Game Theory: The Mathematics of Strategy

Every time you choose a move while knowing that someone else is also choosing, you are playing a game in the mathematical sense. Game theory is the branch of mathematics that studies these strategic situations, and it is used by economists, army planners, football coaches and even biologists. In this article we will build the key ideas from scratch and solve two classic games step by step.

What Counts as a "Game"?

A game in mathematics has three ingredients:

• Players – the decision makers (two companies, two countries, a striker and a goalkeeper).
• Strategies – the choices available to each player.
• Payoffs – the outcome each player receives for every combination of choices, written as numbers.

The crucial feature is interdependence: your best choice depends on what the other player does, and their best choice depends on what you do. This is what separates game theory from ordinary optimisation, where you simply maximise a function on your own.

The subject was put on a firm footing by the mathematician John von Neumann, who proved his minimax theorem in 1928, and by von Neumann and the economist Oskar Morgenstern in their 1944 book Theory of Games and Economic Behavior. In 1950 a young Princeton student, John Nash, introduced the idea of equilibrium that now carries his name, and he later shared the 1994 Nobel Prize in Economics for it.

Nash Equilibrium: The Central Idea

A Nash equilibrium is a combination of strategies in which no player can do better by changing their own strategy alone, given what everyone else is doing. It is a "no regrets" position. It does not mean the outcome is good for everyone; it only means nobody has a reason to deviate on their own.

To find one, we usually look for best responses: for each choice of the opponent, which of my strategies gives me the highest payoff? A cell where both players are playing a best response to each other is a Nash equilibrium.

Worked Example 1: The Prisoner's Dilemma

Two suspects, Ali and Bilal, are arrested and questioned in separate rooms. Each can stay Silent or Confess. The prison sentences (in years, so smaller is better) are:

The first number in each pair is Ali's sentence and the second is Bilal's.

Step 1: Ali's best response if Bilal stays silent. Ali compares 1 year (Silent) with 0 years (Confess). Since \(0 < 1\), Ali should confess.

Step 2: Ali's best response if Bilal confesses. Ali compares 10 years (Silent) with 5 years (Confess). Since \(5 < 10\), Ali should again confess.

So confessing is better for Ali whatever Bilal does. We call this a dominant strategy.

Step 3: Bilal's reasoning. The table is symmetric, so by the same comparisons confessing is also dominant for Bilal.

Step 4: The equilibrium. Both confess and each receives 5 years. Check: if Ali alone switches to Silent, his sentence rises from 5 to 10, so he will not switch. The same is true for Bilal. Hence (Confess, Confess) is the Nash equilibrium.

The paradox: if both had stayed silent they would each serve only 1 year. Total years served are \(1 + 1 = 2\) compared with \(5 + 5 = 10\). Individually rational choices lead to a collectively worse result. The same structure explains price wars between petrol stations, over-fishing of shared seas, and arms races between nations.

Mixed Strategies: When Being Predictable Loses

Some games have no equilibrium in which players pick a single fixed move. Think of a penalty kick: if a striker always shoots left, the goalkeeper will always dive left. The solution is to randomise, choosing each option with a certain probability. This is called a mixed strategy, and Nash proved that every finite game has at least one equilibrium once mixed strategies are allowed.

The key trick is the indifference principle: in a mixed equilibrium, each player chooses probabilities that make the opponent indifferent between their options. If the opponent strictly preferred one option, they would always choose it, and the mixing would collapse.

Worked Example 2: The Penalty Kick

Suppose data show these probabilities of scoring:

The striker wants to maximise the scoring probability; the keeper wants to minimise it.

Step 1: Find the keeper's mix. Let the keeper dive left with probability \(q\). The striker's expected scoring probability is:

$$ \text{Shoot Left: } 0.60q + 0.95(1-q) = 0.95 - 0.35q $$

$$ \text{Shoot Right: } 0.90q + 0.70(1-q) = 0.70 + 0.20q $$

Step 2: Make the striker indifferent. Set these equal:

$$ 0.95 - 0.35q = 0.70 + 0.20q \;\Rightarrow\; 0.25 = 0.55q \;\Rightarrow\; q = \frac{5}{11} \approx 0.455 $$

So the keeper should dive left about 45.5% of the time.

Step 3: Find the striker's mix. Let the striker shoot left with probability \(p\). The keeper faces:

$$ \text{Dive Left: } 0.60p + 0.90(1-p) = 0.90 - 0.30p $$

$$ \text{Dive Right: } 0.95p + 0.70(1-p) = 0.70 + 0.25p $$

Step 4: Make the keeper indifferent.

$$ 0.90 - 0.30p = 0.70 + 0.25p \;\Rightarrow\; 0.20 = 0.55p \;\Rightarrow\; p = \frac{4}{11} \approx 0.364 $$

Step 5: The value of the game. Substitute \(q = 5/11\):

$$ 0.95 - 0.35 \times \frac{5}{11} = 0.95 - 0.159 = 0.791 $$

Check with the other expression: \(0.70 + 0.20 \times \frac{5}{11} = 0.70 + 0.091 = 0.791\). Both agree, so in equilibrium about 79.1% of penalties are scored.

Notice something surprising: the striker shoots to his strong side (Left gives 0.95 against a wrong dive) only about 36% of the time. Because the keeper knows it is the striker's strength, overusing it would be punished. Studies of professional penalty data have found that real players mix in roughly the way this theory predicts.

Where Game Theory Is Used

• Business: firms decide prices, advertising budgets and when to enter markets by modelling rivals' responses.
• Auctions: governments sell mobile-phone spectrum licences through auctions designed with game theory; the 2020 Nobel Prize in Economics went to Paul Milgrom and Robert Wilson for this work.
• Biology: John Maynard Smith used "evolutionarily stable strategies" to explain why animals often fight with ritual displays instead of deadly attacks.
• Politics and diplomacy: models of deterrence and negotiation are built on repeated games.
• Computer science: online advertising auctions and network routing protocols are analysed as games between many self-interested agents.

Did You Know?

• The film A Beautiful Mind (2001) is based on the life of John Nash, though its famous bar scene does not correctly describe a Nash equilibrium.
• In repeated prisoner's dilemma tournaments run by political scientist Robert Axelrod around 1980, the winning strategy was the very simple "Tit for Tat": cooperate first, then copy your opponent's last move.
• Rock–paper–scissors has a single equilibrium: play each option with probability \(\tfrac{1}{3}\).

Key Takeaways

• A game consists of players, strategies and payoffs, where each player's best choice depends on the others.
• A Nash equilibrium is a set of strategies where no player gains by changing their choice alone.
• The prisoner's dilemma shows how individually rational choices can produce a worse outcome for everyone.
• Mixed strategies are found by choosing probabilities that make the opponent indifferent between their options.
• Game theory guides real decisions in economics, sport, biology, auctions and international relations.

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