Introducing Game Theory: A Graphic Guide (Graphic Guides)

by Ivan Pastine

Game theory is a set of tools used to help analyze situations where an individual’s best course of action depends on what others do or are expected to do. Game theory allows us to understand how people act in situations where they are interconnected.

Game theory is useful whenever there is strategic interaction, whenever how well you do depends on the actions of others as well as your own choices.

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.

creating simplistic structures, called models. Models are simple enough to analyze but still capture some important feature of the real-world problem.

quickly learn to reason via backward induction: you can figure out your opponent’s response to your possible actions and take that into consideration before making your own move.

very simplified models, which are called games.

Rationality refers to players understanding the setup of the game and exercising the ability to reason.

I also need a second level of knowledge: I have to know that you know that I am rational. I need a third level of knowledge as well: I have to know that you know that I know that you know I am rational.

Keynes’ Beauty Contest,

likens investment in financial markets to a newspaper competition in which readers have to choose the “prettiest face”; the readers who choose the most frequently chosen face win.

If you can predict the behaviour of the average investor, you can make a killing.

iterative elimination of dominated strategies.

A strategy is dominated if it

is worse than another strategy

regardless of what other ...

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.

in financial markets we observe bubbles – excessively inflated prices – even if all participants are rational. This is because of a lack of common knowledge of rationality.

Often players do not know the actions of the other players when they make their own decisions. Games with this feature are called simultaneous-move games.

strategic interaction.

strategic (or normal) form of the game. The strategic form of a game is a table which is also known as a payoff matrix.

The payoff numbers in the table represent the total payoff that the players get from each outcome: in a particular outcome a player might benefit directly, as well as indirectly from hurting or helping others. The payoff numbers include everything they care about.

Nash equilibrium

he applied it to mathematical analysis of games in general,

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

This is a simultaneous-move game: even if the prisoners’ decisions are not literally simultaneous, we can think of them that way because the players are in separate rooms and so neither player knows the other’s decision when making his own choice.

In the Prisoners’ Dilemma, both players confess in the Nash equilibrium.

Pareto efficient

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

One solution to this problem is to lower the transmission power of both routers so that they are no longer within range of one another. But if only one router is using low power, then its signal is overwhelmed by the high-power router.

If both routers are part of the same network then it is possible to force them both to use low-power mode to minimize the conflict. Most routers have “advanced” settings, which are there to force them to cooperate with other routers on the network rather than to compete aggressively for resources.

tragedy of the commons,

“commons” has evolved to encompass any shared resource.

The Nash equilibrium of the game, therefore, is global nuclear build-up.

The Nash equilibrium outcome in the nuclear arms race is not Pareto efficient because both countries would be better off if neither engaged in nuclear build-up.

although there is benefit to cooperative behaviour, individual incentives encourage conflict.

the Roommate Game. This game both emphasises the theory’s broad applications and provides a framework to think about how social norms can help to overcome individual incentives for excessive conflict.

The Nash equilibrium of the game is {don’t clean up, don’t clean up} because if either expects her roommate not to clean up, the best response is not to clean up.

free-rider problem

One way out of the free-rider problem is to change the payoffs in the payoff matrix.

impose a moral cost when engaged in non-cooperative behaviour

a moral cost can change the equilibrium

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

International cooperation on environmental protection is like the Roommate Game writ large.

One way to view environmental activism is as an effort to change the social norms.

In games with multiple Nash equilibria, the concept of Nash equilibrium by itself does not provide us with sufficient tools to predict what will happen.

The game might seem pretty mundane and based on outdated stereotypes, but it’s a useful illustration because the same form of incentives appear in many different situations.

coordination failure due to misaligned expectations. In this case the game theorist would observe an “out-of-equilibrium” outcome