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class="flex items-center justify-center border border-black/10 bg-green-800" style="width:22px;height:22px"></div><div class="flex items-center justify-center border border-black/10 bg-green-800" style="width:22px;height:22px"></div><div class="flex items-center justify-center border border-black/10 bg-green-800" style="width:22px;height:22px"></div><div class="flex items-center justify-center border border-black/10 bg-green-800" style="width:22px;height:22px"></div><div class="flex items-center justify-center border border-black/10 bg-green-800" style="width:22px;height:22px"></div><div class="flex items-center justify-center border border-black/10 bg-green-800" style="width:22px;height:22px"></div><div class="flex items-center justify-center border border-black/10 bg-green-800" style="width:22px;height:22px"></div><div class="flex items-center justify-center border border-black/10 bg-green-800" style="width:22px;height:22px"></div></div></div></div><div class="absolute inset-0 bg-gradient-to-b from-green-950/40 to-green-900/40"></div><div class="relative z-10 max-w-4xl mx-auto text-center"><h1 class="text-3xl sm:text-4xl lg:text-5xl font-bold leading-tight tracking-tight mb-6 text-white [text-shadow:0_0_60px_rgba(0,0,0,1),0_0_30px_rgba(0,0,0,1),0_2px_4px_rgba(0,0,0,1)]">The Expanded Othello AI Arena</h1><p class="text-lg sm:text-xl text-white mb-4 leading-relaxed [text-shadow:0_0_40px_rgba(0,0,0,1),0_0_16px_rgba(0,0,0,1),0_1px_3px_rgba(0,0,0,1)]">Evaluating Intelligent Systems Through Constrained Adaptation to Unseen Conditions</p><div class="flex flex-wrap justify-center gap-x-6 gap-y-2 text-white text-base mt-8 mb-4 [text-shadow:0_0_24px_rgba(0,0,0,1),0_1px_3px_rgba(0,0,0,1)]"><a href="https://scholar.google.com/citations?user=XIIINK4AAAAJ&amp;hl=en" target="_blank" rel="noopener noreferrer" class="hover:text-white underline underline-offset-2 decoration-slate-500 hover:decoration-white transition-colors">Byunghwa Yoo</a><a href="https://sundong.kim/" target="_blank" rel="noopener noreferrer" class="hover:text-white underline underline-offset-2 decoration-slate-500 hover:decoration-white transition-colors">Sundong Kim</a><a href="https://scholar.google.com/citations?user=YBYE93sAAAAJ&amp;hl=en" target="_blank" rel="noopener noreferrer" class="hover:text-white underline underline-offset-2 decoration-slate-500 hover:decoration-white transition-colors">Kyung-Joong Kim</a></div><div class="flex justify-center mb-10"><div class="inline-flex items-center gap-2 bg-amber-500/20 border border-amber-400/40 rounded-full px-4 py-1.5 text-amber-300 text-sm font-medium [text-shadow:0_0_16px_rgba(0,0,0,1)]"><span class="w-2 h-2 rounded-full bg-amber-400 inline-block"></span>Transactions on Machine Learning Research (TMLR) · June 2026</div></div><div class="flex flex-wrap justify-center gap-4"><a href="https://openreview.net/forum?id=WXKQtqPC2d" target="_blank" rel="noopener noreferrer" class="inline-flex items-center gap-2 bg-white 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1<path stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M9 12h6m-6 4h6m2 5H7a2 2 0 01-2-2V5a2 2 0 012-2h5.586a1 1 0 01.707.293l5.414 5.414a1 1 0 01.293.707V19a2 2 0 01-2 2z"></path></svg>Read Paper</a><a href="https://github.com/blakeyoo/ExpandedOthello-AEC" target="_blank" rel="noopener noreferrer" class="inline-flex items-center gap-2 bg-gray-900 hover:bg-gray-800 text-white font-medium px-6 py-2.5 rounded-lg transition-colors text-sm"><svg class="w-4 h-4" fill="currentColor" viewBox="0 0 24 24"><path d="M12 0C5.374 0 0 5.373 0 12c0 5.302 3.438 9.8 8.207 11.387.599.111.793-.261.793-.577v-2.234c-3.338.726-4.033-1.416-4.033-1.416-.546-1.387-1.333-1.756-1.333-1.756-1.089-.745.083-.729.083-.729 1.205.084 1.839 1.237 1.839 1.237 1.07 1.834 2.807 1.304 3.492.997.107-.775.418-1.305.762-1.604-2.665-.305-5.467-1.334-5.467-5.931 0-1.311.469-2.381 1.236-3.221-.124-.303-.535-1.524.117-3.176 0 0 1.008-.322 3.301 1.23A11.509 11.509 0 0112 5.803c1.02.005 2.047.138 3.006.404 2.291-1.552 3.297-1.23 3.297-1.23.653 1.653.242 2.874.118 3.176.77.84 1.235 1.911 1.235 3.221 0 4.609-2.807 5.624-5.479 5.921.43.372.823 1.102.823 2.222v3.293c0 .319.192.694.801.576C20.566 21.797 24 17.3 24 12c0-6.627-5.373-12-12-12z"></path></svg>Environment</a></div></div></section><section id="demo" class="bg-white py-20 px-6"><div class="max-w-6xl mx-auto"><div class="flex flex-col items-center justify-center min-h-80 py-16 text-center space-y-6"><div class="text-6xl font-black tracking-tight text-slate-300 select-none">???</div><div class="space-y-4 max-w-xl"><p class="text-2xl font-bold text-gray-900 leading-snug">You don&#x27;t know how to win.</p><p class="text-xl font-bold text-gray-900 whitespace-nowrap">How many games will you need to figure out how to win?</p><p class="text-sm text-gray-500 leading-relaxed">The winning condition is hidden — this isn&#x27;t standard Othello. No hints. Just play and figure it out.</p></div><button class="px-8 py-3 bg-indigo-600 hover:bg-indigo-500 text-white font-semibold rounded-lg transition-colors text-sm">Accept the Challenge →</button></div></div></section><section id="challenge" class="py-20 px-6 border-b border-gray-100 bg-gray-50"><div class="max-w-3xl mx-auto"><h2 class="text-2xl font-bold mb-1 text-gray-900">The Challenge</h2><p class="text-gray-500 text-sm mb-10">When the Objective is Unknown</p><div class="space-y-6 text-gray-700 leading-8 text-[1.05rem]"><p>The capacity for rapid strategic adaptation to a novel, unseen environment is regarded as one of the major properties of artificial general intelligence (AGI). Novelty can take many forms — a new task, an unfamiliar geometry, or a changed set of rules. But one dimension remains central and largely unaddressed: the objective of the environment itself. If what counts as success is unknown, how should an AI — or a human — figure it out, and adapt quickly enough to the environment?</p><p>The difficulty lies in the nature of the signal. When the objective is unknown, the only evidence is the outcome — what happens at the very end of each attempt. Nothing along the way indicates whether a given approach is on the right track. This is compounded when the result is also shaped by another party — one whose responses to every decision further obscure the objective.</p><p>We propose the Expanded Othello AI Arena as a benchmark built around this challenge. It is not a test of how well an agent plays Othello. It is a test of how quickly one can identify a hidden objective and develop an effective strategy in an expanded, non-standard variant of Othello with undisclosed winning conditions.</p></div></div></section><section id="environment" class="py-20 px-6 border-b border-gray-100 bg-white"><div class="max-w-3xl mx-auto"><h2 class="text-2xl font-bold mb-1 text-gray-900">The Environment: The Expanded Othello</h2><p class="text-gray-500 text-sm mb-10">Enviroment formalization and the design rationale</p><div class="space-y-6 text-gray-700 leading-8 text-[1.05rem]"><p>Evaluating how an AI discovers a latent objective requires going beyond what traditional meta-learning environments offer. Visual or structural variation — changes to inputs, layouts, or level appearances — leaves the fundamental rules of the environment intact. What is needed instead is an environment where the rules themselves change: where what counts as winning is not given, and must be inferred. Moreover, AGI demands more than single-agent adaptation. Deriving an objective from an environment that includes another agent — one whose actions are shaped by the same hidden rules — is a qualitatively harder and more realistic test of general intelligence.</p><p>We select Othello as the substrate for the environment to test such abilities because it satisfies these requirements. Its terminal state produces a continuous disc occupancy ratio ρ ∈ [0, 1] — making the winning condition directly parameterizable through a single threshold, and therefore variable in ways that go beyond structural change. It is zero-sum and adversarial by construction, requiring adaptation against an opponent whose decisions co-determine the outcome. And its flipping mechanics — high global state-change dynamics from simple local rules — mean that board geometry directly shapes which strategies are viable, making spatial layout a meaningful variable rather than mere decoration.</p><p>The environment space is therefore formalized as<!-- --> <span class="font-mono bg-gray-100 px-1.5 py-0.5 rounded text-sm">ℰ = ℒ × 𝒞</span>. Each instance<!-- --> <span class="font-mono bg-gray-100 px-1.5 py-0.5 rounded text-sm">E = (L, C)</span> <!-- -->is defined by two independently varying components.<!-- --> <strong>L</strong> is the board geometry — the size and shape of the board, along with the placement of non-interactive obstacles. It is fully observable, and represents the structural dimension of variation fam
