Static Games and Nash Equilibrium

Static Games and Nash Equilibrium

What Is Game Theory?

Roger Myerson defined game theory as "the mathematical study of conflict and cooperation among rational decision-makers." The key features of this definition are worth unpacking. Rational means that each player maximizes her utility given her beliefs about what others will do — the same assumption we made in Lectures 1 and 2. Conflict and cooperation captures the central feature that distinguishes game theory from ordinary decision theory: the payoff each player receives depends not only on her own choice, but on the choices of others. When my payoff depends on your actions and vice versa, we have strategic interaction, and ordinary individual optimization is insufficient to predict behavior.

Game theory is the right framework for analyzing electoral competition (candidates choose platforms, voters respond), legislative bargaining, international conflict, and many other political economy phenomena.

Taxonomy of Games

Games can be classified along two dimensions:

Timing: A game is static (or simultaneous) if all players choose their actions at the same time without observing each other's choices. A game is dynamic (or sequential) if players move in some order and later movers can observe what earlier movers chose. This lecture covers static games exclusively.

Information: A game has complete information if all players know all other players' payoff functions. It has incomplete information if some players have private information about payoffs. This lecture and the next cover complete information games.

The Formal Definition of a Static Game

A static game of complete information has three components:

Players: A set $N = \{1, 2, \ldots, n\}$ of players who will make decisions.

Actions: Each player $i$ has an action set $A_i$, and chooses an action $a_i \in A_i$. The collection of all players' choices $(a_1, a_2, \ldots, a_n)$ is called an action profile. It is standard to write $a_{-i} = (a_1, \ldots, a_{i-1}, a_{i+1}, \ldots, a_n)$ for the actions of all players except $i$.

Payoffs: Each player $i$ has a payoff function $u_i(a_i, a_{-i})$ that maps every action profile to a real number representing player $i$'s utility from that outcome.

Best Response and Nash Equilibrium

Given a static game, we need a solution concept — a prediction of how rational players will behave. We build up to Nash equilibrium in two steps.

A player's best response to a given profile of opponents' actions is the action that maximizes her payoff. Formally, $a_i^* \in A_i$ is a best response to $a_{-i}$ if:

$u_i(a_i^*, a_{-i}) \geq u_i(a_i, a_{-i}) \quad \text{for all } a_i \in A_i$

A best response is not a prediction of behavior by itself — it tells player $i$ what to do given what she believes others are doing. To close the model, we need a consistency condition on beliefs.

A Nash equilibrium is an action profile $(a_1^*, a_2^*, \ldots, a_n^*)$ in which every player is simultaneously best-responding to all other players' choices:

$u_i(a_i^*, a_{-i}^*) \geq u_i(a_i, a_{-i}^*) \quad \text{for all } a_i \in A_i \text{ and all } i \in N$

The defining property is mutual best response: each player's action is optimal given the others' actions, and no player can gain by deviating unilaterally. A Nash equilibrium is self-enforcing: if players arrive at a Nash equilibrium through any process — communication, convention, or learning — none has an incentive to deviate.

The standard algorithm for finding Nash equilibria in games with finite action sets is: (1) for each player and each possible action of the opponents, find the player's best response; (2) look for action profiles where every player is best-responding simultaneously.

Prisoner's Dilemma

Two suspects, Player 1 and Player 2, are held in separate rooms. Each can either Remain Silent (S) or Testify against the other (T). The payoffs represent years in prison (negative is worse):

$\begin{array}{c|cc} & S & T \\ \hline S & -2,\,-2 & -20,\,0 \\ T & 0,\,-20 & -5,\,-5 \end{array}$

Let us find the best responses. If Player 2 plays S, Player 1 gets $-2$ from S and $0$ from T, so Player 1 prefers T. If Player 2 plays T, Player 1 gets $-20$ from S and $-5$ from T, so Player 1 still prefers T. In both cases, Player 1's best response is T, regardless of what Player 2 does. We say T is a dominant strategy for Player 1. By symmetry, T is also a dominant strategy for Player 2.

The unique Nash equilibrium is (T, T), with payoffs $(-5, -5)$. But notice: both players would be better off with (S, S), which yields $(-2, -2)$. The individually rational outcome is collectively inferior. This is the tragedy of the Prisoner's Dilemma: rational individual behavior leads to a Pareto-inferior outcome.

