Chance constrained games
Prof. Abdel Lisser ([email protected])
Game theory is a formal discipline which studies situations of competition and cooperation between several involved players. Game theory has a large number of applications
ranging from strategic questions in warfare to understanding economic competition, from
robotics to social networks, from cloud/distributed computing to network security; and
this list is certainly not exhaustive.
A milestone in the history of game theory is the paper of John von Neumann [5] where
he proved the famous minimax theorem for zero-sum games. He showed that there exists
a mixed strategy saddle point equilibrium for a two player zero sum matrix game. Later,
John Nash [4] showed that there always exists a mixed strategy Nash equilibrium for an nplayer general sum game with finite number of actions for each player. There two seminal
results in game theory considered that the players’ payoffs are deterministic. However,
there can be practical cases where the players’ payoffs are better modeled by random
variables following certain distributions and as a result players compete in a stochastic
Nash game. The wholesale electricity markets are the good examples that capture this
situation.
One way to study stochastic Nash games is by using expected payoff criterion. Ravat and Shanbhag [7] considered stochastic Nash games using expected payoff criterion.
Jadamba and Raciti [2] used a variational inequality approach on probabilistic Lebesgue
spaces to study stochastic Nash games. The expected payoff criterion is more appropriate
for the cases where the decision makers are risk neutral. A risk averse payoff criterion
based on chance constraint programming [1, 6] has also received some attention in electricity market [3]. These games are called chance-constrained games. The main topic of
this thesis is the stochastic Nash game problems where each player is interested in the
payoffs that can be obtained with certain confidence. In this case, chance constraints will
be used to model these games.
In non-cooperative games each player is interested in maximizing his payoff which also
depends upon the strategies of other players. Therefore, the players seek for a Nash
equilibrium strategy profile from where there is no incentive for any player to deviate
unilaterally.
We consider n-player finite strategic games with random payoffs. Let Ai be finite
action set of player i. We denote the random payoff vector of player i by
Qn(ri (a))a∈A where
a = (a1 , a2 , · · · , an ) is an action profile of all the players and A = i=1 Ai denote the
set of all action profiles of the players. Let τi be a mixed strategy of player i which is a
probability distribution over action set Ai . We denote the set of all
Q mixed strategies of
player i by Xi , and the set of all mixed strategy profiles by X = ni=1 Xi . For a mixed
strategy profile τ = (τ1 , τ2 , · · · , τn ) ∈ X the payoff of player i is given by
ri (τ ) =
n
XY
τj (aj )ri (a).
a∈A j=1
For each τ , the payoff of each player i is given by a random variable ri (τ ). We consider the
case where each player is interested in the payoffs which can be obtained with a certain
confidence. Let αi ∈ [0, 1] be a confidence level of player i and α = (αi )ni=1 denotes a
confidence level vector. For a given strategy profile τ ∈ X, and a given confidence level
1
vector α, the payoff of player i is given by
uαi i (τ ) = sup{γi |P (ri (τ ) ≥ γi ) ≥ αi }.
∗
) is called a Nash equilibrium of a
For an α ∈ [0, 1]n , a strategy profile τ ∗ = (τi∗ , τ−i
chance-constrained game if for each i = 1, 2, · · · , n, the following inequality holds
∗
∗
uαi i (τi∗ , τ−i
) ≥ uαi i (τi , τ−i
), ∀ τi ∈ Xi .
We define the set of best response strategies of player i against a given strategy profile
τ−i of other players by
BRiαi (τ−i ) = {τ̄i ∈ Xi |uαi i (τ̄i , τ−i ) ≥ uαi i (τi , τ−i ), ∀ τi ∈ Xi } .
Let P(X) be a power set of X. Define a set valued map G : X → P(X) such that
n
Q
G(τ ) =
BRiαi (τ−i ). Then, it is easy to see that a Nash equilibrium of a chancei=1
constrained game is a fixed point of G.
