Dependencies

The game has \(24\) action spaces. Thirteen are available from round 1 (preprinted); the rest are revealed one per round. Each space can be occupied by at most one dwarf per round.

Sheep convert to \(2\) food, wild boar to \(3\), cattle to \(4\). Donkeys have a superlinear pairing bonus: \(1\) donkey gives \(1\) food, but pairs give \(3\) food (a \(+1\) bonus per pair).

The \(8\) archetypes are:

1. FurnRush (furnishing rush): maximize furnishing bonus points.

2. WeapRush (weapon rush): forge early, exploit expeditions.

3. PeaceFarm (peaceful farming): grain/vegetable engine.

4. MineHeavy (mining heavy): ore and ruby mines.

5. AnimHusb (animal husbandry): large pastures, breeding.

6. RubyEcon (ruby economy): ruby mining and conversion.

7. PeaceCave (peaceful cave engine): peaceful interior development.

8. Balanced (balanced): diversified portfolio.

\(\text{beerParlorGold}(g) = \lfloor g / 2 \rfloor \times 3\) for \(g\) grain.

\(\text{bestResponse}(s)\) is the strategy \(t\) that maximizes \(M_{\text{toFin}(t),\text{toFin}(s)}\), i.e. the best reply to an opponent playing \(s\).

The personal player board has \(24\) spaces total (Theorem 1.8).

The bonus points of many furnishings depend on game state. We define a \(\texttt{BonusContext}\) structure with \(14\) named fields: number of dwarfs, armed dwarfs, adjacent dwellings, dogs, sheep, etc. This replaces a 17-positional-argument function signature.

Each furnishing has a bonus function \(\text{furnishingBonusPoints} : \text{FurnishingId} \to \text{BonusContext} \to \mathbb {N}\) that computes conditional bonus points based on game state.

Two strategies conflict if they compete for the same scarce action spaces or resources. Same-strategy matchups always conflict.

The game has seven resource types (wood, stone, ore, gold, grain, vegetables, food), four animal types (dogs, sheep, wild boar, cattle, where dogs are non-farm animals), and two crop types (grain, vegetables).

Dogs act as sheepdogs: \(n\) dogs allow housing \(n+1\) sheep in living spaces (no pasture needed).

Strategy \(s\) dominates strategy \(t\) in score estimates if \(\text{maxScoreEstimate}(s) \ge \text{maxScoreEstimate}(t)\) and \(\text{minScoreEstimate}(s) \ge \text{minScoreEstimate}(t)\) with at least one strict inequality.

\(\text{feedingCost}(d, o) = 2d + o\) where \(d\) is the number of adult dwarfs and \(o\) the number of offspring.

A weapon is forged at strength \(s \in [1,8]\) by spending \(s\) ore. After each expedition, the weapon’s strength increases by \(1\), up to a cap of \(14\).

Each furnishing has: a name, category, resource costs (wood, stone, ore, gold, grain, vegetables, food), base victory points, and a bonus function. Categories are: dwelling, cooking, working, parlor, storage, and special.

Converting \(n\) gold to food yields \(\max (0, n-1)\) food. This is lossy: \(1\) gold is wasted as overhead.

In the 2-player game, harvests occur on a fixed schedule across 12 rounds. Round 3 always triggers a normal harvest (Theorem 7.4).

Each player starts with \(2\) unarmed dwarfs. Player 1 receives \(1\) food; Player 2 receives \(2\) food. All other resources and animals start at zero. The initial dwarf count is \(\text{initialDwarfCount} = 2\).

The number of loot options available depends on weapon strength. Higher strength unlocks more valuable loot items.

A pair \((a, b)\) is a Nash equilibrium if neither player can unilaterally improve: \(a = \text{bestResponse}(b)\) and \(b = \text{bestResponse}(a)\).

A small pasture holds \(2\) animals; a large pasture holds \(4\). Each stable doubles the capacity of its pasture. A large pasture with two stables holds \(16\) animals.

