# PSuU & Social Choice

## Motivation

Parameter Selection under Uncertainty is an Optimization technique with the aim to find the *best* set of parameters for a system of multiple components in an environment where multiple (partially conflicting) goals exist. This procedure offers solutions (in form of parameter ranges) where trade-offs of different system realizations are adequately accounted for.

However, such systems typically have various stakeholders which might have different *preferences* over such realzations; in other words: have a different ranking of system goals and their importance. Given a set of articulated orderings by multiple individuals, how to select the final parameter value ranges provided by PSuU?

One feasible approach would be to treat the space of stakeholder preference relations as an input to a mechanism which would output the selection, utilizing the concept of *Social Welfare* and *Social Choice*.

## Formulation

We define the process of PSuU as a mapping from a space of $K$ metrics and associated $P$ parameter values for $N$ simulation runs to a criterion $C(I)$ suggesting for each parameter $j = 1, \dots, P$ an optimal range $[ \underline{p_j}, \overline{p_j}]$.

In addition to that there might be specified a preference ordering $\succsim$ over parameters when emphasis is laid on the importance of some system goals over others. In a situation where multiple stakeholders $i=1, \dots, S$ exist, their preference ordering $\succsim_i$ might vary for each of them.

Assuming the set of $P$ parameters as the set of alternatives $A$, we can denote by $L$ the set of linear orders on $A$ such that $\succsim_i \in L$ for each stakeholder $i$.

It would be desireable to be able to reach consensus on the importance of system goals and preferences over parameters in such a situation.

## Social Welfare and Social Choice

Following the notation above we can introduce the *social welfare function* $F$ as one aggregating preferences of all voters into a common preference:

$$
F: L^S \to L
$$

This offers a possibility to arrive at a consensual preference ordering $\succsim$ for all stakeholders finding an agreeable outcome of the parameter selection problem described above.

As a sidenote, consider the case when we were interested in choosing the most relevant parameter out of all stakeholders' orderings we would also consider the *social choice function* $f$:

$$
f: L^S \to A
$$

Note, that this goes against the ambition to select parameter values in a setting with trade-offs between multiple conflicting but relevant goals.

## Desireable Properties and Impossibility

Desireable properties of a social welfare fuction are:
- *unanimity*: If $\succsim \hat{=}\succsim_1 = \dots = \succsim_S$ identical for all stakeholders, the social welfare function selects exactly this ordering: $F(\succsim_1 = \dots = \succsim_S) = \succsim$
- *non-dictatorship*: There exists no stakeholder $i \in S$ that would prescribe the outcome of $F(\cdot)$ independent of all other stakeholders' preferences. That is $\nexists i \in S: F(\succsim_1 = \dots = \succsim_S) = \succsim_i$ valid $\forall \succsim_1 = \dots = \succsim_S \in L$
- *independence of irrelevant alternatives*: The social preference between any two alternatives depends only on the voters' preferences between them.

For more than two alternatives it is impossible to find a social welfare function that satisfies all the above properties. This is caputured by the following central theorem:

**Theorem (*Arrow*):** Every social welfare function over a set of more than 2 candidates $(|A| \geq 3)$ that satisfies unanimity and independence of irrelevant alternatives is a dictatorship.

This result limits our design space for the targeted mechanism for parameter selection as described above. Luckily there are ways out, by introducing additional features.

## Mechanisms with Money

By using preference relations only no emphasis is laid on the magnitude of the differences between alternatives. This can be achieved by introducing a quantifying unit *money* which also can be used to conduct transfers between players and/or the mechanism.

To extend the formulation, we introduce a *valuation function* $v_i$ encoding the preferences of player $i$:
$$
v_i: A \to \mathbb{R}
$$
by assigning a value $v_i(a) \in \mathbb{R}$ to the choice of the alternative $a$. Furthermore each player aims to maximuize their *utility*
$$
u_i = v_i(a) + m
$$
where $m$ is the amount of money this player holds.

