Network Markets for Homogeneous Goods
Project leaders: Alexander Vasin
Consultants: Francisco Marhuenda, Polina Vasina.
Abstract. The
project aims to study network markets for electricity, gas and other
homogeneous goods. One problem under consideration is to develope a method for evaluation of
imperfect competition impact on the network auctions. Another task is to study the market including
vesting contracts between producers and consumers besides the day before auction. For every model,
we aim to find Nash equilibrium outcome and compare it with Walrasian outcome.
1. Setting of the problem.
The main objects of this project are as follows. Construction of a game-theoretic model for
the electricity market including vesting contracts under uncertain demand at the first stage and a
network supply function auction at the second stage. Evaluation of market power of market agents
depending on characteristics of the market, in particular, for the network electricity auction .
Development of methods for analysis and computation of Nash equilibria. Obtaining
recommendations on the market rules in order to reduce the market price and the loss of the social
welfare.
Importance of the study. On the 1st of November 2003 the competitive segment was
introduced on Wholesale Russian Electricity Market. It allows to sell up to 15 % of working capacity
(or to buy up to 30% of hour electricity consumption) at marginal node prices or through the system of
vesting contracts. During last 17 months the market has developed significantly, the number of
participants of bargains has grown from 14 till 90, and now there are up to 23 sellers and up to 57
buyers (some participants act as producers and as consumers simultaneously), including such
companies as Sverdlovenergo, Tumenenergo, Rosenergoatom, Lenenergo, Novgorodenergo,
Permskaya GRES, Transnefteservis Verhnevolzhskih GES Cascade. The daily consumption volume of
the competitive sector reaches 185 mln kWth, that is about 11 % of the total consumption of European
Russia and Ural. About 25% of this value is sold via vesting contracts. Besides participants of
regulated wholesale market, the large consumers of retail market, where regulated prices are
significantly higher than in the wholesale market, gradually enter the competitive sector. In May of
2005 it is planned to spread the market area on the territory of Siberia. The actual model of the market
should change in 2006: the regulated sector of market will be transformed into the market of longlasting regulated vesting contracts.
The behavior of agents on Russian electricity market as well as conditions of its functioning
are not yet settled. For more developed markets the data (see Baldick et al., 2000) shows that large
firms actively use market power for lifting prices and their revenue in comparison to a competitive
equilibrium. In our previous papers (Vasin et al., 2003, 2005) we obtain a method for evaluations of
the Nash equilibrium deviation from the competitive equilibrium for a model of the local auction
1
under the current market rules and also for the Vickrey auction with reserve prices. The following
table shows competitive equilibrium price and the expected deviation from it depending on demand
elasticity, market structure and the auction type for the Central economical zone of Russia.
Table. Walrasian, Cournot,Vickrey and modified Vickrey prices for the Central economical
zone of Russia, market structures with 3 and 5 companies,  - the slope of the demand line.
p


~p
p5* ~
p
p3* ~
p
pV 5 ~
p
pV 3 ~
p
pV 5 ~
p p ~
V3 p
0.1
0,1
135
4.24
5.65
1.59
2.19
0.51
0.62
0.2
0,2
150
2.45
3.10
1.49
1.92
0.44
0.57
0.4
0,4
172.5
1.56
1.87
1.49
1.76
0.42
0.49
0.6
0,6
219.67
1.15
1.34
1.30
1.46
0.33
0.38
However, the current plan of the electricity market developement does not include
mechanisms similar to Vickrey auction. According to this plan, the number of large generating
companies is about 70. Thas, if we proceed from the spot market model, the expected deviation from
the Walrasian price andthe welfare loss are sufficiently small. However, for more accurate estimation
of possible deviation it is necessary to investigate the standard supply function auction with account
of network structure of the market. In contrast to many other countries, this structure is essential for
price determination at the Russian market due to rather large transmission losses and transmission
restrictions on some parts of the network. In our paper (2005) we made an analysis of Nash equilibria
for a supply function auction on the simple network with two nodes. Recently we generalized these
results for a linear network where all nodes are on one line. Our discussion with RAO UES experts
shows that an important problem is to develope an efficient method for Cournot equilibrium
computation for networks with cicles. The difficulty is that electricity networks are more complicated,
they work under Kirhgoff laws. In particular, transmitting capacities are limited not on the edges but
on cross sections of the network.