The payoff matrix \(M : \text{Fin}\, 8 \to \text{Fin}\, 8 \to \mathbb {N}\) assigns \(M_{i,j}\) as the expected score of archetype \(i\) when the opponent plays archetype \(j\). Entries are derived from score estimates adjusted for contention and synergy effects. The matrix is indexed via \(\text{toFin}\) and \(\text{ofFin}\) conversions between \(\text{StrategyArchetype}\) and \(\text{Fin}\, 8\).

Each begging marker costs \(3\) points. Each unused board space costs \(1\) point.

A player state tracks: the list of dwarfs, resource supplies (wood, stone, ore, gold, grain, vegetables, food, rubies), animal counts, board tile placements (mountain spaces, forest spaces, meadows, fields, pastures, mines), installed furnishing tiles, and accumulated begging markers.

\(\sigma (s) = \sum _{j=0}^{7} M_{\text{toFin}(s),j}\), the total payoff of strategy \(s\) summed across all opponent choices.

Each farm animal (sheep, wild boar, cattle) scores \(1\) point per animal. For each of the three farm animal types not represented at all, the player loses \(2\) points. Dogs score \(0\) directly but enable sheep capacity.

We define several reference points:

• \(\text{theoreticalMinScore} = -28\) (max penalties, zero positive).

• \(\text{doNothingScore} = -55\) (skip all actions for 12 rounds).

• \(\text{theoreticalMaxSum} = 202\) (sum of maxed individual categories).

• \(\text{practicalCeiling} = 140\) (attainable upper bound).

• \(\text{practicalFloor} = 60\) (competent-play lower bound).

Grain scores \(\lfloor (n+1)/2 \rfloor \) for \(n\) grain. Vegetables score \(1\) per vegetable.

For each archetype \(s\), we define:

• \(\text{maxScoreEstimate}(s)\): the ceiling score achievable under favorable matchups.

• \(\text{minScoreEstimate}(s)\): the floor score under unfavorable matchups.

These are derived from analysis of board access, contention, and tile synergies.

Small pastures score \(2\) each, large pastures \(4\) each. Ore mines score \(3\), ruby mines score \(4\). Each dwarf scores \(1\). Gold scores \(1\) per gold. Rubies score \(1\) per ruby.

\(\text{socialWelfare}(a,b) = M_{\text{toFin}(a),\text{toFin}(b)} + M_{\text{toFin}(b),\text{toFin}(a)}\). The Nash welfare is \(\text{socialWelfare}(\textsc{FurnRush}{},\textsc{FurnRush}{})\). The social optimum is the maximum social welfare over all pairs.

There are \(48\) unique furnishing tiles.

The total score is the sum of all positive scoring components minus all penalties, plus furnishing bonus points.

A weapon has integer strength in \([1,14]\). A dwarf optionally carries a weapon. Each player begins with \(2\) dwarfs (Definition 1.4).

\(\text{accumulatedValue}(r, n) = r \cdot n\) for accumulation rate \(r\) and \(n\) rounds of waiting.

With one growth at round 4, total placements are \(47\). This finite action budget prevents any exploit from running unbounded.

For every archetype \(s\), \(\text{minScoreEstimate}(s) {\gt} \text{doNothingScore}\).

\(\forall \, i\, j,\; M_{i,j} {\gt} 0\).

The maximum payoff asymmetry is \(70\): \(\forall \, i\, j,\; |M_{i,j} - M_{j,i}| \le 70\). Among non-\(\textsc{FurnRush}{}\) pairs, asymmetry is at most \(35\).

\(\text{beerParlorGold}(20) = 30\). Even with \(20\) grain (an extreme stockpile), the Beer Parlor produces at most \(30\) gold.

\(\text{beerParlorGold}(10) = 15\) and \(\text{beerParlorGold}(4) = 6\). In practice, a farming player might accumulate \(\sim 10\) grain, yielding \(15\) gold, not an unbounded engine.

For \(n {\gt} 0\), \(\text{penaltyBegging}(n) \ge 3\). In fact \(\text{penaltyBegging}(n) = 3n\) (linear).

\(\forall \, s,\; \text{bestResponse}(s) = \textsc{FurnRush}{}\). No matter what the opponent does, \(\textsc{FurnRush}{}\) is the best reply.

