Math Symbols,
in Plain English

Read One Piece at a Time

Suppose a game gives you 3 points each round. Call the number of rounds r.

1. Name the input: an input is the number or choice you put into a rule. Here, r is how many rounds you play.
2. Name the result: “P of r” means your total points for that many rounds.
3. Read the instruction: multiply the number of rounds by 3. The multiplication sign is left out in “3r.”
4. Try a number: for 4 rounds, the total is 3 × 4 = 12 points.

Always check the nearby definition. The same letter can mean something different on another page. Here P means points; elsewhere it might stand for probability. Capital and lowercase letters can also have different meanings.

Letters and Small Labels

\(x\), \(n\), or another letterIn words: a named number

A variable is a name for a number that can change or is not yet known. If n is the number of players, “n = 8” means there are 8 players. A constant is a number we hold fixed in that calculation.

\(u_i\)In words: “u sub i,” or “player i’s points”

The small lower label is a subscript. It tells you which item you mean. If i is 2, u with the label 2 means player 2’s points. It does not mean “u times 2.” Other labels name what we are measuring: R for the row player, C for the column player, or “outer” for an outer road route.

\(s_{-i}\)In words: “everyone else’s choices”

In these game formulas, the label “−i” means all the players except player i. It is a label, not an instruction to subtract a player.

\(x^*\)In words: “x star”

The star marks a special value chosen by the explanation, such as an equilibrium choice or a best result. An equilibrium in these games is a situation where no player gains by changing only their own choice. That situation does not have to be the best result for the group. The star is a label, not multiplication; check what it marks in that formula.

NE and SOIn words: two different kinds of result

NE stands for Nash equilibrium, the no-gain-from-switching-alone situation just described. SO stands for social optimum: the best result for the group according to the model’s chosen measure. In the traffic example, that measure is everyone’s total driving time, so SO means the plan with the lowest total. It need not give each person their shortest trip.

\(q_i^t\)In words: “team i’s share at review t”

In the coordination experiment, the raised t labels a review number. It is not a power, which indicates repeated multiplication. The lower i labels the team. At review 3, the raised 3 simply names that review; it does not ask you to multiply anything. The local definition tells you when a raised mark is a label.

\(x^T\)In words: “x transpose”

The raised capital T means to turn a column of numbers into a row, or a row into a column. It changes their arrangement, not their values. The tables and vectors example shows how this is used.

\(\tau\) and \(\varepsilon\)In words: “tau” and “epsilon”

These are Greek letters used as names. In the playground, tau (rhymes with “cow”) names a traffic toll, and epsilon (“EP-sih-lon”) names a chance of a mistake. They work like familiar letters such as x or y; the definition beside the formula tells you what each one means.

\(\bar u\) and \(\hat q\)In words: “u bar” and “q hat”

The line above u marks an average. When every number counts equally, add the numbers and divide by how many there are: the average of 2 and 4 is (2 + 4) ÷ 2 = 3. The little roof above q marks an estimate: a value worked out from the information available, which may differ from what actually happens. See the longer average example.

Familiar Arithmetic in a New Shape

\(\frac{6}{3}\)In words: “six divided by three”

The fraction bar means divide the entire top by the entire bottom. The top is called the numerator; the bottom is the denominator. In 6 over 3, the numerator is 6 and the denominator is 3, so the answer is 2. For (4 + 2) over 3, add the top first, then divide: 6 ÷ 3 = 2. The denominator cannot be zero: dividing by zero does not give a single numerical answer.

\(3x\), \(xy\), or \(x\cdot y\)In words: multiplication

Numbers and variables written next to one another often mean multiply. If x is 2 and y is 5, then “xy” means 2 × 5 = 10. A centered dot can also mean multiply. A function is a named rule that takes a number or choice and gives a result. In a function such as “P(r),” the parentheses hold what you put into the rule. See functions and their inputs.

\(x^2\)In words: “x squared”

The small raised 2 is an exponent. With an exponent of 1, 2, 3, and so on, it tells you how many copies of the number to multiply together. Each number being multiplied is a factor. For example, 3 squared means 3 × 3 = 9: two factors, each equal to 3. Cubed means three copies multiplied together: 2 cubed is 2 × 2 × 2 = 8. An expression such as x squared is called a power. Any number other than zero raised to the zero power equals 1.

\(2(3+1)\)In words: “two times the quantity three plus one”

Parentheses keep a group together. Work inside them first: 3 + 1 = 4, then 2 × 4 = 8. Next do powers, then multiplication and division from left to right, then addition and subtraction from left to right. Square or curly brackets can also group a calculation.

