Causal Conceptions of Fairness and their Consequences

Hamed Nilforoshan · Johann Gaebler · Ravi Shroff · Sharad Goel

**Concept Graph & Resume using Claude 3.5 Sonnet | Chat GPT4o | Llama 3:**

graph LR
classDef fairness fill:#d4f9d4, font-weight:bold, font-size:14px
classDef utility fill:#f9d4d4, font-weight:bold, font-size:14px
classDef counterfactual fill:#d4d4f9, font-weight:bold, font-size:14px
classDef mathematical fill:#f9f9d4, font-weight:bold, font-size:14px
A[Causal Conceptions of

Fairness and their

Consequences] --> B[Causal fairness:

fairness via

causal reasoning. 1] A --> C[Counterfactual equalized

odds:

equal rates in

counterfactual scenarios. 2] A --> D[Conditional principal

fairness:

independence given

outcomes, covariates. 3] A --> E[Path-specific fairness:

limits effects

along causal

paths. 4] A --> F[Counterfactual predictive

parity:

equal success rates

if accepted. 5] A --> G[Utility-maximizing policies:

maximize utility

within constraints. 6] B --> H[Pareto dominance:

preferred by

all decision

makers. 7] H --> I[Prevalence:

generalizes sets

to infinite

dimensions. 8] I --> J[U-fine distributions:

utilities with

a density. 9] J --> K[Threshold policies:

accept above

a utility

threshold. 10] H --> L[Multiple threshold

policies:

group-specific utility

thresholds. 11] C --> M[Budget-exhausting policies:

use full

available budget. 12] M --> N[Consistent utilities:

agree on sign,

ordering within

groups. 13] N --> O[Overlapping utilities:

positive decisions

in each

stratum. 14] O --> P[Splitting utilities:

all positive or

negative decisions. 15] D --> Q[Counterfactuals:

potential outcomes

in hypothetical

scenarios. 16] Q --> R[Path-specific counterfactuals:

intervening along

certain paths. 17] R --> S[Structural equations:

define causal

relationships

mathematically. 18] A --> T[Causal DAG:

graph of

causal

relationships. 19] T --> U[Total variation

norm:

metric for

comparing measures. 20] U --> V[Pushforward measure:

measure induced

by function

application. 21] V --> W[Radon-Nikodym derivative:

density of one

measure with

another. 22] A --> X[Absolute continuity:

zero measure

assignment

preservation. 23] X --> Y[Universal measurability:

measurable sets

under all

Borel measures. 24] Y --> Z[Shy sets:

measure zero

sets in

infinite dimensions. 25] Z --> AA[Probe:

subspace to

detect shy

sets. 26] A --> AB[Markov chains:

systems transitioning

between

states. 27] AB --> AC[Recurrent classes:

states where

systems get

trapped. 28] AC --> AD[Beta distribution:

probability distribution

on 0,1. 29] AD --> AE[Infra-marginality:

persistent group

differences despite

equality. 30] class B,C,D,E,F fairness class G,H,I,J,K,L utility class M,N,O,P,Q,R counterfactual class S,T,U,V,W,X,Y,Z,AA,AB,AC,AD,AE mathematical

Fairness and their

Consequences] --> B[Causal fairness:

fairness via

causal reasoning. 1] A --> C[Counterfactual equalized

odds:

equal rates in

counterfactual scenarios. 2] A --> D[Conditional principal

fairness:

independence given

outcomes, covariates. 3] A --> E[Path-specific fairness:

limits effects

along causal

paths. 4] A --> F[Counterfactual predictive

parity:

equal success rates

if accepted. 5] A --> G[Utility-maximizing policies:

maximize utility

within constraints. 6] B --> H[Pareto dominance:

preferred by

all decision

makers. 7] H --> I[Prevalence:

generalizes sets

to infinite

dimensions. 8] I --> J[U-fine distributions:

utilities with

a density. 9] J --> K[Threshold policies:

accept above

a utility

threshold. 10] H --> L[Multiple threshold

policies:

group-specific utility

thresholds. 11] C --> M[Budget-exhausting policies:

use full

available budget. 12] M --> N[Consistent utilities:

agree on sign,

ordering within

groups. 13] N --> O[Overlapping utilities:

positive decisions

in each

stratum. 14] O --> P[Splitting utilities:

all positive or

negative decisions. 15] D --> Q[Counterfactuals:

potential outcomes

in hypothetical

scenarios. 16] Q --> R[Path-specific counterfactuals:

