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# System prepended metadata

title: 'Theory, Heuristics and Fairness Directions'

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## Fairness

About how we can relate the project to a social justice or fairness setting: how about a fair future workload and fair lookahead into future tasks that affect income directly. E.g., if the employer suddenly changes the workload or reduces shift hours then the stochastic employee will be left with less utility since they face sudden unexpected financial shock. But if they are given a fair advance warning with a lookahead then they'd know what events are coming up and then they can have better utility in the face of the financial crisis. This also relates to fair workload balance satisfaction.

- About why we need advance notice & the the laws demanding such a thing to actually happen: https://www.epi.org/publication/irregular-work-scheduling-and-its-consequences/

- All kinds of unfairness in *advance notice* in data: https://www.ilo.org/wcmsp5/groups/public/---ed_protect/---protrav/---travail/documents/publication/wcms_619044.pdf  (e.g., pages 14, 15, 16, 17, 18  aka 23, 24, 25, 26 of pdf are very interesting).

- Again shows unfairness in advance notice in data with consequences: https://shift.hks.harvard.edu/its-about-time-how-work-schedule-instability-matters-for-workers-families-and-racial-inequality/


**From the EPI article**: "Adopting recommendations from the Retail Workers Bill of Rights, the San Francisco Board of Supervisors has enacted new protections for hourly workers in retail chain stores, to require employers to provide more advance notice in setting and changing work schedules to make them more predictable. The protections include providing priority access to extra hours of work—if and when available—to those employees who explicitly request such hours, which could be a model way to help alleviate the chronic underemployment in the U.S. labor market."


"The experiences in cities could inform elements of reforms of the Fair Labor Standards Act (FLSA) at the national level, such as the proposed The Schedules that Work Act (H.R. 5159)."


**From the ILO inequality in advance notice article**: Variability in total hours matters a great deal if workers’ total income varies as a result. The Federal Reserve’s Survey of Household Economics and Decision-Making asked whether a respondent, along with any partner or spouse, had income that varied from month to month in the past year. Among employees and consultants/contractors ages 18-65, 8.4% said that their income often varied quite a bit from one month to the next, and another 25.0% said that there were some unusually high or low months although their income was otherwise roughly the same in most months. Of these two groups with variable income, 51% said that an irregular work schedule contributed to their income instability. The proportion who cited an irregular work schedule greatly exceeded the proportion who mentioned seasonal employment and periods of unemployment (as well as bonuses, commissions, investment income, and other).


Lambert, Fugiel and Henly (2014) also report the length of advance notice for the same group of young hourly workers in 2011, comparing them to their nonhourly counterparts. (Table 2 and Figure 9) While unpredictability was widely distributed among all categories of workers represented in the table, workers paid by the hour, and part-time workers, were more likely to have one week of notice or less. Those paid by salary or something else were more likely to get at least a month of notice, as were full-time workers (I will discuss the distributions by gender, race and ethnicity shortly).




**Note:** in that ILO article there're tons of other data statictics on unfair advance notice for different education, racial, job sector, age groups, financial sectors, industries, etc.




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-- Some ineteresting articles on relevant topics:



Algorithmic-governed works causing burn out: https://bmcpsychology.biomedcentral.com/articles/10.1186/s40359-023-01402-0



Routine work schedule instability and precarity (no AI element in this article): https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7730535/


Algorithms, precarity, inequality in advance notices, and job schedule instability: https://shift.hks.harvard.edu/its-about-time-how-work-schedule-instability-matters-for-workers-families-and-racial-inequality/


AI-optimized workflows are not always the best: https://venturebeat.com/ai/why-ai-optimized-workflows-arent-always-best-for-business/

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**Fairness scenarios we could explore**


1- Simple stochatic vs deterministic agent (no AI): We just say that a stochatic agent in the workforce getting no advance notice compared to a deteminitic with e.g., 5 events advance notice, is unfair since the second agent yields a higher utility.

2- Simple stochatic vs deterministic but with linking to the inequility in advance notice data mentioned above (no AI): that is, we say the stochastic agent which is usually treated unfairly in terms of advance notice is usually from one of these marginalized worker or industries according to staticitics in the ILO and  'harvard . edu' article for example.


3- AI in workforce management and scheduling: *with the same lookahead* an AI shift scheduling tool might try to fairly assign two agents a sequence of jobs. It assigns the first person {A, B, C} and the second person {B, C, A} while these may seem fair statistically and by individual fairness measures, they are not fair since one agent yields a different utility from the other.


4- AI repeating past biases in workload scheduling by using past data in which marginalized people were not given proper advance notice. (could not find an article on this)

5-AI trying to maximize the employer utility by making the schedule as efficient as possible but being as impersonal as possible at the same time as well and not giving people advance notice of work reductions, etc. (could not find an article on this).

5-AI replacing humans which results in work reduction due to automation: This has tons of articles but this is a gradual process and although it directly impacts income since people become unemployed I just can't properly link it to the advance notice concept.

