System Model: Game Components

Formal Definition of Multi-Player Strategic Games

1. Formal Game Definition

Strategic (Normal) Form Game
$$G = \langle N, \{S_i\}_{i \in N}, \{u_i\}_{i \in N} \rangle$$
$N$
Set of Players
$\{S_i\}$
Strategy Sets
$\{u_i\}$
Payoff Functions
Why "Strategic Form"?

Also called normal form, this representation captures games where players choose strategies simultaneously. All relevant information is encoded in the strategy sets and payoff functions.

2. Core Components

Players $(N)$

The set of rational decision-makers in the game.

$$N = \{1, 2, \ldots, n\}$$
  • Each player has preferences
  • Players are indexed $i \in N$
  • $n = |N|$ is the number of players

Example: Two competing coffee shops, $N = \{\text{Shop A}, \text{Shop B}\}$

Strategies $(S_i)$

Available actions/choices for each player $i$.

$$S_i = \{s_i^1, s_i^2, \ldots, s_i^m\}$$
  • Each player has their own strategy set
  • Strategies can be pure or mixed
  • $|S_i|$ = number of options for player $i$

Example: Each shop can set $S_i = \{\text{Low Price}, \text{High Price}\}$

Action Profile $(s)$

A combination of strategies, one from each player.

$$s = (s_1, s_2, \ldots, s_n) \in S$$
  • $S = S_1 \times S_2 \times \cdots \times S_n$ (Cartesian product)
  • $s_{-i}$ = strategies of all players except $i$
  • $s = (s_i, s_{-i})$ notation is common

Example: $s = (\text{Low}, \text{High})$ — Shop A undercuts, Shop B prices high

Payoff Function $(u_i)$

Maps action profiles to utility values for player $i$.

$$u_i: S \rightarrow \mathbb{R}$$
  • $u_i(s)$ = player $i$'s utility when profile is $s$
  • Higher is better (utility maximization)
  • Represents preferences over outcomes

Example: $u_A(\text{Low}, \text{High}) = \$800$, $u_B(\text{Low}, \text{High}) = \$200$ (daily profit)

Information Structure

What each player knows about the game and other players.

Complete Information

All players know $N$, $S_i$, and $u_i$ for all players

Incomplete Information

Some payoff functions or types are unknown

3. Key Assumptions

Rationality

Each player acts to maximize their expected utility.

  • Players have well-defined preferences (utility functions)
  • Players can compute and choose optimal strategies
  • No "mistakes" or irrational behavior
Common Knowledge of Rationality

Everyone knows that everyone is rational, and everyone knows that everyone knows, ad infinitum.

  • I know you're rational
  • You know that I know you're rational
  • I know that you know that I know you're rational...
Common Knowledge of the Game

All players know the structure of the game: $N$, $S_i$, $u_i$ for all players.

  • No hidden players or secret strategies
  • Payoff matrices are common knowledge
  • Rules of the game are understood by all
Simultaneous Moves

In strategic form games, all players choose strategies simultaneously.

  • No player observes others' choices before deciding
  • Equivalent to: all choose in secret, then reveal
  • Sequential games require extensive form

4. Problem Formulation: From English to Game Theory

Why Problem Formulation Matters

The most challenging part of game theory is translating a real-world scenario described in natural language into a formal game-theoretic model. This skill is essential for applying game theory to actual problems.

Step-by-Step Formulation Process
1
Identify the Players ($N$)

Ask: "Who are the decision-makers?"

  • Look for actors making strategic choices
  • Must have their own goals/objectives
  • Usually explicitly mentioned in the problem
2
List Available Strategies ($S_i$)

Ask: "What choices does each player have?"

  • Enumerate all possible actions
  • Each player may have different options
  • Must be mutually exclusive choices
3
Determine Outcomes ($S$)

Ask: "What happens for each combination?"

