dsic_agent_one

formal source

 168
 169/-- **DSIC for agent 1.** Symmetric to agent 0. Note tie-breaking
 170goes to agent 0, so agent 1's wins occur strictly when `b₁ > b₀`. -/
 171theorem dsic_agent_one (v₁ b₀ : ℝ) (b₁' : ℝ) :
 172    utility₁ v₁ b₀ b₁' ≤ utility₁ v₁ b₀ v₁ := by
 173  unfold utility₁
 174  by_cases h_truth : b₀ ≥ v₁
 175  · -- Truthful: agent 1 loses (since b₀ ≥ v₁), utility = 0.
 176    rw [if_pos h_truth]
 177    by_cases h_dev : b₀ ≥ b₁'
 178    · -- Deviation also loses, utility = 0.
 179      rw [if_pos h_dev]
 180    · -- Deviation wins (b₁' > b₀), utility = v₁ - b₀.
 181      -- Since v₁ ≤ b₀, we get utility ≤ 0.
 182      rw [if_neg h_dev]
 183      linarith
 184  · -- Truthful: agent 1 wins (since v₁ > b₀), utility = v₁ - b₀ > 0.
 185    rw [if_neg h_truth]
 186    by_cases h_dev : b₀ ≥ b₁'
 187    · -- Deviation loses, utility = 0 ≤ v₁ - b₀.
 188      rw [if_pos h_dev]
 189      push_neg at h_truth
 190      linarith
 191    · -- Deviation also wins, utility = v₁ - b₀ (same).
 192      rw [if_neg h_dev]
 193
 194/-! ## §4. The pivot identity and σ-conservation -/
 195
 196/-- **PIVOT IDENTITY.** Under truthful bidding, the winner's payment
 197equals the loser's valuation: the externality the winner imposes on
 198the displaced agent. -/
 199theorem pivot_identity (v₀ v₁ : ℝ) :
 200    -- If agent 0 wins (v₀ ≥ v₁), they pay v₁.
 201    (v₀ ≥ v₁ → payment₀ v₀ v₁ = v₁) ∧