Why in a duel the player who starts wins 93% of the games

Imagine a game in which two equally strong players face the same map. Nobody cheats, the dice are fair, the rules are the same for both. The only difference: one of them is on the move first. And this single difference decides 93 games out of 100.

That is exactly what Principalities^[1] in Audamir look like for two equally strong AI opponents. No reliable figure on the order of play was found for the original game, so this is a measurement of this game. The number is unexpected, and at first glance one hardly wants to believe it. That is why it is worth saying how we arrived at it, and even more what we did with it.

A suspicion that a feeling cannot refute

In a game with dice everything can be blamed on luck. The dice fell badly. The opponent's dice always fall well. A feeling is not an argument, and when a game is made so that it can be balanced, one has to measure.

The rules we are going to measure are simple and come from the original game:

• Each tile holds a stack of dice (at most 8).
• The attacker and the defender both roll all the dice of their stack. The higher total wins, on a tie the defender wins.
• After a win all dice but one move onto the captured tile. After a loss the attacker is left with a single one.
• At the end of the turn the player receives new dice according to the size of their largest connected region, and they are distributed at random among their tiles.

The last rule is the hidden core of the game. Whoever conquers, grows. Whoever grows, receives more dice. Whoever receives more dice, keeps conquering. In games such feedback is called a snowball.

Measurement: equal players, different order

To tell the effect of skill from the effect of the order of play, the players have to be made equal. In the tool tools/fairness.js all the players are equally strong AI opponents (level 2, "the average player"). The tool counts two things: the share of wins by seat number and the share of wins of the one who started that game. A measurement takes less than three seconds. The result today (20 October 2026), medium map, the tile shape "Random shapes", 400 games per measurement:

With two players the starter wins 93.3%, with three 61.3%, with four 41.8% (the fair share is 25%). With eight players it is 15.8%, and the difference from the fair 12.5% is lost in the noise. The average game with two players lasts 8 rounds, with eight almost 20.

Why it is so

This is my explanation, not a measurement. The first player attacks on a map on which nobody has yet had time to distribute reinforcements. Every tile they conquer enlarges their largest region, so at the end of the turn they receive more dice. The second player starts with smaller holdings and a smaller gain. With two players the feedback is never interrupted, because there is no third player to stand in the winner's way. With eight players the first player immediately meets seven others, so the advantage dissolves.

What we tried to fix

The first idea is the oldest one: to even it out. In chess White moves first, and White's score (a draw counts as half a point) is around 52 to 56% in the statistics. In Go White receives extra points for moving second, the so-called komi, and its value has been adjusted over the history: under the Japanese rules it rose from 5.5 to 6.5 points in 2002, and today it is 6.5 or 7.5 depending on the rules. We tried the same here: we gave the second player starting dice.

Three rows, one problem. A fix that brings 95% down to 56% brings it to 9% with a further 25% of dice. A game for two is as sensitive as a knife edge. A compensation that would work for all maps and all levels of players would be fragile, and the player would also feel it as an injustice in the opposite direction.

What we did instead

In Principalities a random player starts. The advantage of the first move does not disappear, it is only shared fairly: every seat wins roughly its share. With four players it comes out at 24.8 / 22.8 / 28.0 / 24.5%. And when a player chooses a game for two, the game says so openly: a duel is decided mostly by who starts.

That last part is deliberate. Better to admit a limitation than to hide it behind complicated compensation.

Skill shows where there are enough decisions

With a random start one can also measure something that interests the player more: how much it matters how they play. In place of the player, level 2 (average) and level 3 (good) play against the AI opponents:

The average player wins roughly the fair share. With more players the good player wins roughly two to four times more than the fair share. With two players, however, skill is almost indistinguishable (52% against 55%), because the order decides the duel. The more players, the more decisions, and the more room in which one can play better.

(The table is from the measurement of 18 Oct: 800 games per cell, an error of about ±3 percentage points, the tile shape "Random shapes".)

Three levels of artificial intelligence and an experiment that did not work

The opponents have three levels of strength: Weak attacks even at bad odds and chooses almost at random, Normal attacks at favourable odds and tries to join territories, Strong also counts on retaliation, that is, on the risk to both the source and the captured tile. The strength was not set by eye, it was measured by a tournament:

• Strong beats Normal in 71.5% of the games one against one (800 games today; 73% in the documentation).
• In a tournament of all three levels at once Weak wins 5.8%, Normal 31.3% and Strong 62.8%.

A fourth level, "expert", was never made. We tried automatically tuned weights and Monte Carlo simulations that play out the rest of the turn together with the opponent's retaliation. They did not outdo the Strong level. We measured about 50%, that is, nothing. The game is to a large extent about luck and the reinforcements are distributed at random, so the additional "wisdom" has nothing to act upon.

Decision: there will be no such level. When a difference cannot be measured, you have nothing to offer and nothing to fix. It is one of the most useful rules that working with measurement has brought.

What I take from it

1. A suspicion is cheap, a measurement is even cheaper. Four hundred games take less than three seconds and the argument about luck is over.
2. Do not fix a sensitive thing with an extra count. If a small change reverses the result, it is not a number you can adjust, it is a lever. Replace the lever with a random draw.
3. Tell the player the truth about the game. A two-player duel with a clear advantage of the first move is fine, as long as the player knows it.
4. Know when to stop. The attempt at an "expert" could be ended with a single number, about 50%.

In the next article we look at how this measurement grew into a whole system: the difficulty of every setting of the game is today expressed in points, and the points come from a simulation.

Sources and notes

• Measurements: node tools/fairness.js --classic N --games 400, node tools/tournament.js 400 2,3 and node tools/tournament.js 300 1,2,3, run on 20 October 2026 (Node, 16 threads).
• Decisions and the table of skill: docs/DESIGN.md §3 and §19, the procedure of the measurement docs/TUNING.md §7 and §8.
• Chess: the score of White is about 52 to 56% (a draw counts as half a point) according to the overview First-move advantage in chess; outright wins of White are lower, about 35 to 40%, and in elite games draws become more frequent. Go: komi 6.5 (Japanese and Korean rules) and 7.5 (Chinese and American rules), in Japan from 5.5 to 6.5 in September 2002, according to Komi (Go).