# On a Robust Approach for Stochastic Equilibrium Problems

Lead Research Organisation:
University of Southampton

Department Name: School of Mathematics

### Abstract

Stochastic programming has been extensively used by operation researchers, economists and various decision makers/practitioners to model optimal decision making in economics, management, engineering, transportation networks and the environment. When a decision problem involves not only uncertainty, but also several

decision makers who are in a competitive relationship, it becomes a stochastic game. An important approach in understanding such a game is to look at the equilibrium outcomes. These are the set of possible outcomes at

the end of competition, given that each player seeks to optimize their own payoff.

A fundamental issue in stochastic programming and equilibrium concerns the representation of

uncertainty. Many of the models in the literature assume complete knowledge of the distributions of random variables (representing the uncertainty). In

many practical cases, however, such distributions are not known precisely and have to be either estimated from historical data or constructed using

subjective judgements. The available information is often insufficient to give confidence in the distribution identified. In the absence of full information on the underlying distribution, it may still be possible to

identify a set of possible probability distributions within which the true distribution lies. While a robust optimization approach to this problem is

based on making the decision that would be appropriate given the worst probability distribution in the set of possible distributions, robust analysis of stochastic equilibrium is to look into worst equilibrium outcomes given the incomplete information of the underlying stochastic elements and robust design

requires one to set out optimal policy/parameters which accommodate any worst equilibrium outcomes.

The project is proposed to develop a mathematical framework that allows one to carry out robust anaysis of a stochastic equilibrium problem with incomplete information on the underlying uncertainty, identify optimal policy/design which accommodate the worst possible equilibrium outcomes, develop efficient numerical methods for solving the new mathematical models and apply apply them to some interesting practical problems in economics and engineering with a particular focus on energy industry.

decision makers who are in a competitive relationship, it becomes a stochastic game. An important approach in understanding such a game is to look at the equilibrium outcomes. These are the set of possible outcomes at

the end of competition, given that each player seeks to optimize their own payoff.

A fundamental issue in stochastic programming and equilibrium concerns the representation of

uncertainty. Many of the models in the literature assume complete knowledge of the distributions of random variables (representing the uncertainty). In

many practical cases, however, such distributions are not known precisely and have to be either estimated from historical data or constructed using

subjective judgements. The available information is often insufficient to give confidence in the distribution identified. In the absence of full information on the underlying distribution, it may still be possible to

identify a set of possible probability distributions within which the true distribution lies. While a robust optimization approach to this problem is

based on making the decision that would be appropriate given the worst probability distribution in the set of possible distributions, robust analysis of stochastic equilibrium is to look into worst equilibrium outcomes given the incomplete information of the underlying stochastic elements and robust design

requires one to set out optimal policy/parameters which accommodate any worst equilibrium outcomes.

The project is proposed to develop a mathematical framework that allows one to carry out robust anaysis of a stochastic equilibrium problem with incomplete information on the underlying uncertainty, identify optimal policy/design which accommodate the worst possible equilibrium outcomes, develop efficient numerical methods for solving the new mathematical models and apply apply them to some interesting practical problems in economics and engineering with a particular focus on energy industry.

### Planned Impact

The research outcomes can be applied to for decision making problems which involve uncertainty and competition/conflicting factors. Here we list a couple of application areas/beneficiaries.

1. Regulators and or market designers. Take energy industry for an example. Over the past two decades, optimization and game theory have been effectively used to study various issues such as market power and competitiveness in deregulated electricity industry in many part of the world. The research outcomes in this project may be directly or indirectly used by market designers/regulators to carry out worst equilibrium analysis of a future market under the incomplete information of market uncertainty at present. They can also be used as a mathematical model to simulate policies which may effectively avoid or accommodate worst possible market/competition equilibrium scenarios (similar to stress test in finance industry). Note that this is not a robust game theory model which has already been investigated in the literature, it is the robust mathematical analysis model for a stochastic game, which is completely new and particularly suitable for policy makers of energy industry (which involves substantial uncertainty in future).

