Institute for Transport StudiesFACULTY OF ENVIRONMENT

Cooperative and Non-Cooperative Equilibrium Problems with Equilibrium Constraints: Applications in Economics and TransportationAndrew Koh17 Online World Conference on Soft ComputingAnywhere on Earth, (10-21 December 2012)a.koh@its.leeds.ac.uk

Presentation Outline• Background: Hierarchical Optimization (Bilevel Programming

Problem and Equilibrium Problem with Equilibrium Constraint)

• Behaviour of Leaders in EPECs1. Cooperative Multiobjective Optimization2. Non cooperative Nash Equilibrium

• The research question: How to move from Non Cooperative to Cooperative Solution

• Our Proposal: The Collusion Path• Examples• Conclusions and Further Research

Hierarchical Optimization Problem and Advances

BiLevel Programming Problem (BLPP)

• Special Session on BLPP at IEEE Congress on Evolutionary Computation in 2012

Equilibrium Problem with Equilibrium Constraint (EPEC)

Main Difference vs BLPP:Multiple Leaders

Behaviour of Leaders in EPEC

Fully Competitive Non Cooperative EPEC(NCEPEC)

Followers Behavior represented by some variational inequality parameterized in leaders’ variables

Co-operate Multiobjective EPEC (MOPEC)

Evolutionary Algorithms (EA) are particularly suitable methods for EPECs

Cooperative Case

If Leaders cooperate we have a MultiObjective Problem with Equilibrium Constraint (MOPEC)

Key solution concept: Pareto DominationCompare two chromosomes a & b: a Pareto Dominates b if a is just as good as b in all objectives AND a is strictly better than b in at least one objective generate Pareto Front• Many Evolutionary Algorithms available for this task• We use Multiobjective Self Adaptive Differential Evolution

(MOSADE by Huang et al 2007)

Non Cooperative Case

• In competitive case, each leader maximizes his own utility function, and is bound by the Variational Inequality

• This is a Non Cooperative EPEC (NCEPEC)• Nash Dominance concept in Lung and Dumitrescu (2012)

can be used to solve this problem

Nash Dominance

Nash Equilibrium means that each leader has no incentive to deviate from his current strategyIn Evolutionary Algorithms fitness plays a key role and fitter chromosomes should be allowed to breed.How to compare fitness for Nash Equilibrium?? – comparing chromosomes c and d, then we say c is fitter if:there are fewer players using c that can increase their profit by deviating to d compared to the number that can increase profit when playing d and deviating to c. The very definition of Nash Equilibrium is usedLung and Dumitrescu (2010) were the first to propose this concept.

Nash Domination and NDEMO Algorith

• The NCEPEC Solution is NOT PARETO OPTIMAL

• It is possible to make someone else better off without making anyone worse off.

• NDEMO Algorithm implementing Nash Domination criteria is shown below

• NDEMO uses Differential Evolution to create a new trial vector

Research Question

• How to move from Solution of the Nash Equilibrium Problem to the Pareto Front of the cooperative problem?

• In other words, each leader takes into account a proportion (α) of the objective of the opponents

• Doing this it is a “signal” to the opponents that there is an intention to collude because the resulting decision variable (x) will change

• We call this the “collusion path” that maps NCEPEC solution to MOPEC solution (the Pareto Front)

Example 1: Competition in Production of Homogenous Good

• 5 producers of which 2 are leaders• The remaining 3 followers take the leaders production

quantities as given and play Nash game amongst themselves

• The follower’s problem is parameterised in the leaders’ decisions variables

• Two MOPEC solutions by deterministic method discussed in Mordukhovich et al (2008)

• NCEPEC solution in Koh (2012) • Next slide shows “transmission mechanism” as to how

leader’s variables affect followers

Transmission Mechanism for Example 1

Leader’s Production

Quantities Q

Non Linear Complementarity Problem among Followers

Leader’s Objective Function

Resulting Follower’s Quantity vector (y)

Resulting Non linear Complementarity Problem is in effect a binding active constraint on the leader’s

Example 1: Follower’s problem

• The followers problem is a Non Linear Complementarity Problem (NCP) as follows:

• In order words for a given tuple of the leaders variables, the solution (sol) of the NCP gives the follower’s variables.