1iliar from traditional meta-learning.<!-- --> <strong>C</strong> is the victory condition — latent and never directly revealed, it must be implicitly inferred through interaction. Leveraging Othello&#x27;s natural disc occupancy ratio ρ, the winning criteria is varied across environments through a single parameterization <em>K</em>, which determines the threshold that separates victory from over-dominance draw conditions, i.e., the winning condition is not simply defined via majority of the discs; it has to stay in a specific ratio region, or strategic draw is often needed for prventing opponent win. For detailed definition of <em>K</em>, please refer to our<!-- --> <a href="https://openreview.net/forum?id=WXKQtqPC2d" target="_blank" rel="noopener noreferrer" class="underline text-gray-500 hover:text-gray-800">paper</a>.</p></div></div></section><section id="benchmark" class="py-20 px-6 border-b border-gray-100 bg-gray-50"><div class="max-w-5xl mx-auto"><h2 class="text-2xl font-bold mb-1 text-gray-900">The Be
1nchmark</h2><p class="text-gray-500 text-sm mb-12">56 environments · 7 layouts × 8 win conditions</p><div class="space-y-6 text-gray-700 leading-8 text-[1.05rem] mb-16"><p>The goal of the benchmark is to measure how quickly an AI adapts to an environment with a latent objective — a capability we define as<!-- --> <strong>Skill-Acquisition Efficiency (SAE)</strong>. To quantify this, we evaluate the performance an agent achieves when interacting with a given environment under a strictly limited number of interactions.</p><p>Concretely, the interaction budget per environment is fixed at<!-- --> <strong>2,000 games</strong>. Within this budget, the agent learns through self-play by default; pre-training on prior environments is permitted. To enable systematic and quantified evaluation, we define a standard benchmarking suite of 56 environments — the Cartesian product of 8 victory conditions (C) and 7 board layouts (L) — designed to provide comprehensive coverage across diverse structural and objective-level challenges. The conditions are as follows:</p></div><h3 class="text-lg font-semibold text-gray-900 mb-6">Layout Space — <span class="font-normal text-gray-500">7 board geometries</span></h3><div class="grid grid-cols-4 gap-4 mb-16 sm:grid-cols-7"><div class="flex flex-col items-center gap-2"><img src="/board_standard.svg" alt="Standard 8×8" class="w-full rounded border border-gray-200"/><p class="text-xs text-center text-gray-600 leading-tight">Standard 8×8</p><p class="text-xs text-center text-gray-400">8×8</p></div><div class="flex flex-col items-center gap-2"><img src="/board_no_corners.svg" alt="No Corners" class="w-full rounded border border-gray-200"/><p class="text-xs text-center text-gray-600 leading-tight">No Corners</p><p class="text-xs text-center text-gray-400">8×8</p></div><div class="flex flex-col items-center gap-2"><img src="/board_partial_c.svg" alt="Partial C-Squares" class="w-full rounded border border-gray-200"/><p class="text-xs text-center text-gray-600 leading-tight">Partial C-Squares</p><p class="text-xs text-center text-gray-400">8×8</p></div><div class="flex flex-col items-center gap-2"><img src="/board_x_squares.svg" alt="X-Squares" class="w-full rounded border border-gray-200"/><p class="text-xs text-center text-gray-600 leading-tight">X-Squares</p><p class="text-xs text-center text-gray-400">8×8</p></div><div class="flex flex-col items-center gap-2"><img src="/board_random1.svg" alt="Random Board 1" class="w-full rounded border border-gray-200"/><p class="text-xs text-center text-gray-600 leading-tight">Random Board 1</p><p class="text-xs text-center text-gray-400">12×10</p></div><div class="flex flex-col items-center gap-2"><img src="/board_random2.svg" alt="Random Board 2" class="w-full rounded border border-gray-200"/><p class="text-xs text-center text-gray-600 leading-tight">Random Board 2</p><p class="text-xs text-center text-gray-400">6×8</p></div><div class="flex flex-col items-center gap-2"><img src="/board_random3.svg" alt="Random Board 3" class="w-full rounded border border-gray-200"/><p class="text-xs text-center text-gray-600 leading-tight">Random Board 3</p><p class="text-xs text-center text-gray-400">10×10</p></div></div><h3 class="text-lg font-semibold text-gray-900 mb-2">Condition Space — <span class="font-normal text-gray-500">8 victory conditions</span></h3><div class="flex items-center gap-4 mb-6 text-xs text-gray-500"><span class="flex items-center gap-1.5"><span class="inline-block w-3 h-2 rounded-sm bg-emerald-500"></span> Win</span><span class="flex items-center gap-1.5"><span class="inline-block w-3 h-2 rounded-sm bg-red-400"></span> Lose</span><span class="flex items-center gap-1.5"><span class="inline-block w-3 h-2 rounded-sm bg-gray-300"></span> Draw</span><span class="text-gray-400">· x-axis = disc ratio (0% → 100%)</span></div><div class="grid grid-cols-1 sm:grid-cols-2 gap-8"><div class="space-y-3"><p class="text-xs font-semibold text-gray-400 uppercase tracking-widest mb-3">Majority</p><div class="bg-gray-50 border border-gray-200 rounded-lg p-4"><div class="flex items-center justify-between mb-1"><span class="font-medium text-gray-900 text-sm">Majority</span><span class="font-mono text-xs text-gray-400 bg-white border border-gray-200 px-2 py-0.5 rounded">K &gt; 1.0</span></div><p class="text-xs text-gray-500 mb-2">More discs wins — standard Othello.</p><div class="mt-2 relative h-2 w-full rounded-full bg-gray-100 overflow-hidden"><div class="absolute h-2 bg-red-400" style="left:0%;width:50%"></div><div class="absolute h-2 bg-emerald-500" style="left:50%;width:50%"></div></div><div class="relative h-4 mt-1"><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:0%">0<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:50%">50<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:100%">100<!-- -->%</span></div></div><div class="bg-gray-50 border border-gray-200 rounded-lg p-4"><div class="flex items-center justify-between mb-1"><span class="font-medium text-gray-900 text-sm">Majority &lt; 80%</span><span class="font-mono text-xs text-gray-400 bg-white border border-gray-200 px-2 py-0.5 rounded">K = 0.8</span></div><p class="text-xs text-gray-500 mb-2">Win with majority, but disc ratio must stay below 80%.</p><div class="mt-2 relative h-2 w-full rounded-full bg-gray-100 overflow-hidden"><div class="absolute h-2 bg-gray-300" style="left:0%;width:20%"></div><div class="absolute h-2 bg-red-400" style="left:20%;width:30%"></div><div class="absolute h-2 bg-emerald-500" style="left:50%;width:30%"></div><div class="absolute h-2 bg-gray-300" style="left:80%;width:20%"></div></div><div class="relative h-4 mt-1"><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:0%">0<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:20%">20<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:50%">50<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:80%">80<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:100%">100<!