The Prisoner's Dilemma arises in many political economy settings — arms races between countries, free-rider problems in public goods provision, and failure of international climate agreements all share this structure.

Fighting Countries

Countries A and B each decide whether to Fight (F) or Not Fight (NF). If neither fights, both get 0. If one fights and the other does not, the fighter gets a payoff reflecting territorial or geopolitical gain, while the non-fighter gets 0. If both fight, both incur costs: $(-1, -1)$.

Specifically, suppose:

$\begin{array}{c|cc} & NF & F \\ \hline NF & 0,\,0 & 0,\,2 \\ F & 3,\,0 & -1,\,-1 \end{array}$

Best responses: If B plays NF, A gets 0 from NF and 3 from F — A prefers F. If B plays F, A gets 0 from NF and $-1$ from F — A prefers NF. So A's best response to NF is F, and A's best response to F is NF. By checking B symmetrically: B's best response to NF is F, and B's best response to F is NF.

The Nash equilibria are the profiles where both players are best-responding:

• (F, NF): A plays F (BR to NF), B plays NF (BR to F). Payoffs $(3, 0)$. This is a NE.
• (NF, F): A plays NF (BR to F), B plays F (BR to NF). Payoffs $(0, 2)$. This is a NE.

There are two Nash equilibria, with different payoffs for each player. The theory does not predict which one will be selected — multiple equilibria is a genuine limitation of Nash equilibrium as a solution concept.

Coordination Game (Protest)

Two citizens, 1 and 2, each decide whether to Protest (P) or Stay Home (H). If both protest, each gets a payoff of 2 from the collective action. If only one protests, that person faces a penalty of $-10$ (repression, arrest) while the other gets 0. If both stay home, both get 0.

$\begin{array}{c|cc} & P & H \\ \hline P & 2,\,2 & -10,\,0 \\ H & 0,\,-10 & 0,\,0 \end{array}$

Best responses: If Citizen 2 plays P, Citizen 1 gets 2 from P and 0 from H — prefers P. If Citizen 2 plays H, Citizen 1 gets $-10$ from P and 0 from H — prefers H. So Citizen 1's best response matches whatever Citizen 2 does.

There are two Nash equilibria: (P, P) with payoffs $(2, 2)$ and (H, H) with payoffs $(0, 0)$. The (P, P) equilibrium is Pareto superior — both citizens prefer it. But if either citizen believes the other will stay home, staying home is individually optimal. This is a coordination problem: the bad equilibrium (H, H) is self-enforcing even though (P, P) is better for everyone. Protests, revolutions, and collective action more generally have this structure. Coordination devices — public speeches, shared calendars, common knowledge of a focal point — can shift expectations toward the good equilibrium.

A 3×3 Matrix Example

Consider two players with three actions each: Top (T), Middle (M), Bottom (B) for Player 1, and Left (L), Center (C), Right (R) for Player 2. The payoff matrix is:

$\begin{array}{c|ccc} & L & C & R \\ \hline T & 1,\,2 & 3,\,1 & 4,\,2 \\ M & 2,\,3 & 1,\,2 & 2,\,1 \\ B & 1,\,1 & 2,\,4 & 3,\,3 \end{array}$

Finding Nash equilibria requires checking all $3 \times 3 = 9$ cells. Player 1's best responses: given L, Player 1 gets 1, 2, or 1 from T, M, B — best response is M. Given C, Player 1 gets 3, 1, or 2 — best response is T. Given R, Player 1 gets 4, 2, or 3 — best response is T. Player 2's best responses: given T, Player 2 gets 2, 1, or 2 — best responses are L and R. Given M, Player 2 gets 3, 2, or 1 — best response is L. Given B, Player 2 gets 1, 4, or 3 — best response is C.

Now find mutual best responses: (M, L) — Player 1 plays M (BR to L), Player 2 plays L (BR to M). Both are best-responding. NE with payoffs $(2, 3)$. (T, R) — Player 1 plays T (BR to R), Player 2 plays R (BR to T, since L also works but we check R specifically). Player 2's BR to T includes R. Both best-responding. NE with payoffs $(4, 2)$. Two Nash equilibria exist.

Matching Pennies and the Limits of Pure Strategies

Two players each secretly place a penny, Heads (H) or Tails (T). Player 1 wins if they match; Player 2 wins if they differ.