\(-2\)In words: “negative two”

A minus sign before a number says it is below zero. In a points game it can mean a loss. Subtracting a larger cost from a smaller reward can produce a negative result: 3 − 5 = −2.

Compare Two Amounts

\(=\), \(\ne\), and \(\approx\)In words: equal, not equal, and approximately equal

3 + 2 = 5 says both sides have the same value. 3 ≠ 5 says they differ. The wavy sign in 1 ÷ 3 ≈ 0.33 says the answer is close, often because we rounded it.

\(>\) and \(<\)In words: greater than and less than

5 > 3 means “5 is greater than 3.” Read 3 < 5 as “3 is less than 5.” The wide, open end faces the larger number.

\(\ge\) and \(\le\)In words: greater than or equal to, and less than or equal to

The extra line allows a tie. “Points ≥ 3” includes exactly 3 points. “Cost ≤ 5” includes exactly 5. Expert text may call these weak inequalities; “weak” just means equality is allowed.

\(0\le p\le1\)In words: “p is between zero and one, including both ends”

This joins two comparisons: p is at least 0 and at most 1. For example, 0.25 fits; 1.2 does not.

Functions Are Named Rules

A function is a named rule that takes an input and gives a result. The input is the number or choice you put into the rule. Read \(C(z)\) as “C of z”: the cost when the input is z. Here the parentheses hold the input; they do not mean “C times z.”

For example, if the rule is \(C(z)=2z+5\), then an input of 3 gives 2 × 3 + 5 = 11. We write \(C(3)=11\).

Some rules use more than one input. Read \(u_R(r,c)\) as “the row player’s points when the row choice is r and the column choice is c.” The comma separates the inputs. You can find the answer by looking up that row and column in a points table.

Try reading a points table in Why Are We Stuck?

“min” and “max” Pick a Number

min means choose the smallest number listed. max means choose the largest. So \(\min\{3,7\}=3\) and \(\max\{3,7\}=7\). The braces keep the choices together.

They can also keep an answer inside an allowed range: the values from the smallest permitted amount to the largest. Here the range runs from 0 through 10. Read the inside instruction first:

1. Inside: choose the larger of 0 and x. This stops the answer from going below 0.
2. Outside: choose the smaller of 10 and that answer. This stops it from going above 10.
3. Try it: an input of −3 gives 0; an input of 6 gives 6; an input of 14 gives 10.

Expert text sometimes calls this “clipping” a value. In the traffic experiment, it prevents the number of shortcut drivers from being negative or larger than the total number of drivers.

The Large Σ Means “Add These Up”

Capital sigma (“SIG-muh”) is a compact way to write a long addition. The label below tells you where to start. The number above tells you where to stop, including that number.

Read it: “Add x sub i, for i from 1 through 3.” If the three scores are 2, 4, and 6, this means 2 + 4 + 6 = 12. The letter i is just a counter that takes the values 1, 2, and 3.

The upper number on a summation sign is a stopping label, not an exponent. If the lower label says “j ≠ i,” add the listed items except the one labeled i. Two summation signs mean repeat the addition for two lists: for example, add every cell across each row, then add the row totals.

To find a usual average, add the values and divide by how many there are. For these scores, 12 ÷ 3 = 4.

Chance and Expected Values

\(p=0.25=25\%\)In words: “a one-in-four chance”

A probability is a chance written from 0 to 1. Multiply by 100 to turn it into a percentage. Zero means impossible under the rules; 1 means certain. A 25% chance does not promise exactly one success in every four tries.

\(1-p\)In words: “the chance of the other outcome”

When two outcomes cover all possibilities and cannot both happen, their chances add to 1. If the chance of continuing is 0.75, the chance of stopping is 1 − 0.75 = 0.25.

\(P(X=3)\) or \(\Pr(X=3)\)In words: “the chance that X equals three”

Here X names a result that can vary by chance. A fair die has the same chance of landing on each face. For a fair six-sided die, the chance of rolling a 3 is 1 ÷ 6. “Pr” is another way to write probability.

\(E[X]\) or \(\mathbb{E}[X]\)In words: “the expected value of X”

This is the average predicted by the possible results and their chances over many repetitions. Multiply each possible result by its chance, then add. A fair coin has a 50% chance of heads and a 50% chance of tails. If it pays 4 points for heads and 0 for tails, the expected score is 0.5 × 4 + 0.5 × 0 = 2 points. One flip still pays either 4 or 0.

\(pq\)In words: “p times q,” when choices are independent

If one event has chance 0.5 and an independent event has chance 0.2, the chance of both is 0.5 × 0.2 = 0.1. Independent means learning one result does not change the other’s chance. Do not multiply the chances this way without that assumption.

