Mathematical Modelling 101 – Intro to Game Theory

This post is going to just be a very brief introduction to what Game Theory is, how it works and some basic terminology. In later posts I will get more advanced and cover how it can be applied to Cultural Evolution.

What is Game Theory?

Game theory is a branch of applied mathematics most commonly used in Economics. However, it can be very successfully applied to other social sciences as well as Evolutionary Biology. It gives both descriptive answers (what people do) and prescriptive answers (what people should do) in a given game.

Why is this relevant?

Game Theory is a very good tool in predicting outcomes, not only in the very simple games covered in this post, but also in predicting the outcomes of evolutionary strategies and of predicting outcomes for signalling games which can inform us on human and animal communicative strategies.  Running iterated games over populations can introduce interesting qualifications to these very simple ideas as well and explain some things which may, at first, appear maladaptive, as such is a very useful tool in bypassing our intuitions. It is of these things that the next few posts will explore.

What is a Game?

A game needs 3 things before game theory can be applied:

1. Players

(Note the plural, when there is only one player the theory applied would be decision theory.)

2. Strategies

3. Payoffs

How does Game Theory work?

A simple example of a coordination game:

The Driving Game

left right
left 100, 100 0, 0
right 0, 0 100, 100

n (no. of players) = 2

Imagine two cars driving along a road towards each other. Each driver has the choice of whether to drive on the right or the left. The table represents all possible outcomes. If both players decide to drive on the left (or right) then they will pass each other and each get a payoff of 100, however if the players decide to drive with one on the right and the other on the left then they will collide and both players get a payoff of 0.

What should the players do? In the situation of a one off game where the players are not aware of the other’s decision the choice is 50/50. In an iterated game, however, the tendency will be to choose either left or right and stick with it, this will allow the other driver to co-ordinate so both players get the maximum payoff. Once both players are either playing left every time or right every time neither will switch because both know that switching would probably result in quite a nasty crash. These strategies, where nobody has anything to gain from switching, are know as Nash Equilibria (after John Nash who proposed it). So the driving game has 2 Nash Equilibria (left, left and right, right). (It has 3 actually but I’ll get to that in a later post).

Nash Equilibria

So, as I’ve stated above, a Nash Equilibrium is a situation in which neither player can benefit from changing their strategy. Nash Equilibria are self-enforcing, every game has at least one Nash Equilibrium, although some have more than one, in these instances one of the equilibria may be inefficient (we’ll see why in a later post).


The Love Game