Another important task is to study the impact of vesting contracts on the market Nash
equilibrium. According to the current plan, such contracts are to play an important role in price
stabilization. However, this assumptions is dautful and requires investigation. In contrast to the day
before auction, the contracts are made under significant uncertainty on the future supply-demand
relation. Papers in this sphere (Дьякова, 2003, Wolfram, 1999) consider a local market with linear
supply theoretical and empirical functions that is not in accordance with Russian wholesale market
rules and cost structure of energetic companies. We plan to study a market with step cost functions and
network structure.
2
2. Literature Review.
Our paper Vasin et al (2005) provides the last findings on existence and properties of the
Nash equilibrium for the Cournot oligopoly and the model of competition via supply
functions. In every case, the underlying market includes a fixed finite number of producers
that are heterogeneous in production capacities and non-decreasing marginal costs of
production. Consumers do not play any active role in the models. Their behavior is
characterized by the demand function that is the common knowledge.
We show that there exists a unique Nash equilibrium in the Cournot model for any nonincreasing demand function with the non-decreasing demand elasticity under mild
assumptions on the demand asymptotics as the price tends to infinity. We develop a
descriptive method for computation of the Cournot outcome under any affine demand
function and piece-wise constant marginal costs of producers. In the general case, we obtain
an explicit upper estimate of the deviation of the Cournot outcome from the Walrasian
outcome proceeding from the demand elasticity and the maximal share of one producer in the
total supply at the Walrasian price.
Amir (1996) and Amir and Lambson (2000) study existence and uniqueness of the Nash
equilibrium in the Cournot model for logconvex and logconcave inverse demand functions.
(Note that D 1 (v) is concave (convex) iff p | D ( p) | increases (decreases) in p .) Thus, the
first property is stronger than increasing of the demand elasticity while the second may hold
or not hold in our case. A typical example of the demand function with increasing elasticity
that does not meet the both properties is the demand for a necessary good with the low
elasticity for low prices and the high elasticity for high prices, such that consumers prefer
some substitute.
We consider a model where the market price is determined by the sealed bid auction, and
producers set multi-step supply functions as their strategies. We show that, besides the
Cournot outcome, there exist other Nash equilibria. For any such equilibrium the cut price lies
between the Walrasian price and Cournot price. Vice versa, for any price between the
Walrasian price and the Cournot price, there exists the corresponding equilibrium. However,
we show that only the Nash equilibrium corresponding to the Cournot outcome is stable with
respect to some adaptive dynamics under general conditions.
This result echoes Moreno and Ubeda (2002) who obtained a similar proposition for a
two-stage model where at the first stage producers choose production capacities, and at the
second stage they compete by setting the reservation prices. The difference is that in our
3
model the Cournot type equilibrium always exists under fixed production capacities since the
agents set the production volumes as well as the reservation prices.
Our results differ from Klemperer and Meyer (1989) who study competition with
arbitrary supply functions reported by producers. Under similar conditions, they obtain an
infinite set of Nash equilibria corresponding to all prices above the Walrasian price. Our
constraint that permits only non-decreasing step functions is reasonable in context of
development of electricity markets.
The estimates of the Cournot outcome deviation from competitive equilibrium as well
as the results of calculations for the concrete market show that market price in the supply
function auction can essentially (3-5 times) exceed the Walrasian price under the current
market organization. Thus, investigation of alternative variants of auction organization is of
great theoretical and practical interest. Our report(2005) considers Vickrey auction with
reserve prices. In such auction the cut-off price and production volumes are determined in the
same way as in the standard supply function auction. However, the good obtained from a
producer is paid at the reserve prices. The marginal price is a minimum of the marginal cost
of the same output for other producers and the marginal reserve price of this output for
consumers. The marginal cost is calculated on the basis of reported supply functions, but in
this case reporting the actual costs and production capacities is a weakly dominating strategy.