Percentage pointsIn words: a difference between percentages

Moving from 40% to 60% adds 20 percentage points. It is a 50% increase relative to the starting value, because 20 ÷ 40 = 0.5. The coordination control uses percentage points: +20 means add 20 directly to the percentage.

Try the ticket-bag example in Be Unpredictable.

Pairs, Lists, and Tables

A pair such as (3, 5) lists two values in order. In a playground payoff table, the first is the row player’s points and the second is the column player’s points. The comma is not division.

A vector is an ordered list. For example, \(p=(0.5,0.25,0.25)\) can mean a 50% chance of rock, 25% of paper, and 25% of scissors, in that order. Those chances add to 1.

A raised capital T means transpose: turn a row into a column, or a column into a row. For example, \((2,4,6)^T\) puts 2, 4, and 6 vertically instead of side by side. The values stay the same. In the expert formula \(x^TAy\), x and y list the two players’ chances and A is the points table. Read it as: “Multiply each outcome’s points by its chance, then add them all.”

A matrix is a rectangular table of numbers. In the expert rock-paper-scissors table, each row is one of your choices and each column is one of your opponent’s. The entry gives your points: 1 for a win, 0 for a tie, −1 for a loss. The row and column labels explain how to look up an entry.

Read \(A_{ij}\) as “the entry of table A in row i, column j.” If the rows and columns are numbered starting with 1, \(A_{23}\) is row 2, column 3. It does not mean “A times twenty-three.”

For expected points, multiply each cell’s points by the chance of landing in that cell, then add the results. The compact matrix formula is shorthand for adding those results.

Lists of Choices and Rules With Cases

\(\{A,B\}\)In words: “the set containing A and B”

A set is a collection of allowed items. Curly braces can mark that collection. Check the context: braces around a calculation may simply keep a group together.

\(r\in\{A,B\}\)In words: “r is one of A or B”

The symbol ∈ means “belongs to.” Here the row player must choose one of those two actions. The symbol ∉ means “does not belong to.”

“arg max”In words: “which choice gives the largest result?”

Suppose choice A pays 3 points and choice B pays 5. The maximum payoff is the number 5; the arg max is the choice B that earns it. If both pay 5, both choices belong in the answer. “Arg min” asks which choice gives the smallest result.

\(BR_i\)In words: “player i’s best responses”

“BR” abbreviates best response: the choice or choices giving a player the most points when the other players’ choices stay fixed. A tie can give more than one best response.

\(u_E(\text{enter}\mid a)\)In words: “the new business’s points for entering, given action a”

An entrant is the new business deciding whether to start competing with an existing business. The vertical bar means given: work out the result assuming the existing business takes action a. If that action is “share,” it accepts competition from the new business; look up the points for that situation. The bar is not a division sign.

A tall brace beside several rowsIn words: “use the row whose condition fits”

This is a piecewise rule. For example: charge 0 if a child is under 5; charge 3 if the child is 5 or older. Check the age, then use the matching row. Do not add all the rows together.

“max min”In words: “choose the best of the worst cases”

First find each plan’s lowest possible score. Then pick the plan with the highest of those low scores. If plan A’s worst score is −2 and plan B’s is 0, this rule chooses B. The order matters; follow the explanation beside the formula.

Prime Marks Describe Change

Calculus is a branch of math that studies change and how small amounts add up. Here we only need its language for change. You can play the experiments and follow their number examples without studying calculus.

A graph can show a rule as a picture: moving right increases the input, and the height shows the result. Its slope describes how much the height changes compared with how far you move right. A positive slope rises, a negative slope falls, and a slope of zero is level at that point. A graph line that bends is a curve; its slope can change from one place to another.

\(L'(z)\)In words: “L prime of z”

The small stroke is a prime mark. Here it means a derivative: the rate of change, or slope, at one input value. It describes how fast the result L changes as the input z increases a tiny amount. A positive rate means the result is increasing there; a negative rate means it is decreasing.

\(L''(z)\)In words: “L double prime of z”

This is the second derivative: how the slope itself changes as the input grows. When it is positive throughout the values we are checking, the slope keeps getting larger. The result might still be falling, but less steeply. If the slope changes from negative to positive, the result changes from falling to rising, giving a lowest point there.

A small example: if a cost rule is \(L(z)=3z+2\), its rate of change is 3: each increase of 1 in the input adds 3 to the cost. Other rules can have rates that vary. The traffic expert section uses these marks to explain how it finds the best shared plan. It also checks the endpoints: the smallest and largest allowed inputs. If the allowed inputs run from 0 through 10, those endpoints are 0 and 10.

