Consider a perfectly competitive problem with F non-cooperative players simultaneously making decisions to minimize their respective objective functions while treating prices and competitors’ decisions exogenously. In a specialization of (1.1), let player i solve the problem
minimize
xi∈Ki
fi(xi; x−i, p), (2.5)
where the vector p ∈ RL represents L commodity prices and for each p, the mapping fi( • ; x−i, p) : U → R is a continuously differentiable, convex function defined on some
open convex proper superset U of Ki , {x ∈ Rn+i | Aix ≥ bi} with Ai ∈ Rmi×ni and
sufficient for optimality: 0 ≤ xi ⊥ ∇xifi(xi; x−i, p) − A T i λi ≥ 0 0 ≤ λi ⊥ Aixi− bi ≥ 0, (2.6)
where λi is the Lagrange multiplier vector for the constraint Aixi− bi ≥ 0.
In a perfectly competitive framework, players may both buy commodities (such as re- sources) from supply markets and sell commodities (the products) in demand markets; their purchases and sales determine the commodity prices described by the vector p. Unlike the classical Arrow-Debreu general equilibrium problem [7], a partial equilibrium problem stipulates the way that prices are derived. One such stipulation is through an econometric model wherein prices are determined by explicit inverse demand/supply functions. With xi` denoting player i’s production for market `, an example of such a function is a linear
inverse supply function given by
p`(x) = P0` − P0` Q0` F X i = 1 xi`, (2.7)
where x , (xi)Fi = 1 and the scalar P0` and the ratio
− P0 `
Q0 `
represent the price-intercept and the slope of the function for commodity `, respectively. Inverse demand functions are defined similarly. For perfectly competitive games, it is important to note that even when (2.7) is specified, it is not substituted into the objective function of (2.5) because p is taken to be exogenous by each player. Rather, the inverse supply function is substituted for p in each player’s KKT optimality conditions (2.6).
anisms; namely, p is postulated to satisfy the following market clearing condition expressed by a complementarity condition between the price and excess demand/supply:
0 ≤ p ⊥ g(p ; x) ≥ 0, (2.8)
where g is a vector-valued, continuous excess demand/supply function. As such, g(p ; x) simply requires supply to be greater than or equal to a given demand. The concept of an auction arises because a non-binding market clearing constraint (i.e., supply is greater than demand) implies a zero market price. Furthermore, each player realizes the same commodity price if it is positive. If the function g( • ; x) is integrable such that g(p ; x) = ∇pθ(p ; x) for some scalar function θ, then (2.8) is equivalent to the first-order optimality
conditions of the optimization problem in price:
minimize
p ≥ 0 θ(p ; x).
Therefore, in this (price-integrable) case, a game with prices determined by (2.8) can be modeled as an extended game with one additional player, the price player. More generally, if this (price-) integrability condition does not hold, then the same game is an instance of the recently introduced class of (distributed) multi-agent optimization problems with equilibrium constraints (MOPECs), where the complementarity condition is generalized to a variational inequality with p as the primal variable and x as an exogenous variable (but endogenous to the overall MOPEC). The class of MOPECs was recently introduced by Michael Ferris without a detailed analysis. In this chapter, this framework is not explicitly addressed but rather mentioned as a direction for further research. For related MOPEC models, the interested reader is referred to [24] and the slides of a presentation available at http://www.cs.wisc.edu/~ferris/talks/chicago-mar.pdf.
When price is determined by a market clearing condition, the function g(p ; x) must in- corporate demand for the commodity. This demanded quantity can be specified either exogenously as in (2.8) or endogenously depending on the type of consumer behavior pos- tulated. For the exogenous case, consumers demand a fixed amount of the commodity regardless of price, a situation of perfect price inelasticity. The third pricing mechanism considered here follows the formulation of (2.8) but replaces the exogenously specified de- mand in g(p ; x) with an endogenous quantity. This modification is achieved by modeling consumers as choosing the demanded quantity to maximize consumer surplus, which is defined as the integral of the consumer demand function less commodity price from zero to the quantity demanded. Since perfectly competitive equilibrium models are of interest, commodity prices should be treated as exogenous in the consumer optimization problem. Thus, perfectly competitive games with endogenous demand can be modeled with F + 2 players (i.e., F producers, 1 consumer, and 1 price player); for a model of this type, see Section 2.7.
