Social Utility: Fehr-Schmidt Inequity Aversion

@cite{fehr-schmidt-1999} @cite{houlihan-kleiman-weiner-hewitt-tenenbaum-saxe-2023}

@cite{fehr-schmidt-1999} model agents who care about fairness, not just material payoff. An agent's utility depends on the difference between their own payoff and others' payoffs:

U_i = v_i − α · max(0, v_j − v_i) − β · max(0, v_i − v_j)

where α ≥ 0 captures disadvantageous inequity aversion (DIA — disliking getting less than others) and β captures advantageous inequity aversion (AIA — disliking getting more than others).

@cite{houlihan-kleiman-weiner-hewitt-tenenbaum-saxe-2023} use this as the base utility in their inverse planning model of emotion prediction. Observers infer agents' α and β weights from observed choices, then use those inferred social preferences to compute emotion appraisals.

Connection to RSA

In politeness models (@cite{yoon-etal-2020}), the "social utility" term is a primitive kindness weight φ. Fehr-Schmidt decomposes social utility into two structurally distinct components (AIA, DIA), each with its own behavioral signature. This decomposition predicts which emotions arise: DIA-weighted appraisals drive envy; AIA-weighted appraisals drive guilt.

Core Utility Function

def Core.fehrSchmidt (vSelf vOther α β : ℚ) : ℚ

Fehr-Schmidt inequity aversion utility.

U = v_self − α · max(0, v_other − v_self) − β · max(0, v_self − v_other)

• α (DIA): penalty for disadvantageous inequality (I got less)
• β (AIA): penalty for advantageous inequality (I got more)

Typically 0 ≤ β ≤ α: people dislike being behind more than being ahead.

Equations

• Core.fehrSchmidt vSelf vOther α β = vSelf - α * max 0 (vOther - vSelf) - β * max 0 (vSelf - vOther)

def Core.disadvantageousInequality (vSelf vOther : ℚ) : ℚ

Disadvantageous inequality: how much worse off I am than the other.

Equations

• Core.disadvantageousInequality vSelf vOther = max 0 (vOther - vSelf)

def Core.advantageousInequality (vSelf vOther : ℚ) : ℚ

Advantageous inequality: how much better off I am than the other.

Equations

• Core.advantageousInequality vSelf vOther = max 0 (vSelf - vOther)

theorem Core.fehrSchmidt_decompose (vSelf vOther α β : ℚ) : fehrSchmidt vSelf vOther α β = vSelf - α * disadvantageousInequality vSelf vOther - β * advantageousInequality vSelf vOther

Fehr-Schmidt decomposes into three additive terms.

Special Cases

theorem Core.fehrSchmidt_selfish (vSelf vOther : ℚ) : fehrSchmidt vSelf vOther 0 0 = vSelf

A purely selfish agent (α = β = 0) maximizes own payoff.

theorem Core.fehrSchmidt_equal (v α β : ℚ) : fehrSchmidt v v α β = v

Equal payoffs produce no inequity penalty.

theorem Core.di_zero_when_equal (v : ℚ) : disadvantageousInequality v v = 0

When payoffs are equal, DI = 0.

theorem Core.ai_zero_when_equal (v : ℚ) : advantageousInequality v v = 0

When payoffs are equal, AI = 0.

theorem Core.di_ai_complementary (vSelf vOther : ℚ) : disadvantageousInequality vSelf vOther = 0 ∨ advantageousInequality vSelf vOther = 0

DI and AI are complementary: exactly one is positive.

Monotonicity

theorem Core.fehrSchmidt_mono_α (vSelf vOther α₁ α₂ β : ℚ) (hα : α₁ ≤ α₂) : fehrSchmidt vSelf vOther α₂ β ≤ fehrSchmidt vSelf vOther α₁ β

Higher α increases DIA penalty (weakly).

theorem Core.fehrSchmidt_mono_β (vSelf vOther α β₁ β₂ : ℚ) (hβ : β₁ ≤ β₂) : fehrSchmidt vSelf vOther α β₂ ≤ fehrSchmidt vSelf vOther α β₁

Higher β increases AIA penalty (weakly).

Value Function

@cite{houlihan-kleiman-weiner-hewitt-tenenbaum-saxe-2023} apply a value function ν to raw monetary payoffs to capture diminishing marginal utility. For their purposes, ν is a sign-adjusted logarithm. We keep the utility function generic over any monotone transform.

def Core.fehrSchmidtV (ν : ℚ → ℚ) (vSelf vOther α β : ℚ) : ℚ

Composed Fehr-Schmidt: apply a value function to raw payoffs before computing inequity penalties.

Equations

• Core.fehrSchmidtV ν vSelf vOther α β = Core.fehrSchmidt (ν vSelf) (ν vOther) α β

theorem Core.fehrSchmidtV_id (vSelf vOther α β : ℚ) : fehrSchmidtV id vSelf vOther α β = fehrSchmidt vSelf vOther α β

When ν is the identity, fehrSchmidtV reduces to fehrSchmidt.