* Might be relevant
    * 

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**Concerns and Questions**:
- Where exactly AI is coming into the picure? I am still not clear. All the proposed scenarios seem vague. Do we really need this for a FAccT submission? Because I feel forcefully adding the AI element is ruining a complete and nice storyline.
- Who are we addressing this unfairness to? The policy-maker? The employer? AI developer?
    - To advocates
- Cannot really do group fairness.
- IFP with return rate on assets is more like how a household would consume. Its conceptually more inclined that way. 
- Should we even do this story spinning and link this to a workplace use case? Or should we just do more theory with simple online consumption lookahead we had and add theory to that for FORC?



## Lookahead utility comparison

Let $D_R,D_Y,x_0$ be the return distribution, income distribution and initial assets. Let $\tau$ be the lookahead.

Let $R_t$ be the returns coming from a distribution $D_R$ where each element is samples from the interval $[a,b]$ for some parameters $b,a > 1$, let $\pi_{t,0}$ be the consumption function at time point $t$ with lookahead $0$, and let $\pi_{t,1}$ be the consumption function at time point $t$ with lookahead $1$, could we claim that at the time of death $T$,
$\Delta_T = \sum_{t=1}^T f_t(\pi_{t,1}(x_{t,1}))-f_t(\pi_{t,0}(x_{t,0})) \ge \epsilon(x_0,D_R,D_Y,\dots)$ 
for some $\epsilon$ (that depends on initial income and other parameters). Here $f_t(\pi_{t,i}(x_{t,i}))=\sum_{j=t}^T \beta^{(j-t)}u(\pi_{t,i}(x_{j-1,i})) \; \text{for} \; i\in \{0,1\}$

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**Questions**:
- Is $f_t(\pi_{t,i}(x_{t,i}))=\sum_{j=t}^T \beta^{(j-t)}u(\pi_{t,i}(x_{j-1,i})) \; \text{for} \; i\in \{0,1\}$ the function we should consider or should we look at $f_t(\pi_{t,i}(x_{t,i}))=\sum_{j=1}^t \beta^{(j-1)}u(\pi_{j,i}(x_{{j-1,i}})) \; \text{for} \; i\in \{0,1\}$
- Given that the difference between lookahead 0 and 1 is small in the simulations, is there any hope in this kind of an approach? 
- Would the viable approach be to argue the gap would be to go for an upper bound instead of a lower bound?
- Places to look at to get an intuition: RL algorithms with finite or infinite horizons?
    - [Addressing infinite-horizon optimization in MPC via Q-learning](https://www.sciencedirect.com/science/article/pii/S2405896318326478)
    - [Efficient model-based reinforcement learning for approximate online optimal control](https://www.sciencedirect.com/science/article/pii/S0005109816303272)
    - [Finite Horizon Q-learning: Stability, Convergence, Simulations and an application on Smart Grids](https://arxiv.org/abs/2110.15093)
    - The problem is the idea of horizons seem slightly different in the two cases
- Stochastic vs Deterministic
    - [Deterministic Approximation for Stochastic Control Problems](https://epubs.siam.org/doi/10.1137/S0363012993254540)
    - [Deterministic Approximations to Stochastic Production Problems](https://www.jstor.org/stable/170650)




#### Start

Let $R_t=1$ for all $t$ and $Y_t \in \{Y-\delta, Y+\delta\}$ for known $Y,\delta$. Let $\tilde{V}_T(X)$ be the total utility gained from consuming $X$ in $T$ time points in a stochastic manner. 

Now consider the case of Lookahead 0. Let $f_0(X,T)$ be the total utility with lookahead $0$. Then, we can see that this gives us,

$\begin{align}f_0(X,T) = \max_{c_1,c_2} u(c_1)+\beta u(c_2)+\beta^2 \left[\frac{1}{4}\tilde{V}_{T-2}(S+2\delta)+\frac{1}{4}\tilde{V}_{T-2}(S)+\frac{1}{4}\tilde{V}_{T-2}(S-2\delta)+\frac{1}{4}\tilde{V}_{T-2}(S)\right]\end{align}$

where $S=X-c_1-c_2+2Y$

Similarly, let $f_1(X,T)$ be the total utility with lookahead $1$. Assuming our income at the next time point was $Y+\delta$, we can see that this gives us,

$\begin{align}f_1(X,T) = \max_{c_1,c_2} u(c_1)+\beta u(c_2)+\beta^2 \left[\frac{1}{2}\tilde{V}_{T-2}(S+2\delta)+\frac{1}{2}\tilde{V}_{T-2}(S)\right]\end{align}$

Then we can see that, letting $c_1,c_2$ be the solution to $f_0(X,T)$,

$f_1(X,T)-f_0(X,T) \ge \frac{\beta^2}{4} \left[\tilde{V}_{T-2}(S+2\delta)-\tilde{V}_{T-2}(S-2\delta)\right]$

