Institute for Transport Studies
FACULTY OF ENVIRONMENT
Cooperative and Non-Cooperative
Equilibrium Problems with Equilibrium
Constraints: Applications in Economics
and Transportation
Andrew Koh
17 Online World Conference on Soft Computing
Anywhere 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 EPECs
1. Cooperative  Multiobjective Optimization
2. 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
Co-operate
Multiobjective EPEC
(MOPEC)
Fully Competitive
Non Cooperative EPEC
(NCEPEC)
Followers Behavior
represented by some
variational inequality
parameterized in leaders’
variables
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 Domination
Compare 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 strategy
In 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 used
Lung 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
Resulting
Follower’s
Quantity
vector (y)
Leader’s
Objective
Function
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
Solutions reported Mordukhovich et al (2007) (Deterministic Method):
Solution 1
Solution 2
Profit Leader 1
978.89
840.86
Profit Leader 2
410.97
485.63
These solutions lie on the Pareto Front
marked by
NCEPEC solution shown as ×
It lies INSIDE Pareto Front
NOT PARETO OPTIMAL
NCEPEC Solution:
Profit Leader 1: 950.56
Profit 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
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
• 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)
Example 2: Pareto Front
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
• 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)
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 economics
policy implications in regulation of competitors
 Observe moves by players in the market
But 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