-- -->%</span></div></div><div class="bg-gray-50 border border-gray-200 rounded-lg p-4"><div class="flex items-center justify-between mb-1"><span class="font-medium text-gray-900 text-sm">Majority &lt; 60%</span><span class="font-mono text-xs text-gray-400 bg-white border border-gray-200 px-2 py-0.5 rounded">K = 0.6</span></div><p class="text-xs text-gray-500 mb-2">Narrow window — ratio must land between 50% and 60%.</p><div class="mt-2 relative h-2 w-full rounded-full bg-gray-100 overflow-hidden"><div class="absolute h-2 bg-gray-300" style="left:0%;width:40%"></div><div class="absolute h-2 bg-red-400" style="left:40%;width:10%"></div><div class="absolute h-2 bg-emerald-500" style="left:50%;width:10%"></div><div class="absolute h-2 bg-gray-300" style="left:60%;width:40%"></div></div><div class="relative h-4 mt-1"><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:0%">0<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:40%">40<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:50%">50<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:60%">60<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:100%">100<!-- -->%</span></div></div><div class="bg-gray-50 border border-gray-200 rounded-lg p-4"><div class="flex items-center justify-between mb-1"><span class="font-medium text-gray-900 text-sm">Majority / Blitz<span class="text-gray-400 font-normal"> (10 turn limit)</span></span><span class="font-mono text-xs text-gray-400 bg-white border border-gray-200 px-2 py-0.5 rounded">K &gt; 1.0</span></div><p class="text-xs text-gray-500 mb-2">Standard rules with a hard turn limit.</p><div class="mt-2 relative h-2 w-full rounded-full bg-gray-100 overflow-hidden"><div class="absolute h-2 bg-red-400" style="left:0%;width:50%"></div><div class="absolute h-2 bg-emerald-500" style="left:50%;width:50%"></div></div><div class="relative h-4 mt-1"><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:0%">0<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:50%">50<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:100%">100<!-- -->%</span></div></div></div><div class="space-y-3"><p class="text-xs font-semibold text-gray-400 uppercase tracking-widest mb-3">Minority</p><div class="bg-gray-50 border border-gray-200 rounded-lg p-4"><div class="flex items-center justify-between mb-1"><span class="font-medium text-gray-900 text-sm">Minority</span><span class="font-mono text-xs text-gray-400 bg-white border border-gray-200 px-2 py-0.5 rounded">K &lt; 0</span></div><p class="text-xs text-gray-500 mb-2">Fewer discs wins — inverse Othello.</p><div class="mt-2 relative h-2 w-full rounded-full bg-gray-100 overflow-hidden"><div class="absolute h-2 bg-emerald-500" style="left:0%;width:50%"></div><div class="absolute h-2 bg-red-400" style="left:50%;width:50%"></div></div><div class="relative h-4 mt-1"><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:0%">0<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:50%">50<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:100%">100<!-- -->%</span></div></div><div class="bg-gray-50 border border-gray-200 rounded-lg p-4"><div class="flex items-center justify-between mb-1"><span class="font-medium text-gray-900 text-sm">Minority &gt; 40%</span><span class="font-mono text-xs text-gray-400 bg-white border border-gray-200 px-2 py-0.5 rounded">K = 0.4</span></div><p class="text-xs text-gray-500 mb-2">Win with minority, but disc ratio must stay above 40%.</p><div class="mt-2 relative h-2 w-full rounded-full bg-gray-100 overflow-hidden"><div class="absolute h-2 bg-gray-300" style="left:0%;width:40%"></div><div class="absolute h-2 bg-emerald-500" style="left:40%;width:10%"></div><div class="absolute h-2 bg-red-400" style="left:50%;width:10%"></div><div class="absolute h-2 bg-gray-300" style="left:60%;width:40%"></div></div><div class="relative h-4 mt-1"><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:0%">0<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:40%">40<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:50%">50<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:60%">60<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:100%">100<!-- -->%</span></div></div><div class="bg-gray-50 border border-gray-200 rounded-lg p-4"><div class="flex items-center justify-between mb-1"><span class="font-medium text-gray-900 text-sm">Minority &gt; 20%</span><span class="font-mono text-xs text-gray-400 bg-white border border-gray-200 px-2 py-0.5 rounded">
1K = 0.2</span></div><p class="text-xs text-gray-500 mb-2">Win with minority, but disc ratio must stay above 20%.</p><div class="mt-2 relative h-2 w-full rounded-full bg-gray-100 overflow-hidden"><div class="absolute h-2 bg-gray-300" style="left:0%;width:20%"></div><div class="absolute h-2 bg-emerald-500" style="left:20%;width:30%"></div><div class="absolute h-2 bg-red-400" style="left:50%;width:30%"></div><div class="absolute h-2 bg-gray-300" style="left:80%;width:20%"></div></div><div class="relative h-4 mt-1"><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:0%">0<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:20%">20<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:50%">50<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:80%">80<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:100%">100<!-- -->%</span></div></div><div class="bg-gray-50 border border-gray-200 rounded-lg p-4"><div class="flex items-center justify-between mb-1"><span class="font-medium text-gray-900 text-sm">Minority / Blitz<span class="text-gray-400 font-normal"> (10 turn limit)</span></span><span class="font-mono text-xs text-gray-400 bg-white border border-gray-200 px-2 py-0.5 rounded">K &lt; 0</span></div><p class="text-xs text-gray-500 mb-2">Inverse rules with a hard turn limit.</p><div class="mt-2 relative h-2 w-full rounded-full bg-gray-100 overflow-hidden"><div class="absolute h-2 bg-emerald-500" style="left:0%;width:50%"></div><div class="absolute h-2 bg-red-400" style="left:50%;width:50%"></div></div><div class="relative h-4 mt-1"><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:0%">0<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:50%">50<!-- -->%</span><span class="absolute text-[10px] text-gray-400 -translate-x-1/2" style="left:100%">100<!-- -->%</span></div></div></div></div></div></section><section id="how-to-use" class="py-20 px-6 border-b border-gray-100 bg-white"><div class="max-w-3xl mx-auto"><h2 class="text-2xl font-bold mb-1 text-gray-900">How to Use</h2><p class="text-gray-500 text-sm mb-6">Installation &amp; Quick Start</p><p class="text-base text-gray-600 mb-12 leading-relaxed">Please check<!-- --> <a href="https://github.com/blakeyoo/ExpandedOthello-AEC" target="_blank" rel="noopener noreferrer" class="inline-flex items-center gap-1.5 bg-gray-900 hover:bg-gray-800 text-white font-medium px-3 py-0.5 rounded-md transition-colors text-sm align-middle"><svg class="w-3.5 h-3.5" fill="currentColor" viewBox="0 0 24 24"><path d="M12 0C5.374 0 0 5.373 0 12c0 5.302 3.438 9.8 8.207 11.387.599.111.793-.261.793-.577v-2.234c-3.338.726-4.033-1.416-4.033-1.416-.546-1.387-1.333-1.756-1.333-1.756-1.089-.745.083-.729.083-.729 1.205.084 1.839 1.237 1.839 1.237 1.07 1.834 2.807 1.304 3.492.997.107-.775.418-1.305.762-1.604-2.665-.305-5.467-1.334-5.467-5.931 0-1.311.469-2.381 1.236-3.221-.124-.303-.535-1.524.117-3.176 0 0 1.008-.322 3.301 1.23A11.509 11.509 0 0112 5.803c1.02.005 2.047.138 3.006.404 2.291-1.552 3.297-1.23 3.297-1.23.653 1.653.242 2.874.118 3.176.77.84 1.235 1.911 1.235 3.221 0 4.609-2.807 5.624-5.479 5.921.43.372.823 1.102.823 2.222v3.293c0 .319.192.694.801.576C20.566 21.797 24 17.3 24 12c0-6.627-5.373-12-12-12z"></path></svg>Environment</a> <!-- -->for the detailed guidelines.</p><div class="mb-12"><h3 class="text-base font-semibold text-gray-900 mb-4">Installation</h3><pre class="bg-gray-900 text-gray-100 rounded-lg px-5 py-4 text-sm font-mono overflow-x-auto leading-relaxed"><span class="text-gray-500 select-none"># Install directly from GitHub<!-- -->