\(\text{blacksmithingParlorGold}(5, 5) = 10\). With \(5\) rubies and \(5\) ore, the Blacksmithing Parlor produces \(10\) gold (\(1\) ruby \(+ 1\) ore \(\to 2\) gold \(+ 1\) food per activation).

\(\text{totalBoardSpaces} = 24\).

Round 1 has \(13\) available action spaces with \(4\) dwarfs to place. Utilization is \(30\% \).

Over \(7\) harvest phases with breeding, total food output is \(35\).

\(\text{furnishingBonusPoints}(\text{broomChamber}, \{ \text{numDwarfs} := 5\} ) = 5\).

Cattle gives strictly more food than wild boar and sheep per animal.

Cattle’s minimum required strength exceeds the maximum initial weapon strength, meaning cattle can only be obtained through expeditions.

\(\text{practicalCeiling} {\lt} \text{theoreticalMaxSum}\), i.e., \(140 {\lt} 202\).

For every strategy \(i\), there exists an opponent \(j \ne i\) such that \(M_{i,j} {\gt} M_{i,i}\). Mirror matchups are never optimal.

\(\text{doNothingScore} = -55 {\lt} 0\).

\(\text{dogSheepCapacity}(1) = \text{smallPastureCapacity} = 2\). One dog matches a small pasture for sheep housing.

\(\text{dogSheepCapacity}(n) = n + 1\) and \(\text{dogSheepCapacity}(10) = 11\). Even \(10\) dogs only house \(11\) sheep, making this a weak scaling path compared to stabled pastures.

\(\text{dogSheepCapacity}(1) = 2\), \(\text{dogSheepCapacity}(3) = 4\), and the function is strictly monotone.

For \(n {\lt} 15\), \(\text{dogSheepCapacity}(n) {\lt} 16\). Dogs are inferior to a fully stabled large pasture for mass animal housing.

\(\text{donkeyFoodValue}(2) = 3\) and \(\text{donkeyFoodValue}(2) {\gt} 2 \cdot \text{donkeyFoodValue}(1)\), confirming the superlinear pairing bonus.

If \(s {\lt} 14\), an expedition raises strength to \(s+1\). At \(s = 14\), strength stays at \(14\).

Starting from strength \(8\), it takes \(6\) expeditions to reach \(14\). Starting from strength \(1\), it takes \(13\).

\(\text{feedingCost}(2,0) - \text{startingFoodP1} = 3\). Every player faces a 3-food deficit at the first harvest.

\(\text{feedingCost}(5, 0) = 10\) and \(\text{feedingCost}(5, 1) = 11\).

\(\text{furnishingBonusPoints}(\text{foodChamber}, \{ \text{grainCount} := 4, \text{vegCount} := 4\} ) = 8\).

Player 1’s starting food (\(1\)) is strictly less than the first harvest feeding cost (\(4\)): \(1 {\lt} \text{feedingCost}(2, 0)\).

\(\text{numGoodFoodSpaces} = 2\) and \(\text{initialDwarfCount} \ge \text{numGoodFoodSpaces}\). The first player to act claims the best food space, giving them a structural advantage.

The feeding cost at the initial dwarf count exceeds the starting food for both players. This forces every archetype to allocate early actions to food acquisition.

For all strategies \(s\), \(\text{maxScoreEstimate}(\textsc{FurnRush}{}) \ge \text{maxScoreEstimate}(s)\).

\(M_{0,0} = 85\), \(M_{0,1} = 130\), \(M_{0,2} = 135\). The \(\textsc{FurnRush}{}\) mirror (\(85\)) is the worst \(\textsc{FurnRush}{}\) outcome, showing the contention cost of both players racing for the same furnishings.

\(\text{dominates}(\textsc{FurnRush}{}, \textsc{PeaceCave}{})\).

\(\textsc{FurnRush}{}\) simultaneously dominates \(\textsc{PeaceFarm}{}\), \(\textsc{RubyEcon}{}\), \(\textsc{PeaceCave}{}\), and \(\textsc{WeapRush}{}\) in score estimates.

\(\text{dominates}(\textsc{FurnRush}{}, \textsc{PeaceFarm}{})\).

