Strategy Sets in Strategic-Form Games 🎮
students, imagine two people playing rock-paper-scissors at the same time. No one waits to see the other person’s move first. Each player must choose a plan before the game starts, and that plan determines what they will do in every situation that could come up. That idea is the heart of a strategic-form game, also called a normal-form game. In this lesson, you will learn how to describe each player’s available strategies, how strategies are different from single actions, and how to list every strategy in a finite game.
What you will learn
By the end of this lesson, students, you will be able to:
- Define a strategy set for a player.
- Distinguish actions from complete strategies.
- List all strategies in a finite game.
Understanding strategy sets helps you read game tables, predict possible outcomes, and analyze what each player can do. This is useful in economics, sports, politics, and everyday decisions like choosing when to arrive at a store or whether to wait for a sale 🛒.
What is a strategy set?
In a strategic-form game, each player has a set of possible strategies. A strategy is a complete plan of action that tells a player what to do in every possible situation they might face in the game. The strategy set for a player is the collection of all strategies available to that player.
This is more specific than just saying what move a person might make right now. A strategy includes the whole plan, not just one step.
For example, suppose a player is in a game where they may have to respond to different possible choices by the other player. A complete strategy must say what the player would do in each case. If there are two possible situations, then a strategy must give an action for both situations.
In math notation, the strategy set for player $i$ is often written as $S_i$. If player $i$ has strategies $s_{i1}, s_{i2}, \dots, s_{ik}$, then we can write:
$$S_i = \{s_{i1}, s_{i2}, \dots, s_{ik}\}$$
The full list of strategy sets for all players is what defines the game in normal form.
Actions versus strategies
A very important idea in game theory is the difference between an action and a strategy.
An action is a single move made at one point in the game. A strategy is the full plan that tells a player which action to take in every possible case.
Think about a student deciding how to react in class. If the teacher asks a question, the action might be “raise hand” or “stay quiet.” But a strategy is bigger than that. It might be “raise hand if I know the answer, stay quiet if I do not.” That is a complete plan.
Here is another way to compare them:
- Action = one choice in one situation.
- Strategy = a rule for choosing actions across all possible situations.
In games with only one decision point and no uncertainty about what will happen next, an action and a strategy can look the same. But in games where players may need to plan ahead, strategies are more detailed.
For example, in a simple simultaneous-move game like rock-paper-scissors, each strategy is just one action because the player only chooses once. But in a larger game with more stages, a strategy may include a decision for every possible future branch. That is why strategy sets are so important in game theory 📘.
Finite games and how to list strategies
A finite game is a game with only a limited number of strategies for each player. When a game is finite, you can list every strategy one by one.
To list all strategies in a finite game, follow these steps:
- Identify each player.
- Write down every action or plan available to that player.
- Make sure each listed strategy is complete.
- Count how many strategies there are.
If a player has $m$ strategies and another player has $n$ strategies, then the number of possible strategy profiles is:
$$m \times n$$
A strategy profile is one combination of strategies, one for each player. It does not mean a payoff yet; it just means the chosen strategies together.
Example 1: Rock-paper-scissors
In rock-paper-scissors, each player chooses one of three actions:
- Rock
- Paper
- Scissors
Because the game is simultaneous and one-shot, each action is also a complete strategy. So each player’s strategy set is:
$$S_i = \{R, P, S\}$$
For two players, the number of strategy profiles is:
$$3 \times 3 = 9$$
The nine profiles are:
- $(R, R)$
- $(R, P)$
- $(R, S)$
- $(P, R)$
- $(P, P)$
- $(P, S)$
- $(S, R)$
- $(S, P)$
- $(S, S)$
Each ordered pair shows Player 1’s strategy first and Player 2’s strategy second.