In absence of information on production costs the guaranteed value of total profit reaches its
maximum at the corresponding Nash equilibrium.
Our results generalize the results of Ausubel and Cramton (1999) who studied Vickrey
auction on trading a divisible good. In their model the players are consumers. Moreover, we
show that the specified outcome corresponds to the so-called truthful equilibrium for the
menu auction introduced in the paper by Bernheim and Whinston (1986), see also Bolle,
2004. At this equilibrium each producer obtains the profit equal to the increase of the total
welfare of all participants of the auction due to his participation in the auction. However, the
construction of this equilibrium in the specified papers needs the complete information on
4
consumers’ reserve prices (in our case, on production costs). In framework of the Vickrey
auction, the equilibrium in dominant strategies realizes this outcome under any actual cost
functions and private information of each participant on his function.
Our calculations for the Central Economic Region of Russia show that Vickrey
auction price for consumers exceeds the Walrasian price only 1,5 times (to compare with 3,55 times for the standard auction). However, such increase seems to be also rather essential.
Besides, there exists reasonable arguing that participants of the auction typically would not
reveal their actual costs, that is, the specified equilibrium in dominant strategies is not realized
(see Rothkopf et al., 1990). The main argument is that reporting the actual costs gives an
advantage to the auctioneer (and also to other economic partners) in the further interactions
with this producer.
The situation differs significantly if the marginal costs and the maximal capacity of
each generator are a common knowledge, and the uncertainty relates to a decrease of
capacities due to breakdowns and repairs. In this case current information on the working
capacities is weakly correlated with the future state, and the specified argument against
revealing the actual costs turns out to be invalid. Moreover, the common information may be
used for redistribution of the total income in favor of consumers. We specify the rule for
calculation of reserve prices with account of such information. This rule provides the maximal
guaranteed value of the total profit of consumers. Under this rule, reporting the actual
producer’s characteristics stays his dominant strategy, and the total welfare still reaches the
maximum.
The second part of our paper (2005) considers a simple network market – the market
with two nodes. As above, each local market is characterized by the demand function and the
finite set of producers with non-decreasing marginal costs. For every producer his strategy is
supply function that determines his supply of the good depending on the price. The markets
are connected by the transmitting line with the fix share of loss under transmission. Under
given strategies of producers, the network administrator first computes the cut prices for the
separated markets. If the ratio of the prices is sufficiently close to one then transmutation is
unprofitable with account of the loss. In this case, the outcome is determined by the cut prices
for isolated markets. Otherwise the network administrator sets the flow to the market with the
higher cut price (for instance market 2). This flow reduces the supply and increases the cut
price at the market 1. Simultaneously it increases the supply and reduces the cut price at the
market 2. The network administrator determines the transmitted volume so that the ratio of
5
the final cut prices corresponds to the loss coefficient. Thus, he acts as if perfectly
competitive intermediaries transmit the good from one market to the other. It is easy to show
that such strategy maximizes the total welfare if the reported supply functions correspond to
the actual costs.
First we consider the Cournot competition model for this market. Our study shows that
there exist three possible types of Nash equilibrium: 1) an equilibrium with zero flow between
the markets and the ratio of the prices close to 1; 2) an equilibrium with a positive flow and
the ratio of the prices corresponding to the loss coefficient; 3) an equilibrium with a flow
equal to a transmitting capacity of the line. We develop a method how to compute them. We
study under what conditions the local equilibrium is a real Nash equilibrium. For the market
with constant marginal costs and affine demand functions, we determine the set of Nash
equilibria depending on the parameters.
A generalization of these results for the network with general structure is an important
and non-trivial problem.