Mechanisms with money are able to choose a social alternative **and** determine monetary payments to be made by different players.

A *direct revelation mechanism* is a social choice function $f: V_1 \times \cdots \times V_n \to A$ and a vector of payment functions $p_1, \dots, p_n$ where $p_i:V_1 \times \cdots \times V_n \to \mathbb{R}$ is the amount that player $i$ pays.

A mechanism $(f, p_1, \dots, p_n)$ is called *incentive compatible* if for every player $i$, every $v_1 \in V_1, \dots, v_n \in V_n$ and every $v_i' \in V_i$, if we denote $a = f(v_i, v_{-i})$ and $a' = f(v_i', v_{-i})$, then
$$
v_i(a) - p_i(v_i, v_{-i}) \geq v_i(a') - p_i(v_i', v_{-i})
$$

Players prefer to *tell the truth* to maximize their utility.

## Vickrey-Clarke-Groves Mechanisms

VCG mechanisms are incentive compatible.

A mechanism $(f, p_1, \dots, p_n)$ is called a Vickrey-Clarke-Groves (VCG) mechanism if:
- $f(v_1, \dots, v_n) \in \mathrm{argmax}_{a \in A} \sum_i v_a(a)$; that is $f$ maximizes the social welfare, and
- for some functions $h_1, \dots h_n$, where $h_i:V_{-i} \to \mathbb{R}$ (i.e. $h_i$ does not depend on $v_i$), we have that for all $v_1 \in V_1, \dots, v_n \in V_n: p_i(v_1, \dots, v_n) = h_i(v_{-i})- \sum_{j \neq i} v_j (f(v_1, \dots,v_n))$

The last term in the above definition means that each player is paid an amount equal to the sum of the values of all other players. When this term is added to his own value, the sum becomes exactly the social welfare of $f(v_1, \dots, v_n)$. Thus this mechanism aligns all players' incentives with the social goal of maximizing social welfare, which is exactly achieved by telling the truth.

## Discussion

- Is there a (known & incentive compatible) mechanism (e.g. VCG) that would provide a weighted ordering of alternatives?
- Can the above structure be applied in a situation where there is only one stakeholder, but the existence of multiple goals implies different parameter preferences?

## Alignment with existing PSuU process

### Standard Process

[Parameter Selection under Uncertainty (PSuU)](https://medium.com/block-science/how-to-perform-parameter-selection-under-uncertainty-976931ba7e5d) is a method using modeling and simulation tools to decide on system parameters starting from system goals and requirements. Goals are represented with *metrics* or *KPIs* and values of these metrics pin down trade-offs of different system realizations $x \in X$. Given a weighting $W = \{w_1, \dots, w_n \}$ of goals $G = \{g_1, \dots, g_n \}$ and associated metrics $f_{g_1}(x), \dots, f_{g_n}(x)$, it allows to find "reasonable" and "stable" parameter ranges justifying a selection.

The process consists of several steps:
1. System goals identification
2. Control parameter identification
3. Environmental parameter identification
4. Metric identification
5. Simulation
6. Optimal parameter selection

In the example case of $n=3$ a visualization of trade-offs, given *fixed* weights $w_1, w_2, w_3$ is demonstrated below, showing regions of stable parameter values creating a *baisin of attraction*.


### Welfare interpretation

Main idea is to create a **social welfare analogue** to the system goals & the way they are aggregated. There exists a rich body of work in Microeconomics, Welfare Economics & Social Choice Theory that can be leveraged for this purpose.

In this literature individual preferences are aggregated into a sociatal preference while accounting for all challenges associated with such an aggregation (like impossibility, dictatorship, etc.). System goals $G$ (and associated metrics) can be interpreted as local objective functions (individual welfare functions) $f_i$ for each participant $i$. In the above triangle example each vertex being an individual welfare piece that is beeing added up and the baisin of attraction indicates under which circumstances there is confidence about stability of the soultion. In other words, how small pertubations in the weighting between the different individual welfare functions translate into changes of the whole solution.