2</span><span>pip install git+https://github.com/blakeyoo/ExpandedOthello-AEC.git</span></pre><p class="text-xs text-gray-400 mt-2">Requires Python ≥ 3.9 · numpy · numba · pettingzoo · gymnasium</p></div><div class="mb-12"><h3 class="text-base font-semibold text-gray-900 mb-2">Load a Benchmark Environment</h3><p class="text-sm text-gray-500 mb-4">56 preset environments are available — the Cartesian product of 7 board layouts and 8 win conditions.</p><pre class="bg-gray-900 text-gray-100 rounded-lg px-5 py-4 text-sm font-mono overflow-x-auto leading-relaxed"><span class="text-indigo-400">from</span> expanded_othello <span class="text-indigo-400">
2import</span> load_preset
3
4<span class="text-gray-500 select-none"># Load one of 56 official environments (index 0–55)<!-- -->
5</span>env = load_preset(0)
6
7<!-- -->obs, _ = env.reset()
8<span class="text-indigo-400">while</span> env.agents:
9<!-- -->    agent = env.agent_selection
10<!-- -->    action_mask = obs[<span class="text-amber-400">&quot;action_mask&quot;</span>]
11<!-- -->    action = your_agent.act(obs, action_mask)
12<!-- -->    env.step(action)
13<!-- -->    obs, reward, term, trunc, _ = env.last()</pre></div><div><h3 class="text-base font-semibold text-gray-900 mb-2">Custom Environment</h3><p class="text-sm text-gray-500 mb-4">Define your own board geometry and win condition beyond the preset suite.</p><pre class="bg-gray-900 text-gray-100 rounded-lg px-5 py-4 text-sm font-mono overflow-x-auto leading-relaxed"><span class="text-indigo-400">from</span> expanded_othello <span class="text-indigo-400">import</span> make_env
14
15<!-- -->env = make_env(
16<!-- -->    board_size=(<span class="text-amber-400">10</span>, <span class="text-amber-400">10</span>),
17<!-- -->    obstacles=[(<span class="text-amber-400">0</span>, <span class="text-amber-400">0</span>), (<span class="text-amber-400">9</span>, <span class="text-amber-400">9</span>)],
18<!-- -->    win_cond=<span class="text-amber-400">0.6</span>,
19<!-- -->    n_turns=<span class="text-amber-400">-1</span>,
20<!-- -->)</pre></div></div></section><section id="results" class="py-20 px-6 border-b border-gray-100 bg-gray-50"><div class="max-w-5xl mx-auto"><h2 class="text-2xl font-bold mb-1 text-gray-900">Possible Approaches</h2><p class="text-gray-500 text-sm mb-12">What works, what doesn&#x27;t, and where the open problems are</p><div class="space-y-6 text-gray-700 leading-8 text-[1.05rem] mb-10"><p>To demonstrate the benchmark&#x27;s utility, we introduce a neuroevolutionary Adaptive Minimax as a foundational baseline. It combines two components: a population of positional value networks meta-evolved across diverse board layouts (<strong>PosNet</strong>), providing spatial priors that transfer to unseen geometries; and an adaptive utility-weighting mechanism that evolves within each environment to infer the latent winning condition (<strong>WeightNet</strong>). Together, they allow the agent to adapt both spatially and strategically from terminal outcomes alone.</p><p>In broad regimes — standard, inverse, and blitz conditions — the Adaptive Minimax converges rapidly, often reaching high win rates within the first generation of updates. Notably, it achieves win rates comparable to MCTS-100 despite MCTS having direct simulator access to the latent threshold K during search. The contrast with PPO is stark: trained under the same 2,000-game budget, PPO&#x27;s win rate curves plateau early and remain largely flat — sparse terminal rewards provide insufficient gradient signal for meaningful adaptation within the budget, even when training is extended to 10,000 games.</p></div><figure class="mb-10"><img src="/MiniMaxPPO.png" alt="Convergence speed of Adaptive Minimax and PPO" class="w-full rounded-lg border border-gray-200"/><figcaption class="mt-3 text-xs text-gray-400 leading-relaxed">Convergence speed of Adaptive Minimax (left) and PPO (right) under increasing training budgets, measured by win rate against Random (top) and MCTS-100 (bottom). Each curve is averaged over the 7 environments sharing the same victory condition C. For the Minimax, each generation requires 100 games.</figcaption></figure><div class="space-y-6 text-gray-700 leading-8 text-[1.05rem] mb-14"><p>Where the Adaptive Minimax falls short is in narrow-interval regimes: at K = 0.6 and K = 0.4, win rates collapse and outcomes are dominated by draws. The agent learns to avoid losing, but cannot control its terminal disc ratio with the precision required to enter the admissible interval consistently. These regimes remain an open challenge for all tested approaches.</p></div><div class="flex items-center gap-4 mb-4 text-xs text-gray-500"><span class="flex items-center gap-1.5"><span class="inline-block w-3 h-2 rounded-sm bg-emerald-500"></span> Win</span><span class="flex items-center gap-1.5"><span class="inline-block w-3 h-2 rounded-sm bg-gray-300"></span> Draw</span><span class="flex items-center gap-1.5"><span class="inline-block w-3 h-2 rounded-sm bg-red-400"></span> Lose</span></div><h3 class="text-lg font-semibold text-gray-900 mb-1">Adaptive Minimax</h3><p class="text-gray-500 text-sm mb-6">averaged over 7 layouts per condition · trained with 2,000 games of self
20-play per environment</p><div class="mb-4"><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 text-xs font-semibold text-gray-400 uppercase tracking-widest mb-3 px-4"><span>Condition</span><span>vs Random</span><span>vs PPO</span><span>vs MCTS-100</span></div><div class="space-y-3"><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 items-center rounded-lg px-4 py-3 bg-gray-50 border border-gray-200"><span class="text-sm text-gray-800">Majority (K &gt; 1.0)</span><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:95.7%"></div><div class="h-full bg-gray-300" style="width:1.4%"></div><div class="h-full bg-red-400" style="width:2.9%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">95.7<!-- -->%</span><span class="text-gray-400">1.4<!-- -->%</span><span class="text-red-400">2.9<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:85.7%"></div><div class="h-full bg-gray-300" style="width:0%"></div><div class="h-full bg-red-400" style="width:14.3%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">85.7<!-- -->%</span><span class="text-gray-400">0<!-- -->%</span><span class="text-red-400">14.3<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:78.6%"></div><div class="h-full bg-gray-300" style="width:0.7%"></div><div class="h-full bg-red-400" style="width:20.7%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">78.6<!-- -->%</span><span class="text-gray-400">0.7<!-- -->%</span><span class="text-red-400">20.7<!-- -->%</span></div></div></div><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 items-center rounded-lg px-4 py-3 bg-gray-50 border border-gray-200"><span class="text-sm text-gray-800">Majority &lt; 80% (K = 0.8)</span><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:54.3%"></div><div class="h-full bg-gray-300" style="width:35.7%"></div><div class="h-full bg-red-400" style="width:10%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">54.3<!-- -->%</span><span class="text-gray-400">35.7<!-- -->%</span><span class="text-red-400">10<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:64.3%"></div><div class="h-full bg-gray-300" style="width:28.6%"></div><div class="h-full bg-red-400" style="width:7.1%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">64.3<!-- -->%</span><span class="text-gray-400">28.6<!-- -->%</span><span class="text-red-400">7.1<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:44.3%"></div><div class="h-full bg-gray-300" style="width:20%"></div><div class="h-full bg-red-400" style="width:35.7%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">44.3<!