Game Theory: Continued Examples and Algorithm

Game Theory: Continued Examples and Algorithm

Review: What Nash Equilibrium Means

Before working through more examples, it is worth consolidating the core idea from Lecture 5. A Nash equilibrium is a profile of actions where every player is simultaneously best-responding. The critical test for any candidate equilibrium $(a_1^*, a_2^*)$ is: given that Player 2 plays $a_2^*$, is $a_1^*$ Player 1's best response? And given that Player 1 plays $a_1^*$, is $a_2^*$ Player 2's best response? If both answers are yes, we have a Nash equilibrium. If either answer is no, we do not.

A useful way to organize this check is the following algorithm for finding Nash equilibria in finite matrix games:

1. Go through each column. For each fixed action of Player 2, identify Player 1's best response(s) — the row(s) that maximize Player 1's payoff in that column. Mark them.
2. Go through each row. For each fixed action of Player 1, identify Player 2's best response(s) — the column(s) that maximize Player 2's payoff in that row. Mark them.
3. Nash equilibria are the cells where both players' choices are marked — that is, where both players are simultaneously best-responding.

This algorithm is mechanical and reliable. Let us apply it to the games from Lecture 5 more carefully, and then develop the Public Goods Game fully.

Revisiting the Coordination Game

In the Protest game, two citizens each choose to Protest (P) or Stay Home (H):

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

Step 1 (Player 1's best responses by column): Given Citizen 2 plays P, Citizen 1 gets 2 from P and 0 from H — best response is P. Mark (P, P). Given Citizen 2 plays H, Citizen 1 gets $-10$ from P and 0 from H — best response is H. Mark (H, H).

Step 2 (Player 2's best responses by row): Given Citizen 1 plays P, Citizen 2 gets 2 from P and 0 from H — best response is P. Mark (P, P). Given Citizen 1 plays H, Citizen 2 gets $-10$ from P and 0 from H — best response is H. Mark (H, H).

Step 3: Both (P, P) and (H, H) are doubly marked. These are the two Nash equilibria.

Notice: (P, H) and (H, P) are not equilibria. In (P, H), Citizen 1 gets $-10$ — she would switch to H. In (H, P), Citizen 2 gets $-10$ — she would switch to H. Neither is stable.

Revisiting Matching Pennies

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

Player 1's best responses: given Player 2 plays H, Player 1 gets 1 from H vs. $-1$ from T — best response is H. Given Player 2 plays T, Player 1 gets $-1$ from H vs. 1 from T — best response is T. Player 2's best responses: given Player 1 plays H, Player 2 gets $-1$ from H vs. 1 from T — best response is T. Given Player 1 plays T, Player 2 gets $1$ from H vs. $-1$ from T — best response is H.

Doubly-marked cells: In (H, H): Player 1's mark is here, but Player 2's best response to Player 1's H is T, not H — Player 2's mark is not here. In (H, T): Player 1's mark is at H (given T), and Player 2's mark is here (T given H). But wait — is this doubly marked? Player 1's best response to T is T (Player 1's mark is at T, not H, when Player 2 plays T). So (H, T) is not doubly marked.

Going through each cell: (H,H) has Player 1's mark (BR to H is H) but not Player 2's (BR to H is T). (H,T) has Player 2's mark (BR to H is T) but not Player 1's (BR to T is T). (T,H) has Player 2's mark (BR to T is H) but not Player 1's (BR to H is H). (T,T) has Player 1's mark (BR to T is T) but not Player 2's (BR to T is H). No cell is doubly marked — no pure-strategy Nash equilibrium exists. This is consistent with the intuition that any fixed pattern in this zero-sum game can be exploited.

The Public Goods Game: Full Treatment

Let us work through the Public Goods Game from first principles. Two players, A and B, each start with $20 and simultaneously choose contribution levels $x_A, x_B \in [0, 20]$. The payoff function for player $i$ is:

$u_i(x_A, x_B) = \underbrace{(20 - x_i)}_{\text{keep}} + \underbrace{1.5 \cdot \frac{x_A + x_B}{2}}_{\text{public good}}$

The first term is the money player $i$ keeps after contributing. The second term is the value of the public good — the average contribution scaled by 1.5 — which both players receive equally.

To find Player A's best response, we simplify her payoff:

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

Player A's payoff is a linear function of $x_A$ with coefficient $-0.25$. Since this coefficient is negative, $u_A$ is strictly decreasing in $x_A$ for any fixed $x_B$. Therefore, Player A's best response is $x_A^* = 0$, regardless of what B contributes.

The same calculation for Player B gives:

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

Again, $u_B$ is decreasing in $x_B$, so B's best response is $x_B^* = 0$.

The unique Nash equilibrium is $(x_A^*, x_B^*) = (0, 0)$.

To see why this is collectively inefficient, consider what happens if both contribute the maximum $x_A = x_B = 20$:

$u_i(20, 20) = 0 + 1.5 \cdot \frac{40}{2} = 30$

Each player receives $30. In the Nash equilibrium, each keeps $20. Full contribution would give each player $10 more. But contributing $20$ is not individually rational: if B contributes $20$, A's payoff from contributing $x_A$ is:

$u_A(x_A, 20) = 20 - 0.25 x_A + 0.75 \cdot 20 = 35 - 0.25 x_A$

This is maximized at $x_A = 0$, giving A a payoff of $35 — much better than the $30 she would get by contributing fully. B, reasoning symmetrically, also contributes 0. The Nash equilibrium delivers $(20, 20)$ rather than $(30, 30)$.

This is precisely the Prisoner's Dilemma logic in a continuous setting. Each player's dominant strategy is free-riding, but if both free-ride they end up worse off than if both had contributed. Individual rationality conflicts with collective welfare.

What Nash Equilibrium Tells Us (and What It Doesn't)

Nash equilibrium has important strengths as a solution concept:

Self-enforcement: In a Nash equilibrium, no player wants to deviate given what others are doing. It is stable in the sense that, if players ever coordinate on it, no one has an incentive to leave.

Correct beliefs: A Nash equilibrium can be interpreted as a situation where every player correctly anticipates what the others will do and acts optimally given those correct anticipations.

But Nash equilibrium also has limitations:

Multiple equilibria: When a game has more than one Nash equilibrium — as in the Fighting Countries and Coordination games — the theory does not predict which equilibrium will be reached. This is a genuine indeterminacy that requires additional theory (focal points, communication, history, institutional context) to resolve.

No account of dynamics: Nash equilibrium is a static concept. It says nothing about whether players will converge to equilibrium from arbitrary starting points, how long convergence takes, or what happens when players repeatedly interact and adjust.

Despite these limitations, Nash equilibrium is the workhorse of modern game theory and political economy. It provides clean, testable predictions in many settings, and the qualitative insights it delivers — about the logic of cooperation failures, coordination problems, and strategic competition — are robust and widely applicable.

Bridge to Electoral Competition

The games in Lectures 5 and 6 have all featured abstract players and actions. Starting in Lecture 7, we apply this framework directly to electoral competition. The players are political candidates. Their actions are policy platforms $x_i \in [-1, 1]$. The payoffs depend on electoral outcomes — who wins and what policy gets implemented — which in turn depend on voter preferences as characterized by the Median Voter Theorem.

The key insight bridging social choice theory and game theory is this: the Median Voter Theorem tells us that the candidate closer to $v^M$ wins. This converts the candidates' optimization problems into best-response problems, and we can find Nash equilibria in platforms. What those equilibria look like — whether candidates converge to the center or diverge to the extremes — depends critically on what candidates care about, which is the subject of Lectures 7 through 9.