In the above example
- the weights were fixed, however this does not necessarily have to be the case:
- so far the trade-off between goals was assumed to be known (a weighting $(w_1, w_2, w_3)$ explicitely prescribed) & put into relation to the investigation of the (trivial) edge cases $(1,0,0), (0,1,0)$ and $(0,0,1)$

- individual system goals $G$ were assumed to be known and codified via the metrics - generalizing this into individual welfare functions is more challenging, as:
- however individual preferences might be unknown. In social science we have the initial task of illiciting preferences from participants & there exist mechanisms which achieve this in an incentive-compatible way. (truth-telling)
- preferences can also be dynamic - subject to change over time

What could be layered in with the suggested approach is **another price discovery mechanism** that could be used to build out the triangle (or more generally an *$n$-dim simplex*)

### Welfare Theorems

**First Welfare Theorem**

The First Welfare Theorem states that, under certain conditions, competitive markets lead to Pareto-efficient outcomes (resources are allocated in such a way that it is impossible to make one individual better off without making someone else worse off). In other words, the economy is operating at its maximum potential efficiency.
The key conditions for the First Welfare Theorem to hold are:

- Perfect Competition: There are many buyers and sellers, and no individual or firm has the power to influence prices.
- Complete Markets: All goods and services are traded in markets, and there are no missing markets.
- No Externalities: There are no external costs or benefits associated with production or consumption.
- No Information Asymmetry: All market participants have access to the same information.

If these conditions are met, then the competitive equilibrium (where supply equals demand in all markets) will be Pareto-efficient. However, the First Welfare Theorem does not address issues of equity or fairness; it only guarantees efficiency.

**Second Welfare Theorem**

The Second Welfare Theorem builds upon the First Welfare Theorem and addresses the issue of distribution or equity. It states that, under certain conditions, any Pareto-efficient outcome can be achieved through the redistribution of resources. In other words, if the initial allocation of resources is not Pareto-efficient, it is possible to redistribute resources in such a way that everyone can be made better off without making anyone worse off.
The key conditions for the Second Welfare Theorem to hold are:

- The same conditions as the First Welfare Theorem (perfect competition, complete markets, no externalities, and no information asymmetry).

- The possibility of lump-sum transfers: It must be possible to redistribute resources without any distortionary taxes or subsidies.

The Second Welfare Theorem highlights that, while competitive markets can achieve efficiency, policymakers can use redistribution mechanisms (such as taxes and subsidies) to address concerns about equity and achieve any desired distribution of resources, as long as they can use lump-sum transfers.

## Formulation as Lagrangian (constrained optimization)

**First Theorem**

The First Welfare Theorem is concerned with the conditions under which competitive markets lead to Pareto efficiency. We can represent this as follows:

Goal is to maximize:
$$
\sum U_i(x_i)
$$
Subject to:
$$
\sum p_ix_i \leq \sum p_ix_i^0 \ \ \ \ \ \forall i \\
\sum x_i = X
$$

Where:

- $\sum U_i(x_i)$ is the sum of utility functions of all individuals, representing the overall welfare.
- $x_i$ is the bundle of goods consumed by individual $i$.
- $p_i$ is the price vector for goods for individual $i$.
- $x_i^0$ represents the initial allocation of goods.
- $X$ is the total endowment of goods.

We can use a Lagrange multiplier to represent the trade-off between efficiency and equity:

$$
\mathcal{L} = \sum U_i(x_i) - \sum λ_i(p_ix_i - p_ix_i^0) - μ(X - \sum x_i)
$$

Where:

- $λ_i$ is the Lagrange multiplier (lambda weight) associated with the budget constraint for individual i.
- $μ$ is a Lagrange multiplier associated with the aggregate resource constraint.

