The Trust Machine

How to Read the Symbols

The symbols below are short names for points, rounds, and chances. You can use the game without memorizing them. The math reading guide has more examples.

Letters and small labels

A letter stands for a value. A small letter below it is a label: \(u_i\), read “u for player i,” means that player’s points. The letter \(i\) identifies a player; it is not a number to multiply by. Lowercase \(u\) means points in one round, uppercase \(U\) means a total, and a bar above \(u\) means an average. Read about letters and labels.

Inputs in parentheses

\(u_i(t)\), read “u for player i at round t,” asks for that player’s points in round \(t\). Here the parentheses identify the round; they do not mean multiplication. Read about functions.

Comparison and multiplication

\(>\) means “greater than,” and \(\le\) means “less than or equal to.” Letters or numbers written next to each other usually mean multiply: \(2R\) means 2 times \(R\), and \(w_kU_k\) means weight times total points. Read about comparisons.

Addition, division, and powers

\(\sum\), the capital Greek letter sigma, means “add these terms.” The labels below and above it say where to start and stop. A fraction bar means divide. A raised number is a power: \(p^2\) means \(p\) times \(p\). The first term in the round-count sum below is 1 because round 1 always happens, even if the chance of continuing is zero. Read about sums and powers.

Chance and expected value

A probability is a chance written from 0 to 1: 0.9 means 90%. \(\mathbb{E}[L]\), read “expected L,” is the average number of rounds the chances predict over many encounters. The square brackets identify what we are averaging; they do not mean multiply. \(\varepsilon\), read “epsilon,” is the Greek letter used here for the chance of an accidental action flip. Read about probability and averages.

Repeated Prisoner’s Dilemma

Let \(C\) denote cooperation and \(D\) defection. A payoff is the points awarded to a player. The stage-game payoffs are \(R=3\) for mutual cooperation, \(S=0\) for cooperating against defection, \(T=5\) for defecting against cooperation, and \(P=1\) for mutual defection. Each cell below lists (row payoff, column payoff).

Read the table by finding your row and the other player’s column. In \((3,3)\), the comma separates the row player’s points from the column player’s points; it does not ask you to do arithmetic.

Thus \(T > R > P > S\) reads “5 is greater than 3, which is greater than 1, which is greater than 0.” So \(D\) strictly dominates \(C\) in the stage game: defecting pays more whichever choice the other player makes. Also \(2R > T+S\) says “two times 3 is greater than 5 plus 0.” An encounter adds the points actually earned, with a point in a later round counting just as much as one now. For player \(i\), let \(u_i(t)\) be the points in round \(t\), and let \(L\) be the number of rounds actually played:

Read it aloud: “Total points for player i equal the sum of that player’s points from round 1 through round L. Average points equal total points divided by rounds.” For 10 rounds earning 3 each, this means add ten 3s to get 30, then divide by 10 to get 3.

A “horizon” is how long an encounter lasts. For a known horizon, \(L=H\), where \(H\) is the selected round count. With continuation probability \(p\), the first round always occurs and each later round requires a fresh draw to continue. For example, \(p=0.9\) means a 90% chance of another round after each round. The expected round count is \(1/(1-p)\), meaning “subtract p from 1, then divide 1 by that result,” without a cap; the implemented cap gives:

Read it aloud: “The expected number of rounds is the sum of the chances of reaching rounds 1 through 200.” In the sum, \(t\) counts the round, so the first three terms are \(p^0\), \(p^1\), and \(p^2\). At a 90% continuation chance, those are 1, 0.9, and 0.81. The fraction is a shortcut for adding all 200 terms; \(p^{200}\) means multiply 200 copies of \(p\). The comparison at the end says that \(p\) can be anywhere from 0 through 0.95, including both ends.

Each intended action flips independently with mistake probability \(\varepsilon\). Both policies act from completed rounds, then observe actual actions without learning whether a flip occurred. Tit for tat begins with \(C\) and copies the opponent’s last actual action; its forgiving variant chooses \(C\) after \(D\) with the selected probability \(f\). The coin policy independently chooses \(C\) with probability \(1/2\). The final-round variant chooses \(D\) on round \(H\) only with a known horizon.

Tournament ranking pools points and rounds rather than averaging encounter averages. For each strategy, each player appearance \(k\) contributes points \(U_k\) and rounds \(L_k\), weighted by \(w_k=1\) in cross-strategy matches and \(w_k=1/2\) for each copy in a self-match:

Read it aloud: “Add the weighted point totals, add the weighted round counts, then divide the first sum by the second.” Here \(k\) labels each player appearance in the tournament, and \(\sum_k\) means add across all those appearances. \(U_k\) is its point total, \(L_k\) is its round count, and \(w_k\) is its weight. A weight of \(1/2\) counts half of that copy’s points and rounds, so two copies in a self-match together count as one pairing.

Every unordered pairing runs the selected number of trials. Longer encounters contribute more rounds to the pooled score. These fixed-policy simulations neither solve for equilibrium nor predict human behavior. In the separate theoretical game with no mistakes, a commonly known finite horizon, payoff maximization, and common knowledge of rationality, backward induction yields defection every round. Playing against a fixed cooperative policy has different assumptions. Rankings depend on the policy population, noise, horizon, scoring, and sampled random sequence.

Primary reading: Robert Axelrod and William Hamilton, The Evolution of Cooperation (1981). This six-strategy tournament illustrates repeated interaction; it does not reproduce their tournament.