2. Public sectors. Government decisions in areas such as tax, transportation and the environment always have an impact on economy, industry and general public. Optimization and game theoretic models have been well used to study the policy impacts. However, deterministic or classical stochastic models might not be adequate to address the impact of uncertainty in these decision making problems. For example, a commitment to the Kyoto global warming pact on the emissions of carbon dioxide will result in both long term and short term actions and interactions in the industries (e.g. developing new energy such as wind power, improving energy efficiency such as smart grid, buying emission quota from others in the short term, moving companies to developing countries, all of which have further impact on national economy and quality of public services). Our robust stochastic equilibrium model may provide a new mathematical tool for policy makers to carry out some feasibility analysis to avoid disastrous consequences. Similar applications can also be found in transportation design where the interaction between service providers and/or customers may be mathematically described as a stochastic game. While the existing models can provide some business insights, a robust stochastic equilibrium model might be useful for a long term planning (e.g. capital investment) where the information on future uncertainty is often incomplete.

1. Regulators and or market designers. Take energy industry for an example. Over the past two decades, optimization and game theory have been effectively used to study various issues such as market power and competitiveness in deregulated electricity industry in many part of the world. The research outcomes in this project may be directly or indirectly used by market designers/regulators to carry out worst equilibrium analysis of a future market under the incomplete information of market uncertainty at present. They can also be used as a mathematical model to simulate policies which may effectively avoid or accommodate worst possible market/competition equilibrium scenarios (similar to stress test in finance industry). Note that this is not a robust game theory model which has already been investigated in the literature, it is the robust mathematical analysis model for a stochastic game, which is completely new and particularly suitable for policy makers of energy industry (which involves substantial uncertainty in future).

2. Public sectors. Government decisions in areas such as tax, transportation and the environment always have an impact on economy, industry and general public. Optimization and game theoretic models have been well used to study the policy impacts. However, deterministic or classical stochastic models might not be adequate to address the impact of uncertainty in these decision making problems. For example, a commitment to the Kyoto global warming pact on the emissions of carbon dioxide will result in both long term and short term actions and interactions in the industries (e.g. developing new energy such as wind power, improving energy efficiency such as smart grid, buying emission quota from others in the short term, moving companies to developing countries, all of which have further impact on national economy and quality of public services). Our robust stochastic equilibrium model may provide a new mathematical tool for policy makers to carry out some feasibility analysis to avoid disastrous consequences. Similar applications can also be found in transportation design where the interaction between service providers and/or customers may be mathematically described as a stochastic game. While the existing models can provide some business insights, a robust stochastic equilibrium model might be useful for a long term planning (e.g. capital investment) where the information on future uncertainty is often incomplete.

## People |
## ORCID iD |

Huifu Xu (Principal Investigator) |

### Publications

Huifu Xu (Author)
(2016)

*CVaR Approximations for Minimax and Robust Convex Optimization*
Liu Y
(2014)

*Entropic Approximation for Mathematical Programs with Robust Equilibrium Constraints*in SIAM Journal on Optimization
Sun H
(2016)

*Convergence Analysis for Distributionally Robust Optimization and Equilibrium Problems*in Mathematics of Operations ResearchDescription | We have proposed a number of distributionally robust models for analysing stochastic games and equilibrium problems when some or all players do not have complete information on the underlying uncertainty. Various numerical schemes have been developed for solving the resulting mathematical models. The models have interesting practical applications. For example, in a recent follow up research, we have studied Stackelberg robust followers model for hierarchical competition in a supply chain management where a buyer of certain products not only invests in its own capacity to supply a uncertain market but also outsources a certain amount of market supplies to multiple competing suppliers who invest in capacity for the buyer's orders. The new model indicates that the buyer has more incentives to invest in capacity whereas the suppliers are less to do so in the case when the latter are confronted with more demand uncertainty. This finding suggests that the buyer can benefit from outsourcing to the suppliers who are the potential competitors in the product market when the suppliers have no complete information on market demand uncertainty. By allowing the different degrees of informational completeness on uncertainty between the players in different hierarchies, the study provides a novel approach to operations and supply chain literature on vertical relationships and the management of global supply chains where many well-established insights are derived from stochastic Nash equilibrium schemes. |

Exploitation Route | The models and numerical schemes may be used by academic researchers in the area of robust optimization, stochastic programming and game theory. They may also be used by practitioners in operational research, economists, research scientists in management sciences when they deal with practical problems involving stochastic equilibrium problems such as transportation networks, market competition and structural design in engineering. |

Sectors | Creative Economy,Energy,Financial Services, and Management Consultancy,Manufacturing, including Industrial Biotechology,Transport |