3. Research Methodology. We propose to develop methods of Nash equilibrium computation
for a model of competition via supply functions and Vickrey auction with reserve prices and to obtain
the estimates of for the Nash equilibrium deviation from the competitive equilibrium, proceeding from
the following models (see also Vasin et al., 2003).
The basic model of the local market.
Consider a market with a homogenous good and a finite set of producers A . Each
producer a is characterized by his cost function C a (v) with the non-decreasing marginal cost
for v  [0,V a ], where V a is his production capacity. The precise form of C a (v) is his
private information. The practically important case is where the marginal cost is a step
function: C a (0)  0, C a ' (v)  cia for v  (Via1 ,Via ), i  1,..., n, V0a  0, Vna  V a . Consumers’
behavior is characterized by the demand function D ( p ) which is continuously differentiated,
decreases in p and is known to all agents.
Consider a model of Cournot competition for this market. Then a strategy of each
producer a is his production volume v a  [0,V a ] . Producers set these values simultaneously.


Let v  (v a , a  A) denote a strategy combination. The market price p (v ) equalizes the

demand with the actual supply: p(v )  D 1 (  v a ). The payoff function of producer a
aA


determines his profit f a (v )  v a p(v )  C a (v a ). Thus, the interaction in the Cournot model
6
 
corresponds to the normal form game С  A, [0,V a ], f a (v ), v   [0,V a ], a  A , where
aA
[0,V a ] is a set of strategies a  A .
Recall basic definitions.
For a game   A, S a , f a( s ), s  S , a  A  , strategy combination s *  ( s a* , a  A) is
Nash equilibrium (NE) if f a ( s * )  f a ( s * || s a ) for any a , R a .
Combination (v a* , a  A) of production volumes is Cournot equilibrium (CE) if it is a NE
in the game  C .
Combination (v~ a , a  A) of production volumes is a Walrasian equilibrium (WE) and ~p
def
is a Walrasian price of the local market if, for any a , v~ a  s a ( ~
p)  Arg max (v a ~
p  c a (v a )) ,
va
 v~
a
 D( ~
p) .
a
Let (v a *, a  A) denote Nash equilibrium production volumes and p *  D 1 (  v a *) be
aA
the corresponding price. A necessary and sufficient condition for this collection to be Nash
equilibrium is that, for any a ,
p *  Arg
max





p D 1   vb* V a , D 1   vb*  
b

a
b

a





D p  Dp  v p  C D p  Dp  v .
*
a*
a
*
a*
Then the F.O.C. for Nash equilibrium is
v a*  ( p*  C a' (v a* ) | D' ( p* ) | , for any a s.t. C a ' (0)  p *
(1)
v a*  0 if C a' (0)  p* ,
(2)
where C a' (v)  [Ca' (v), Ca' (v)] in the break points of the marginal cost function.
The supply function auction.
Consider the following closed bid auction: every producer a  A simultaneously sends to
the auctioneer his reported supply (r-supply) function R a ( p) that determines the amount of
the good this producer is ready to sell at price p, p  0 . Below we assume that R a ( p) is a
non-decreasing step function with a limited number of steps. The combination of r-supply
functions determines the total r-supply R( p)   R a ( p) and the cut price c~( Ra , a  A)
a
~ )  R( c
~) .
that meets condition D(c
7
In order to define the payoff functions, we should consider two cases. Let
def
def
~
~

R ( p)  sup R( p), R ( p)  inf R( p) . If Ra  (c )  D(c ) then each producer sells the

reported volume Ra (c~) at the cut price. Otherwise, first each producer sells Ra (c~) , and
then
the
residual
demand
D(c~)  R (c~)
is
distributed
among
producers
with
~
~
Ra (c )  Ra (c ) according to some rationing rule.
Note that there are three possible types of Nash equilibria for S : a) those Nash equilibria
for which
~
 ~
R (c )  D(c ) (Nash equilibria without rationing), b) those for which
D(c~ )  ( R (c~), R (c~))
(Nash
equilibria
with
rationing),
c)
those
for
which
D(c~)  R (c~)  R (c~) (Nash equilibria with a barrier, see Figure 2).