intervening along

certain paths. 17] R --> S[Structural equations:

define causal

relationships

mathematically. 18] A --> T[Causal DAG:

graph of

causal

relationships. 19] T --> U[Total variation

norm:

metric for

comparing measures. 20] U --> V[Pushforward measure:

measure induced

by function

application. 21] V --> W[Radon-Nikodym derivative:

density of one

measure with

another. 22] A --> X[Absolute continuity:

zero measure

assignment

preservation. 23] X --> Y[Universal measurability:

measurable sets

under all

Borel measures. 24] Y --> Z[Shy sets:

measure zero

sets in

infinite dimensions. 25] Z --> AA[Probe:

subspace to

detect shy

sets. 26] A --> AB[Markov chains:

systems transitioning

between

states. 27] AB --> AC[Recurrent classes:

states where

systems get

trapped. 28] AC --> AD[Beta distribution:

probability distribution

on 0,1. 29] AD --> AE[Infra-marginality:

persistent group

differences despite

equality. 30] class B,C,D,E,F fairness class G,H,I,J,K,L utility class M,N,O,P,Q,R counterfactual class S,T,U,V,W,X,Y,Z,AA,AB,AC,AD,AE mathematical

**Resume: **

**1.-** Causal fairness: Algorithmic fairness definitions based on causal reasoning about the effects of protected attributes and decisions on outcomes.

**2.-** Counterfactual equalized odds: Requires equal true/false positive rates across groups in counterfactual scenarios where decisions are altered.

**3.-** Conditional principal fairness: Decisions should be independent of protected attributes given potential outcomes and reduced covariates.

**4.-** Path-specific fairness: Limits effects of protected attributes on decisions along certain causal paths while allowing effects along others.

**5.-** Counterfactual predictive parity: Among rejected applicants, proportion who would have succeeded if accepted should be equal across groups.

**6.-** Utility-maximizing policies: Decision policies that maximize a utility function while satisfying fairness constraints and budget limitations.

**7.-** Pareto dominance: A policy Pareto dominates another if it's preferred by all decision makers with a certain class of utility functions.

**8.-** Prevalence: Generalizes "full measure" sets to infinite-dimensional spaces, used to characterize generic properties of probability distributions.

**9.-** U-fine distributions: Probability distributions where utilities have a density, avoiding issues with atoms in the utility distribution.

**10.-** Threshold policies: Decision rules that accept individuals above a threshold value of utility.

**11.-** Multiple threshold policies: Decision rules with group-specific thresholds on utility.

**12.-** Budget-exhausting policies: Decision rules that use the full available budget (e.g. admit the maximum allowed number of students).

**13.-** Consistent utilities: A set of utility functions that agree on the sign and relative ordering of utilities within groups.

**14.-** Overlapping utilities: When a non-trivial threshold policy results in some positive decisions within each stratum of reduced covariates.

**15.-** Splitting utilities: When a threshold policy results in all positive or all negative decisions within each stratum of reduced covariates.

**16.-** Counterfactuals: Potential outcomes under hypothetical scenarios, like outcomes if decisions or protected attributes were different.

**17.-** Path-specific counterfactuals: Outcomes when intervening on protected attributes only along certain causal paths.

**18.-** Structural equations: Mathematical functions defining causal relationships between variables in a causal model.

**19.-** Causal DAG: Directed acyclic graph representing causal relationships between variables.

**20.-** Total variation norm: Metric for comparing probability measures, used to define convergence of measures.

**21.-** Pushforward measure: Measure induced on the output space by applying a function to a measure on the input space.

**22.-** Radon-Nikodym derivative: Density of one measure with respect to another, used to define densities of probability distributions.

**23.-** Absolute continuity: Property where one measure assigns zero to any set assigned zero by another measure.

**24.-** Universal measurability: Sets measurable under all finite Borel measures, used to define shy sets.

**25.-** Shy sets: Generalization of measure zero sets to infinite-dimensional spaces.

**26.-** Probe: Finite-dimensional subspace used to detect shy sets.

**27.-** Markov chains: Mathematical systems transitioning between states with probabilities depending only on the current state.

**28.-** Recurrent classes: Sets of states in a Markov chain that the system eventually gets trapped in.

**29.-** Beta distribution: Continuous probability distribution defined on the interval [0,1], used to model utilities in examples.

**30.-** Infra-marginality: Issue where group differences persist even after equalizing decision rates due to within-group heterogeneity.

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