What it means
In the game, each player secretly puts a penny down showing heads or tails, and then both reveal. If the two coins match, the first player wins a dollar from the second, and if they differ, the second player wins a dollar from the first.
This is a zero-sum game, meaning that one player's win is exactly the other's loss, so the total of the two payoffs is always zero. There is no choice that is best regardless of what the other does, because any predictable pattern can be exploited by an opponent.
The solution is a mixed strategy, in which each player picks heads or tails at random with equal probability. At that point neither player can improve by changing their approach, which game theorists call a Nash equilibrium (a state in which no player gains by switching strategy alone).
Economists use the example because it is the simplest case of a game with no equilibrium in fixed choices. More complex games, such as price wars and bidding contests, can be understood by first spotting the matching pennies element within them.
Business situations often look like matching pennies. A retailer and a rival may try to guess each other's promotion dates, or a tax authority may decide which firms to audit while firms decide how aggressively to report, and predictable behaviour gives the other side an advantage.
The game is a simplification, since real contests have different stakes, memory and information. It nevertheless teaches an important idea: in competitive situations with no stable best choice, unpredictability has value, and randomising can be a rational strategy.
In practice
Real-world examples.
Example
Two rival supermarkets each choose whether to run a big promotion on Friday or Saturday. One chain profits if it picks the same day as its rival, because it can pull shoppers away, while the rival profits by picking the other day, so each tries to keep the other guessing.
Example
A tax authority decides which of two sectors to audit this year, while a company decides in which sector to aggressively claim deductions, and each side suffers if the other correctly predicts its choice. If the authority guesses right, the company loses, and if it guesses wrong, the company gains.
Example
A goalkeeper and a penalty taker choose left or right. The taker wins if they choose differently from the keeper, so both benefit from being unpredictable, and sports economists often use this example when teaching the game. A keeper who always dives the same way would soon be punished.
Formula
Calculation
Expected payoff = Sum of (Probability of outcome x Payoff of outcome)
Suppose each player picks heads or tails with a probability of 0.5, and the winner takes $1. The chance that the coins match is 0.5 and the chance that they differ is 0.5.
For the first player, expected payoff = (0.5 x $1) + (0.5 x -$1) = $0.50 - $0.50 = $0. The game is fair, and over many rounds each player expects to break even. Over 1,000 rounds, the typical gain or loss would be small compared with the $1,000 total staked. If the second player always chose heads, the first player would notice and always choose heads too, winning $1 every round, so predictability would cost money.Case study
Seen in the real world.
Parkside Beverages is an illustrative, fictional drinks company that kept launching promotions on the first Monday of every month. Its main competitor learned the pattern and ran a bigger promotion one week earlier, drawing customers away before Parkside's campaign began.
The marketing director recognised the situation as a matching pennies game. Any predictable timing could be countered, so the company needed to vary its schedule, with a random mix of early, mid and late-month promotions.
After the change, the competitor could no longer pre-empt every campaign, and Parkside's average market share during promotional periods rose by about a percentage point. The marketing director shared the idea with the sales team, who agreed to hold the schedule confidential until two weeks before each launch. In this illustrative story the lesson was that when your rival is trying to guess your move, a fixed routine can be a costly habit.
Watch out
Common mistakes.
- Believing there is a clever fixed strategy that always wins, when any pattern can be exploited by an informed opponent.
- Treating the game as a model of every business conflict, when many situations involve cooperation and different payoffs.
- Assuming that random means careless, when a mix of choices is a deliberate strategy that follows from analysing the other side's incentives.
Questions
People also ask.
Why is matching pennies a zero-sum game?
Because whatever one player wins, the other loses, so the gains and losses cancel out and the total payoff to both players is always zero.
What is a mixed strategy?
It is a plan that chooses among options at random with set probabilities, rather than always picking the same one.
Is there a pure strategy equilibrium?
No, because in any pure choice one player would want to change after seeing the other's, which is why the solution is to randomise.
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