In poker, every dollar you win comes from someone else’s loss. In chess, one player wins and one loses. In tennis, your point is your opponent’s lost opportunity. These are zero-sum games—pure competition where one player’s gain equals another’s loss.
Understanding zero-sum games changes how you compete, negotiate, and think about conflict. Let’s explore this fundamental concept.
What is a Zero-Sum Game?
A zero-sum game is a situation where the total gains and losses always sum to zero. Whatever one player wins, another must lose.
The defining property:
Player A's gain + Player B's loss = 0
Or equivalently:
Sum of all payoffs = 0
Example: A simple betting game
| Opponent: Heads | Opponent: Tails | |
|---|---|---|
| You: Heads | (+1, -1) | (-1, +1) |
| You: Tails | (-1, +1) | (+1, -1) |
Notice: In every cell, the payoffs sum to zero. Your +1 is their -1, and vice versa.
Classic Examples of Zero-Sum Games
1. Poker
When you win a pot, that money came from other players. The total amount of money doesn’t change—it just moves from losers to winners.
2. Chess
One player wins (+1), one loses (-1), or it’s a draw (0, 0). The total is always zero.
3. Sports Matches
Your team’s victory is the other team’s defeat. Points, games, and tournaments are fundamentally zero-sum (ignoring the fun everyone has playing).
4. Market Share Competition
If there are only two companies and 100 customers, every customer you gain is one your competitor loses. (Though in reality, markets often grow, making it non-zero-sum.)
5. Elections
In single-winner elections, your victory is your opponent’s loss. Only one person can be president, mayor, or team captain.
between players"] C --> C1["Win/Loss/Draw
adds to zero"] D --> D1["Your victory is
their defeat"] E --> E1["Fixed customers
split between firms"] F --> F1["Single winner
multiple losers"] style A fill:#4c6ef5 style B1 fill:#ff6b6b style C1 fill:#ff6b6b style D1 fill:#ff6b6b style E1 fill:#ff6b6b style F1 fill:#ff6b6b
Zero-Sum vs. Non-Zero-Sum
The distinction between zero-sum and non-zero-sum games is crucial:
Zero-Sum: Pure Competition
- Total payoff = 0
- One player’s gain = Other’s loss
- No possibility for mutual benefit
- Purely adversarial
Examples: Chess, poker, tennis
Non-Zero-Sum: Mixed Motives
- Total payoff can be positive or negative
- Both players can win (or both can lose)
- Cooperation can create value
- Mix of competition and cooperation
Examples: Business negotiations, Prisoner’s Dilemma, trade agreements
My gain = Your loss"] B --> B2["Examples:
Chess, Poker, Tennis"] C --> C1["Mixed motives
Both can win/lose"] C --> C2["Examples:
Trade, Prisoner's Dilemma"] B --> D["No cooperation
benefits"] C --> E["Cooperation possible
Create value together"] style A fill:#4c6ef5 style B fill:#ff6b6b style C fill:#51cf66 style D fill:#ff6b6b style E fill:#51cf66
Key insight: In zero-sum games, there’s no point in cooperation. You should be purely strategic and competitive. In non-zero-sum games, finding ways to cooperate can benefit everyone.
The Minimax Strategy
In zero-sum games, the optimal strategy is called the minimax strategy: minimize the maximum loss your opponent can inflict on you.
Or equivalently, from your opponent’s perspective, they’re trying to minimize the maximum gain you can achieve.
outcome for you"] B --> B2["Assume opponent plays
optimally against you"] C --> C1["Minimize your
maximum gain"] C --> C2["Maximize their own
minimum gain"] B2 --> D["Conservative strategy
Guaranteed security level"] C2 --> D style A fill:#4c6ef5 style B fill:#51cf66 style C fill:#ffd43b style D fill:#ae3ec9
Example:
| Opponent: Left | Opponent: Right | |
|---|---|---|
| You: Up | (+3, -3) | (-2, +2) |
| You: Down | (+1, -1) | (+2, -2) |
Your analysis:
- If you play Up: worst case is -2 (if opponent plays Right)
- If you play Down: worst case is +1 (if opponent plays Left)
Minimax choice: Play Down—it guarantees at least +1, while Up risks -2.
Opponent’s analysis:
- If they play Left: worst case is -3 (losing 3 to you)
- If they play Right: worst case is -2 (losing 2 to you)
Their minimax choice: Play Right—it minimizes their maximum loss.
Result: (Down, Right) = (+2, -2)
Matching Pennies: A Perfect Zero-Sum Game
Two players each have a penny. They simultaneously choose heads or tails:
- If they match (both heads or both tails): Player 1 wins
- If they don’t match: Player 2 wins
| Player 2: Heads | Player 2: Tails | |
|---|---|---|
| Player 1: Heads | (+1, -1) | (-1, +1) |
| Player 1: Tails | (-1, +1) | (+1, -1) |
Key insight: There’s no pure strategy Nash Equilibrium!
- If P1 plays Heads, P2 should play Tails
- But then P1 should switch to Tails
- But then P2 should switch to Heads
- And so on…
The solution: Both players should randomize 50-50 between heads and tails.
50% Heads, 50% Tails style P1 fill:#ff6b6b style P2 fill:#ffd43b
This is why poker players bluff randomly and chess players vary their openings—to be unpredictable.
