Game Theory: Zero-Sum Games
Game Theory can help us find the ...
- best decision in a competitive situation, or
- fairest decision in a cooperative situation
... where the outcome for each player depends on their decision and the decisions of other players.
In a zero-sum game, whatever one player gains, another player loses by exactly the same amount.
Total Gain + Total Loss = 0
Zero-sum games often appear in:
- competitive games and sports
- auctions and bidding
- some types of trading and investing
A Simple Coin Game
Let's look at a very simple example.
We have two players: Alice and Bob.
- Alice and Bob each have a penny
- They secretly choose to turn their penny to either Heads or Tails
- They reveal them at the same time
- If the pennies match (HH or TT), Alice wins
- If the pennies don't match (HT or TH), Bob wins
Whatever one player wins, the other loses. No money is created or destroyed.
Example: Payoff Table
Each cell shows (Bob, Alice) winnings:
In every possible outcome, Bob's gain plus Alice's gain equals 0. That's why this is called a zero-sum game.
The "Game" Part of Game Theory
Because this is based on choice, a player can develop a strategy.
If Bob notices Alice always picks Heads, he will start picking Tails to win every time.
A simple fix is to use a Mixed Strategy: pick Heads or Tails randomly (roughly half the time each), which stops the other player from spotting a pattern.
Zero-Sum vs Non-Zero-Sum Games
Not all games are zero-sum.
- Zero-sum game: one player's gain is exactly another player's loss
- Non-zero-sum game: players can both gain, or both lose
For example:
- Chess is zero-sum: one winner, one loser
- Teamwork at work is non-zero-sum: everyone can benefit
Real-Life Zero-Sum Examples
- Chess: one player wins, the other loses, a tie also results in a zero sum
- Simple bets: money won by one person is lost by another
Nash Equilibrium (Zero-Sum Idea)
In a zero-sum game, a Nash Equilibrium happens when each player is using their best strategy, given what the other player is doing.
If both players are playing their best strategies, neither can improve their result by changing alone.
This idea helps players choose strategies that protect them from the worst possible outcome.
Conclusion
In a zero-sum game, whatever one player gains, another player loses.
Once we understand this idea, it becomes much easier to spot truly competitive situations and think more clearly about the best moves to make.