Prime marks have other uses elsewhere, such as marking a different version of a value. Read the local definition before assuming a prime always means a derivative.

See the traffic example worked out with ordinary arithmetic.

Update a Chance After New Evidence

Evidence is information you can use to judge a claim. Conditional probability means a chance worked out using specified information. In the signaling experiment, a badge is evidence about a product’s hidden quality, but it is not proof.

\(P(H\mid S)\)In words: “the chance of high quality, given a signal”

Here H means the product has high quality, and S means it shows the signal, such as a badge. The upright bar means given: look only at products showing the signal. It is not division. \(P(S\mid H)\) asks a different question: among high-quality products, how many show the signal?

Prior, likelihood, and posteriorIn words: a starting chance, an evidence rate, and an updated chance

A prior is the chance before this new observation. A likelihood tells you how often this evidence would appear under a particular possibility. A posterior is the chance after using the observation. Bayes’ rule is a way to calculate that update by comparing the cases that could have produced the evidence.

Count a small example: imagine 100 products. Sixty have high quality and 40 have ordinary quality. Suppose 80% of high-quality products show a badge, while 20% of ordinary products do.

1. High quality with a badge: 60 × 0.8 = 48 products.
2. Ordinary quality with a badge: 40 × 0.2 = 8 products.
3. All products with a badge: 48 + 8 = 56.
4. Among those 56, the high-quality share is 48 ÷ 56, or about 86%.

The starting chance was 60%; the updated chance after seeing a badge is about 86%. If no product could show the badge under the model, we would have zero cases to divide by. That conditional chance is undefined: these rules do not supply an answer. It is not a zero-percent chance.

Try counting signals. In the advice experiment, the same idea lets you count only the rounds when you received a particular recommendation.

Read Rules That Repeat Each Round

\(x_t\) and \(x_{t+1}\)In words: “x this round” and “x next round”

The small lower label t counts rounds. If t is 3, \(x_t\) means the value in round 3 and \(x_{t+1}\) means the value in round 4. The label is not multiplication. \(x_{i,t}\) uses two labels: player i, round t.

\(\alpha\)In words: “alpha,” a learning fraction in these experiments

This Greek letter, pronounced “AL-fuh,” can say how much of the gap to a target you move in one round. With alpha equal to 0.25, you move one quarter of the way. Its meaning is chosen by the author; alpha does not always mean learning.

In words: keep part of your old value, then add part of this round’s target, named \(T_t\). If your old value is 40, the target is 20, and alpha is 0.25, the next value is 0.75 × 40 + 0.25 × 20 = 35. That is five units closer to 20.

\(\delta\)In words: “delta,” the fraction of a reward that counts after waiting one round

Pronounced “DEL-tuh,” this letter is a discount factor in the bargaining experiment. A factor is a number you multiply by. If delta is 0.9, a reward of 10 next round counts like 9 now. A reward of 10 two rounds away counts like 10 × 0.9 × 0.9 = 8.1 now. This describes the model’s treatment of waiting, not a fee charged to anyone.

\(\mu\)In words: “mu,” a switching chance in the evolution experiment

Pronounced “myoo,” this letter names the chance of switching to the other strategy after the copying step. The expert word is mutation. Here it means a change in behavior; it does not require a biological change. Check the local definition: another author could use mu for something else.

\(\bar f\)In words: “f bar,” an average score in the evolution experiment

The short line above the letter marks an average here. A weighted average gives each group a share of the calculation that matches its size. If 75% of players score 4 and 25% score 8, the average is 0.75 × 4 + 0.25 × 8 = 5. The weights, 0.75 and 0.25, add to 1.

Try repeated guesses. A rule that updates the previous result is also called a recurrence. A fixed point is a value the rule leaves unchanged: putting it in gives the same value back. That does not by itself mean nearby values will move toward it.

Count Teams and Joining Orders

A coalition is a group working together. Suppose the full group is \(N=\{A,B,C\}\). The capital letter N names this set of people; on another page it might instead count people. Read its local definition.

\(S\subseteq N\)In words: “S is a team drawn from N”

The symbol means is a subset of: every member of S also belongs to N. The team could be smaller than the full group or could include everyone.

\(\varnothing\) and \(|S|\)In words: “the empty set” and “the number of members of S”

The empty set contains nobody. Here two upright bars around a set count its members: if \(S=\{A,C\}\), then \(|S|=2\). This differs from the single “given” bar in a probability. Bars around a number can have another meaning, so always check what is inside them.

