INTERACTIVE COMPANION · TOURNAMENT BENCH · GAME THEORY

The Tournament Bench

A live model accompanying “Running Axelrod's Tournament Again, With Noise This Time”

Two benches sit below. The first runs a round robin over whichever strategies you check, at whatever noise you set, and fills in the payoff matrix and the leaderboard live. The second takes that same matrix and lets the strategies breed, so you can watch tit for tat hold its own in a clean room and lose it as soon as the moves start landing wrong.

Everything runs in your browser, in plain JavaScript, with a seeded random number generator so the same settings always give the same answer. The strategies are the same fifteen the article used and the code is a direct translation of the Python.

Model 1. The round robin

Payoffs are Axelrod's: T = 5 for defecting against a cooperator, R = 3 for mutual cooperation, P = 1 for mutual defection, S = 0 for cooperating into a defector. A match is the number of rounds you set. Every checked strategy plays every other one and also plays a copy of itself, which is how Axelrod scored his entrants against their twins. A strategy's score is its mean payoff per round averaged over all the opponents in the room, including its twin.

Noise is an execution error. Each player's intended move is flipped with the probability you set, and the flipped move is what both players see. Neither player is told that an error happened.

Noise 0.000 Start at 0 and read the leaderboard. Then walk it up slowly. Generous tit for tat leads until about 0.02, contrite tit for tat takes over around 0.05, and somewhere near 0.20 always defect climbs to the top of a tournament nobody is winning. Repetitions 60 Each repetition is one complete round robin. At noise 0 the deterministic cells never move no matter how many you run, so the matrix is exact there. Drop to 5 and watch the leaderboard jump around. Push it to 400, the setting the article used, to separate the top three, and expect a second of waiting each time you move it that far. Rounds per match 200 200 is Axelrod's length and the article's default. Pull it down to 50 with noise at 0 and tit for tat takes first place outright, because every exploiter gets its free lunch only once and a short match does not dilute it. Push it to 400 and the forgiving variants pull ahead again. Highlight The highlighted row of the matrix is outlined and its bar is filled solid, so you can follow one strategy while you change the room around it.

Who is in the room

winner: – tit for tat: – TFT vs TFT: – matches run: –

–

Three cells you can check with a pencil. Set noise to 0 and any number of repetitions.
TFT against TFT = 3.000. Both cooperate on round one and copy a cooperation forever after, so every round pays R = 3.
TFT against ALLD = 0.995. Tit for tat cooperates into a defector once for S = 0, then mirrors defection for 199 rounds at P = 1, so (0 + 199) / 200 = 0.995.
ALT against TFT = 2.515. Round one is mutual cooperation, R = 3. After that the alternator takes T = 5 on its hundred defecting rounds and S = 0 on its ninety-nine remaining cooperating rounds, so (3 + 500 + 0) / 200 = 2.515.
All three are exact, because neither strategy in any of those pairs carries a coin. They are printed in the raw output of the article's analysis script with the same values.

Why the leaderboard is a statement about the room

Press Nice only with the noise at zero. Every score in the leaderboard reads exactly 3.000. There is no winner: all eight of those strategies cooperate with all the others on every round, so the tournament ends in an eight-way tie at the maximum and the sort order breaks it arbitrarily. A room with nobody willing to punish anything cannot tell its members apart. Now nudge the noise up one step and watch the tie come apart.

Press No exploiters and grim trigger climbs. Press Drop Joss with noise at zero and the gap between generous tit for tat and tit for tat almost vanishes, because that gap was one opponent. Joss cooperates like tit for tat and then defects at random one time in ten. Tit for tat echoes the defection, Joss echoes the echo, and the two of them fall into a feud neither one is trying to have. Generous tit for tat lets a defection go one time in three, which is enough to break the loop.

A round robin score is an average over whoever else is in the room. Change the room, change the answer. This is the single largest weakness of the tournament as a way of learning anything, and it is why the second bench matters.

Model 2. Letting them breed

Replicator dynamics take the payoff matrix and make the strategies compete for population rather than for points. Every strategy starts at some frequency, each one earns the average payoff its opponents give it, and the ones earning more than the population average grow while the rest shrink:

x_i(t+1) = x_i(t) · f_i(t) / φ(t), f_i = ∑_j M_ij x_j, φ = ∑_i x_i f_i

The chart below is a stacked area: the height of each band is that strategy's share of the population, and the whole stack sums to one at every generation. Nothing here mutates and nothing here is spatial. The only thing happening is that earning more means having more descendants.

Noise 0.010 Drag this and the run restarts on a freshly measured matrix. At exactly 0 the nice strategies coexist forever because they all score 3 against each other and nothing can separate them. One tenth of a per cent breaks that tie. Above 0.005 contrite tit for tat takes the whole population. Speed 6 generations per frame At noise 0.002 the sweep takes more than a thousand generations, so push the speed up to see the end of it. At noise 0.05 it is over in about 150, so pull the speed down to 1 and watch tit for tat rise, peak and get eaten. Starting mix Try starting from always defect at 80 per cent with noise at 0.01. The defectors eat the cooperators first, then run out of food, and the reciprocators come back from almost nothing. Then try always cooperate at 80 per cent and watch the opposite order.

generation: – leader: – mean payoff: – tit for tat: – total frequency: –

Watch the last readout. Frequencies are never renormalised after the update, so the total staying at 1.000000 is the arithmetic checking itself rather than a constraint being imposed. In the article's run the largest departure from one over four thousand generations at ten noise levels was 5.6 parts in ten million billion, which is two and a half units in the last binary place of a double.

What this bench cannot tell you

It cannot tell you anything about cooperation in living things. It is a game with two moves, four payoffs, fixed partners and no way to walk away, and every strategy in it was written by a person who already knew the rules. The result is about the model.

The noise here is misimplementation: the error happens to the move and both players see it. Under misperception, where the move is played correctly and only the opponent's copy of it is corrupted, contrite tit for tat would lose its whole mechanism, because it works by knowing its own defection was an accident. That version is not implemented here and we did not run it in the article either.

And the population is infinite, well mixed, and free of mutation. A finite population would drift, a structured one would let clusters of cooperators survive where a well mixed one cannot, and a mutating one would keep reintroducing the strategies this model lets go extinct for good.