An Introduction to Game Theory: Week 3

To begin, there are three main concerns about the mathematical modelling to determine human behaviour, as Game Theory does. The first is that it shouldn’t work because free will negates any effort to predict behaviour. However, while this is true, it is not an issue. Although people have free will, they are of course attracted to the behaviour that best suits them, exactly what Nash Equilibrium calculates. Secondly, some suggest we don’t need mathematical modelling because you can just ask the person why they behaved as they did. However, there are two problems with this. They may lie about what their motives were. Furthermore, they may not be able to articulate their reasons for why they behaved as they did. Thus, game theory uncovers the mechanism behind such intuitive behaviour. Thirdly, some state it simply doesn’t work. In many cases, however, it does. There are swathes of evidence, if you’d like to ‘google’ it! Of course, it is not always accurate. In such cases, however, it does provide a useful starting block on which to base analysis on, to answer very general questions, questions such as ‘what policies will the republican and democrat parties (political parties in the USA) implement?’, for example.

So, what exactly are pay-offs, as I’ve mentioned in the previous two weeks? Well, a pay-off could signify the profit accumulated, satisfaction felt, or any other benefit. How are numbers assigned to a particular pay-off? This varies depending on the situation. If there is no uncertainty (i.e. no random events, a thunderstorm for example, which can affect the outcome) it is very simple. Simply rank the outcomes from best to worst and assign numbers to reflect that. The bigger the pay-off, the better the outcome. The distribution does not have to be linear, as long as a bigger number equates to a better outcome. When there is uncertainty, assigning pay-offs to particular outcomes is more tricky. The pay-off depends on the player’s attitude to risk. To take an example, a player is given two options. The player can take a lottery, with ½ a chance of winning £10, or he can settle for a guaranteed £5. Here, the average payment is £5 for each. Thus, we have to represent the player’s attitude to risk to determine which option will be chosen. So, we assume the player gets ‘utility’ rom the outcome, and they aim to maximise their expected utility. If the player likes taking risks, his expected utility gained will be higher if he chooses the lottery, and vice versa. To sum up, pay-off has to be adjusted, in situations in which uncertainty is present, to account for the player’s attitude to risk.

What does it mean to be rational, exactly? Well, a player is rational if they are three things; aware of all possible events, able to assign (objective or subjective) probabilities to each event and can maximise expected utility. However, with games involving human interaction, there is a problem, as I have previously mentioned. Repeated assumptions of others behaviour leads to the problem of infinite regression. And this hierarchy of sophisticated reasoning (my belief about your belief of my belief…) can lead to surprising outcomes.

First though, there are strategies that are obviously ‘bad’ or ‘good.’ Some strategies are better than another, irrespective of what the other player does. To take the game of ‘Prisoners Dilemma’ as an example, defection is always a better strategy than cooperation. Thus, we say defection ‘strictly dominates’ cooperation. If a strategy is never worse than another strategy, but is occasionally equally as good as that strategy, we say it ‘weakly dominates.’

A situation in which common knowledge of rationality is present can lead to a Nash Equilibrium being reached, even without correct beliefs of behaviour. Common knowledge of rationality means that the players are rational, they know the other players are rational, and so on. To prove the above statement, let’s use Hoteling’s location game, in which there are 100 places to set up shop in a street, and the Nash Equilibrium is to take up residence in the middle of the street, at point 50. The further from the middle of the street you set up shop, the worse your strategy is (as shown in week 1). Now, if players (A &B) are hyper rational, they will choose Nash Equilibrium. A won’t choose the end points, as those are dominated strategies. 1 and 99 may be good strategies, if B is irrational and chooses an end point. However, B knows A is rational, and as such will not choose an end point. Thus, now 1 and 99 are dominated strategies, and 2 and 98 are the next proposed strategies. This cycle continues, until you reach the Nash Equilibrium at point 50. So, to conclude, common knowledge of rationality, even when without correct beliefs of behaviour, can lead to a Nash Equilibrium.

On the other end of the scale, what happens when players are not smart, and have low rationality? A Nash Equilibrium can still be reached, albeit more slowly. Overtime, players gain experience, and observe others. Thus, they try various actions to find a better strategy: trial and error adjustment. By repeated elimination of strictly dominated strategies, as they gain experience, a Nash Equilibrium is reached. So, surprisingly, low-rationality adjustment leads to a Nash Equilibrium using the same basic logic as in a hyper rational situation.

Does Game Theory apply even in zero-intelligence situations? It can. While humans use trial-and error adjustment to reach a Nash Equilibrium, biological evolution involves a similar process, revolving around the ‘survival of the fittest’ concept. A mutation is the equivalent of a ‘trial.’ To formulate biological evolution as a game, firstly you have each animal, carrying a particular gene, as the players. Thus, a gene is a player. The strategies are a particular physical or behavioural characteristic, or a gene that causes the particular characteristic. As such, in this game, players and strategies correspond. The pay-off is equal to the number of off-spring produced. Eventually, the dominant strategy (characteristic) comes to dominate the population, until only those with the dominant strategy survive. At this point a Nash Equilibrium is reached. No player can profit from deviating to other strategies (when the best strategy is dominant).