Introducing Game Theory: A Graphic Guide (Graphic Guides)

by Ivan Pastine

Because game theory can help analyze any environment where a person’s best action depends on others’ behaviour, it has proven useful in a wide variety of fields.

Game theory is the study of strategic interaction. Strategic interaction is also the key element of most board games, which is where it gets its name. Your decision affects the other player’s actions and vice versa. Much of the jargon of game theory is borrowed directly from games. The decision makers are called players. Players make a move when they make a decision.

In human interaction, for instance, it’s not just our decisions, but also our expressions, our tone of voice and our body language that influence others.

We can circumvent this complexity by creating simplistic structures, called models. Models are simple enough to analyze but still capture some important feature of the real-world problem. A cleverly chosen simple model can help us learn something useful about the complex real-world problem.

One feature of complex board games like chess is that the more skilled the players are, the more frequently the game ends with a draw. How can we explain this observation?

Once players learn to reason via backward induction, all noughts & crosses games are likely to end in a draw.

its aim is to improve our understanding of interactions between people, companies, countries, animals, etc., when the actual problems are too complex to fully understand.

The creation of a useful model is both a science and an art. A good model is simple enough to allow us to fully understand the incentives motivating players. At the same time, it must capture important elements of reality, which involves creative insight and judgement to determine which elements are most relevant.

Game theory usually assumes rationality

Common knowledge of rationality is a more subtle requirement. Not only do we both have to be rational, but I have to know that you are rational.

Game theorists solve the Guessing Game in a similar fashion using iterative elimination of dominated strategies.

Human behaviour is probably better approximated by bounded rationality. That is, human rationality is limited by the tractability of the decision problem (how easy it is to manage), the cognitive limitations of our minds, the time available in which to make the decision, and how important the decision is to us.

The idea of Nash equilibrium is both simple and powerful: in equilibrium each rational player chooses his or her best response to the choice of the other player. That is, he or she chooses the best action given what the other player is doing.

One of the features of Nash equilibrium is that it is regret free.

The Nash equilibrium is also a rational expectations equilibrium.

The most well-known game theory paradox is the Prisoners’ Dilemma.

The game illustrates the difficulty of acting together for common or mutual benefit given that people pursue self-interest.

An outcome is Pareto efficient if there is no other potential outcome where somebody is better off and nobody is worse off.

The network router problem is closely related to the tragedy of the commons, a concept conceived by William Forster Lloyd

farmers may be acting in self-interest, contrary to the best interest of the whole group, and depleting the feeding potential of the common land.

One way out of the free-rider problem is to change the payoffs in the payoff matrix. Early childhood parental involvement and schooling can impose a moral cost when engaged in non-cooperative behaviour (such as leaving dishes in the sink).

But in their social interaction, a moral cost can change the equilibrium by encouraging both girls to behave more cooperatively.

There is a Pareto improvement in the outcome of the Roommate Game due to moral values;

The belief that there will be a bank run is a self-fulfilling expectation: the expectation itself causes the bank run.

What makes it interesting for game theory is that it has no equilibrium where players behave predictably. If a player is predictable, the other player will exploit that and win. So, players try to be unpredictable; the game does not have a pure-strategy Nash equilibrium.

mixed-strategy Nash equilibrium. This means that in equilibrium the players randomize over possible pure strategies: “rock”, “paper” and “scissors”.

critics of mixed strategies argue that randomization is not a reasonable description of human behaviour.

Harsanyi points out that even if players play pure strategies, if they are slightly uncertain about each other’s payoffs, from the outside they will seem as if they are randomizing between actions.

from the individual’s point of view the other player’s chance of choosing a particular action is exactly the probability we get in a mixed-strategy Nash equilibrium without uncertainty about payoffs.

thinking about more realistic settings with repeated interaction, where players play the same game again and again.

With an infinite horizon, backward induction does not unravel cooperation from the last round, since there is no certain last round.

Both players playing the grim strategy can be a Nash equilibrium in a repeated Prisoners’ Dilemma type of game if players are patient enough (if they are able to resist the temptation of a high payoff today in order to be able to collect cooperative payoffs in the future). In this case, punishment for defection can deter the players from non-cooperative actions.

if collusion breaks down, both players have an incentive to renegotiate, ignore the deviation and simply start colluding all over again.

The experiment’s results were broadly in line with game theory. Cooperative outcomes were often recorded as long as the end of the game was not in sight. But as time was running out and the end of the game drew near, players started to defect and mutual coordination broke down.

They often think of people or animals as being socially or genetically programmed to engage in certain behaviours, which may or may not be based on reason.

This long-run evolutionary “steady state” with the proportion of hawks in the population equal to 5/6 is called an evolutionarily stable equilibrium. It is an equilibrium which is stable in the sense that if we add a small number of animals with different conditioning, evolutionary forces will eventually restore the equilibrium.

But some games have more than one evolutionarily stable equilibrium.

Oddly enough, the evolutionarily stable proportion of hawk-types (5/6) is also equal to the equilibrium probability in the mixed-strategy Nash equilibrium of the game if the animals were choosing their strategies rationally.

Subgame perfection implies that players are forward-looking. They do the best they can at each decision node they encounter without either holding grudges or developing goodwill for past actions.

There are many games where making an early commitment puts a player at a disadvantage.

This is called the time-inconsistency problem: the decision maker does not find it optimal to follow the original action plan.

Financial markets often use collateral as a commitment device.

Due to the difficulty of coming up with a commitment device, the poor stay poor, while the rich get richer.

microcredit (small loans) in a pooled manner – loans to a connected group of people rather than to an individual.

There are also situations where players have imperfect information about the game tree: players’ past actions may be either unobservable or imperfectly observable.

Every person on the committee has transitive preferences and votes sincerely. But when acting as a group the committee’s preferences are non-transitive – whatever it chooses, the group will always think that another option is better.

Arrow’s Impossibility Theorem makes a lot of the strange behaviour we see in committee meetings and in parliaments more understandable. For instance, in committee work we frequently see the same issue coming up over and over again.

The development of game theory as a discipline has provided an extensive toolkit which allows us to explore conflict and cooperation in much greater depth.

players often have to choose from continuous options. In these situations, the game theory logic is exactly the same, but the presentation becomes more mathematical.