Fundamentals of Game Theory: Players, Strategies, Payoffs

When Your Success Depends on Others' Decisions

Most life tasks are solved individually: how much to study, how to allocate your budget, when to go to bed. But a whole class of decisions is fundamentally different—their outcome depends not only on your actions, but also on the actions of other people. A driver on the road, a company setting prices, a country at international negotiations, a startup in a competitive market—all are engaged in strategic interaction.

Game theory is a mathematical science of decision-making in situations of interdependence. The foundations were laid by John von Neumann and Oskar Morgenstern in the book "Theory of Games and Economic Behavior" (1944). John Nash in 1950 proposed the concept of equilibrium, and in 1994 he received the Nobel Prize in Economics for it.

Formal Definition of a Strategic Game

A strategic game in normal form is specified by three components:

• N — a finite set of players {1, 2, ..., n}
• Sᵢ — the set of strategies available to player i; the strategic profile S = S₁ × S₂ × ... × Sₙ
• uᵢ: S → ℝ — the payoff function for each player i

Each player i chooses a strategy sᵢ ∈ Sᵢ, and their payoff uᵢ(s₁, s₂, ..., sₙ) depends on the strategies of all participants. Each player seeks to maximize their own payoff, knowing that others are doing the same.

Classification of Games

By sum of payoffs: zero-sum games—one player's gain equals another's loss (chess, poker); nonzero-sum games—mutual gain or loss is possible (negotiations, trade).

By information: complete information—everyone knows all payoff functions; incomplete—the opponent’s type or payoffs are unknown.

By time: static (simultaneous choice) and dynamic (sequential).

Iconic Examples with Payoff Matrices

Prisoner's Dilemma — the most famous example. Two suspects in different cells. Each can Remain Silent (S) or Betray (B). Payoffs are in years of imprisonment (negative: the fewer, the better):

Reading the matrix: in cell (row, column) the payoffs (Player 1, Player 2) are given. For example, if P1 is silent and P2 betrays: P1 gets −5 years, P2 is freed (0 years).

For Player 1: regardless of P2's action, "Betray" is better: if P2 is silent (0 > −1); if P2 betrays (−3 > −5). "Betray" is a strictly dominant strategy. The same logic applies to P2. The equilibrium: (Betray, Betray) → both get −3 years, although (Silent, Silent) would give −1 each. The central paradox: individual rationality leads to a collectively irrational outcome.

"Chicken" Game: two drivers speed toward each other. Swerve = lose face ("chicken"). Not swerving = risk of catastrophe.

Here, there are two Nash equilibria in pure strategies: (Swerve, Not Swerve) and (Not Swerve, Swerve). Who yields? The answer depends on reputation, history, etc.

Strategies: Pure and Mixed

Pure strategy — a deterministic choice of one action. For example, "always play Rock."

Mixed strategy σᵢ — a probabilistic distribution over pure strategies. Example: σᵢ = (1/3, 1/3, 1/3) means an equal chance of playing Rock, Scissors, Paper.

Expected payoff with mixed strategies: $ uᵢ(σ) = \sum_{s} \left[\prod_{j} \sigma_{j}(s_{j})\right] \cdot uᵢ(s) $

Introducing mixed strategies fundamentally broadens the analysis: Nash’s theorem guarantees the existence of equilibrium in mixed strategies for any finite game.

Numerical Example: Rock–Scissors–Paper

Payoff matrix (Player 1, Player 2), R=Rock, S=Scissors, P=Paper:

There is no Nash equilibrium in pure strategies: for any fixed action of the opponent, there is a better response, but that too has a better response—a cycle.

Equilibrium in mixed: σ* = (1/3, 1/3, 1/3) for each player. Check: with σ₂ = (1/3,1/3,1/3), the expected payoff from R = 0·(1/3) + 1·(1/3) + (−1)·(1/3) = 0, and the same for S and P—the player is indifferent to their choice. Thus, σ* is indeed an equilibrium.

Applications of Game Theory

Economics: oligopoly (Cournot, Bertrand), auctions, international trade. Biology: evolutionary stability, altruism and kin selection. Computer science: algorithmic game theory, security (attacker–defender). Political science: elections, international negotiations, nuclear deterrence. Sociology: collective action, emergence of norms.

Game Theory in Strategic Management and Government

Game theory has become a standard tool of strategic consulting and government regulation. The "five forces of Porter" method implicitly describes competitive equilibrium in an industry: the intensity of rivalry, bargaining power of buyers and suppliers, threat of substitutes—all these are parameters of a multilateral game. BCG and McKinsey use game-theoretic models when analyzing price wars, mergers and acquisitions, competitors’ responses to strategic initiatives. In government, regulators apply concepts of dominance and equilibrium to assess the consequences of antitrust measures: if a merger of two major players shifts the market equilibrium from Bertrand to Cournot, prices will rise even without explicit collusion. Game theory in international negotiations describes diplomatic bargaining as multistage games with incomplete information: parties signal their interests through behavior, not just through declarations. The Nobel Prize in Economics has been awarded for game theory three times (1994—Nash, Harsanyi, Selten; 2005—Aumann, Schelling; 2012—Roth, Shapley), reflecting its central role in modern economic science.

Assignment: Write the "Stag Hunt" game in matrix form: two hunters can jointly hunt a stag (payoff 4 each) or individually hunt a hare (payoff 1 each). If one hunts stag alone—0. Find all Nash equilibria in pure strategies. Explain why both coordination outcomes are possible, though one is "better" than the other.