# (751h) Fairness Measures for Decision-Making and Conflict Resolution

- Conference: AIChE Annual Meeting
- Year: 2019
- Proceeding: 2019 AIChE Annual Meeting
- Group: Computing and Systems Technology Division
- Session:
- Time:
Friday, November 15, 2019 - 10:06am-10:24am

*total utility*(the sum of individual utilities). This allocation paradigm, often known as the classical utilitarian approach or as the

*social welfare*approach, is intuitive but might yield allocations that are not fair. The lack of fairness is the result of inherent solution degeneracies (multiple allocations can yield the same total utility) and to the extreme sensitivity of this approach to subsystem scales. For example, in the case of wholesale electricity markets, the market is cleared by solving a social welfare problem that determines demand and supply allocations to various stakeholders (utility companies and power producers) [1]. Here, allocations tend to favor large stakeholders over small ones and multiplicity of solutions are often encountered when many market participants are present (due to large numbers of degrees of freedom).

In the field of game theory, the utility allocation problem has been viewed as a bargaining game between stakeholders. Nash [2] first provided an axiomatic approach to obtain solutions to the bargaining problem. These axioms include Pareto optimality, symmetry, affine invariance, and independence of irrelevant alternatives. Nash also proved that there exists a utility allocation scheme that satisfies these axioms (what is now known as the Nash solution). A generalization of Nash's scheme is the proportional fairness scheme, which has been widely used to allocate bandwidth in telecommunication networks [3]. Fairness measures have also been widely used to quantify income inequality [4]. We observe that the ultimate goal of a fairness measure is to *shape an allocation distribution* in a desirable way. As such, the utility allocation problem can also be interpreted as a stochastic programming problem in which one seeks to find allocations that shape distribution of outcomes (in stochastic programming the outcome distribution is shaped by using a risk measure) [5, 6]. As in the case of fairness measures, axioms have been proposed in the stochastic programming literature to study the selection of suitable risk measures [7].

In this talk, we will present concepts of fairness from the perspectives of game theory, economics, statistics, and engineering by using an axiomatic approach [8]. The axiomatic approach lists a set of properties (such as Pareto optimality, symmetry, affine invariance, independent of irrelevant alternatives, and restricted monotonicity) that a fairness measure should ideally satisfy. We will mathematically analyze the axiomatic properties of different measures such as social welfare, Nash solution, Kalai-Smorodinsky solution, max-min fairness, alpha-fairness, superquantile, and entropy [9, 10] allocation schemes. Our work reveals significant deficiencies in the social welfare allocation approach and highlights interesting and desirable properties and connections between Nash [2] and entropy allocation approaches. These fundamental connections can guide the selection of suitable measures for utility allocation in complex-decision making environments. We will conclude the presentation with a case study on using different fairness measures to balance soil phosphorus concentration in the upper Yahara watershed region in the State of Wisconsin. This case study reveals that solution degeneracy (or not satisfying the axiom of symmetry) in the social welfare approach results in multiple areas within the study area with disproportionate amounts of phosphorus.

References:

[1] Zavala, V. M., Kim, K., Anitescu, M., & Birge, J. (2017). A stochastic electricity market clearing formulation with consistent pricing properties. *Operations Research*, *65*(3), 557-576.

[2] Zukerman, M., Tan, L., Wang, H., & Ouveysi, I. (2005). Efficiency-fairness tradeoff in telecommunications networks. *IEEE Communications Letters*, *9*(7), 643-645.

[3] Nash Jr, J. F. (1950). The bargaining problem. *Econometrica: Journal of the Econometric Society*, 155-162.

[4] Venkatasubramanian, V. (2017). *How Much Inequality Is Fair?: Mathematical Principles of a Moral, Optimal, and Stable Capitalist Society*. Columbia University Press.

[5] Dowling, A. W., Ruiz-Mercado, G., & Zavala, V. M. (2016). A framework for multi-stakeholder decision-making and conflict resolution. *Computers & Chemical Engineering*, *90*, 136-150.

[6] Hu, J., & Mehrotra, S. (2012). Robust and stochastically weighted multiobjective optimization models and reformulations. *Operations research*, *60*(4), 936-953.

[7] Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (1999). Coherent measures of risk. *Mathematical finance*, *9*(3), 203-228.

[8] Moulin, H. (1991). Axioms of cooperative decision making. No. 15. *Cambridge university press*.

[9] Venkatasubramanian, V. & Luo, Y. (2018). How much income inequality is fair? Nash bargaining solution and its connection to entropy,â€ *arXiv preprint arXiv:1806.05262*.

[10] Cowell, F. A. & Kuga, K. (1981). Additivity and the entropy concept: an axiomatic approach to inequality measurement. *Journal of Economic Theory*, 25(1), 131â€“143.