\(S\cup\{i\}\)In words: “team S with person i added”

The cup-shaped symbol means union: combine the listed members without counting anyone twice. If S contains A and person i is B, the new team contains A and B. \(S\setminus\{i\}\), read “S without i,” means remove person i from the team.

\(v(S\cup\{i\})-v(S)\)In words: “the extra value from adding person i”

The function v gives a team’s total value in points. If the team can make 8 points without you and 13 with you, your marginal contribution to that team is 13 − 8 = 5. “Marginal” means the change caused by adding one more person or unit.

\(3!=3\times2\times1=6\)In words: “three factorial equals six”

The exclamation mark is an instruction called factorial, not excitement. For a positive whole number, multiply it by every smaller positive whole number down to 1. Three people have six possible joining orders: ABC, ACB, BAC, BCA, CAB, and CBA. An order is also called a permutation. By definition, 0! = 1: there is one way to order an empty list.

The Shapley value, pronounced “SHAPE-lee,” averages your extra contribution over all joining orders. It is one rule for sharing a group’s value. An equal split simply divides the total by the number of people. Those rules answer different questions, so their answers can differ.

Walk through all six joining orders.

Read Preferences and “Who Knows What”

\(A\succ B\)In words: “A is preferred to B”

The symbol describes a person’s ranking of choices, not a comparison of their point totals. A strict ranking puts every choice in order without ties. In matching, a blocking pair is two people who each prefer one another to their current partners. A stable matching has no such pair under the stated preferences.

\(K_A F\)In words: “A knows fact F”

F is a statement, such as “the lamp is on.” K says “knows,” and the small A names the person. This is an instruction about knowledge, not multiplication. In this model, A knows F only when F is true in every situation A still considers possible.

\(E F\) and \(E^2F\)In words: “everyone knows F” and “everyone knows that everyone knows F”

Here E means everyone in the group knows. The raised 2 repeats that knowledge instruction twice; it is not ordinary squaring. For two people, EF requires both \(K_A F\) and \(K_B F\). Here E does not mean the expected-value notation E[X] used with numbers.

\(C F\)In words: “F is common knowledge”

Everyone knows F, everyone knows everyone knows F, and that pattern continues for every further level. It is stronger than everyone individually receiving the news. The experiment’s public announcement assumes everyone hears it and knows that everyone hears it, with that certainty continuing at every further level. A finite chain of private “message received” replies does not establish all those levels.

Explore the possible situations in the knowledge experiment. Its possible world is one complete description of a situation that might have happened. It is a reasoning tool, not a claim about another universe.

Translate the Words Around the Math

Some hard parts of an equation are the surrounding words. These meanings apply to the playground’s bargaining, voting, and rule-design experiments.

Payoff and incentive

A payoff is how a player rates an outcome, represented by points. An incentive is a reason to favor one choice, such as a higher payoff. Points represent only what the model counts; they need not capture all a real person cares about.

Outside option and surplus

An outside option is what you can get if no agreement happens. A surplus is the extra value available beyond the stated comparison. If an agreement makes 20 points and the two outside options total 12, the extra value to share is 20 − 12 = 8.

Allocation, transfer, and cost

An allocation decides who receives an item or how a total is divided. A transfer moves points or money between people. A payment of 3 subtracts 3 from the payer and adds 3 to the receiver, so it cancels when adding both totals. A real resource cost, such as time spent working, need not cancel that way.

Incentive compatibility and truthful reporting

A rule is incentive compatible under its stated assumptions when a player cannot improve their payoff by supplying a false report instead of the truth. A true report can tie another report. A weakly best choice allows such ties; a strictly best choice scores higher than every alternative being compared. Check which players, reports, and assumptions a result covers.

Efficiency, fairness, and the core

Efficiency in these games usually means achieving the largest total value under the model’s rules. That does not decide how fairly the value is shared. A proposed team split is in the core when it shares the full value and no person or smaller team can obtain a larger total by leaving and working alone. Some games have no split meeting all these conditions.

Majority, tie, and voting cycle

A majority is more than half, not merely the largest share. A tie is an equal result. A voting cycle can occur when separate head-to-head votes favor A over B, B over C, and C over A. These group results can circle even when each voter has a clear ranking.

Compare rules for assigning one item. A mechanism is a rule that turns participants’ choices or reports into an outcome and any payments. Mechanism design asks which rules encourage the behavior needed for a chosen goal.

Take the Words Back to the Formula

Ask three questions: What does each letter name? What operation does each symbol ask me to do? What would happen if I put in a small number?

If a result measures minutes, points, or people, keep that unit with it. If a page changes a letter’s meaning, use the new definition. You can always return to the experiment’s “Work It Out, Step by Step” section before opening the expert details.