( ) ( ( ) ) ( ( ) ) ( ) ( )
{ , : , 0 , , 0 , } .
min"
" F x y y G x y y H x y y x y y
y
≤ = ∈ Ψ
(3)This problem is the bilievel-programming problem or the leader’s problem. The function
F is called the upper level objective and the functions G and H are called the upper level
constraint functions. Strongly speaking, this definition of the bilevel programming problem is valid only in case when the lower level solution is uniquely determined for each possible y.The quotation marks have been used to express this uncertainty in the definition of the bilievel programming problem in case of non-uniquely determined lower level optimal solutions. If the lower level problem has at most one (global) optimal solution for all values of the parameter, the quotation marks can be dropped and the familiar notation of an optimization problem arises.
The bilevel programming problem demonstrates that applications in economics, in engineering, medicine, ecology etc. have often inspired mathematicians to develop new theories and to investigate new mathematical models. The bilevel programming problem in its original formulation goes back to H.v. Stackelberg (1934), introduced a special case of such problems when he investigated real market situations. This particular formulation is called a Stackelberg game which we will give a briefly state in the following part.
3.2 Game Theory
Two classical models in the theory of oligopoly are those of Cournot (1838) and Bertrand (1883). In both models the equilibrium concept is the non-cooperative equilibrium of Nash (1950) in Game theory. Game theory (hereafter GT) is a powerful tool for analyzing situation in which the decisions of multiple agents affect each agent’s payoff. As such, GT deals with interactive optimization problems. While many economists in the past few centuries have worked on what can be considered game-theoretic models, John von Neumann and Oskar Morgenstern are formally credited as the fathers of modern game theory. Their classic book
“Theory of Games and Economic Behavior” written by von Neumann and Morgenstern (1944), summarizes the basic concepts existing at that time. GT has since enjoyed an explosion of developments, including the concept of equilibrium by Nash (1950), games with imperfect information by Kuhn (1953), cooperative games by Aumann (1959) and Shubik
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(1962) and auctions by Vickrey (1961). Citing Shubik (2002), “By the late 1980s, game theory in the new industrial organization has taken over… game theory has proved its success in many disciplines.”
The essential elements of a game are players, actions, payoffs, and information. These are collectively known as the rules of the game, and the modeler’s objective is to describe a situation in terms of the rules of a game so as to explain what will happen in that situation.
Trying to maximize their payoffs, the players will devise plans known as strategies that pick actions depending on the information that has arrived at each moment. The combination of strategies chosen by each player is known as the equilibrium. Given an equilibrium, the modeler can see what actions come out of the conjunction of all the players’ plans, and this tells him the outcome of the game.
To predict the outcome of a game, the modeler focuses on the possible combination of strategy s, since it is the interaction of the different players’ strategies that determines what happens. The distinction between strategy combinations, which are sets of strategies, and outcomes, which are sets of values of whichever variables are considered interesting, is a common source of confusion. Often different strategy combinations lead to the same outcome.
Predicting what happens consists of selecting one or more strategy combination as being the most rational behavior for all player i acting to maximize his payoff π . That is, an i equilibrium
(
*2 *)
* 1
*
s
,s
,...,s
ns
= is a strategy combination consisting of a best strategy for each of the n players in the game. If no player has incentive to deviate from his strategy given that the other players do not deviate, the strategy combinations
* is known as a Nash equilibrium. Formally,(
i i) (
i i i)
ii
s s s s s
i
, *, * ≥ ' , * , ∀ '∀
π
−π
− wheres
*−1 refers to the strategies chosen by the other players except player i.3.2.1 Stackelberg Games
The investigation of bilevel programming problems is strongly motivated by (real world) applications. In his monograph about market economy, Stackelberg (1934) used by the first
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time an hierarchical model to describe real market situations. This model especially reflects the case that different decision makers try to realize best decisions on the market with respect to their own, generally different objectives and that they are often not able to realize their decisions independently but are forced to act according to a certain hierarchy. We will first consider the simplest case of such a situation where there are only two acting decision makers.
Then, this hierarchy divides the two decision makers in one which can handle independently on the market (the so-called leader) and in the other who has to act in a dependent manner (the follower). A leader is able to dictate the selling prices or to overstock the market with his products but in choosing his selections he has to anticipate the possible reactions of the follower since his profit strongly depends not only on his own decision but also on the response of the follower. On the other hand, the choice of the leader influences the set of possible decisions as well as the objectives of the follower who thus has to react on the selection of the leader.
It seems to be obvious that, if one decision maker is able to take on an independent position (and thus to observe and utilize the reactions of the dependent decision maker on his decisions) then he will try to make good use of this advantage (in the sense of making higher profit). The problem he has to solve is the so-call Stackelberg game, which can be formulated as follows: Let X and Y denote the set of admissible strategies x and y of the follower and of the leader, respectively. Assume that the values of the choices are measured by means of the functions
f
L(x , y
) andf
F(x , y
), denoting the utility functions of the leader resp. the follower. Then, knowing the selection y of the leader the follower has to select his best strategy x(y) such that his utility function is maximized on X:} : ) , ( { max :
) ( )
( y y Arg f x y x X
x
Fx
∈
= Ψ
∈
(4)Being aware of this selection, the leader solves the Stackelberg game for computing his best selection:
)}
( , : ) , ( {
"
max
" f
Fx y y Y x y
x
∈ ∈ Ψ
(5)If there are more than one person on one or both levels of the hierarchy, then these are assumed to search for an equilibrium (as e.g. a Nash or again a Stackelberg equilibria)
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between them.
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4 THE MODEL OF CHANNEL COMPETITION GAME
The model we demonstrate is mentioned in N. Singh and X. Vives’ (1984) paper and here we extend it into a two-echelon manufacturer-retailer market.