If one has seen the movie “Beautiful Mind”, one will never forget the scene where John Nash (role played by Russell Crowe) finds the best strategy to guarantee a pleasant night to him and all his friends. In fact, his thinking revolutionized the science of strategy. Since its beginning, Game Theory (a multidisciplinary field) has become an intrinsic part of Microeconomics, and we will now take a look into what it is about.
Please enjoy the video below (until 5:30):
The Initial Game
There are two players (player 1 – Big Guy; and player 2 – Small Guy) – please do not mind the typo (Smal) in the diagrams. Each player has two possible strategies (either they ‘split’ or they ‘steal’). There are £13.600 to be split, taken entirely by one of the players or simply to vanish (we get the payoffs from here). Both players know the possible strategies of the other player and the respective payoffs, so we are in the presence of a complete information game. Given this, here is the initial game:
By definition, a Nash Equilibrium is the optimal outcome of a game. In a NE, no individual can receive incremental benefit from changing his strategy, assuming the other player remains constant in his strategy.
So, is there any NE in this game? Where?
Let’s imagine Big Guy plays ‘split’. If so, Small Guy is better off playing ‘steal’ and taking the whole money. Now, if Big Guy plays ‘steal’, the Small Guy is indifferent between ‘split’ or ‘steal’ since he will get nothing in both outcomes. Because of this, one can say that for the Small Guy, ‘steal’ is a weakly dominant strategy. The same applies to the Big Guy (try thinking about that on your own) and, thus, given that ‘steal’ is a WD strategy for both players, (‘steal’,’steal’) is a NE.
Now look at the bottom-left outcome (Big Guy plays ‘split’, Small Guy plays ‘steal’):
- Given that Big Guy is playing ‘split’, could the Small Guy be better off by changing from ‘steal’ to ‘split’? No, his payoff decreases.
- Given that Small Guy is playing ‘steal’, could the Big Guy be better off by changing from ‘split’ to ‘steal’? No, his payoff is zero in both outcomes.
So, again, we found a NE = (‘split’,’steal’). Try on your own to understand why (‘steal’,’split’) is also a NE.
As we can see, there are three NE in this game. The only outcome that is not a NE is shadowed in yellow.
The (‘split’,’split’) outcome can be considered the fairest and most rational one but, in fact, it is the less likely to happen. Why? Because if a player knows the other is going to play ‘split’, he has a great incentive to deviate from his strategy, play ‘steal’ and take all the money. So, it is clear that the game is set up in the way that both players have the incentive to ‘steal’ and, in the end, making it funnier to watch.
“Small Guy, I want you to trust me 100% that I am going to pick the ‘steal’ ball” – said the Big Guy
Well, as we saw in the video, players ended up reaching the only outcome that is not a NE. Why was that? It only happened because of the previous statement that, after all, changed the whole game.
After such statement, the Small Guy is facing an entirely different game. Moreover, the Big Guy added “I want you to do ‘split’ and I promise I will split the money with you after the show” to which the crowd reacted with laughter. In any case, if Big Guy is playing ‘steal’, the Small Guy knows he will get nothing independently of what he plays. But with the promise made by the Big Guy, the Small Guy can attribute a probability (p) to the truthfulness of the statement. If Big Guy is actually speaking the truth, playing ‘split’ is the obvious choice for the Small Guy (hence, he will get a positive payoff instead of picking ‘steal’ and getting zero for sure).
To everyone’s amusement, Big Guy was lying. He did not play ‘steal’ as he said he would. Instead, he played ‘split’ knowing that he had forced the Small Guy to desperately pick ‘split’.
One can infer that, in spite of lying, the Big Guy was being truthful when saying they would split the money. He could easily go for the ‘steal’ and keep the money for himself, but he did not (of course, the pressure of being broadcasted in live television counts).
Strategic behaviour can be quite fun, right?