-- -->%</span><span class="text-gray-400">20<!-- -->%</span><span class="text-red-400">35.7<!-- -->%</span></div></div></div><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 items-center rounded-lg px-4 py-3 bg-amber-50 border border-amber-200"><span class="text-sm text-gray-800">Majority &lt; 60% (K = 0.6)<span class="ml-2 text-[10px] font-semibold text-amber-600 bg-amber-100 px-1.5 py-0.5 rounded uppercase tracking-wide">narrow</span></span><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:19.3%"></div><div class="h-full bg-gray-300" style="width:73.6%"></div><div class="h-full bg-red-400" style="width:7.1%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">19.3<!-- -->%</span><span class="text-gray-400">73.6<!-- -->%</span><span class="text-red-400">7.1<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:0%"></div><div class="h-full bg-gray-300" style="width:92.9%"></div><div class="h-full bg-red-400" style="width:7.1%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">0<!-- -->%</span><span class="text-gray-400">92.9<!-- -->%</span><span class="text-red-400">7.1<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:13.6%"></div><div class="h-full bg-gray-300" style="width:67.9%"></div><div class="h-full bg-red-400" style="width:18.6%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">13.6<!-- -->%</span><span class="text-gray-400">67.9<!-- -->%</span><span class="text-red-400">18.6<!-- -->%</span></div></div></div><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 items-center rounded-lg px-4 py-3 bg-amber-50 border border-amber-200"><span class="text-sm text-gray-800">Minority &gt; 40% (K = 0.4)<span class="ml-2 text-[10px] font-semibold text-amber-600 bg-amber-100 px-1.5 py-0.5 rounded uppercase tracking-wide">narrow</span></span><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:18.6%"></div><div class="h-full bg-gray-300" style="width:72.9%"></div><div class="h-full bg-red-400" style="width:8.6%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">18.6<!-- -->%</span><span class="text-gray-400">72.9<!-- -->%</span><span class="text-red-400">8.6<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:14.3%"></div><div class="h-full bg-gray-300" style="width:78.6%"></div><div class="h-full bg-red-400" style="width:7.1%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">14.3<!-- -->%</span><span class="text-gray-400">78.6<!-- -->%</span><span class="text-red-400">7.1<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:8.6%"></div><div class="h-full bg-gray-300" style="width:60%"></div><div class="h-full bg-red-400" style="width:31.4%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">8.6<!-- -->%</span><span class="text-gray-400">60<!-- -->%</span><span class="text-red-400">31.4<!-- -->%</span></div></div></div><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 items-center rounded-lg px-4 py-3 bg-gray-50 border border-gray-200"><span class="text-sm text-gray-800">Minority &gt; 20% (K = 0.2)</span><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:77.9%"></div><div class="h-full bg-gray-300" style="width:16.4%"></div><div class="h-full bg-red-400" style="width:5.7%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">77.9<!-- -->%</span><span class="text-gray-400">16.4<!-- -->%</span><span class="text-red-400">5.7<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:64.3%"></div><div class="h-full bg-gray-300" style="width:14.3%"></div><div class="h-full bg-red-400" style="width:21.4%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">64.3<!-- -->%</span><span class="text-gray-400">14.3<!-- -->%</span><span class="text-red-400">21.4<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:45%"></div><div class="h-full bg-gray-300" style="width:24.3%"></div><div class="h-full bg-red-400" style="width:30.7%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">45<!-- -->%</span><span class="text-gray-400">24.3<!-- -->%</span><span class="text-red-400">30.7<!-- -->%</span></div></div></div><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 items-center rounded-lg px-4 py-3 bg-gray-50 border border-gray-200"><span class="text-sm text-gray-800">Minority (K &lt; 0.0)</span><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:94.3%"></div><div class="h-full bg-gray-300" style="width:2.1%"></div><div class="h-full bg-red-400" style="width:3.6%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">94.3<!-- -->%</span><span class="text-gray-400">2.1<!-- -->%</span><span class="text-red-400">3.6<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:92.9%"></div><div class="h-full bg-gray-300" style="width:0%"></div><div class="h-full bg-red-400" style="width:7.1%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">92.9<!-- -->%</span><span class="text-gray-400">0<!-- -->%</span><span class="text-red-400">7.1<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:55.7%"></div><div class="h-full bg-gray-300" style="width:3.6%"></div><div class="h-full bg-red-400" style="width:40.7%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">55.7<!-- -->%</span><span class="text-gray-400">3.6<!-- -->%</span><span class="text-red-400">40.7<!-- -->%</span></div></div></div><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 items-center rounded-lg px-4 py-3 bg-gray-50 border border-gray-200"><span class="text-sm text-gray-800">Majority / Blitz (K &gt; 1.0)<span class="ml-2 text-[10px] text-gray-400 font-normal">10 turn limit</span></span><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:90%"></div><div class="h-full bg-gray-300" style="width:6.4%"></div><div class="h-full bg-red-400" style="width:3.6%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">90<!-- -->%</span><span class="text-gray-400">6.4<!-- -->%</span><span class="text-red-400">3.6<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:85.7%"></div><div class="h-full bg-gray-300" style="width:14.3%"></div><div class="h-full bg-red-400" style="width:0%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">85.7<!-- -->%</span><span class="text-gray-400">14.3<!-- -->%</span><span class="text-red-400">0<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:56.4%"></div><div class="h-full bg-gray-300" style="width:32.1%"></div><div class="h-full bg-red-400" style="width:11.4%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">56.4<!-- -->%</span><span class="text-gray-400">32.1<!-- -->%</span><span class="text-red-400">11.4<!-- -->%</span></div></div></div><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 items-center rounded-lg px-4 py-3 bg-gray-50 border border-gray-200"><span class="text-sm text-gray-800">Minority / Blitz (K &lt; 0.0)<span class="ml-2 text-[10px] text-gray-400 font-normal">10 turn limit</span></span><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:84.3%"></div><div class="h-full bg-gray-300" style="width:12.1%"></div><div class="h-full bg-red-400" style="width:3.6%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">84.3<!-- -->%</span><span class="text-gray-400">12.1<!-- -->%</span><span class="text-red-400">3.6<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:78.6%"></div><div class="h-full bg-gray-300" style="width:14.3%"></div><div class="h-full bg-red-400" style="width:7.1%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">78.6<!-- -->%</span><span class="text-gray-400">14.3<!-- -->%</span><span class="text-red-400">7.1<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:55.7%"></div><div class="h-full bg-gray-300" style="width:10.7%"></div><div class="h-full bg-red-400" style="width:33.6%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">55.7<!-- -->%</span><span class="text-gray-400">10.7<!-- -->%</span><span class="text-red-400">33.6<!-- -->%</span></div></div></div></div></div><p class="text-xs text-gray-400 mb-14">Adaptive Minimax results are averaged across layouts after 2,000 training episodes (trained with terminal game outcomes alone). Full per-layout breakdown available in the paper.</p><h3 class="text-lg font-semibold text-gray-900 mb-1">PPO</h3><p class="text-gray-500 text-sm mb-6">averaged over 7 layouts per condition · trained with 2,000 games of self