The optimal allocation will occur when the marginal utilities of consumption for all individuals are equal, and the budget constraints and resource constraints are satisfied:

$$
\frac{\partial \mathcal{L}}{\partial x_i} = 0 \ \ \ \ \forall i \\
\sum p_ix_i = \sum p_ix_i^0 \\
\sum x_i = X
$$

This result is derived by setting
$$
\frac{\partial \mathcal{L}}{\partial x_i} = 0 \ \ \ \ \forall i \\
\frac{\partial \mathcal{L}}{\partial \lambda_i} = 0 \ \ \ \ \forall i \\
\frac{\partial \mathcal{L}}{\partial \mu} = 0
$$
## References

This document serves as an outline of the research arc to get a better grip on the existing literature that can be applied to this kind of problems. This work could directly be translated into an extension of the PSuU process adding a different perspective on this methodology.

- [Microeconomics](https://www.pearson.com/store/p/advanced-microeconomic-theory/P200000004552/9780273731917):
- [Algorithmic Game Theory](https://www.cambridge.org/core/books/algorithmic-game-theory/0092C07CA8B724E1B1BE2238DDD66B38): Nisan, N., Roughgarden, T., Tardos, E., & Vazirani, V. (Eds.). (2007). Algorithmic Game Theory. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511800481
- [PSUU](https://medium.com/block-science/how-to-perform-parameter-selection-under-uncertainty-976931ba7e5d)
- [TEGG presentation](https://www.youtube.com/watch?v=9C2xgMSi3Fs&ab_channel=TEAcademy): Shorish, Jamsheed: Aligning Interests: Reward Mechanisms and the Engineering of Token Incentives

