makers might do.  A canonical example of such reasoning can be found in basic economic modeling.  A supplier decides a price to sell a good and a potential customer decides whether or not to buy the good.  Of course, the benefit to the supplier depends on the price they set and the buyer’s decision to buy, while the buyer’s benefit depends on the price paid if they choose to buy the good.  In a sense, the seller and the buyer are interacting because one’s decision affects the other’s benefit.

In this course, we will delve into the basic mathematical tools and definitions for analyzing such interacting decision makers and determining their choices. We’ll also illustrate how to apply such logic to a several stylized but realistic scenarios.

Now I understand what game theory is, but how is it related to complex systems science?

It is easy to argue that social systems consisting of humans are the most complex systems we study.  Furthermore, humans are constantly interacting, making decisions and anticipating how other humans will act.  If we want to understand such systems, we need to understand how the incentives humans face shape their actions. Game theory provides us a tool for doing just that.  Game theory is not particular to complex systems, per se. Instead, it is a tool that can be used to building complex models that have a decision making component.  For example, if one wishes to build a complex model of a stock market, one could use game theory to model each individual’s decision to either buy or sell a stock.

Can you tell us about some recent research in game theory that has interested you?

Currently, I am working on several game theory projects that I find interesting. In one project, I am using game theory to investigate the differences between lobbying and bribing. In another project, I am specifically focusing on the importance of timing in games.  For example, in a factory with a ‘boss’ and a ‘worker’,  how does the frequency with which the boss can observe the worker determine how much effort the worker expends? This general question is called the principal-agent problem in formal game theory.  We won’t be going over this but I’ve included it in the course supplementary materials.  I’m also deeply involved with applying game theory to problems in cyber security.  In fact, we will go over an example in this tutorial!

There are other researchers affiliated with the SFI that are also doing interesting work in game theory.  David Wolpert has used game theory to derive the value of information in games.  Matthew Jackson is an expert in game theory and network formation.

So what else does your tutorial cover?

The tutorial is split into two parts.  In the first part, I focus on the standard definitions that are ubiquitous throughout game theory literature.  This includes definitions of strategies, payoffs, expected utilities and the Nash equilibrium.  The rest of the first part of the tutorial provides several canonical examples of game theory that one would expect to encounter in many first courses on game theory.

In the second part, we talk about dynamic games---a game that takes place over time.  These games are drastically different than standard static (or one-shot) games.  We will go over how to represent dynamic games, solve dynamic games and provide several examples.  In the second part of the tutorial we also introduce the concept of imperfect information, which is simply a case where some elements of the game are unknown to the decision makers.  We will conclude the second part of the tutorial with behavioral game theory, which relaxes the assumption that decision makers always make the right choice.

How quantitative are the tutorials?  Do students get to practice anything?

With the exception of one (optional) example, the tutorials only require basic algebra.  However, this does not mean the math is easy. In fact, a main challenge in game theory is understanding how the relatively simple math accurately represents the game. That said, these tutorials will not focus on building new mathematical tools but be heavy on explaining how the math relates to the elements of the game.

What should a student expect after completing the tutorials?  What should the student do if they want more information after completing the tutorials?

After completing the tutorials, a student should expect to be familiar with many of the game theoretic concepts that one would find in an advanced undergraduate course that uses game theory (microeconomics, multi-agent systems, for example).  We’ll also examine more in-depth some applications of game theory in some later videos in the course.

For a student looking for more information, I have provided a list of resources in the supplementary materials sections of the tutorials.  I’ve included some of the most popular textbooks for learning game theory and most likely to be used in any pure game theory or graduate level microeconomics course.

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