In summary, methods of price determination in a perfectly competitive market model can be separated into three groups as shown in Figure 2.1.
Figure 2.1: Price determination categories
Price
Inverse demand/supply function
Market clearing condition
Endogenousdemand/supply
Exogenousdemand/supply
model is a vector of decisions (x∗, p∗) such that for each producer i = 1, · · · , F ,
fi(xi; x∗−i, p∗) ≥ fi(x∗i; x∗−i, p∗) for all xi∈ Ki
and, for each commodity ` = 1, · · · , L, either p∗` = p`(x∗) (commodity price by inverse
demand/supply function) or
p`g`(p`; p∗−`, x∗) ≥ p∗`g`(p∗, x∗) = 0 for all p` ≥ 0 (commodity price by market clearing).
Concatenating the KKT conditions of the producers’ optimization problems along with the price stipulations, the following nonlinear complementarity formulations are obtained for the perfectly competitive market game:
• when all market prices are determined by inverse demand/supply functions:
0 ≤ xi ⊥ ∇xifi(xi; x−i, p)
p = p(x)− ATi λi ≥ 0 for all i = 1, · · · , F
0 ≤ λi ⊥ Aixi− bi ≥ 0 for all i = 1, · · · , F ;
(2.9)
• when all market prices are determined by the market clearing complementarity condi- tion (2.8): 0 ≤ xi ⊥ ∇xifi(xi; x−i, p) − A T i λi ≥ 0 for all i = 1, · · · , F 0 ≤ λi ⊥ Aixi− bi ≥ 0 for all i = 1, · · · , F 0 ≤ p ⊥ g(p ; x) ≥ 0. (2.10)
Presumably, a mixed model can be stated wherein some prices are determined by inverse demand/supply functions while others are determined by market clearing conditions. An
extended analysis can be made for such a mixed model; for simplicity, only the above two cases wherein all prices are determined by the same method are examined here. An- other remark about the two formulations (2.9) and (2.10) is that while the same notation fi(xi; x−i, p) is used for player i’s objective, in the case of (2.10), this function should
include some form of the constraint function g(p ; x) (see Section 2.6).
A key challenge in establishing the existence of an equilibrium pair satisfying (2.9) or (2.10) is the lack of explicit bounds for the variables x and p. Specialized to capacity expansion models such as those detailed in [78, 245], players choose both production levels and the amount of production capacity to install; such production decisions are captured by the vector x. There are constraints enforcing that production levels are bounded by installed production capacity. In the event that this capacity has no prescribed upper bound, the compactness assumption commonly employed to prove solution existence does not hold for these problems. This is the primary mistake made in the existence proof of [78] where boundedness is claimed through an appeal to objective function properties at Nash equi- libria before the existence of such equilibria is established; such a “proof” is a circular argument and needs to be corrected.
Instead of applying an argument based on explicit boundedness of the decision variables, the treatment in [245] is followed by employing a fundamental NCP existence result to analyze the problems (2.9) and (2.10). Formally, being a specialization of a variational inequality, the NCP attempts to find a vector z such that 0 ≤ z ⊥ F (z) ≥ 0, where F is a given continuous vector function. The following result is drawn from [75, Theorem 2.6.1]. Theorem 2.1. Let F : Rn → Rn be a continuous function. If there exists a constant
C > 0 such that all solutions of 0 ≤ z ⊥ F (z) + zτ ≥ 0 for τ > 0 satisfy kzk ≤ C, then the NCP: 0 ≤ z ⊥ F (z) ≥ 0 has a solution.
Notice that boundedness is assumed in a special way in Theorem 2.1, namely, on the solution sets of the augmented NCPs parametrized by the scalar τ . Rather than analyz- ing (2.9) and (2.10) in their abstract formulation, a concrete application of Theorem 2.1 is provided for the capacity expansion problem with uncertainty, extending the special cases in [78, 245] to broader contexts. Theorem 2.1 is applied by assuming that, corresponding to a sequence {τk} of positive scalars, an unbounded sequence zk exists such that for
each k, 0 ≤ zk ⊥ F (zk) + zkτ
k ≥ 0; a contradiction is then derived.