Normal-Form Games

Normal-Form Games are a fundamental concept in game theory, used to model strategic interactions among rational decision-makers, known as players. A normal-form game specifies the strategic setting by explicitly listing the players, the strategies available to each player, and the payoffs each player receives for every possible combination of strategies chosen by all players. This form is also called the strategic form because it focuses on the strategies rather than the sequential structure of the game.

Definition and Components of Normal-Form Games

A normal-form game is formally defined by a tuple (N, {S_i}, {u_i}) where:

• N is the finite set of players, indexed by i = 1, 2, ..., n.
• S_i is the finite set of strategies available to player i.
• u_i : S_1 × S_2 × ... × S_n → ℝ is the payoff function for player i, assigning a real number payoff to each strategy profile (an n-tuple of strategies, one from each player).

In a normal-form game, each player chooses a strategy simultaneously or without knowledge of the other players’ choices. The outcome depends on the combination of strategies chosen, and players aim to maximize their own payoffs given the strategies of others.

Representation of Normal-Form Games

Matrix Representation for Two-Player Games

Most commonly, normal-form games involving two players are represented using a payoff matrix. Each row corresponds to a strategy of the first player, and each column corresponds to a strategy of the second player. Each cell in the matrix contains an ordered pair representing the payoffs to both players for that strategy profile.

Example of a 2x2 normal-form game matrix:

Here, the first component in the pair is Player 1's payoff, and the second component is Player 2's payoff.

Generalization to n-Player Games

For games with more than two players, the representation becomes more complex and is often described in terms of payoff functions rather than explicit matrices, as enumerating all strategy profiles grows exponentially with the number of players.

Strategies in Normal-Form Games

Pure Strategies

A pure strategy is a specific, deterministic choice of an action from the player's strategy set. Each player selects exactly one pure strategy.

Mixed Strategies

A mixed strategy is a probability distribution over a player's pure strategies. Instead of choosing a single pure strategy, the player randomizes among them according to assigned probabilities. Mixed strategies allow capturing situations where players may want to keep their opponents uncertain.

Formally, a mixed strategy for player i is a vector p_i = (p_i1, p_i2, ..., p_i|S_i|) where each p_ij ≥ 0 and the sum of p_ij over all j equals 1.

Solution Concepts in Normal-Form Games

Nash Equilibrium

A key concept in normal-form games is the Nash equilibrium, a strategy profile from which no player has an incentive to unilaterally deviate. Formally, a strategy profile (s_1*, s_2*, ..., s_n*) is a Nash equilibrium if for every player i and for every alternative strategy s_i ∈ S_i,

meaning that player i's payoff by sticking to s_i* is at least as good as switching to any other strategy s_i, given that other players' strategies remain fixed.

Nash equilibria can be in pure strategies or mixed strategies. The Nash existence theorem guarantees that at least one mixed strategy Nash equilibrium exists in any finite normal-form game.

Dominant Strategies and Dominated Strategies

• A dominant strategy for a player is one that yields a strictly higher payoff than any other strategy, regardless of what strategies other players choose.
• A dominated strategy is one that is worse than some other strategy, no matter what the opponents do, and thus can be eliminated from consideration.

Iterated Elimination of Dominated Strategies

This is a method to simplify normal-form games by successively removing dominated strategies, potentially leading to a reduced game where equilibria are easier to identify.

Applications and Importance of Normal-Form Games

Normal-form games provide a powerful framework for analyzing strategic situations in economics, political science, biology, and computer science. They are used to study competition, cooperation, bargaining, auctions, oligopolies, voting systems, and many other interactive decision-making scenarios.

The normal form helps to analyze the strategic incentives of players when moves are made simultaneously or when the order of moves is irrelevant or unknown. It also serves as the basis for more complex models that incorporate sequential moves (extensive-form games) or incomplete information.

Example: Prisoner’s Dilemma in Normal Form

The Prisoner’s Dilemma is a classic example illustrating strategic interaction and the tension between individual rationality and collective welfare.

• Each player chooses between cooperating or defecting.
• Payoffs represent years in prison (negative values).
• Defecting strictly dominates cooperating for both players.
• The Nash equilibrium is both defecting, even though mutual cooperation yields a better collective outcome.

Mathematical Expression of Payoffs and Strategies

For a player i, the expected payoff under mixed strategies is calculated as:

where p_j(s_j) is the probability that player j selects strategy s_j under their mixed strategy p_j.

Summary of Core Elements

Normal-form games form the backbone of analyzing strategic interactions where simultaneous decisions define outcomes, providing analytical clarity and tools for equilibrium analysis, comparative statics, and predictions of rational behavior.