Experiment 02 Nash Equilibrium
Why Are We Stuck?
Choose a move, inspect the payoff table, and see whether either player can improve by changing alone.
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Start with “Work It Out, Step by Step” below. The optional expert section explains its symbols as you go. For more examples, use the plain-language math guide.
1. Make a Prediction
Would Anyone Want to Move?
A game has two players, each choosing A or B. One player chooses a row; the other chooses a column. Where they meet, the table shows (row player’s points, column player’s points). For example, (3, 5) means 3 points for the row player and 5 for the column player.
More points are better. A Nash equilibrium is a pair of choices where neither player can earn more by switching alone. Keep the other player’s choice the same when checking a switch.
2. Make a Consequential Choice
Choose an Outcome, Then Try to Escape
Choose A or B for each player, then reveal the incentives.
Switching alone is called a “unilateral move.” A “best response” is a choice that earns the most points given the other player’s choice. Tied choices can both be best responses. Try a move even when it loses points, then see why that player might resist it.
| Row player ↓ / Column player → | A | B |
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3. Change One Assumption
Rewrite the Payoffs
A payoff is the points a player receives. Edit one payoff and reveal the outcome again. A tie counts as a reason to stay: an equilibrium only requires that neither player can earn more by switching alone. Shared links preserve these points and starting choices.
Reveal the Incentives
Work It Out, Step by Step
Worked example using the default Prisoner’s Dilemma points. These numbers explain the starting game; they do not update when you edit the table.
In this game, A means cooperate: choose the action that helps both players when both do it. B means defect: choose the other action, which pays you more in this round if the other player’s choice stays the same. Test one switch at a time using subtraction.
- Start at A / A. Each player gets 3 points. Together, they get \(3 + 3 = 6\).
- Let only the row player switch to B. The column player stays at A. The row player’s points rise from 3 to 5, a gain of \(5 - 3 = 2\). That means A / A is not an equilibrium: someone can gain by switching alone.
- Now test B / B. Each player gets 1 point. If the row player alone switches to A, their points drop from 1 to 0: \(0 - 1 = -1\). The minus sign means a loss of 1 point. The column player would also lose 1 by switching alone.
- Compare the result. B / B is an equilibrium because neither player can earn more by switching alone. But its combined score is only \(1 + 1 = 2\), compared with 6 at A / A.
The question “Can I do better by switching alone?” is different from “Could we do better by changing together?” An equilibrium answers the first question.
This explorer checks pure choices: choose A or choose B for sure. A strategy that uses chance, such as a coin flip, is a mixed strategy. Some games have no equilibrium among pure choices but do have one using chance. A tie also matters: if switching changes your points by 0, you have not earned more, so that switch does not rule out an equilibrium.
For experts: formal model and assumptions
How to Read the Symbols
These formulas write the same “Would switching help?” check in shorthand. Read the translations beside them first, or visit the math reading guide for more examples.
- Choices and sets
- \(r\) is the row player’s choice, and \(c\) is the column player’s choice. \(\{A,B\}\) is a set: a collection containing the choices A and B. \(\in\), read “is in,” says a choice belongs to that collection. So \(r,c\in\{A,B\}\) says each player chooses A or B. Read about sets.
- Payoff functions and labels
- \(u_R(r,c)\), read “u for the row player at choices r and c,” looks up the row player’s points for those choices. The small \(R\) labels the row player; \(C\) labels the column player. The parentheses hold the two choices to look up. They do not mean multiplication. A general label \(i\) means either player. Read about functions and labels.
- Best response and arg max
- \(BR\) stands for “best response.” \(\arg\max\), read “arg max,” asks which choice gives the largest score. For example, if A earns 3 and B earns 5, the largest score is 5, but the arg max is the choice B. The text below arg max says which choices to check. A tie can produce two best choices. Read about maximums and best choices.
- Stars and comparisons
- \(r^*\), read “r star,” and \(c^*\), “c star,” label the choices being tested as an equilibrium. The star is a label, not multiplication or a power. \(\ge\) means “greater than or equal to,” so matching the other score is allowed. Read about comparisons.
- Chance, multiplication, and expected points
- \(x\) and \(y\) are each player’s chance of choosing A, written from 0 to 1. A chance of 0.5 means 50%. The remaining chance of choosing B is \(1-x\) or \(1-y\). Letters written together mean multiply: \(xy\) means \(x\) times \(y\). Uppercase \(U_i(x,y)\) means player i’s expected points: the average these chances predict over many plays. Read about probability and averages.
Best Responses and Nash Equilibrium
Let \(r,c\in\{A,B\}\) be the row and column actions. The functions \(u_R(r,c)\) and \(u_C(r,c)\) give their respective payoffs. A best response maximizes a player’s payoff with the other player’s action fixed:
Read it aloud: “Hold the column player’s choice still and find the row choices that earn the most points. Then hold the row player’s choice still and do the same for the column player.” The subscripts on \(BR_R\) and \(BR_C\) tell you whose best choices you are finding.
These are sets, so both actions belong when payoffs tie. A pure Nash equilibrium \((r^*,c^*)\) is a mutual best response. Equivalently:
Read the two checks aloud: “The row player’s current points are at least as high as the points from either row choice, while the column choice stays still. The same is true for the column player.” The pair \((r^*,c^*)\) lists the row choice first and the column choice second. “For every” means check both A and B, not just the better-looking one.
The implementation checks each of the four cells against each player’s other action, using weak inequalities to retain ties. It reports the change from a unilateral switch as alternative payoff minus current payoff. A cell is an equilibrium exactly when both reported changes are at most zero.
The model assumes two players, two actions each, simultaneous choices, known payoffs, and maximization of each player’s own payoff. The switching controls inspect incentives; they do not simulate a learning process or establish convergence.
For mixed strategies, let \(x\) be the probability that the row player chooses A and \(y\) the probability that the column player chooses A. With independent randomization, player \(i\)’s expected payoff is:
Read it aloud: “For each of the four possible outcomes, multiply its chance by its points, then add the four answers.” Independent choices let us multiply the two players’ chances: if each chooses A with chance 0.5, A / A has chance \(0.5\times0.5=0.25\), or 25%. In that example all four outcomes have a 25% chance. Parentheses in \((1-x)(1-y)\) group two numbers to multiply; parentheses after \(u_i\) instead tell us which outcome’s points to look up.
Finite games have a Nash equilibrium allowing mixed strategies. This explorer searches only pure choices and does not calculate mixed equilibria. Thus “no pure equilibrium” does not mean “no equilibrium.”
The combined score \(u_R+u_C\) means add the row player’s points to the column player’s points. It is a teaching measure on a common point scale; meaningful comparisons between people require that scale. An equilibrium need not maximize this sum or be Pareto efficient. “Pareto efficient” means there is no available outcome that helps someone without hurting anyone else. Primary reading: John Nash, Equilibrium Points in N-Person Games (1950).
4. Transfer the Lesson
Two departments use incompatible standards. Each would benefit if both moved to one shared standard, but switching alone is costly. Does their current stable arrangement prove it is the best arrangement?
Reveal a possible answer
No. Stability against one person changing alone says nothing about a coordinated change by both. A shared commitment or a change in switching costs can make a better arrangement reachable. The coordination preset lets you examine this difference.
A Model Is a Place to Start
These small models make the incentives visible. Their results follow from their stated rules; they are not forecasts of how every person or organization behaves. A simulated strategy is a rule, not a personality.
Scenario links save the controls and random seed. To reproduce an interactive run, make the same choices in the same order. Changing a setting restarts the experiment.