\(\text{dominates}(\textsc{FurnRush}{}, \textsc{RubyEcon}{})\).

\(\text{dominates}(\textsc{FurnRush}{}, \textsc{WeapRush}{})\).

\(\forall \, s,\; \sigma (\textsc{FurnRush}{}) \ge \sigma (s)\).

For every \(s \ne \textsc{FurnRush}{}\), either \(\text{maxScoreEstimate}(\textsc{FurnRush}{}) {\gt} \text{maxScoreEstimate}(s)\) or \(\text{minScoreEstimate}(\textsc{FurnRush}{}) {\gt} \text{minScoreEstimate}(s)\).

\(\text{goldToFood}(2) = 1\) and \(\text{goldToFood}(1) = 0\).

If \(a \le b\) then \(\text{scoreGrain}(a) \le \text{scoreGrain}(b)\).

Without growth: \(44\) total dwarf placements. With one growth at round 4: \(47\) placements. With both growths: \(56\) placements.

“Wish for Children” appears at round 4; “Family Life” at round 8. Early growth is available \(4\) rounds before the late option.

Not all strategy pairs are comparable: \(\neg \, \text{dominates}(\textsc{MineHeavy}{}, \textsc{AnimHusb}{})\) and \(\neg \, \text{dominates}(\textsc{AnimHusb}{}, \textsc{MineHeavy}{})\).

The full interaction space is \(8 \times 8 \times 2880 = 184{,}320\) strategy-setup combinations.

The “furnish cavern” loot option is worth \(\sim 4\) points. At maximum strength \(14\), total available loot value is \(\sim 13\) points. Expeditions are valuable but capped.

\(\text{availableLootCount}(1) \le \text{availableLootCount}(8)\).

At strength \(1\): \(3\) loot options. At strength \(8\): \(13\) options. At strength \(14\): \(18\) options.

A weapon can be forged at strength \(8\) (costing \(8\) ore). No weapon can be forged above strength \(8\).

For any animal counts \(a\), \(\text{penaltyMissingAnimals}(a) \le 8\). (At most 4 types missing at 2 points each.)

\(\forall \, i\, j,\; M_{i,j} \le 135\), and \(M_{0,2} = 135\). The best outcome is \(\textsc{FurnRush}{}\) against \(\textsc{PeaceFarm}{}\).

\(\forall \, i\, j,\; M_{i,j} \ge 55\), and \(M_{6,6} = 55\). The worst outcome is a \(\textsc{PeaceCave}{}\) mirror match.

A Miner produces \(10\) ore over \(5\) activations (\(2\) per activation). Ore trading yields at most \(6\) per use (\(3 \times 2\)).

\(\text{isNashEquilibrium}(\textsc{FurnRush}{}, \textsc{FurnRush}{})\).

\((\textsc{FurnRush}{}, \textsc{FurnRush}{})\) is the unique pure Nash equilibrium. If \((a, b)\) is any Nash equilibrium, then \(a = b = \textsc{FurnRush}{}\).

\(\text{nashWelfare} = 170\) (both players score \(85\) in the mirror).

If a player has at least one of each farm animal type, \(\text{penaltyMissingAnimals}(a) = 0\).

Logging yields \(9\) wood after \(3\) rounds vs. \(3\) after \(1\) round: a \(3\times \) payoff for waiting.

Prayer Chamber gives \(8\) bonus points when no dwarfs have weapons, but \(0\) when any dwarf is armed. This creates a hard choice between the peaceful and weapon-based strategies.

\(\text{availableLootCount}(14) - \text{availableLootCount}(8) = 5\). The last \(5\) loot items require strength above \(8\).

\(\text{socialOptimumValue} {\gt} \text{nashWelfare}\): i.e., \(210 {\gt} 170\). The price of anarchy ratio is \(21/17 \approx 1.24\). Rational play sacrifices \(\sim 19\% \) of social welfare.

In the 2-player game, round 3 always triggers a normal harvest.

The row sums satisfy the strict ordering:

\(\textsc{FurnRush}{}\) has the highest aggregate payoff; \(\textsc{RubyEcon}{}\) the lowest.

