Game theory is a concept in applied mathematics that can be used to study and interpret a variety of behaviour.
We’ve all seen examples of it in popular culture – the Oscar winning 2001 film A Beautiful Mind is an account of the life of Nobel laureate John Nash who made important contributions to the subject.
In business and economics game theory is most commonly used to study aspects of decision making – particularly between competing parties in activities that involve bargaining.
Many of the games or strategies involve outcomes where co-operation, or mutual agreement benefit all parties, while over-competitive behaviour punishes all but one of the parties involved.
Game theory can most commonly be seen in business in so-called "price wars", where it would be mutually beneficial for all competing parties that manufacture or sell the same product to agree on a price.
Despite this favourable outcome, all parties enter into a battle to get a competitive edge over their rivals by lowering their prices further.
A commonly expressed facet of game theory is the zero-sum game, or nil-sum game. This is where the sum of all gains for one party, or parties, must equal the sum of all losses suffered by a counterparty, or counterparties.
Game theory can therefore be used to explain the behaviour of stock markets – essentially, a very complex zero-sum game. Single transactions do not necessarily conform to this rule, but the market as a whole does.
Examine some of the games and a hint of the type of behaviour that characterises investor sentiment emerges: holding on to losing positions, failing to maximise profit because of ill-timed exits, and failing to accurately assess the balance between risk and reward.
Let's look at a few of the classic examples of game theory that explain this type of behaviour.
1. The Prisoner's Dilemma
Game theory starts with this decision-making strategy.
Two prisoners – let's call them Convict 1 and Convict 2 – accused of the same crime are held in separate rooms and cannot communicate with each other.
They are each told that if one testifies against the other he can go free. If Convict 1 refuses to testify against Convict 2, but Convict 2 agrees to testify against Convict 1, Convict 1 will get three years in prison while Convict 2 goes free and vice versa.
They are also told that if both testify against the other, they both get a two year sentence, while if both refuse to testify, they both get one year in prison.
So imagine you're one of the prisoners. The best strategy is co-operation: if you both refuse to testify you get just one year each.
Yet research shows that most people will testify against the other – this can be for two reasons:
- In the vain hope that the other refuses and you go free
- For fear that if you stay silent, the other party testifies giving you three years
Even though the reward is greater if you stay silent, you have to be sure that your counterparty will also do so, but this is too great a risk to take.
Market example: In periods of market stress, it would be mutually beneficial for investors to stand together and hold their nerve – then the stress would soon pass. Instead the prevalent behaviour is to crowd for the exits, ensuring heightened stress and greater losses.
Very similar to the Prisoner's Dilemma, except the outcome that is mutually beneficial is the dominant strategy - one that produces the highest payoff. Here's how it works:
If both players co-operate they each get 1 point, if both refuse to co-operate they get 2 points. If you co-operate but your opponent refuses, you get 0 points and your opponent gets 3 points, and vice versa.
Non co-operation produces the best outcome, whether your opponent chooses to co-operate or not. You'll either get 3 points or 2 points.
3. Centipede Game
Two players take it in turns to either "take" or "pass". The game starts with a £2 prize. If player 1 decides to take, the prize is divided equally between the two players. If player 1 passes, the prize is upped to £3 and the decision to take or pass goes to player 2.
If player 2 now takes, he gets £2 and player 1 gets £1. If player 2 passes back to player 1, the prize is raised to £4. Should player 1 now take, the prize is again shared equally, and so on. This continues until the prize pot reaches £100 at which point it is automatically split £50 each.
Game theory suggests that player 1 takes on the very first move. Distrust rather than greed dictates the decision making.
In studies, however, the decision to take comes later on, as the pot increases.
Market example: Markets offer similar choices. Do you take as soon as you hit profit, or hold on until the pot gets bigger? Market timing is a difficult decision that can go the wrong way if you leave it too late.
4. Cournot Duopoly
This came before game theory was recognised as a discipline. Named after French mathematician Augustin Cournot, who devised this explanation of "imperfect competition" in markets in 1838.
Two companies produce identical products. If they co-operate to produce at lower levels, the limited supply will keep prices relatively high.
However, if just one produces at higher levels, the price drop will be limited, and that producer can sell more at a slightly lower price, therefore lifting his profits. The other will be selling the same amount as before at a slightly lower price, lowering his profit.
If they both decide not to co-operate and produce at higher levels, the price drops substantially. They both sell more, but at much lower prices and so their profit suffers.
Market example: This type of behaviour can be observed on a massive scale within the oil cartel OPEC, which regularly sets production quotas to limit price movements in the petroleum markets.
Game theory's relevance to investing should not be overstated, but all investors should have a strategy. If you leave things to dumb chance, you might as well be just feeding your cash into a fruit machine.
If game theory teaches us anything as investors, it should be that it is not only outside events that have an impact on markets, but even other players – doing exactly what we're doing – can have significant effects too.