Rock-Paper-Scissors: Another Zero-Sum Classic
Everyone knows this game:
- Rock beats Scissors
- Scissors beats Paper
- Paper beats Rock
| Opponent: Rock | Opponent: Paper | Opponent: Scissors | |
|---|---|---|---|
| You: Rock | (0, 0) | (-1, +1) | (+1, -1) |
| You: Paper | (+1, -1) | (0, 0) | (-1, +1) |
| You: Scissors | (-1, +1) | (+1, -1) | (0, 0) |
Again, no pure strategy equilibrium!
If you always play Rock, your opponent will play Paper and beat you. The optimal strategy is to randomize equally: 1/3 Rock, 1/3 Paper, 1/3 Scissors.
You lose"] B -->|No, Random| D["Equal probability
for each choice"] D --> E["Optimal Strategy
Expected payoff = 0"] C --> F["Lose on average"] style A fill:#4c6ef5 style C fill:#ff6b6b style D fill:#51cf66 style E fill:#51cf66 style F fill:#ff6b6b
The lesson: In zero-sum games with no dominant strategy, being unpredictable is crucial.
When Games Are NOT Zero-Sum
Many situations that seem zero-sum actually aren’t:
Business Competition
It looks zero-sum (my market share gain is your loss), but:
- Markets can grow
- Innovation creates new value
- Both companies can profit
- Cooperation (standards, R&D sharing) can benefit both
Salary Negotiation
It looks zero-sum (more for me = less for you), but:
- A good deal can improve retention and motivation
- Better performance benefits both parties
- Long-term relationship matters
- Fairness affects future interactions
International Relations
It looks zero-sum (power balance), but:
- Trade makes both countries richer
- Alliances provide mutual security
- Cooperation on global issues (pandemics, climate) benefits all
- War is negative-sum (both lose)
Thinking] --> B[Business] A --> C[Negotiation] A --> D[Politics] B --> B1["Seems: Market share fight
Reality: Markets grow"] C --> C1["Seems: Fixed pie
Reality: Create value"] D --> D1["Seems: Power struggle
Reality: Trade benefits all"] B1 --> E["Non-zero-sum
opportunities"] C1 --> E D1 --> E style A fill:#ff6b6b style E fill:#51cf66
The danger: Treating non-zero-sum situations as zero-sum leads to missed opportunities for mutual benefit.
The Fixed Pie Bias
Humans have a cognitive bias: we tend to see situations as more zero-sum than they really are.
The Fixed Pie Bias: The assumption that any gain for one party must come at the expense of another—ignoring possibilities for expanding the total value.
Example: Salary negotiations
- Zero-sum thinking: “If I get more money, the company loses money”
- Expanding the pie: Consider total compensation (salary, equity, flexibility, training, career path, projects)
Both sides can win by finding creative solutions that provide value to both parties.
purely competitive] B --> C["Miss opportunities to
create value"] C --> D["Worse outcomes
for everyone"] E[Growth Mindset] --> F[Look for ways to
expand the pie] F --> G["Find mutual gains
Create win-win solutions"] style A fill:#ff6b6b style B fill:#ff6b6b style C fill:#ff6b6b style D fill:#ff6b6b style E fill:#51cf66 style F fill:#51cf66 style G fill:#51cf66
Strategic Implications
In true zero-sum games:
- Be purely competitive
- Focus on outsmarting opponents
- Being unpredictable is valuable
- Cooperation doesn’t help
- Study minimax strategies
In non-zero-sum games:
- Look for mutual gains
- Consider cooperation
- Build trust and reputation
- Think long-term
- Create value before dividing it
The key: Correctly identify which type of game you’re in!
Real-World Applications
1. Sports and Competition
Zero-sum: Focus on beating opponents, finding weaknesses, strategic unpredictability.
2. Legal Cases (mostly zero-sum)
Often structured as zero-sum: one party wins, the other loses. But settlement negotiations can be non-zero-sum (both save legal costs and time).
3. Poker
Pure zero-sum: Your edge comes from being less predictable and reading opponents better.
4. Business Strategy
Usually non-zero-sum: Look for ways to grow the market, create new value, or find underserved niches rather than just fighting for market share.
zero-sum?} B -->|Yes| C[Pure Competition] B -->|No| D[Mixed Motives] C --> C1["Be unpredictable
Minimax strategy
No cooperation"] D --> D1["Look for mutual gains
Consider cooperation
Expand the pie"] style A fill:#4c6ef5 style C fill:#ff6b6b style C1 fill:#ff6b6b style D fill:#51cf66 style D1 fill:#51cf66
The Takeaway
Zero-sum games are about pure competition: Your gain is my loss, and vice versa. In these games:
- Be strategic and competitive
- Use minimax reasoning
- Be unpredictable when needed
- Don’t expect cooperation to help
But most of life is NOT zero-sum: Business, relationships, negotiations, and international relations usually have opportunities for mutual benefit.
The mistake: Treating non-zero-sum situations as zero-sum leads to:
- Missed opportunities for mutual gain
- Destructive competition when cooperation would be better
- Pessimism about human relations and progress
The wisdom: Recognize when you’re in a true zero-sum game (compete hard), and when you’re not (look for ways to create value for everyone).
This is part of our Game Theory Series. Understanding zero-sum games helps you recognize when competition is inevitable and when there are opportunities for mutual benefit—a crucial distinction for better decision-making.