Game Theory: Zero-Sum and Positive-Sum Games

Exploring the fundamentals of game theory: from conflict to cooperation

Zero-Sum Game: Matching Pennies

In zero-sum games, one player's gain is exactly balanced by the other player's loss. The total payoff is always zero.

Mathematical Formula: uA + uB = 0

Player Controls

Game Statistics

Rounds: 0
Player Score: 0
Computer Score: 0
Total Score: 0 (should be zero)

Payoff Matrix (Player's View)

Computer: Heads
Computer: Tails
Player: Heads
+1
-1
Player: Tails
-1
+1
Player Wins (+1)
Computer Wins (-1)

Cumulative Scores Chart

Strategy Analysis

Heads Frequency: 0%
Optimal Strategy: Mixed Strategy (50% Heads, 50% Tails)
Nash Equilibrium: Both players randomize with 50% probability

Positive-Sum Game: Prisoner's Dilemma

In positive-sum games, cooperation can create additional value. The total payoff can be greater than zero.

Cooperation Beats Defection: uA(合作) + uB(合作) > uA(背叛) + uB(背叛)

Player A Controls

Game Statistics

Rounds: 0
Player A Score: 0
Player B Score: 0
Total Score: 0 (higher when cooperating)
Cooperation Rate: 0%

Payoff Matrix (Player's View)

B: Cooperate
B: Defect
A: Cooperate
(3, 3)
(0, 5)
A: Defect
(5, 0)
(1, 1)
Mutual Cooperation (3, 3)
Mutual Defection (1, 1)
Temptation (5, 0)/(0, 5)

Cumulative Scores Chart

Strategy Analysis

Nash Equilibrium: Both players defect (1, 1)
Pareto Optimal: Both players cooperate (3, 3)
The Dilemma: Individual rationality leads to collective irrationality

Move History

Zero-Sum vs Positive-Sum Comparison

Compare the characteristics and outcomes of both game types side by side.

Zero-Sum Game

Heads
Tails
Heads
+1 / -1
-1 / +1
Tails
-1 / +1
+1 / -1
Total Payoff 0
Value of Cooperation: None
Nature: Pure Conflict

Positive-Sum Game

Cooperate
Defect
Cooperate
(3, 3)
(0, 5)
Defect
(5, 0)
(1, 1)
Total Payoff 6 (合作时)
Value of Cooperation: Significant
Nature: Conflict & Opportunity

Key Differences

Feature Zero-Sum Positive-Sum
Total Payoff Always zero Can be positive
Cooperation No benefit Can be beneficial
Win-Win Possible Impossible Possible
Classic Example Matching Pennies Prisoner's Dilemma

Nash Equilibrium Visualization

Explore the mixed strategy space and the concept of Nash equilibrium.

Mixed Strategy Space

Strategy Probabilities

50%
50%

Expected Payoff

Player A: 0.00
Player B: 0.00
Nash Equilibrium Point: p = 50%, q = 50%
Why is it an equilibrium?

Neither player can improve their payoff by unilaterally changing strategy. In zero-sum games, Nash equilibrium is the solution to the minimax strategy.

Best Response Functions

Blue line: A's best response to B | Red line: B's best response to A | Intersection is Nash equilibrium

Real-World Applications

Various applications of game theory in the real world.

Zero-Sum Applications

Sports Competition

Win-lose relationships in competitive sports where one side's victory means the other's defeat.

Futures Trading

In short-term trading, buyer and seller payoffs sum to zero (ignoring transaction costs).

Poker

Chip transfer between players with conservation of total amount.

Arms Race

Competition for relative advantage where one side's gain is the other's loss.

Positive-Sum Applications

International Trade

Trade benefits both sides by creating value through comparative advantage.

Climate Change Agreements

Global cooperation can avoid catastrophic losses and achieve win-win outcomes.

Joint Ventures

Companies share resources through cooperation to create greater market value.

Disarmament Negotiations

Both sides reducing armaments saves resources and increases security.

Interactive Example: Trade Game

Two countries decide whether to open trade. Free trade can achieve win-win, but protectionism may offer short-term advantages.

Country B: Open
Country B: Protect
Country A: Open
(4, 4)
Win-Win
(2, 3)
B Exploits
Country A: Protect
(3, 2)
A Exploits
(1, 1)
Lose-Lose

In this game, free trade is the Pareto optimal choice, but the temptation of protectionism may lead to suboptimal outcomes.