20-play per environment</p><div class="mb-4"><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 text-xs font-semibold text-gray-400 uppercase tracking-widest mb-3 px-4"><span>Condition</span><span>vs Random</span><span>vs Adaptive Minimax</span><span>vs MCTS-100</span></div><div class="space-y-3"><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 items-center rounded-lg px-4 py-3 bg-gray-50 border border-gray-200"><span class="text-sm text-gray-800">Majority (K &gt; 1.0)</span><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:78.6%"></div><div class="h-full bg-gray-300" style="width:2.9%"></div><div class="h-full bg-red-400" style="width:18.6%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">78.6<!-- -->%</span><span class="text-gray-400">2.9<!-- -->%</span><span class="text-red-400">18.6<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:14.3%"></div><div class="h-full bg-gray-300" style="width:0%"></div><div class="h-full bg-red-400" style="width:85.7%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">14.3<!-- -->%</span><span class="text-gray-400">0<!-- -->%</span><span class="text-red-400">85.7<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:17.1%"></div><div class="h-full bg-gray-300" style="width:2.1%"></div><div class="h-full bg-red-400" style="width:80.7%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">17.1<!-- -->%</span><span class="text-gray-400">2.1<!-- -->%</span><span class="text-red-400">80.7<!-- -->%</span></div></div></div><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 items-center rounded-lg px-4 py-3 bg-gray-50 border border-gray-200"><span class="text-sm text-gray-800">Majority &lt; 80% (K = 0.8)</span><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:68.6%"></div><div class="h-full bg-gray-300" style="width:10%"></div><div class="h-full bg-red-400" style="width:21.4%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">68.6<!-- -->%</span><span class="text-gray-400">10<!-- -->%</span><span class="text-red-400">21.4<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:7.1%"></div><div class="h-full bg-gray-300" style="width:28.6%"></div><div class="h-full bg-red-400" style="width:64.3%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">7.1<!-- -->%</span><span class="text-gray-400">28.6<!-- -->%</span><span class="text-red-400">64.3<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:15.7%"></div><div class="h-full bg-gray-300" style="width:5%"></div><div class="h-full bg-red-400" style="width:79.3%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">15.7<!-- -->%</span><span class="text-gray-400">5<!-- -->%</span><span class="text-red-400">79.3<!-- -->%</span></div></div></div><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 items-center rounded-lg px-4 py-3 bg-amber-50 border border-amber-200"><span class="text-sm text-gray-800">Majority &lt; 60% (K = 0.6)<span class="ml-2 text-[10px] font-semibold text-amber-600 bg-amber-100 px-1.5 py-0.5 rounded uppercase tracking-wide">narrow</span></span><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:23.6%"></div><div class="h-full bg-gray-300" style="width:52.9%"></div><div class="h-full bg-red-400" style="width:23.6%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">23.6<!-- -->%</span><span class="text-gray-400">52.9<!-- -->%</span><span class="text-red-400">23.6<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:7.1%"></div><div class="h-full bg-gray-300" style="width:92.9%"></div><div class="h-full bg-red-400" style="width:0%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">7.1<!-- -->%</span><span class="text-gray-400">92.9<!-- -->%</span><span class="text-red-400">0<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:2.9%"></div><div class="h-full bg-gray-300" style="width:23.6%"></div><div class="h-full bg-red-400" style="width:73.6%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">2.9<!-- -->%</span><span class="text-gray-400">23.6<!-- -->%</span><span class="text-red-400">73.6<!-- -->%</span></div></div></div><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 items-center rounded-lg px-4 py-3 bg-amber-50 border border-amber-200"><span class="text-sm text-gray-800">Minority &gt; 40% (K = 0.4)<span class="ml-2 text-[10px] font-semibold text-amber-600 bg-amber-100 px-1.5 py-0.5 rounded uppercase tracking-wide">narrow</span></span><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:26.4%"></div><div class="h-full bg-gray-300" style="width:57.9%"></div><div class="h-full bg-red-400" style="width:15.7%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">26.4<!-- -->%</span><span class="text-gray-400">57.9<!-- -->%</span><span class="text-red-400">15.7<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:7.1%"></div><div class="h-full bg-gray-300" style="width:78.6%"></div><div class="h-full bg-red-400" style="width:14.3%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">7.1<!-- -->%</span><span class="text-gray-400">78.6<!-- -->%</span><span class="text-red-400">14.3<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:3.6%"></div><div class="h-full bg-gray-300" style="width:28.6%"></div><div class="h-full bg-red-400" style="width:67.9%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">3.6<!-- -->%</span><span class="text-gray-400">28.6<!-- -->%</span><span class="text-red-400">67.9<!-- -->%</span></div></div></div><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 items-center rounded-lg px-4 py-3 bg-gray-50 border border-gray-200"><span class="text-sm text-gray-800">Minority &gt; 20% (K = 0.2)</span><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:70.7%"></div><div class="h-full bg-gray-300" style="width:9.3%"></div><div class="h-full bg-red-400" style="width:20%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">70.7<!-- -->%</span><span class="text-gray-400">9.3<!-- -->%</span><span class="text-red-400">20<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:21.4%"></div><div class="h-full bg-gray-300" style="width:14.3%"></div><div class="h-full bg-red-400" style="width:64.3%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">21.4<!-- -->%</span><span class="text-gray-400">14.3<!-- -->%</span><span class="text-red-400">64.3<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:6.4%"></div><div class="h-full bg-gray-300" style="width:3.6%"></div><div class="h-full bg-red-400" style="width:90%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">6.4<!-- -->%</span><span class="text-gray-400">3.6<!-- -->%</span><span class="text-red-400">90<!-- -->%</span></div></div></div><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 items-center rounded-lg px-4 py-3 bg-gray-50 border border-gray-200"><span class="text-sm text-gray-800">Minority (K &lt; 0.0)</span><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:75%"></div><div class="h-full bg-gray-300" style="width:2.9%"></div><div class="h-full bg-red-400" style="width:22.1%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">75<!-- -->%</span><span class="text-gray-400">2.9<!-- -->%</span><span class="text-red-400">22.1<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:7.1%"></div><div class="h-full bg-gray-300" style="width:0%"></div><div class="h-full bg-red-400" style="width:92.9%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">7.1<!-- -->%</span><span class="text-gray-400">0<!-- -->%</span><span class="text-red-400">92.9<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:5.7%"></div><div class="h-full bg-gray-300" style="width:0.7%"></div><div class="h-full bg-red-400" style="width:93.6%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">5.7<!-- -->%</span><span class="text-gray-400">0.7<!-- -->%</span><span class="text-red-400">93.6<!