Figure 1.
a)
b)
c)
Proposition 1.3. a) For every Nash equilibrium without rationing, the production volumes
correspond to the local Cournot equilibrium. Vice versa, if (v a , a  A) is a Cournot
equilibrium, then the corresponding Nash equilibrium exists in S .
b) If ( R a , a  A) is a Nash equilibrium such that, D(c~ )  ( R (c~), R (c~)) , then it meets one
of the following conditions:
1) there exists at most one producer b  A such that R b (c~)  S b (c~) (so
p, p * ) .
v a  S a (c~) for any a  b ); the cut price lies in the interval ( ~
8
p.
2) c~  ~
c) For any Nash equilibrium of the type c), the cut price lies in the interval [ ~
p, p* ] . Vise
p [ ~
p, p* ) there exists a Nash equilibrium ( R a , a  A) such that
versa, for any
c~ ( R a , a  A)  p .
Thus, the bound of deviation obtained for the Cournot outcome is valid for any Nash
equilibrium of this auction.
A network market with two nodes
Consider two local markets connected with a transmitting line. Every local market l  1,2
is characterized by the finite set A l of producers, | Al | nl , the cost functions C a (v), a  Al ,
and demand function D l ( p), in the same way as the local market in p.1. Agents’ strategies
( R a ( p), a  Al ) – rsupply functions – are defined as in p.1. For a given strategy combination,
the node cut prices c~ l and transmitted volume q are determined as follows.
Let k  (0, L) be the loss coefficient that shows the share of the lost good (in particular,
the electric power) under transmission from one market to the other, Q is the maximal
amount of the transmitted good. Let c l ( R ), l  1,2 denote the cut prices for isolated markets,
  (1  k ) 1 . If 1  c 2 ( R ) / c 1 ( R )   then q  0 , c~ l ( R )  c l ( R ), l  1,2 , that is, the
markets stay isolated. If c 2 ( R ) / c 1 ( R )   then q (the transmitted volume from market 1 to
market 2) is a solution of the system
If q  Q then q  Q , c~ i are determined from (3),(4) with q  Q , c~ 2  c~1 . The
capacity constraint is binding in this case.
Consider the Cournot competition in this model. Then each producer sets R a ( p)  v a .
The
first-order
conditions
for
the
first
type
outcome
with
prices
p1* , p2* s.t. 1  p2* p1*   , to be a Cournot equilibrium are quite similar to the
conditions (1.1), (1.2) for the local market:
v a *  ( pi*  C a' (v a *)) | Di ' ( pi* ) | , for any a  Ai s.t. C a ' (0)  pi*
(7)
v a *  0 if C a ' (0)  pi* , where C a ' (v)  [C a ' (v), C a ' (v)] in the break points of
the marginal cost function.
Besides that,  v a *  D( pi* ), i  1, 2 .
Ai
9
For the second-type outcome with q  (0, Q), p1*  p2* , the first-order conditions of the
Cournot equilibrium are obtained in a similar way
v a *  ( p1*  C a ' (v a *)) | D1 ' ( p1* )  2 D 2 ' (p1* ) | for any a  A1 s.t. C a * (0)  p1* ,
va*  0
if C a ' (0)  p1* .
Similarly, producers in the market 2 face the demand D 2 (p1 )  1  ( D1 ( p1 )   v a ) , and
A1
v a *  (p1*  C a ' (v a *)) | D 2 ' (p1* )  D1 ' ( p1* ) / 2 |
for any a  A 2
s.t. C a ' (0)  p 2* , v a *  0 if
(13)
C a ' (0)  p 2* .
(14)
The given conditions are necessary but not sufficient for strategy combination to be a
Nash equilibrium.
Consider a local equilibrium of the type 1 (with zero flow between the markets).
