Licence MIASHS · Prolog · pair project
Strategy tournament
Two strategies derived on paper, written as Prolog agents, and entered against the rest of the year. They finished 7th and 8th of 16. Read the log match by match and the ranking inverts.
joue/3.
PDF
The game and the strategy
Payoff matrices derived from the rule, and the equilibrium proposed.
Log
The tournament
636 pages of rounds. The standings are on page 1.
What the standings measure
results/head_to_head.csv.
results/leaderboard.csv.
Put the two together and finishing lower correlates with winning more matches, Spearman +0.62 against final rank, p = 0.011. Both entered agents beat the eventual champion in their direct match, 782 to 435 and 490,335 to 4,772. The tournament did not rank agents by how often they won. It ranked them by how much they scored, and those are different objectives.
Why the scores reach 1062
This version of the game pays a repeated number its own value as a
multiplier: play 5 again and the round pays five times what the last one did, verified
on 5,050 of 5,216 consecutive repeats in the log. One stretch of best_duo
repeating 5 pays 540, which is 9 × 1027 from a
single round.
Both entered agents are stateless: they sample from a fixed distribution and never read the history. Their longest runs are 10 and 9, and they repeat 32.9% and 35.4% of the time, which is exactly what independent sampling gives by chance. Every agent placed above them reached runs of 37 to 100. A fixed mixed strategy cannot exploit a rule that rewards deliberate repetition, and that is the whole of the gap between 7th and 1st.
The equilibrium the agents were built on
Both entered agents sample a fixed mixture over the five picks, and the shipped Prolog
plays what the analysis says it plays. Calling joue/3 2,000,000 times per
agent puts every observed share within 0.0009 of its declared weight, and the picks a
mixture gives no weight never came up once in either agent. So the numbers below
describe the agents that actually ran, not just the vectors on paper.
equilibrium/strategy_payoffs.csv.
Undercutting by one takes the sum, which is the 5, 7 and 9 just above the diagonal.
equilibrium/strategy_payoffs.csv.
That flat top is the whole of the equilibrium condition. A mixture is a best reply
exactly when every pick it uses pays the same and nothing unused pays more, so
[0, 0, 4/9, 2/9, 1/3] has regret 0 while
stage_test carries 0.3337 against that opponent and 0.5684 against itself.
The best reply to stage_test is pick 5, the one pick
stage_test never plays.
stage_test does, against the
same opponent. Every bar is positive, and the shortest is 0.1402. The absolute
figures behind it are tabulated in
the
folder's own write-up. Source:
equilibrium/strategy_payoffs.csv.
stage_test does beat Nash head to head, 3.5552 to 3.1521, which is what
the original analysis claimed. It also earns less than Nash would have earned
against every one of those opponents, by at least 0.1402. Payoff is linear in
the opponent's mixture, so beating stage_test against all five pure picks
settles it for every mixture too: the Nash strategy strictly dominates it. It wins the
match by taking 0.7368 off the opponent while giving up 0.3337 itself, which is the
same distinction the tournament standings turned on.
How to read these numbers
- The strategies were derived for the plain game, where undercutting by one takes the sum. The tournament ran the version with the repetition multiplier. They are not answers to the same question.
- One match per pairing, so a 15-match record is a thin sample. The rank correlation is significant at p = 0.011 across the sixteen, but no single match is.
-
khawa_khawa, the adaptive agent in the same file, appears in zero rounds of the log against 1,805 and 1,813 for the two entrants. It was built alongside them rather than entered, so nothing here measures it.