-- -->%</span></div></div></div><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 items-center rounded-lg px-4 py-3 bg-gray-50 border border-gray-200"><span class="text-sm text-gray-800">Majority / Blitz (K &gt; 1.0)<span class="ml-2 text-[10px] text-gray-400 font-normal">10 turn limit</span></span><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:61.4%"></div><div class="h-full bg-gray-300" style="width:20%"></div><div class="h-full bg-red-400" style="width:18.6%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">61.4<!-- -->%</span><span class="text-gray-400">20<!-- -->%</span><span class="text-red-400">18.6<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:0%"></div><div class="h-full bg-gray-300" style="width:14.3%"></div><div class="h-full bg-red-400" style="width:85.7%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">0<!-- -->%</span><span class="text-gray-400">14.3<!-- -->%</span><span class="text-red-400">85.7<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:16.4%"></div><div class="h-full bg-gray-300" style="width:15%"></div><div class="h-full bg-red-400" style="width:68.6%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">16.4<!-- -->%</span><span class="text-gray-400">15<!-- -->%</span><span class="text-red-400">68.6<!-- -->%</span></div></div></div><div class="grid grid-cols-[1fr_160px_160px_160px] gap-x-4 items-center rounded-lg px-4 py-3 bg-gray-50 border border-gray-200"><span class="text-sm text-gray-800">Minority / Blitz (K &lt; 0.0)<span class="ml-2 text-[10px] text-gray-400 font-normal">10 turn limit</span></span><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex">
20<div class="h-full bg-emerald-500" style="width:65.7%"></div><div class="h-full bg-gray-300" style="width:18.6%"></div><div class="h-full bg-red-400" style="width:15.7%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">65.7<!-- -->%</span><span class="text-gray-400">18.6<!-- -->%</span><span class="text-red-400">15.7<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:7.1%"></div><div class="h-full bg-gray-300" style="width:14.3%"></div><div class="h-full bg-red-400" style="width:78.6%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">7.1<!-- -->%</span><span class="text-gray-400">14.3<!-- -->%</span><span class="text-red-400">78.6<!-- -->%</span></div></div><div><div class="relative h-3 w-full rounded-full overflow-hidden bg-gray-200 flex"><div class="h-full bg-emerald-500" style="width:25%"></div><div class="h-full bg-gray-300" style="width:10%"></div><div class="h-full bg-red-400" style="width:65%"></div></div><div class="flex justify-between text-[10px] text-gray-400 mt-0.5"><span class="text-emerald-600">25<!-- -->%</span><span class="text-gray-400">10<!-- -->%</span><span class="text-red-400">65<!-- -->%</span></div></div></div></div></div><p class="text-xs text-gray-400 mb-14">PPO results are averaged across layouts after 2,000 training episodes (trained with terminal game outcomes alone). Full per-layout breakdown available in the paper.</p><div class="bg-gray-50 border border-gray-200 rounded-xl p-6 text-gray-700 leading-8 text-[1.05rem]"><p>These results demonstrate that the Arena captures two qualitatively distinct adaptation challenges: rapid win-oriented convergence in broad regimes, and fragile, draw-heavy occupancy control in narrow ones. The difficulty gradient exposed here — and the failure of all tested baselines in the narrowest regimes — points to open research problems the Arena is specifically designed to surface.</p></div></div></section><footer class="border-t border-gray-100 py-8 text-center text-sm text-gray-400">© 2026 Byunghwa Yoo, Sundong Kim, Kyung-Joong Kim · GIST Department of AI Convergence</footer></main><!--$--><!--/$-->
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20<script>self.__next_f.push([1,"7:[\"$\",\"div\",null,{\"className\":\"space-y-6 text-gray-700 leading-8 text-[1.05rem]\",\"children\":[[\"$\",\"p\",null,{\"children\":\"Evaluating how an AI discovers a latent objective requires going beyond what traditional meta-learning environments offer. Visual or structural variation — changes to inputs, layouts, or level appearances — leaves the fundamental rules of the environment intact. What is needed instead is an environment where the rules themselves change: where what counts as winning is not given, and must be inferred. Moreover, AGI demands more than single-agent adaptation. Deriving an objective from an environment that includes another agent — one whose actions are shaped by the same hidden rules — is a qualitatively harder and more realistic test of general intelligence.\"}],[\"$\",\"p\",null,{\"children\":\"We select Othello as the substrate for the environment to test such abilities because it satisfies these requirements. Its terminal state produces a continuous disc occupancy ratio ρ ∈ [0, 1] — making the winning condition directly parameterizable through a single threshold, and therefore variable in ways that go beyond structural change. It is zero-sum and adversarial by construction, requiring adaptation against an opponent whose decisions co-determine the outcome. And its flipping mechanics — high global state-change dynamics from simple local rules — mean that board geometry directly shapes which strategies are viable, making spatial layout a meaningful variable rather than mere decoration.\"}],[\"$\",\"p\",null,{\"children\":[\"The environment space is therefore formalized as\",\" \",[\"$\",\"span\",null,{\"className\":\"font-mono bg-gray-100 px-1.5 py-0.5 rounded text-sm\",\"children\":\"ℰ = ℒ × 𝒞\"}],\". Each instance\",\" \",[\"$\",\"span\",null,{\"className\":\"font-mono bg-gray-100 px-1.5 py-0.5 rounded text-sm\",\"children\":\"E = (L, C)\"}],\" \",\"is defined by two independently varying components.\",\" \",[\"$\",\"strong\",null,{\"children\":\"L\"}],\" is the board geometry — the size and shape of the board, along with the placement of non-interactive obstacles. It is fully observable, and represents the structural dimension of variation familiar from traditional meta-learning.\",\" \",[\"$\",\"strong\",null,{\"children\":\"C\"}],\" is the victory condition — latent and never directly revealed, it must be implicitly inferred through interaction. Leveraging Othello's natural disc occupancy ratio ρ, the winning criteria is varied across environments through a single parameterization \",[\"$\",\"em\",null,{\"children\":\"K\"}],\", which determines the threshold that separates victory from over-dominance draw conditions, i.e., the winning condition is not simply defined via majority of the discs; it has to stay in a specific ratio region, or strategic draw is often needed for prventing opponent win. For detailed definition of \",[\"$\",\"em\",null,{\"children\":\"K\"}],\", please refer to our\",\" \",[\"$\",\"a\",null,{\"href\":\"https://openreview.net/forum?id=WXKQtqPC2d\",\"target\":\"_blank\",\"rel\":\"noopener noreferrer\",\"className\":\"underline text-gray-500 hover:text-gray-800\",\"children\":\"paper\"}],\".\"]}]]}]\n"])</script>