Under a sufficiently large increase of production volume by player a , the price in the market
1 reduces to the level p1  p2*  . The further increasing of the volume permits the player to
sell his production also at the market 2. Figure 3 shows the demand function for this producer.
Figure 3.
D
D1 ( p1 )   v b*   ( D 2 (p1 )   v b* )
A1 \ a
p2* 
A2
p1*
p1
The optimal strategy of agent a under the demand D1 ( p1 )   v b*   ( D 2 (p1 )   v b* )
A1 \ a
A2
corresponds to the price p1 s.t.



D1 ( p1 )   v b*   ( D 2 (p1 )   v b* )  v a  ( p1  C a (v a )) | D1 ( p1 )  2 D 2 (p1 ) |
A1 \ a
(20)
A2
10
Proposition 2.2. The local equilibrium point of the type a) is not a Nash equilibrium if and
only
if
for
some
producer
a  A1
the
optimal
price
p1
under
demand
D1 ( p1 )   v b*   ( D 2 (p1 )   v b* ) is less than p2*  and the payoff under this price
A1 \ a
A2
exceeds f a (v * ) or, for some producer a  A 2 the optimal price p2 under demand
D 2 ( p2 )   v b*   ( D1 (p2 )   v b* ) is less than p1*  and the payoff under this price
A2 \ a
A1
exceeds f a (v * ) .
Note. In this proposition we assume that transmission capacity constraint q  Q is not
binding.
The necessary and sufficient condition for strategy combination to be a Nash
equilibrium of the second and third types are obtained in a similar way (see our paper, 2003).
Topics for investigation: a generalization of results for a larger number of nodes;
study of equilibria set structure depending on parameters of the model; study of the network
Vickrey auction.
Vesting Contracts.
We propose to study the following 2-stage model: at the first stage, producers and consumers make
vesting contracts under uncertainty about the future demand liable to exogenous shocks. At the second
stage, the agents take part in the Cournot auction, where the bids of producers are limited with the
residual production capacities and consumers’ behavior is determined by the residual demand
function. There are several variants of the formal description for the firs t stage of the model: Cournot
competition, Bertrand competition, the regulated market (according to the current plan of the market
organization). Kamat and Smuel (2005) consider some models in the similar framework.
Topics for investigation: study different variants of this 2-stage model.
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11
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American Economic Review, 89(4), 805-825
17. А.А. Васин, П.А. Васина, 2005, «Модели конкуренции функций предложения и их
приложение к сетевым аукционам»,ERRC, отчет по проекту РПЭИ R03-1011
18. Kamat, R. and S.O.Shmuel “Two-settement Systems for Electricity Markets under Network
Uncertainty and Market Power”
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4. Topics for the NES students MA dissertations.
1) Analysis of Nash equilibria for three- and four-node cyclic network auctions of supply functions.
2) Evaluation of the deviation of Nash equilibrium from the competitive equilibrium for a network
auction of supply functions.
3) Studying of the network Vickrey auction model.
4) Development of the market model including vesting contracts and the supply function auction.
5) The optimal strategy of an allocated generating company in the network auction.
6) Analysis of the 2-stage market model including vesting contracts and the supply function auction
(several variants depending on the type of competition at the first stage).
5. Brief CVs.
Alexander Vasin
Professor, Operations Research Department,
Faculty of Computational Mathematics and Cybernetics, Moscow State University and New Economic
School, Moscow, Russia
Russia
Russian (native speaker) and English
1-st Tverskaya-Yamskaya, 13, apt 114, Moscow, 125047, Russia
(095) 250-31-98 (home); (095) 939-24-91 (office)
[email protected]
Doctor of Sciences (Doktor Nauk, the degree after Ph.D in Russia) in Applied Mathematics, Academy
of Sciences of the USSR, 1990
Dissertation: "Evolutionary Models and Optimality Criteria of Collective Behavior"
More than 60 publications including
1. A.A.Vasin. "Strong Equilibrium Points in Some Supergames". Moscow University Mathematics
and Mechanics, Allerton Press, New York, N1, 30-39.(1978)
2. A.A.Vasin. "Models of Dynamics of Collective Behavior" (in Russian). Moscow University Press,
Moscow.(1989), 150p.