\(\text{scoreMines}(0,1) {\gt} \text{scoreMines}(1,0)\), i.e., a single ruby mine scores strictly more than a single ore mine.

Rubies can be converted to at least \(2\) food each, serving as an emergency food source.

Converting rubies to gold for State Parlor: net yield is \(12 - 2 = 10\) points after accounting for conversion overhead.

Mining \(8\) rubies over a game yields \(4\) ruby mine activations (at \(2\) rubies each). Those \(8\) rubies can convert to \(16\) food as emergency rations.

\(\text{practicalFloor} - \text{doNothingScore} = 115\). Even in the worst case, \(\textsc{FurnRush}{}\) scores \(115\) points above the do-nothing catastrophe.

Mirror matchups (same archetype vs. same archetype) always conflict for \(\textsc{WeapRush}{}\), \(\textsc{MineHeavy}{}\), \(\textsc{RubyEcon}{}\), and \(\textsc{PeaceCave}{}\).

\(\text{practicalCeiling} - \text{doNothingScore} = 195\).

The 2-player game has \(2880\) distinct initial setups: \(144\) card orderings times \(20\) harvest marker placements.

Arming \(3\) dwarfs at strength \(4\) each costs \(3 \times 4 = 12\) ore. The ore investment caps the weapon snowball effect.

\(\text{socialOptimumValue} = 210\), achieved by pairing strategies that avoid contention.

A stable doubles small pasture capacity (\(2 \to 4\)) and large pasture capacity (\(4 \to 8\)).

\(\text{furnishingBonusPoints}(\text{stateParlor}, \{ \text{numAdjacentDwellings} := 4\} ) = 16\).

\(\text{numStrategies} = 8\).

For every \(s \ne \textsc{FurnRush}{}\), there exists an opponent \(j\) where \(M_{0,j} {\gt} M_{\text{toFin}(s),j}\). The dominance is therefore weak (not just tied everywhere).

\(\text{theoreticalMinScore} = -28\).

The sum of all row sums is \(5595\). The average payoff per cell is \(5595 / 64 = 87\).

The theoretical maximum from combining Beer Parlor, Blacksmithing Parlor, and Hunting Parlor is \(30 + 10 + 20 = 60\) gold. This is a ceiling, not an achievable value, because the required input resources (grain, rubies, ore, boar) compete for action tempo.

\(\text{feedingCost}(2, 0) = 4\).

A large pasture with two stables holds \(4 \times 4 = 16\) animals, the maximum single-space animal capacity.

Both players face a food deficit at the first harvest: \(\text{feedingCost}(2,0) - \text{startingFoodP1} = 3\) and \(\text{feedingCost}(2,0) - \text{startingFoodP2} = 2\). This structural constraint forces every viable strategy to solve the food problem within its first \(3\) actions.

\(\text{practicalCeiling} - \text{practicalFloor} = 80\). The score range of \(\textsc{FurnRush}{}\) spans \(80\) points depending on matchup.

Vegetables give \(2\times \) the food of grain (2 vs. 1).

For all opponents \(j\) and all alternative strategies \(s\),

That is, \(\textsc{FurnRush}{}\) (row \(0\)) weakly dominates every other row.

\(\neg \, \text{dominates}(\textsc{WeapRush}{}, \textsc{FurnRush}{})\).

For all strategies \(s\), \(\text{minScoreEstimate}(\textsc{WeapRush}{}) \ge \text{minScoreEstimate}(s)\).

Weapon Storage gives \(+3\) per armed dwarf: theoretical max is \(15\) (all \(5\) dwarfs armed). Realistically, \(3\) armed dwarfs yield \(9\).

\(\text{furnishingBonusPoints}(\text{weaponStorage}, \{ \text{numArmedDwarfs} := 5\} ) = 15\). (Weapon Storage gives \(+3\) per armed dwarf.)

\(\text{penaltyUnusedSpaces}(24) = 24\).

The Writing Chamber prevents up to \(7\) gold points of negative scoring, regardless of how many penalties exist. \(\text{writingChamberReduction}(24) = 7\), \(\text{writingChamberReduction}(57) = 7\), \(\text{writingChamberReduction}(8) = 7\).