What is an equilibrium?
Definition and explanation
At the heart of it, an equilibrium is simply a system in which all the opposing forces acting on it are balanced, and the system is stable. Think of two tug-of-war teams heaving on the rope with all their might. There’s a whole lot of force going through that rope in opposite directions...but nobody’s going anywhere. There are many examples of equilibria in the natural sciences, economics and game theory.
Examples of equilibria
Nash equilibrium
The Nash equilibrium was devised by a mathematician called John Forbes Nash Junior. This equilibrium describes a situation in a multiplayer game which has come to a standstill.
In this standstill situation no player can make a better move than the one that they are currently making, with the caveat that all other players stick to their guns and don’t change their own strategy.
But since everyone is already playing their best move possible given what everyone else is doing, no one is going to change their strategy: hence we’re locked into a standstill.
Coffee shop example
OK let’s try and make this relatable.
Ever wondered how you can stroll across town looking for a coffee, not find a single shop for ages...and then two Dunkin’ Donuts and a Starbucks are all crammed on the same corner? Surely it would be better for everyone if the shops were more evenly spread?
As it turns out, this would be better for customers; it would minimise the maximum distance that any one person has to walk to a shop (what game theorists would call the ‘Socially Optimal Solution’ - more on this later). This would be okay for the shops too, but it leaves the door open for competition: shops can gain from a change in their strategy (i.e. we are not a Nash Equilibrium).
Let’s see how this works:
Imagine you’re selling coffees out of a kiosk on a high street; the best place for you to be located is in the centre, so that half of the customers are on one side and half are on the other. Now, a competitor comes along and you both agree to move so that you’re each positioned 25% and 75% along the length of the high street respectively (the Socially Optimal Solution). This allows both of you to receive customers from the ends of the high street nearest you, and customers from the centre (in this simplified scenario people are evenly distributed down the street, so you both receive the same number of customers).
But, without you realising it, the other coffee seller has moved slightly closer to the centre, edging towards your stall, and stealing some of your customers from the middle of the high street! In retaliation, you move closer to them, recouping some of your lost customers. They move yet closer to you, and again you are forced to move closer. You realise you can undo all of their dirty work by moving to the other side of them, closer to the centre etc...
We have reached a Nash equilibrium. Both stalls have ended up in the middle of the high street, and it is no longer beneficial for anyone to move. Remember the Socially Optimal Solution from earlier? Unfortunately we are now no longer generating this outcome, because people at the very ends of the high street have to walk all the way to the centre for their cup of delicious caffeine.