## Literature Review

### Political Science

- Dewey and Lippmann
- Original Contributions
- [Holcombe AN. Public Opinion. By Walter Lippmann. (New York: Harcourt, Brace and Company. 1922. Pp. x, 427.). American Political Science Review. 1922;16(3):500-501. doi:10.2307/1943740](https://www.cambridge.org/core/journals/american-political-science-review/article/abs/public-opinion-by-walter-lippmann-new-york-harcourt-brace-and-company-1922-pp-x-427/980CEDE2500673A4A4E17A22E10ACACA)
- [The Phantom Public, Lippmann](https://www.routledge.com/The-Phantom-Public/Lippmann/p/book/9781560006770)
- [DEWEY, JOHN. The Public and Its Problems: An Essay in Political Inquiry. Edited by MELVIN L. ROGERS. Penn State University Press, 2012. http://www.jstor.org/stable/10.5325/j.ctt7v1gh.](https://www.jstor.org/stable/10.5325/j.ctt7v1gh)
- [Dewey, J. (1929). Experience and nature. W W Norton & Co. https://doi.org/10.1037/13377-000](https://psycnet.apa.org/doiLanding?doi=10.1037%2F13377-000)
- [Dewey, J. (1929). The quest for certainty. Minton, Balch.](https://psycnet.apa.org/record/1930-00008-000)
- Reviews:
- [Discussion of Original contributions](https://www.infoamerica.org/teoria_articulos/lippmann_dewey.htm)
- [GRUBE, N., Mass Democracy and Political Governance - The Walter Lippmann-John Dewey Debate](https://brill.com/display/book/edcoll/9789460913457/BP000010.xml)
- [Interesting Dissertation with good overview and motivation + references](https://etda.libraries.psu.edu/files/final_submissions/2519)
- Technocracy
- [problem of knowledge and Role of Experts in modern democracy](https://eric.ed.gov/?id=EJ1000304)
- [Technocracy and the Public Sphere](https://kuleuven.limo.libis.be/discovery/fulldisplay?docid=lirias3969439&context=SearchWebhook&vid=32KUL_KUL:Lirias&search_scope=lirias_profile&adaptor=SearchWebhook&tab=LIRIAS&query=any,contains,LIRIAS3969439&offset=0&lang=en)
- [Governance FAQ](https://a16zcrypto.com/posts/article/governance-faq/), Sections:
- [Andrew Hall](https://www.andrewbenjaminhall.com/)
- Is your project an economic project, or a civic project, or both?
- How do you balance expertise with widespread participation?
- Beyond Token Voting
- https://a16zcrypto.com/posts/article/lightspeed-democracy-what-web3-organizations-can-learn-from-the-history-of-governance/
- [Beyond technocracy, the authority of truth](https://journals.sagepub.com/doi/abs/10.1177/01914537211059510)
- [isaiah berlin (Pluralism)](https://en.wikipedia.org/wiki/Isaiah_Berlin)
- Original Work
- [Published Work](https://en.wikipedia.org/wiki/Isaiah_Berlin#Published_works)
- [Berlin, I., & Williams, B. (1994). Pluralism and Liberalism: A Reply. Political Studies, 42(2), 306-309. https://doi.org/10.1111/j.1467-9248.1994.tb01914.x](https://journals.sagepub.com/doi/abs/10.1111/j.1467-9248.1994.tb01914.x?journalCode=psxa)
- [Berlin, Liberty, Oxford university Press](https://global.oup.com/academic/product/liberty-9780199249893?cc=us&lang=en&)
- Discussions
- [Ferrell, J. Isaiah Berlin: Liberalism and pluralism in theory and practice. Contemp Polit Theory 8, 295–316 (2009). https://doi.org/10.1057/cpt.2009.2](https://link.springer.com/article/10.1057/cpt.2009.2#citeas)
- [Myers E. From Pluralism to Liberalism: Rereading Isaiah Berlin. The Review of Politics. 2010;72(4):599-625. doi:10.1017/S0034670510000550](https://www.cambridge.org/core/journals/review-of-politics/article/abs/from-pluralism-to-liberalism-rereading-isaiah-berlin/32D5128D33BA26FE7D57DD85621E4A78)
- [Crowder, The problem of Value Pluralism; Isiah Berlin and Beyond](https://www.routledge.com/The-Problem-of-Value-Pluralism-Isaiah-Berlin-and-Beyond/Crowder/p/book/9781032085227#:~:text=The%20Problem%20of%20Value%20Pluralism%3A%20Isaiah%20Berlin%20and%20Beyond%20is,voice%20in%20the%20pluralist%20literature.)
- [Max Weber (Pluralism)](https://en.wikipedia.org/wiki/Max_Weber)
- Original Work
- [Bibliography](https://en.wikipedia.org/wiki/Max_Weber_bibliography)
- Discussions
- [Spicer, M. W. (2015). Public Administration in a Disenchanted World: Reflections on Max Weber’s Value Pluralism and His Views on Politics and Bureaucracy. Administration & Society, 47(1), 24-43. https://doi.org/10.1177/0095399714554514](https://journals.sagepub.com/doi/abs/10.1177/0095399714554514)
- [Abraham, Gary A. “Max Weber: Modernist Anti-Pluralism and the Polish Question.” New German Critique, no. 53 (1991): 33–66. https://doi.org/10.2307/488244.](https://www.jstor.org/stable/488244)
- [Steven Seidman, Modernity, Meaning, and Cultural Pessimism in Max Weber, Sociology of Religion, Volume 44, Issue 4, Winter 1983, Pages 267–278, https://doi.org/10.2307/3711610](https://academic.oup.com/socrel/article-abstract/44/4/267/1612555?redirectedFrom=fulltext)
- Pluralism generally
- Miller, Nicholas R. “Pluralism and Social Choice.” The American Political Science Review, vol. 77, no. 3, 1983, pp. 734–47. JSTOR, https://doi.org/10.2307/1957271.
- [Jenson, Jane "Intersections of Pluralism and Social Cohesion: Two Concepts for the Practice of Pluralism"](https://www.pluralism.ca/wp-content/uploads/2019/01/Jane-Jenson-Social-Cohesion.pdf)
- Social planning
- [Huber, Valeska. “Introduction: Global Histories of Social Planning.” Journal of Contemporary History 52, no. 1 (2017): 3–15. https://www.jstor.org/stable/26416510.](https://www.jstor.org/stable/26416510)
- [Manski, CREDIBLE SOCIAL PLANNING UNDER UNCERTAINTY](https://faculty.wcas.northwestern.edu/cfm754/credible_social_planning_under_uncertainty.pdf)
- [Riemer, "Social Planning and Social Organization"](https://www.journals.uchicago.edu/doi/epdf/10.1086/220072)

