Charles Adams – Game Theory: Rock Paper Scissors

This is not another article about Brexit! Just get past the first paragraph and then it takes off.

Charles Adams is professor of physics at Durham University in the UK and blogs at

Cross-posted from progressive pulse

I never really liked the game rock-paper-scissors, because it seemed, well, ‘kind of random’. But there is another story that is rather interesting. This simple game provides us with an abstract framework to walk through complex questions such as whether the supreme court ruling on the prorogation of the UK parliament was appropriate or not.

The essence of the game – the essential feature that allows it to work – is that  no entity – rock paper or scissors – is omnipotent. If rock could crush both paper and scissors then a player with rock can destroy all opponents. In politics, this would correspond to a dictatorship. The dictator has the rock and always wins, until there is a violent revolution. A better way to incentivise the potential of all players in the system – and avoid violent conflict – is to limit the power of the leader. The game of rock-paper-scissors provides an example of how this can be implemented. In rock-paper-scissors power is distributed equally three ways [1]. If two players choose randomly, they both have an equal probability of winning, which in game theory, is known as the Nash equilibrium, after John Nash. It is an odd game as your optimal strategy is to be as random as possible. If your opponent is not perfectly random and shows some bias, like Bart in the Simpsons then it is easy to beat them, as Lisa demonstrates. However, for the purposes of this piece my message is not so much about randomness, as about how to achieve a dynamic equilibrium in a complex multi-agent system.

For example, on the question of how to organise society, we have many competing interests and need some fair mechanism to settle disputes.  The French political philosopher, Montesquieu, suggested that one way to achieve this is an analogous three-way separation of powers, the trias politica. In the trias politica, we have an executive, a legislature and a judiciary, which in the UK are roughly the government (led by the Prime Minister and the cabinet), parliament, and the judges. As an aside, I find it odd that I was not taught about this in school. When Gina Miller emerged from the court room last week and spoke about the separation of powers I suspect that only a very small percentage of the population had any idea what she was talking about. And that is why we need to spend more time playing rock-paper-scissors, watching the Simpsons, and learning about Nash and Montesquieu.

Now, unfortunately, there is not a nice simple one-to-one mapping between the trias politica and rock-paper-scissors. The real world is, of course, much more complex. In the real world, the executive emerges from the legislature, and the executive is dependent on the legislature for its power (see e.g. Peter’s recent posts here and here). But the game does at least provide us with a simple framework. In normal circumstances, we could argue that the executive controls the legislature (by proposing laws) who controls the judiciary (by making laws) so in order to maintain our dynamic equilibrium we have to ensure that the judiciary are able to limit the power of the executive. That was what the supreme court judgement was all about. If you are able to see this in terms of importance of dynamic equilibrium, then you are also more likely to see why the judgement of the supreme court was procedural rather than political, and why it is not such a good idea, as in USA, to allow the executive to choose the judiciary.

Once you are inducted into the world of rock-paper-scissors, you can start to think of other cases of dynamic equilibria where similar principles might be useful. My favourite is the tension between democracy and capitalism. Imagine they are the players in the game where rock-paper-scissors are wealth, political power and freedom. As in the case of executive power, it is easy to recognise that something has to reign in the power of wealth. But that is an story for another day.

[1] Physicists may also be familiar with other fascinating cases of
 tripartite systems with balanced pair-wise interactions such as three spins arranged in an equilateral triangle.
 Physics allows for a completely non-classical solution known as quantum
 entanglement where the binary choice win or lose is replaced by a 
quantum superposition of win and lose.

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