Assume $\tilde{V}_{T-2}(S-2\delta) = u(c)+\beta E_\tilde{Y}\left[\tilde{V}_{T-3}(S-2\delta-c+\tilde{Y})\right]$

Then, $\tilde{V}_{T-2}(S+2\delta) \ge u(c+4\delta)+\beta E_\tilde{Y}\left[\tilde{V}_{T-3}(S-2\delta-c+\tilde{Y})\right]$

Therefore, 
$f_1(X,T)-f_0(X,T) \ge \frac{\beta^2}{4} \left[u(c+4\delta)-u(c)\right]$

Assuming $u(c) = \sqrt{c}$, $u(c+4\delta)-u(c) = \sqrt{c+4\delta}-\sqrt{c} = \frac{4\delta}{\sqrt{c+4\delta}+\sqrt{c}} \ge \frac{4\delta}{(1+\sqrt{2})\sqrt{r}}$ where $r = \max{(c,4\delta)}$. With an appropriate choice of $X$ and $\delta$, we can get $r = 4\delta$ and therefore, $u(c+4\delta)-u(c) \ge \frac{2\sqrt{\delta}}{(1+\sqrt{2})} = 2(\sqrt{2}-1)\sqrt{\delta}$

**Extending to $f_k(X,T)$**

Similar to the case of $f_1,f_0$, we can define the following,

$\begin{align}f_k(X,T) = \max_{c_1,c_2,\dots,c_{k+1}} \sum_{i=0}^k \beta^i u(c_{i+1}) +\beta^{k+1} \left[\frac{1}{2}\tilde{V}_{T-k}(S+2\delta)+\frac{1}{2}\tilde{V}_{T-k}(S)\right]\end{align}$

and

$\begin{align}f_{k-1}(X,T) = \max_{c_1,c_2,\dots,c_{k+1}} \sum_{i=0}^k \beta^i u(c_{i+1}) +\beta^{k+1} \left[\frac{1}{4}\tilde{V}_{T-k}(S+2\delta)+\frac{1}{4}\tilde{V}_{T-k}(S)+\frac{1}{4}\tilde{V}_{T-k}(S-2\delta)+\frac{1}{4}\tilde{V}_{T-k}(S)\right]\end{align}$

where

$S = X-\sum_{i=1}^{k+1}c_i+\sum_{i=1}^{k-1} Y_i+2Y$

We can see that therefore assuming $c$ is the optimal consumption for $\tilde{V}_{T-k}(S-2\delta)$, we get,

$f_k(X,T) - f_{k-1}(X,T) \ge \frac{\beta^{k+1}}{4} \left[u(c+4\delta)-u(c)\right]$

Assuming $c \le 4\delta$, and $u(c) = \sqrt{c}$, we get,

$f_k(X,T) - f_{k-1}(X,T) \ge \frac{\beta^{k+1}}{2}(\sqrt{2}-1)\sqrt{\delta}$

## Idea of Heuristics


Could we come up with a heuristic to come up with the optimal consumption. Let,

$h_{D_R,D_Y,x_0,\tau,t}(\pi)=(1+\epsilon)LA_{D_R,D_Y,x_0,\tau,t}(\pi)EP_{D_R,D_Y,x_0,\tau,t}(\pi)$

where $\pi$ is a consumption function and
$LA_{D_R,D_Y,x_0,\tau,t}(\pi) = \sum_{i=1}^\tau \beta^{i-1} u(\pi(x_{t+i-1})) \; \text{such that} \; x_{t+i} = R_{t+i}(x_{t+i-1}-c_{t+i})+y_{t+i}$ (look-ahead)

and $EP_{D_R,D_Y,x_0,\tau,t}(\pi) = x_{\tau+t+1}$ (end point)

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**Questions**:
- What other parameters or functions would we need in heuristics?
- How should we weight specific functions? Ending up with a lot of assets might not be good if you do not get a lot of overall utlitity by doing so. Since we have diminishing tails with discounting, after a point, even if we had a lot of assets. catching up might be harder.

## Concept Drifts when Distributions Change

Given we are working on the online model, can we claim that at time point $t$, if we change distributions $D_R,D_Y$ to $D_R',D_Y'$, then, $|\sum_{i=t}^T \beta^{t-1} u(\pi_t(x_{i-1})-\sum_{i=t}^T \beta^{t-1} u(\pi_t'(x_{i-1}')| \le f(KL(D_R'||D_R),KL(D_Y'||D_Y))$ for a well-defined function $f$.

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**Questions**:
- Could we have a distribution with smaller KL divergence that can result in divergence assuming $T$ is large enough or since we have discounting, would we get a diminishing tail that would not affect things too much? (Could get some insight through simulations)
- Is this adding any value? How would we add a notion of lookahead to this drift? 
- If we had lookahead, could we claim that the divergence in the no information setting can be converted to a convergence result for small KL divergence settings?
- [Robust Utility Maximization in a Multivariate Financial Market with Stochastic Drift](https://arxiv.org/abs/2009.14559)
