A Brief Introduction to the Basics of Game Theory

Matthew O. Jackson

Stanford University – Department of Economics; Santa Fe Institute; Canadian Institute for Advanced Research (CIFAR)

Abstract:

I provide a (very) brief introduction to game theory. I have developed these notes to provide quick access to some of the basics of game theory; mainly as an aid for students in courses in which I assumed familiarity with game theory but did not require it as a prerequisite.

I provide a (very) brief introduction to game theory. I have developed these notes to provide quick access to some of the basics of game theory; mainly as an aid for students
in courses in which I assumed familiarity with game theory but did not require it as a prerequisite. Of course, the material discussed here is only the proverbial tip of the iceberg, and there are many sources that offer much more complete treatments of the subject.1 Here, I only cover a few of the most fundamental concepts, and provide just enough discussion to get the ideas across without discussing many issues associated with the concepts and approaches. Fuller coverage is available through a free on-line course that can be found via my website: http://www.stanford.edu/jacksonm/

The basic elements of performing a noncooperative 2 game-theoretic analysis are (1) framing the situation in terms of the actions available to players and their payoffs as a function of actions, and (2) using various equilibrium notions to make either descriptive or prescriptive predictions. In framing the analysis, a number of questions become important.

First, who are the players? They may be people, firms, organizations, governments, ethnic groups, and so on. Second, what actions are available to them? All actions that the players might take that could affect any player’s payoffs should be listed. Third, what is the timing of the interactions? Are actions taken simultaneously or sequentially? Are interactions repeated? The order of play is also important. Moving after another player may give player i an advantage of knowing what the other player has done; it may also put player i at a disadvantage in terms of lost time or the ability to take some action. What information do different players have when they take actions? Fourth, what are the payoffs to the various players as a result of the interaction? Ascertaining payoffs involves estimating the costs and benefits of each potential set of choices by all players. In many situations it may be easier to estimate payoffs for some players (such as yourself) than others, and it may be unclear whether other players are also thinking strategically. This consideration suggests that careful attention be paid to a sensitivity analysis.

Once we have framed the situation, we can look from different players’ perspectives to analyze which actions are optimal for them. There are various criteria we can use.

Game Theory: Games in Normal Form

Let us begin with a standard representation of a game, which is known as a normal form game, or a game in strategic form:

  • The set of players is N = {1, . . . , n}.
  • Player i has a set of actions, ai, available. These are generally referred to as pure strategies. This set might be finite or infinite.
  • Let a = a1 × · · · × an be the set of all profiles of pure strategies or actions, with a generic element denoted by a = (a1, . . . , an).
  • Player i’s payoff as a function of the vector of actions taken is described by a function ui : A ! IR, where ui(a) is i’s payoff if the a is the profile of actions chosen in the society.

Normal form games are often represented by a table. Perhaps the most famous such game is the prisoners’ dilemma, which is represented in Table 1. In this game there are two players who each have two pure strategies, where ai = {C,D}, and C stands for “cooperate” and D stands for “defect.” The first entry indicates the payoff to the row player (or player 1) as a function of the pair of actions, while the second entry is the payoff to the column player (or player 2).

Game Theory

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