Part 2: The Cornerstone – Nash Equilibrium in Normal Form Games

If you missed the beginning of our series, start here: Part 1: A Comprehensive Introduction to Game Theory

Welcome back! In Part 1, we mapped out the entire landscape of game theory. Now we roll up our sleeves and start at the very foundation: the strategic (normal) form game and its central solution concept, the Nash Equilibrium.

REFERENCE NOTE

The concepts and mathematics in this post are drawn from Chapters 1–3 of An Introduction to Game Theory by Martin J. Osborne (Oxford University Press, 2003).

1. What Is a Strategic Game?

Before we can solve a game, we need to describe it precisely. A strategic game (or normal form game) is a model for situations where:

A finite set of players each choose a strategy simultaneously and independently.
Every combination of strategies produces a well-defined payoff for each player.

Formal Definition:

A strategic game is a triple \(\langle N, \{A_i\}_{i \in N}, \{u_i\}_{i \in N} \rangle\) consisting of:

A finite set of players \(N = \{1, 2, \ldots, n\}\).
For each player \(i \in N\), a nonempty set of actions \(A_i\) (also called pure strategies).
For each player \(i \in N\), a payoff function \(u_i : A \rightarrow \mathbb{R}\), where \(A = A_1 \times A_2 \times \cdots \times A_n\) is the set of all action profiles.

An action profile is a tuple \(a = (a_1, a_2, \ldots, a_n)\) specifying one action for each player. We write \(a_{-i}\) for the actions of all players except player \(i\).

2. Representing a Game: The Payoff Matrix

For two-player games with finite action sets, the most compact representation is the payoff matrix (also called a bimatrix). Each row corresponds to an action of Player 1, each column to an action of Player 2, and each cell contains the pair \((u_1, u_2)\).

The Prisoner’s Dilemma

Two suspects are interrogated separately. Each can either Cooperate (C) (stay silent) or Defect (D) (betray the other). The payoffs are (years in prison, negated so higher is better):

	C	D
C	(3, 3)	(0, 5)
D	(5, 0)	(1, 1)

Here, \((3,3)\) means both get 3 years of freedom, \((0, 5)\) means Player 1 gets 0 and Player 2 gets 5. The tragedy: both players choose D, landing at \((1,1)\), even though both prefer \((3,3)\).

3. Dominant Strategies

Sometimes one action is unambiguously best for a player — no matter what the opponent does.

Definition: Action \(a_i\) strictly dominates action \(a_i'\) for player \(i\) if: \(u_i(a_i, a_{-i}) > u_i(a_i', a_{-i}) \quad \forall a_{-i} \in A_{-i}\)

A rational player will never play a strictly dominated action. In the Prisoner’s Dilemma, D strictly dominates C for both players: regardless of what the opponent does, defecting always yields a strictly higher payoff.

THE CORE CONCEPT

When every player has a dominant strategy, the outcome is uniquely determined. The result is called a dominant strategy equilibrium — and it is a Nash Equilibrium.

Iterated Elimination of Strictly Dominated Strategies (IESDS): Even when dominant strategies don’t exist for everyone, we can iteratively remove dominated actions. If the process leads to a unique outcome, that outcome is the unique Nash Equilibrium.

4. Best Response and Nash Equilibrium

For most games, no player has a dominant strategy. We need a more general solution concept.

Definition (Best Response): Player \(i\)’s best response to the strategy profile \(a_{-i}\) of the others is: \(B_i(a_{-i}) = \underset{a_i \in A_i}{\text{argmax}} \; u_i(a_i, a_{-i})\)

The best response is the optimal action given what everyone else does. A Nash Equilibrium is a profile where every player is simultaneously playing their best response.

Definition (Nash Equilibrium): A strategy profile \(a^* = (a_1^*, \ldots, a_n^*)\) is a Nash Equilibrium if, for every player \(i \in N\):

\[u_i(a_i^*, a_{-i}^*) \geq u_i(a_i, a_{-i}^*) \quad \forall a_i \in A_i\]

Equivalently: \(a_i^* \in B_i(a_{-i}^*)\) for all \(i\).

THE CORE CONCEPT

In a Nash Equilibrium, no player has a unilateral incentive to deviate. It is a self-enforcing agreement: if all players expect equilibrium play, then playing the equilibrium action is individually rational.

5. Canonical Examples

The Battle of the Sexes

A couple wants to spend an evening together. Player 1 prefers Opera (O), Player 2 prefers Football (F), but both prefer being together over being apart.

	O	F
O	(2, 1)	(0, 0)
F	(0, 0)	(1, 2)

Checking best responses:

If Player 2 plays O: Player 1 prefers O (payoff 2 > 0). ✓
If Player 2 plays F: Player 1 prefers F (payoff 1 > 0). ✓

By symmetry, both (O, O) and (F, F) are Nash Equilibria in pure strategies. There is also a third, mixed strategy equilibrium (covered in Part 3).

The Stag Hunt

Two hunters can cooperate to hunt a Stag (S) (large reward if both cooperate) or individually hunt a Hare (H) (small but certain reward).

	S	H
S	(4, 4)	(0, 3)
H	(3, 0)	(3, 3)

Both (S, S) and (H, H) are Nash Equilibria. The (S, S) equilibrium is Pareto superior (both players prefer it), but (H, H) is risk dominant — it is safe regardless of what the other player does. The Stag Hunt models a deep tension between social optimality and individual risk aversion.

6. Nash’s Existence Theorem

Do Nash Equilibria always exist? Not always in pure strategies — we will see a counterexample in Part 3. But Nash proved something powerful:

Nash’s Theorem (1950): Every finite strategic game (finite players, finite action sets) has at least one Nash Equilibrium in mixed strategies.

The proof uses Kakutani’s fixed-point theorem: define the combined best-response correspondence \(B(a) = B_1(a_{-1}) \times \cdots \times B_n(a_{-n})\). Under mixed strategies, this correspondence is convex-valued and upper-hemicontinuous, so by Kakutani’s theorem, it has a fixed point — which is exactly a Nash Equilibrium.

Visualizing Payoffs

Below is a chart of each player’s payoff under both pure strategy profiles in the Prisoner’s Dilemma, illustrating the tension at the heart of the game.

{
  "title": { "text": "Prisoner's Dilemma: Payoffs by Outcome", "left": "center" },
  "tooltip": { "trigger": "axis" },
  "legend": { "data": ["Player 1 Payoff", "Player 2 Payoff"], "bottom": 0 },
  "xAxis": { "type": "category", "data": ["(C,C)", "(C,D)", "(D,C)", "(D,D)"] },
  "yAxis": { "type": "value", "name": "Payoff" },
  "series": [
    { "name": "Player 1 Payoff", "type": "bar", "data": [3, 0, 5, 1], "itemStyle": {"color": "#2196F3"} },
    { "name": "Player 2 Payoff", "type": "bar", "data": [3, 5, 0, 1], "itemStyle": {"color": "#4CAF50"} }
  ]
}

What’s Next?

We have established the backbone of game theory: the strategic form game and its Nash Equilibrium. But we saw that some games — like Battle of the Sexes — have multiple equilibria, and others have none in pure strategies.

In Part 3, we will introduce Mixed Strategy Nash Equilibrium, learn how to compute it, and see why randomization is a fundamental, rational behavior — not a sign of indecision!

REFERENCE NOTE