3. A.A.Vasin."On the Convergence to Competitive Equilibrium and the Conditions of Pure
Competition". Contributed Paper for IEA X World Congress, Moscow, 1992.
4. E. Boros, V.A. Gurvich, A.A.Vasin."Stable families of coalitions and normal hypergraphs",
Mathematical Social Sciences 34 (1997) 107-123.
5. A.A.Vasin."The Folk theorem for dominance solutions", International Journal of Game Theory
(1999) 28:15-24.
6. A.A.Vasin."On stability of mixed equilibria", Nonlinear Analysis 38 (1999) 793-802.
7. A.A.Vasin. “Tax enforcement and corruption in fiscal administration”, the XII World Congress of
the International Economic Association, Buenos Aires (1999).
8.A.Vasin, E.Panova. Tax collection and corruption in fiscal bodies. EERC working paper N 99-10, 26
p. Moscow (2000)
9.A.Vasin, P.Vasina . The optimal tax enforcement under imperfect taxpayers and auditors. New
Economic School Working paper N 2000/012, 25 p. Moscow (1999).
10.A.Vasin.”Endogenous Determinations of Utility Functions: an Evolutionary Approach”
INTERNATIONAL CONGRESS of MATHEMATICIANS. Game Theory and Applications. Satellite
Conference. Proceeding volume, 785-798 (2002).
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11.A.Vasin.”The Folk Theorems in the Framework of Evolution and Cooperation” The Tenth
International Symposium on Dynamic Games and Applications. Proceedings, volume 2, 852-857
(2002).
Conferences
III Congress of Europian Society for Evolutionary Biology. Debrecen, Hungary, September, 1991, IV
Congress of ESEB, Montpellier, France, 1993.
Meetings of Society for Social Choice and Welfare, Caen, France, 1992., Alicante, 2000
International Economic Association X World Congress, Moscow, Russia, 1992, IEA XI World
Congress, Tunis, 1995, IEA XII World Congress, Buenos Aires, 1999, XIII Congress, Lisbon, 2002.
Game Theory and Applications (Satellite conference of the ICM), Qingdao, 2002.
X International Symposium “Dynamic Games and Applications”, St-Petersburg, 2002
Polina Vasina
Contacts
work phone 128-93-31, email: [email protected]
Working experience
2002- scientific researcher, Computational Center of RAS, dept. of Mathematical Modeling of
Technical Systems
2002- consultant,, Carana corp., project «General Support of Reforming Process in Energetics»
2000 (june-august) – Summer School (Dynamic Systems Project), International Institute of Applied
System Analysis, Laxenburg, Austria
Grants
EERC Grant, 2001
Scientific Degree: Ph D in Applied Mathematics(2002), Thesis title «Game-Theoretical Optimization
Models for Tax System»
Education
1999-2002
1999-2001
1994-1999
Ph D Program, Faculty of Computational Mathematics and Cybernetics, Moscow
State University
Master Program, Faculty of Economics, State University – Higher School of
Economics
Student , Faculty of Computational Mathematics and Cybernetics, Moscow State
University
Francisco Marhuenda Hurtado
Universidad Carlos III, Spain
Position: Full Professor, Department of Economics
Education
Ph D in Mathematics (1985-1990), University of Rochester, New York
Publications: more than 20 including
- “Distribution of Income and Aggregation of Demand”, Econometrica, Vol.63, No 3 (1995), pp.
647-666
- (joint with Miguel Gines) “Efficiency, monotonicity and rationality in economies with public
goods”, Economic Theory, 12, (2), 1998, 423-432
- (joint with I. Ortuno Ortin) “Income taxation, uncertainty and stability”, Journal of Public
Economics, 67, 1998, pp. 285-300
Additional funding: we shall try to get some from RAO UES.
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