Nash Equilibrium

In this post, I introduce a topic of great interest to myself ‘Game Theory’ while also examining the importance of complexity in the replayability of games. I have chosen complexity as for me it is the mental exercise of attempting to preempt my opponent’s decisions that make a game experience enjoyable. An example of a game that does this well is chess, there is no element of chance and the level of complexity means there is no obvious best move. Tic Tac Toe on the other hand again doesn’t involve chance but the number of solutions is so small that it is relatively easy to see what the best possible move is and from then on every game becomes a tie. One of the differences between the two is the number of choices a player is afforded and how in chess that number rises exponentially the more turns a player tries to predict.

Game Theory was developed by Von Neuman and Oskar Morgenstern and then pioneered by John Nash who was made famous in the movie ‘A Beautiful Mind’ (Ross, 2016). Before Nash fell prey to schizophrenia he formulated the idea of Nash Equilibrium which is now one of the fundamental building blocks of game theory (Binmore, 2008).

“This notion captures a steady state of the play of a strategic game in which each player holds the correct expectation about the other players’ behaviour and acts rationally.” (Osborne and Rubinstein, 2016, p.14)

Rationality in game theory is borrowed from its use in economics and is used here to say, that players in a game assess all outcomes and make their decisions based solely of maximising their utility (i.e. maximising their chance of winning) (Ross, 2016). Utility here is another term from economics which is an abstract numerical value assigned to all the possible outcomes of a game for an individual player, where higher values are assigned to more favourable outcomes (Binmore, 2008). So a Nash equilibrium is reached when both players know the best possible strategy for them to win or to stop their opponent from winning resulting in all games looking identical.

Taking the example of tic tac toe it is clear that if both players plan their moves strategically they will reach a Nash Equilibrium and draw every game. Whereas in chess “there are more possible chess games than the number of atoms in the universe” (Kasparov, 2010), meaning even a computer couldn’t come close to calculating every possible solution. The connection I am drawing here is that if a Nash equilibrium can be reached then a game won’t be replayable as the strategies for players involved will be fixed from the start. Or as Roger Caillois puts it: “An outcome known in advance, with no possibility of error or surprise, clearly leading to an inescapable result, is incompatible with the nature of play.” (Caillois, 2001).