Thesis objectives
The candidate will be familiarized with the large and rich literature on stochastic game
theory. The project will start from our recent results and progressively extend them to
the cases defined in the below mentioned first objective of the thesis. The candidate
will benefit from our existing results and experience recently developed on this topic
[9, 10, 11]. We expect to extend the state-of-art results on the chance-constrained games
beyond Elliptical distributions which have already been covered in our previous work
[8]. Optimization methods will be subsequently considered for computing the equilibrium
solutions. The objectives of this project are threefold:
• Existence of Nash equilibrium In this project, we will consider an n-player strategic game where the action set of each player is finite/infinite and the payoff vector of
each player is a random vector. We formulate this problem as a chance-constrained
game by considering that the payoff of each player is defined using a chance constraint. The objective is to show the existence of a Nash equilibrium for a chanceconstrained game in the following cases: independent/dependent random variables,
general probability distributions, cooperative/noncooperative games, static/dynamic
games.
• Computation of Nash equilibrium For the case where the Nash equilibrium is
characterized in the previous step, the candidate will study and use different deterministic and stochastic optimization approaches to calculate the equilibrium solution.
• Applications Recently, the chance-constrained games have received a particular attention in electricity market [3]. The sources of uncertainties could be the customers’
stochastic demand and the presence of wind generators on electricity market. The
studied chance-constrained game framework in this project will be applied to analyze
these games.
The proposal involves methods from various areas: chance constraints, stochastic optimization, probability theory, convex optimization, etc. We do not expect the candidate to
master and contribute to each of these topics. However, we do request strong background
in mathematics namely probability theory, and also convex optimization together with
good programming skills.
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Required background
The candidate should have the following backgrounds:
• A master degree in computer science with strong background in mathematics or
master degree in applied mathematics with experience in convex optimization and
probability theory.
• Good programming skills.
References
[1] A. Charnes, W. W. Cooper, Deterministic equivalents for optimizing and satisficing
under chance constraints, Operations Research 11 (1) (1963) 18–39.
[2] B. Jadamba, F. Raciti, Variational inequality approach to stochastic Nash equilibrium
problems with an application to Cournot oligopoly, Journal of Optimization Theory
and Application 165 (3) (2015) 1050–1070.
[3] M. Mazadi, W. D. Rosehart, H. Zareipour, O. P. Malik, and M. Oloomi. Impact of
wind integration on electricity markets: A chance-constrained Nash Cournot model.
International Transactions on Electrical Energy Systems, 23(1):83–96, 2013.
[4] J. F. Nash. Equilibrium points in n-person games. Proceedings of the National Academy
of Sciences, 36(1):48–49, 1950.
[5] J. V. Neumann, On the theory of games, Math. Annalen 100 (1) (1928) 295–320.
[6] A. Prékopa, Stochastic Programming, Springer, Netherlands, 1995.
[7] U. Ravat, U. V. Shanbhag, On the characterization of solution sets of smooth and
nonsmooth convex stochastic Nash games, SIAM Journal of Optimization 21 (3) (2011)
1168–1199.
[8] V. V. Singh, O. Jouini, and A. Lisser. Existence of Nash equilibrium for chanceconstrained games. Operations Research Letters, 44:640–644, 2016.
[9] Vikas Vikram Singh, Oualid Jouini, and Abdel Lisser. Distributionally robust chanceconstrained games: existence and characterization of Nash equilibrium. Optimization
Letters, pages doi:10.1007/s11590–016–1077–6, 2016.
[10] Vikas Vikram Singh, Oualid Jouini, and Abdel Lisser. Equivalent nonlinear complementarity problem for chance-constrained games. Electronic Notes in Discrete Mathematics, 55:151–154, 2016.
[11] Vikas Vikram Singh, Oualid Jouini, and Abdel Lisser. Operations Research and
Enterprise Systems, chapter Solving Chance-Constrained Games Using Complementarity Problems, pages 52–67. Communications in Computer and Information Science,
Springer, 2017.
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