### Institutional Economics

- [Adkisson, R. V. (2010). The original institutionalist perspective on economy and its place in a pluralist paradigm. International Journal of Pluralism and Economics Education, 1(4), 356. doi:10.1504/ijpee.2010.037976](https://www.inderscienceonline.com/doi/abs/10.1504/IJPEE.2010.037976)
- John F. O’Neill "Pluralism and Economic Institutions", page 83 in [OTTO NEURATH’S ECONOMICS IN CONTEXT](https://link.springer.com/content/pdf/10.1007/978-1-4020-6905-5.pdf#page=83)
- [Claassen, R. J. G. (2009). Institutional pluralism and the limits of the market. Politics, Philosophy & Economics, 8(4), 420-447. https://doi.org/10.1177/1470594X09345479](https://journals.sagepub.com/doi/abs/10.1177/1470594X09345479)
- [Amitava Krishna Dutt (2014) Dimensions of Pluralism in Economics, Review of Political Economy, 26:4, 479-494, DOI: 10.1080/09538259.2014.950461](https://www.tandfonline.com/doi/abs/10.1080/09538259.2014.950461)
- [Janice Peterson (2013) Economics Education after the Crisis: Pluralism, History, and Institutions, Journal of Economic Issues, 47:2, 401-410, DOI: 10.2753/JEI0021-3624470213](https://www.tandfonline.com/doi/abs/10.2753/JEI0021-3624470213)
- [Kyong Hee Yu: Institutional Pluralism, Organizations, and Actors: A Review](https://compass.onlinelibrary.wiley.com/doi/abs/10.1111/soc4.12269)
-