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20<script>self.__next_f.push([1,"9:[\"$\",\"section\",null,{\"id\":\"how-to-use\",\"className\":\"py-20 px-6 border-b border-gray-100 bg-white\",\"children\":[\"$\",\"div\",null,{\"className\":\"max-w-3xl mx-auto\",\"children\":[[\"$\",\"h2\",null,{\"className\":\"text-2xl font-bold mb-1 text-gray-900\",\"children\":\"How to Use\"}],[\"$\",\"p\",null,{\"className\":\"text-gray-500 text-sm mb-6\",\"children\":\"Installation \u0026 Quick Start\"}],[\"$\",\"p\",null,{\"className\":\"text-base text-gray-600 mb-12 leading-relaxed\",\"children\":[\"Please check\",\" \",[\"$\",\"a\",null,{\"href\":\"https://github.com/blakeyoo/ExpandedOthello-AEC\",\"target\":\"_blank\",\"rel\":\"noopener noreferrer\",\"className\":\"inline-flex items-center gap-1.5 bg-gray-900 hover:bg-gray-800 text-white font-medium px-3 py-0.5 rounded-md transition-colors text-sm align-middle\",\"children\":[[\"$\",\"svg\",null,{\"className\":\"w-3.5 h-3.5\",\"fill\":\"currentColor\",\"viewBox\":\"0 0 24 24\",\"children\":[\"$\",\"path\",null,{\"d\":\"M12 0C5.374 0 0 5.373 0 12c0 5.302 3.438 9.8 8.207 11.387.599.111.793-.261.793-.577v-2.234c-3.338.726-4.033-1.416-4.033-1.416-.546-1.387-1.333-1.756-1.333-1.756-1.089-.745.083-.729.083-.729 1.205.084 1.839 1.237 1.839 1.237 1.07 1.834 2.807 1.304 3.492.997.107-.775.418-1.305.762-1.604-2.665-.305-5.467-1.334-5.467-5.931 0-1.311.469-2.381 1.236-3.221-.124-.303-.535-1.524.117-3.176 0 0 1.008-.322 3.301 1.23A11.509 11.509 0 0112 5.803c1.02.005 2.047.138 3.006.404 2.291-1.552 3.297-1.23 3.297-1.23.653 1.653.242 2.874.118 3.176.77.84 1.235 1.911 1.235 3.221 0 4.609-2.807 5.624-5.479 5.921.43.372.823 1.102.823 2.222v3.293c0 .319.192.694.801.576C20.566 21.797 24 17.3 24 12c0-6.627-5.373-12-12-12z\"}]}],\"Environment\"]}],\" \",\"for the detailed guidelines.\"]}],[\"$\",\"div\",null,{\"className\":\"mb-12\",\"children\":[[\"$\",\"h3\",null,{\"className\":\"text-base font-semibold text-gray-900 mb-4\",\"children\":\"Installation\"}],[\"$\",\"pre\",null,{\"className\":\"bg-gray-900 text-gray-100 rounded-lg px-5 py-4 text-sm font-mono overflow-x-auto leading-relaxed\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-gray-500 select-none\",\"children\":[\"# Install directly from GitHub\",\"\\n\"]}],[\"$\",\"span\",null,{\"children\":\"pip install git+https://github.com/blakeyoo/ExpandedOthello-AEC.git\"}]]}],[\"$\",\"p\",null,{\"className\":\"text-xs text-gray-400 mt-2\",\"children\":\"Requires Python ≥ 3.9 · numpy · numba · pettingzoo · gymnasium\"}]]}],[\"$\",\"div\",null,{\"className\":\"mb-12\",\"children\":[[\"$\",\"h3\",null,{\"className\":\"text-base font-semibold text-gray-900 mb-2\",\"children\":\"Load a Benchmark Environment\"}],[\"$\",\"p\",null,{\"className\":\"text-sm text-gray-500 mb-4\",\"children\":\"56 preset environments are available — the Cartesian product of 7 board layouts and 8 win conditions.\"}],[\"$\",\"pre\",null,{\"className\":\"bg-gray-900 text-gray-100 rounded-lg px-5 py-4 text-sm font-mono overflow-x-auto leading-relaxed\",\"children\":[[\"$\",\"span\",null,{\"className\":\"text-indigo-400\",\"children\":\"from\"}],\" expanded_othello \",[\"$\",\"span\",null,{\"className\":\"text-indigo-400\",\"children\":\"import\"}],\" load_preset\\n\\n\",[\"$\",\"span\",null,{\"className\":\"text-gray-500 select-none\",\"children\":[\"# Load one of 56 official environments (index 0–55)\",\"\\n\"]}],\"env = load_preset(0)\\n\\n\",\"obs, _ = env.reset()\\n\",[\"$\",\"span\",null,{\"className\":\"text-indigo-400\",\"children\":\"while\"}],\" env.agents:\\n\",\"    agent = env.agent_selection\\n\",\"    action_mask = obs[\",[\"$\",\"span\",null,{\"className\":\"text-amber-400\",\"children\":\"\\\"action_mask\\\"\"}],\"]\\n\",\"    action = your_agent.act(obs, action_mask)\\n\",\"    env.step(action)\\n\",\"    obs, reward, term, trunc, _ = env.last()\"]}]]}],[\"$\",\"div\",null,{\"children\":[[\"$\",\"h3\",null,{\"className\":\"text-base font-semibold text-gray-900 mb-2\",\"children\":\"Custom Environment\"}],[\"$\",\"p\",null,{\"className\":\"text-sm text-gray-500 mb-4\",\"children\":\"Define your own board geometry and win condition beyond the preset suite.\"}],[\"$\",\"pre\",null,{\"className\":\"bg-gray-900 text-gray-100 rounded-lg px-5 py-4 text-sm font-mono overflow-x-auto leading-relaxed\",\"children\":[\"$L17\",\" expanded_othello \",\"$L18\",\" make_env\\n\\n\",\"env = make_env(\\n\",\"    board_size=(\",\"$L19\",\", \",\"$L1a\",\"),\\n\",\"    obstacles=[(\",\"$L1b\",\", \",\"$L1c\",\"), (\",\"$L1d\",\", \",\"$L1e\",\")],\\n\",\"    win_cond=\",\"$L1f\",\",\\n\",\"    n_turns=\",\"$L20\",\",\\n\",\")\"]}]]}]]}]}]\n"])</script>
20<script>self.__next_f.push([1,"a:[\"$\",\"section\",null,{\"id\":\"results\",\"className\":\"py-20 px-6 border-b border-gray-100 bg-gray-50\",\"children\":[\"$\",\"div\",null,{\"className\":\"max-w-5xl mx-auto\",\"children\":[[\"$\",\"h2\",null,{\"className\":\"text-2xl font-bold mb-1 text-gray-900\",\"children\":\"Possible Approaches\"}],[\"$\",\"p\",null,{\"className\":\"text-gray-500 text-sm mb-12\",\"children\":\"What works, what doesn't, and where the open problems are\"}],[\"$\",\"div\",null,{\"className\":\"space-y-6 text-gray-700 leading-8 text-[1.05rem] mb-10\",\"children\":[[\"$\",\"p\",null,{\"children\":[\"To demonstrate the benchmark's utility, we introduce a neuroevolutionary Adaptive Minimax as a foundational baseline. It combines two components: a population of positional value networks meta-evolved across diverse board layouts (\",[\"$\",\"strong\",null,{\"children\":\"PosNet\"}],\"), providing spatial priors that transfer to unseen geometries; and an adaptive utility-weighting mechanism that evolves within each environment to infer the latent winning condition (\",[\"$\",\"strong\",null,{\"children\":\"WeightNet\"}],\"). Together, they allow the agent to adapt both spatially and strategically from terminal outcomes alone.\"]}],[\"$\",\"p\",null,{\"children\":\"In broad regimes — standard, inverse, and blitz conditions — the Adaptive Minimax converges rapidly, often reaching high win rates within the first generation of updates. Notably, it achieves win rates comparable to MCTS-100 despite MCTS having direct simulator access to the latent threshold K during search. The contrast with PPO is stark: trained under the same 2,000-game budget, PPO's win rate curves plateau early and remain largely flat — sparse terminal rewards provide insufficient gradient signal for meaningful adaptation within the budget, even when training is extended to 10,000 games.\"}]]}],[\"$\",\"figure\",null,{\"className\":\"mb-10\",\"children\":[[\"$\",\"img\",null,{\"src\":\"/MiniMaxPPO.png\",\"alt\":\"Convergence speed of Adaptive Minimax and PPO\",\"className\":\"w-full rounded-lg border border-gray-200\"}],[\"$\",\"figcaption\",null,{\"className\":\"mt-3 text-xs text-gray-400 leading-relaxed\",\"children\":\"Convergence speed of Adaptive Minimax (left) and PPO (right) under increasing training budgets, measured by win rate against Random (top) and MCTS-100 (bottom). Each curve is averaged over the 7 environments sharing the same victory condition C. For the Minimax, each generation requires 100 games.\"}]]}],[\"$\",\"div\",null,{\"className\":\"space-y-6 text-gray-700 leading-8 text-[1.05rem] mb-14\",\"children\":[\"$\",\"p\",null,{\"children\":\"Where the Adaptive Minimax falls short is in narrow-interval regimes: at K = 0.6 and K = 0.4, win rates collapse and outcomes are dominated by draws. The agent learns to avoid losing, but cannot control its terminal disc ratio with the precision required to enter the admissible interval consistently. These regimes remain an open challenge for all tested approaches.\"}]}],[\"$\",\"div\",null,{\"className\":\"flex items-center gap-4 mb-4 text-xs text-gray-500\",\"children\":[[\"$\",\"span\",null,{\"className\":\"flex items-center gap-1.5\",\"children\":[[\"$\",\"span\",null,{\"className\":\"inline-block w-3 h-2 rounded-sm bg-emerald-500\"}],\" Win\"]}],[\"$\",\"span\",null,{\"className\":\"flex items-center gap-1.5\",\"children\":[[\"$\",\"span\",null,{\"className\":\"inline-block w-3 h-2 rounded-sm bg-gray-300\"}],\" Draw\"]}],[\"$\",\"span\",null,{\"className\":\"flex items-center gap-1.5\",\"children\":[[\"$\",\"span\",null,{\"className\":\"inline-block w-3 h-2 rounded-sm bg-red-400\"}],\" Lose\"]}]]}],[\"$\",\"h3\",null,{\"className\":\"text-lg font-semibold text-gray-900 mb-1\",\"children\":\"Adaptive Minimax\"}],[\"$\",\"p\",null,{\"className\":\"text-gray-500 text-sm mb-6\",\"children\":\"averaged over 7 layouts per condition · trained with 2,000 games of self-play per environment\"}],\"$L21\",\"$L22\",\"$L23\",\"$L24\",\"$L25\",\"$L26\",\"$L27\"]}]}]\n"])</script>
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