  • List all strategy profiles: $S = S_1 \times S_2 \times \cdots \times S_n$
  • Each profile leads to a specific outcome
  • Total outcomes = product of all strategy counts
4
Assign Payoffs ($u_i$)

Ask: "How much does each player value each outcome?"

  • Assign numerical utilities to each outcome
  • Higher values = preferred outcomes
  • Use monetary values, years in prison, preferences, etc.
Worked Example: Advertising Battle
TechX
Smartphone Company
PhoneCo
Smartphone Company
Problem Statement (English)

"Two smartphone companies, TechX and PhoneCo, are deciding whether to launch an expensive advertising campaign for the holiday season. If both advertise, they split the market equally but spend heavily, earning $2M each. If neither advertises, they save costs and earn $4M each. If only one advertises, the advertiser captures most of the market ($6M) while the other loses customers ($1M)."

Visual Breakdown
Both Advertise
$2M each
Split market, high costs
Neither Advertises
$4M each
Save costs, share market
Only One Ads
$6M vs $1M
Advertiser wins big
The Dilemma
What to do?
Each fears being undercut
Game Theory Formulation (Click to expand/collapse)
1
Players
$$N = \{\text{TechX}, \text{PhoneCo}\}$$
2
Strategies
$$S_i = \{\text{Advertise}, \text{Don't Advertise}\}$$
3
Total Outcomes
$$|S| = 2 \times 2 = 4 \text{ outcomes}$$
4
Payoff Matrix
PhoneCo
Advertise Don't
TechX Advertise 2M, 2M 6M, 1M
Don't 1M, 6M 4M, 4M
TechX | PhoneCo
Complete Formal Game
$$G = \langle N, \{S_i\}_{i \in N}, \{u_i\}_{i \in N} \rangle$$

Ready for game-theoretic analysis!

Common English → Game Theory Mappings
English Phrase Game Theory Element Example
"Two companies compete..." Players: $N = \{\text{Company A}, \text{Company B}\}$ Firms, countries, individuals
"Each can choose to..." Strategies: $S_i = \{\text{Option 1}, \text{Option 2}, ...\}$ High/Low price, Cooperate/Defect
"If X happens, then Y gets..." Payoffs: $u_i(s) = \text{value}$ Profit, utility, years in jail
"Without knowing what the other does..." Simultaneous game (normal form) Secret bidding, pricing decisions
"After seeing the other's choice..." Sequential game (extensive form) Responding to competitor's move
Common Pitfalls to Avoid
  • Incomplete strategies: Make sure you've listed ALL available options for each player
  • Inconsistent payoffs: Verify payoffs are from each player's perspective, not absolute values
  • Missing outcomes: Ensure you've considered all $|S_1| \times |S_2| \times \cdots$ combinations
  • Confusing players with strategies: Players are who, strategies are what they can do

5. Payoff Matrix Representation

For 2-player games, we represent payoffs in a matrix (also called bimatrix):

Generic 2-Player Game
Player 2
Strategy L Strategy R
Player 1 Strategy U $a_1$, $a_2$ $b_1$, $b_2$
Strategy D $c_1$, $c_2$ $d_1$, $d_2$
Player 1's payoff | Player 2's payoff
Reading the Matrix
  • Rows = Player 1's strategies
  • Columns = Player 2's strategies
  • Each cell contains: (Player 1's payoff, Player 2's payoff)
  • Example: If P1 plays U and P2 plays L → payoffs are $(a_1, a_2)$
Generic 3-Player Game

For 3+ players, we use multiple matrices — one for each strategy of the additional player(s)