• NCP is a special case of the more general Variational Inequality Constraint (Karamardian, 1972)

Example 1: Problem Formulation

• Cooperative Form (MOPEC)

• Non Cooperative Form (NCEPEC)

Solution using MOSADE(Huang et al 2007)

NCP solved using PATH(Ferris and Munson 2000)

Solution using NDEMO(Koh 2012)

NCP solved using PATH(Ferris and Munson 2000)

Example 1: Pareto Front and NCEPEC Solution

Solution 1 Solution 2Profit Leader 1 978.89 840.86

Profit Leader 2 410.97 485.63

NCEPEC solution shown as × It lies INSIDE Pareto Front NOT PARETO OPTIMAL

Solutions reported Mordukhovich et al (2007) (Deterministic Method):

These solutions lie on the Pareto Front marked by

NCEPEC Solution:Profit Leader 1: 950.56Profit Leader 2: 414.72

Example 1: Mapping Pareto Front to NE

• Repeatedly run NDEMO for different α between 0 and 1

• Equivalent to Leader 1 taking into account a “proportion”(α) of the profit function of Leader 2 and vice versa

• Again PATH Solver is used to solve NCP

Example 1: Collusion Path

• Applying NDEMO iteratively assuming a parameter α between 0 and 1

• chart the “collusion” path between the NCEPEC solution (α =0) and the Pareto Front (α =1)

• Note that we cannot trace the entire Pareto Front using this method but we can locate a point on it

• This is a form of “tacit collusion” via signalling behavior

Example 2 : Transportation Example

• Leaders are the cities that compete. • The policy variable of each city is the toll level• The equilibrium constraint is “Wardrop’s Equilibrium” (a

Variational Inequality)How tolls affect leader shown through the mechanism below:

Example 2: Transportation Network

Transportation Network for Example 2 (directed graph)• Line down the middle separates city jurisdiction• City 1 controls network to the left and City 2 controls

network to the right (we call these sub networks)• Each city can set a toll on the dotted links shown in

each sub network• Toll affects route choice which in turns affect leaders

objective function (social welfare of travel in each sub network)

Example 1: Problem Formulation

• Cooperative Form (MOPEC)

• Non Cooperative Form (NCEPEC)

Solution using MOSADE(Huang et al 2007)

VI can be solved by traffic assignment algorithm once tolls are input

Solution using NDEMO(Koh 2012)

The leaders set the tolls x1 for city I and x2 for city II and the users routing constraint specified by Wardrop’s equilibrium is represented as an always active VI constraint

Example 2: Pareto Front

• Due to the scale of the axis the NCEPEC solution is very close to the MOPEC solution

• However NCEPEC solution still lies inside the Pareto Frontier and hence not Pareto Optimal (see next slide)

The solution discussed in Zhang et al (2010) is shown as by on the Pareto Front (PF) one of many possible solutions that lie on PF

Example 2: Collusion Path

• The NCEPEC solution is illustrated here with × (also reported in Zhang et al 2010)

• Repeated application of NDEMO algorithm for all α varying between 0 and 1

• We can trace the “collusion path” that forms the locus of points that links the NCEPEC solution (α =0) to the MOPEC solution (α =1)

Further Research

Demonstrated possibility of tacit collusion and signalling behaviour with applications in transportation and economicspolicy implications in regulation of competitors

Observe moves by players in the marketBut a lot of further research questions:• Is the collusion stable?• Is it possible to spot the collusive moves?• Alternative algorithms for NCEPECS?

Final Words

THANK YOU FOR YOUR ATTENTION!

Please send questions/comments to : a.koh@its.leeds.ac.uk