Elements of Strategic Games

What Makes a Situation "Strategic"?

Before we dive into the formal definitions, let's clarify what we mean by a "strategic" situation. In essence, it's any scenario where the outcome of your decision depends not only on your own choice but also on the choices of other rational, aware individuals.

The following video provides an excellent introduction to this core idea, contrasting strategic settings with non-strategic ones you might have encountered in classical economics.

This concept of mutual awareness and interdependent outcomes is the central problem that game theory is built to solve. To analyze such problems rigorously, we need to abstract them into a mathematical object.

The Formal Definition of a Strategic Game

The standard model for a static, simultaneous-move game is called a strategic form game. Given your background in applied mathematics, you'll find the formal definition to be a familiar way of structuring a problem. A strategic form game is defined as a triplet.

Let's break down each of these three components.

1. Players ($N$ or $I$)

This is the most straightforward component: the set of decision-makers in the game. We typically denote the set of players by $N=\{1,2,...,n\}$, where $n$ is the total number of players.

2. Strategy Sets ($S_i$)

For each player $i\in N$, there is a non-empty set $S_i$ of available actions or strategies. A strategy $s_i\in S_i$ is a complete plan of action for player $i$. In the static games we're considering now, a strategy is simply a single action chosen from the set of possibilities. As we'll see later in the course when we discuss sequential games, the concept of a "strategy" can become more complex, encompassing a plan for how to act in every possible contingency.

A strategy profile, denoted $s=(s_1,s_2,...,s_n)$, is a tuple specifying one strategy for each of the $n$ players. It represents one possible outcome of the game. The set of all possible strategy profiles is the Cartesian product of the individual strategy sets: $S=S_1\times S_2\times ...\times S_n$.

The following video segment illustrates these ideas with a practical example called the "grade game."

Strategy sets can be finite, like $\{\text{Alpha, Beta}\}$ or $\{\text{Rock, Paper, Scissors}\}$, or they can be infinite. For example, in a price-setting game, a firm's strategy set could be any non-negative real number, $S_i=[0,\infty )$. We will explore such games when we analyze Cournot and Bertrand competition models.

3. Payoff Functions ($u_i$)

For each player $i$, the payoff function $u_i:S\to \mathbb{R}$ assigns a numerical value (a utility) to every possible strategy profile $s\in S$. This number represents the desirability of that outcome for player $i$.

This is a critical point: a player's payoff depends not only on their own strategy but on the entire strategy profile—that is, on the combination of choices made by all players. The core assumption of rational choice is that each player $i$ will choose their strategy $s_i$ to maximize their own payoff $u_i(s)$, given their beliefs about the strategies of the other players.

The concept of a payoff function is what breathes life into the game. Two games with identical players and strategy sets can be completely different if their payoff functions vary. The next video segments demonstrate this powerfully.

This illustrates that to define a game, it is not enough to know who is playing and what they can do. We must also know what they want, which is precisely what the payoff functions describe.

An Example: Matching Pennies

Let's solidify these concepts by formally defining a classic game, "Matching Pennies," using the notation we've established.

In this example:

• Players: $N=\{1,2\}$
• Strategy Sets: $S_1=S_2=\{\text{Heads, Tails}\}$
• Payoff Functions: These are defined by the matrix. For example, $u_1(\text{Heads, Tails})=-1$ and $u_2(\text{Heads, Tails})=1$. The matrix is simply a convenient way to represent the output of the payoff functions for all four possible strategy profiles.

For a handy reference, the document you just read contains a useful summary table of the standard notation used in strategic form games.

Conclusion

In this lesson, we have established the formal foundation for analyzing strategic games. We moved from an intuitive understanding of a "strategic situation" to a precise mathematical definition.

The key takeaways are:

• A strategic form game is formally defined by three components: a set of players ($N$), a collection of strategy sets ($\{S_i{\}}_{i\in N}$), and a collection of payoff functions ($\{u_i{\}}_{i\in N}$).
• A strategy ($s_i$) is a complete plan of action for a player. A strategy profile ($s$) specifies a strategy for every player and corresponds to a unique outcome.
• The payoff function ($u_i(s)$) is crucial. It assigns a numerical utility to each outcome for each player and defines the players' objectives. Changing the payoffs, even when actions and outcomes remain the same, creates an entirely new game.

With this formal structure in place, we are now ready to focus on the primary tool for representing and analyzing these games. In the next lesson, we will delve deeper into the normal form representation, learning how to construct and interpret the payoff matrices for games with any number of players and strategies.