### Social Choice
- David Wolpert and William Macready
- original work
- [(No Free lunch theorems)](https://en.wikipedia.org/wiki/No_free_lunch_theorem)
- :exclamation: [D. H. Wolpert and W. G. Macready, "No free lunch theorems for optimization," in IEEE Transactions on Evolutionary Computation, vol. 1, no. 1, pp. 67-82, April 1997, doi: 10.1109/4235.585893.](https://ieeexplore.ieee.org/abstract/document/585893)
- [Macready, W.G. and Wolpert, D.H. (1996), What makes an optimization problem hand?. Complexity, 1: 40-46. https://doi.org/10.1002/cplx.6130010511](https://onlinelibrary.wiley.com/doi/abs/10.1002/cplx.6130010511)
- [Wolpert, What is important about the No Free Lunch theorems?, arxiv](https://arxiv.org/abs/2007.10928)
- [Wolpert, D.H. (2013) "What the no free lunch theorems really mean", Ubiquity, Volume 2013, December 2013,](https://dl.acm.org/doi/10.1145/2555235.2555237)
- Discussion
- [Whitley, Darrell, and Jean Paul Watson. "Complexity theory and the no free lunch theorem." In Search Methodologies, pp. 317–339. Springer, Boston, MA, 2005.](https://www.researchgate.net/publication/226085645_Complexity_Theory_and_the_No_Free_Lunch_Theorem)
- Kenneth Arrow (Impossibility)
- Original Work
- [Arrow Kenneth, Social Choice and Individual Values](https://cowles.yale.edu/sites/default/files/2022-09/m12-2-all.pdf)
- [Arrow, Extended Sympathy and the Possibility of Social Choice](https://www.jstor.org/stable/1815907)
- [ARROW, KENNETH J. “Current Developments in the Theory of Social Choice.” Social Research 44, no. 4 (1977): 607–22. http://www.jstor.org/stable/40971168.](https://www.jstor.org/stable/40971168)
- [Arrow, Kenneth J. "An extension of the basic theorems of classical welfare economics." Proceedings of the second Berkeley symposium on mathematical statistics and probability. Vol. 2. University of California Press, 1951.](https://projecteuclid.org/ebooks/berkeley-symposium-on-mathematical-statistics-and-probability/Proceedings-of-the-Second-Berkeley-Symposium-on-Mathematical-Statistics-and/chapter/An-Extension-of-the-Basic-Theorems-of-Classical-Welfare-Economics/bsmsp/1200500251.pdf)
- Discussion
- [Frohock, Fred M., 1980. "Rationality, Morality, and Impossibility Theorems," American Political Science Review, Cambridge University Press, vol. 74(2), pages 373-384, June.](https://ideas.repec.org/a/cup/apsrev/v74y1980i02p373-384_16.html)
- [Blaug, M. (2007). The fundamental theorems of modern welfare economics, historically contemplated. History of political economy, 39(2), 185-207.](https://coin.wne.uw.edu.pl/mbrzezinski/teaching/HE4/BlaugWelfareTheorems2007.pdf)
- Aumann
- Microeconomics (Mascalell etc)
- here?
- Welfare Theorems
- [Sen, Amartya. “On Weights and Measures: Informational Constraints in Social Welfare Analysis.” Econometrica 45, no. 7 (1977): 1539–72. https://doi.org/10.2307/1913949.](https://www.jstor.org/stable/1913949)
- [Campbell, Donald E. “A Generalization of the Second Theorem of Welfare Economics for Nonconvex Economies.” International Economic Review 29, no. 2 (1988): 201–15. https://doi.org/10.2307/2526662.](https://www.jstor.org/stable/2526662)

### Optimization

- Multidimensional (multiobjective) Optimization (MOO) https://en.wikipedia.org/wiki/Multi-objective_optimization
- An Introduction to Multiobjective Optimization Techniques January 2011 In book: Optimization in Polymer ProcessingChapter: 3Publisher: Nova Science PublishersEditors: António Gaspar-Cunha, José António Covas
- Kaisa Miettinen (1999). Nonlinear Multiobjective Optimization. Springer. ISBN 978-0-7923-8278-2.
- Ching-Lai Hwang; Abu Syed Md Masud (1979). Multiple objective decision making, methods and applications: a state-of-the-art survey. Springer-Verlag. ISBN 978-0-387-09111-2.
- Nguyen, Hoang Anh; van Iperen, Zane; Raghunath, Sreekanth; Abramson, David; Kipouros, Timoleon; Somasekharan, Sandeep (2017). "Multi-objective optimisation in scientific workflow". Procedia Computer Science. 108: 1443–1452.
- Matthias Ehrgott (1 June 2005). Multicriteria Optimization. Birkhäuser. ISBN 978-3-540-21398-7. Retrieved 29 May 2012
- Jürgen Branke; Kalyanmoy Deb; Kaisa Miettinen; Roman Slowinski (21 November 2008). Multiobjective Optimization: Interactive and Evolutionary Approaches. Springer. ISBN 978-3-540-88907-6. Retrieved 1 November 2012.
- Convex Optimization
- [Stephen Boyd and Lieven Vandenberghe, Cambridge University Press, *Convex Optimization*](https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf)

## Further research

- Social Choice & Welfare Economics
- Constrained Optimization
-
- Stochasticity
- Point ranges (mech w/o money)