Example 2: A simple classroom game
Suppose two students, Player 1 and Player 2, must each choose whether to Study or Not Study before a quiz. Each player has the strategy set:
$$S_1 = \{\text{Study}, \text{Not Study}\}$$
$$S_2 = \{\text{Study}, \text{Not Study}\}$$
Each player has $2$ strategies, so there are:
$$2 \times 2 = 4$$
strategy profiles. They are:
- $(\text{Study}, \text{Study})$
- $(\text{Study}, \text{Not Study})$
- $(\text{Not Study}, \text{Study})$
- $(\text{Not Study}, \text{Not Study})$
This type of listing is the starting point for a payoff table in strategic-form analysis.
Why complete strategies matter
students, complete strategies matter because a player’s choice may need to cover situations that never actually happen. That sounds strange at first, but it is essential for analyzing games correctly.
Imagine a game where one player might have to react after seeing the other player’s action. Even if that second situation does not happen in the actual play, the strategy must still specify what the player would do. Why? Because game theory studies the entire decision plan, not just the path that occurred.
This is especially important in extensive games, but the idea still helps in strategic-form games because every strategy profile must be well-defined before outcomes are analyzed.
A strategy set must include only strategies that are allowed by the rules of the game. If a player is only allowed to choose from three actions, then the strategy set cannot contain a fourth one. The set must be complete and accurate.
Comparing strategy sets across players
Different players can have different numbers of strategies. For example, one player may have two strategies while another has three.
Suppose:
$$S_1 = \{A, B\}$$
$$S_2 = \{X, Y, Z\}$$
Then Player 1 has $2$ strategies, Player 2 has $3$ strategies, and there are:
$$2 \times 3 = 6$$
strategy profiles.
The profiles are:
- $(A, X)$
- $(A, Y)$
- $(A, Z)$
- $(B, X)$
- $(B, Y)$
- $(B, Z)$
This list helps you build a payoff matrix. Each profile corresponds to one outcome in the table.
The size of a strategy set affects how complex a game becomes. More strategies mean more possible combinations, which makes analysis richer but also more work to organize.
Common mistakes to avoid
Here are a few common mistakes students make when learning strategy sets:
- Confusing an action with a strategy.
- Forgetting that a strategy must be complete.
- Listing outcomes instead of strategies.
- Mixing up the order of players in a strategy profile.
- Thinking a player’s strategy set is the same as the other player’s strategy set when it is not.
A good habit is to ask: “Is this a single move, or is this a full plan?” If it is only one move, it is an action. If it covers every possible case, it is a strategy.
Real-world connection
Strategy sets appear in real life whenever people must choose plans in advance. A company choosing a pricing rule, a driver choosing whether to take a highway or side streets, or a student deciding whether to study early or cram later all use strategy-like thinking.
For example, a delivery app might set a policy such as “offer a discount during lunch hours” or “do not offer a discount.” Those are strategies because they are complete plans. The outcome depends on what competitors do, how customers respond, and the rules of the market.
This is why strategy sets are a foundation for understanding competition. Before you can compare payoffs or identify best responses, you must know what choices each player actually has.
Conclusion
students, strategy sets are the building blocks of strategic-form games. A player’s strategy set is the list of all complete plans that the player can use. A strategy is not just one action; it is a full rule for what to do in every possible situation. In finite games, you can list every strategy and count the possible strategy profiles by multiplying the number of strategies each player has. Once you know the strategy sets, you are ready to study payoffs, best responses, and pure-strategy outcomes. 🎯
Study Notes
- A strategy set is the set of all strategies available to one player.
- A strategy is a complete plan of action, not just one move.
- An action is a single choice made in one situation.
- In a one-shot simultaneous game, an action may also be a strategy.
- In more complex games, a strategy must specify what to do in every possible case.
- A finite game has only a limited number of strategies for each player.
- If Player 1 has $m$ strategies and Player 2 has $n$ strategies, there are $m \times n$ strategy profiles.
- A strategy profile is one combination of strategies, one from each player.
- Rock-paper-scissors has $3$ strategies for each player: $\{R, P, S\}$.
- To list strategies in a finite game, identify each player, write every allowed strategy, and count the total combinations.