Why Everyday Decisions Are a Game: Intro to Game Theory & Nash Equilibrium

In everyday life we constantly make decisions—what to eat, whether to exercise, which movie to watch—yet these seemingly simple choices involve a complex game‑theoretic process.

When multiple participants are involved, each person's outcome depends not only on their own decision but also on the decisions of others; this interaction is precisely what game theory studies.

Game theory examines strategic interactions among decision‑makers, called players, who choose from a set of possible actions, or strategies, to achieve their objectives.

Strategies are classified as pure strategies , where a player consistently selects the same action, and mixed strategies , where a player randomizes among actions with specific probabilities.

For example, in rock‑paper‑scissors a player may always choose rock (pure strategy) or select rock, paper, or scissors each with probability 1/3 (mixed strategy). Mixed strategies make a player’s behavior unpredictable, increasing expected payoff.

In a two‑player game with payoff matrix U , the element U[i][j] represents the payoff to player A when A chooses strategy i and player B chooses strategy j. The expected payoffs for mixed strategies are calculated by weighting these payoffs with the chosen probabilities.

A central concept is the Nash equilibrium , a set of strategies where no player can improve their expected payoff by unilaterally deviating.

For finite games, Nash’s theorem guarantees at least one mixed‑strategy Nash equilibrium.

Consider a soccer penalty‑kick scenario: the kicker can shoot left, centre, or right, and the goalkeeper can dive left, centre, or right. The payoff matrix assigns scores (e.g., +3 for a goal, –3 for a save) to each combination of actions.

Let the kicker’s probabilities for left, centre, right be p_L, p_C, p_R and the goalkeeper’s probabilities be q_L, q_C, q_R . The expected payoff for the kicker is E_K = p·U·qᵀ , and similarly for the goalkeeper.

At equilibrium, the kicker cannot change his probability distribution to increase his expected payoff.

At equilibrium, the goalkeeper cannot change his probability distribution to increase his expected payoff.

Solving the equilibrium conditions (e.g., via linear programming or Lagrange multipliers) yields the optimal mixed‑strategy probabilities for both players.