$\begin{array}{c|cc} & H & T \\ \hline H & 1,\,-1 & -1,\,1 \\ T & -1,\,1 & 1,\,-1 \end{array}$

This is a zero-sum game: the payoffs always sum to zero. Check each cell: (H,H) — Player 2 can deviate to T and get 1 instead of $-1$. (H,T) — Player 1 can deviate to T and get 1. (T,H) — Player 1 can deviate to H. (T,T) — Player 2 can deviate to H. No cell is a Nash equilibrium. Matching Pennies has no pure-strategy Nash equilibrium.

This is not a failure of the theory — it is an accurate description of the strategic situation. In spy movies and poker, pure strategies invite exploitation: any fixed pattern can be anticipated and countered. The resolution involves mixed strategies (randomizing over actions), but mixed strategy equilibria are beyond the scope of this course.

Nash Equilibrium: The Good and the Bad

Nash equilibrium has two great virtues. First, it is self-enforcing: in a Nash equilibrium, no player wants to deviate, so it is stable once reached. Second, it embodies correct conjectures: each player's action is optimal given the actual actions of others, not given mistaken beliefs.

Nash equilibrium also has well-known limitations. When multiple equilibria exist — as in the Fighting Countries and Coordination games — the theory provides no sharp prediction. Different equilibria may favor different players, and the theory is silent on which will be selected. Additionally, Nash equilibrium says nothing about the process by which players arrive at equilibrium: it is a static consistency condition, not a dynamic story of learning or adjustment.

Continuous Action Spaces

So far we have considered games where each player has finitely many discrete actions. Many economic and political applications have continuous action spaces: a candidate chooses a platform $x \in [-1, 1]$, a firm sets a price $p \in [0, \infty)$, a bidder names a bid $b \in [0, \bar{b}]$.

In these settings, the Nash equilibrium concept is the same — mutual best response — but we find best responses by calculus rather than by inspecting a finite matrix. Each player's best response becomes a best-response function $a_i^*(a_{-i})$ that maps opponents' actions to the optimal action. Nash equilibria are fixed points: $(a_1^*, a_2^*)$ where $a_1^* = \text{BR}_1(a_2^*)$ and $a_2^* = \text{BR}_2(a_1^*)$ simultaneously.

Public Goods Game

As an application with continuous actions, consider two players (A and B) each starting with $20. Each simultaneously chooses how much to contribute to a public good: $x_i \in [0, 20]$. The payoff is:

$u_i(x_A, x_B) = (20 - x_i) + 1.5 \cdot \frac{x_A + x_B}{2}$

The first term is the money A keeps; the second is her share of the public good, which equals 1.5 times the average contribution. Let us compute A's payoff in terms of actions:

$u_A(x_A, x_B) = 20 - x_A + 0.75(x_A + x_B) = 20 - 0.25 x_A + 0.75 x_B$

The coefficient on $x_A$ is $-0.25 < 0$. Regardless of what B contributes, A's payoff decreases in her own contribution. So A's best response is $x_A^* = 0$ for any $x_B$. By symmetry (the problem is identical for B), $x_B^* = 0$.

The unique Nash equilibrium is $(x_A^*, x_B^*) = (0, 0)$ — complete free-riding. No public good is provided.

But if both contributed $x_A = x_B = 20$, each would receive $0 + 0.75 \cdot 40 = 30 > 20$. Full contribution dominates free-riding in terms of total welfare. Yet contributing $20$ is not individually rational: if B contributes 20, A gets $20 - 0.25 \cdot 0 + 0.75 \cdot 20 = 35$ by free-riding vs. $30$ by contributing fully. Free-riding is always better for the individual — this is the Prisoner's Dilemma structure applied to a continuous setting.

The Public Goods Game captures the logic of tax evasion, underinvestment in infrastructure, and failure to maintain common resources — the "tragedy of the commons."

Looking Ahead

The game-theoretic toolkit developed in this lecture — players, actions, payoffs, best responses, Nash equilibrium — provides the foundation for the electoral competition models of Lectures 7 through 9. In those models, the players are candidates, actions are policy platforms, and voters determine payoffs through the electoral outcome. The Median Voter Theorem tells us how voters behave; Nash equilibrium will tell us what candidates choose.