Player 3 chooses X
Player 2
L R
Player 1 U $a_1$, $a_2$, $a_3$ $b_1$, $b_2$, $b_3$
D $c_1$, $c_2$, $c_3$ $d_1$, $d_2$, $d_3$
Player 3 chooses Y
Player 2
L R
Player 1 U $e_1$, $e_2$, $e_3$ $f_1$, $f_2$, $f_3$
D $g_1$, $g_2$, $g_3$ $h_1$, $h_2$, $h_3$
Player 1 | Player 2 | Player 3
3-Player Game Structure
  • Each matrix corresponds to one strategy of Player 3
  • Total outcomes: $|S_1| \times |S_2| \times |S_3| = 2 \times 2 \times 2 = 8$
  • Each cell now has 3 payoffs: $(u_1, u_2, u_3)$
  • For $n$ players, we need $|S_3| \times |S_4| \times \cdots \times |S_n|$ matrices

6. Interactive: Coffee Shop Pricing Game

The Pricing Battle

Two coffee shops across the street from each other must decide their pricing strategy

Shop A
"Café Azure"
Shop B
"Bean Boulevard"
Click any cell to see what happens!
Shop B's Choice
Low Price ($3) High Price ($5)
Shop A's Choice Low ($3)
$400, $400
Click me!
$800, $200
Click me!
High ($5)
$200, $800
Click me!
$600, $600
Click me!
Shop A's Daily Profit | Shop B's Daily Profit

Explore different scenarios:

Game Theory Insight

This is a Prisoner's Dilemma structure! Both shops would be better off if they both priced high ($600 each), but each has an incentive to undercut. The Nash Equilibrium is (Low, Low) — neither can improve by changing alone.

7. Example: Prisoner's Dilemma Structure

The Scenario

Two suspects are arrested. They're held separately and offered a deal:

  • If both stay silent → both get 1 year
  • If one confesses and the other is silent → confessor goes free, other gets 5 years
  • If both confess → both get 3 years
Formal Components
  • N = {Prisoner 1, Prisoner 2}
  • S₁ = S₂ = {Silent, Confess}
  • u₁, u₂ = (see payoff matrix)
  • Info: Complete, simultaneous moves
Prisoner 2
Silent Confess
Prisoner 1 Silent -1, -1 -5, 0
Confess 0, -5 -3, -3
Key Observation

This game has all the components of a strategic form game clearly defined. In Topic 3, we'll analyze it to find dominant strategies, and in Topic 4, we'll find its Nash Equilibrium.

8. Build Your Own Game

Custom 2×2 Game Builder

Create your own strategic game by editing all fields below

S₁
S₂
S₁
S₂
Payoff Matrix
Enter payoff values for each outcome
Player 2
Cooperate Defect
Player 1 Cooperate
,
,
Defect
,
,
Player 1's Payoff | Player 2's Payoff
Load Classic Game
Formal Game Definition

Game: G = ⟨ N, {S₁, S₂}, {u₁, u₂} ⟩

Players: N = {Player 1, Player 2}

Strategies:

  • S₁ = {Cooperate, Defect}
  • S₂ = {Cooperate, Defect}

Action Profiles: |S| = |S₁| × |S₂| = 2 × 2 = 4

9. 3-Player Financial Scenario

The Investment Decision

Three investors must independently decide whether to invest in a Safe bond or a Risky startup

Investor A
"Alpha Capital"
Investor B
"Beta Ventures"
Investor C
"Gamma Holdings"
Current Scenario: All Safe
Investor A
+$50K
Safe Bond
Investor B
+$50K
Safe Bond
Investor C
+$50K
Safe Bond
All investors play it safe with guaranteed returns. Stable but no big wins.
Payoff Rules
  • Safe Bond: Always returns +$50K (guaranteed)
  • Risky Startup (alone): Returns +$200K (you capture the market)
  • Risky (2 investors): Split market, each gets +$80K
  • Risky (all 3): Overcrowded, each gets only +$30K
Formal 3-Player Game

Players: N = {Investor A, Investor B, Investor C}

Strategies: $S_i$ = {Safe, Risky} for each player $i$

Action Space: |S| = 2 × 2 × 2 = 8 possible outcomes

Key Insight: This is a congestion game — the more players choose Risky, the less each earns!

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