## I Introduction

The recent past has seen the stupendous growth of market mechanisms involving buyers and sellers in markets as complex as commodity markets (like electricity) or service markets like transportation. Unlike in old-school markets, the price that suppliers see in these modern markets is itself determined by a fairly complicated decision problem. For example, in the electricity industry, the price is determined by the independent system operator such a way so as to procure electricity at the lowest cost for the consumer. In on-demand transportation, platforms such as Uber, Lyft or Ola determine prices conceivably based on a variety considerations such, demand arrival, supply availability, tolls, etc. This is leading to novel decision problems on the supply side for which classical analysis fails to apply. Moreover, the complexity of these mechanisms also means that this analysis is mathematically non-trivial. The goal of this paper is to develop foundational tools for claiming the existence of equilibria in such games that arise in such formulations by introducing a new class of games, called price-coupling games.

In microeconomics, players on the supply side are generally modeled in two ways, as competitive players and as strategic players. In the formulation as competitive players, the players are called price takers, which means that the players take their decision assuming the price is exogenously given [2], [3]. This is indeed true in markets where there is no player that has enough market power to affect prices. While in the formulation as strategic players, the players anticipate the price and make their decision. Thus, while making decisions, the players consider the price of the quantity to be determined as a function of their decisions [4], [5]. We call the first formulation as the price-taking formulation and the latter as the price-anticipative formulation.

In this paper, we consider the existence of equilibria in these formulations for the class of games called price-coupling games we introduce in this paper. In such games, the objective function of players has the following form. It is a summation of two terms, the first term which depends on the price and the other term which is independent of the price. The form of the term which is dependent on the price is same for all players and the form of the other term which is independent of price can be different for different players. The price itself is determined by another player called the price-determining player who has his own optimization problem. As stated above, the study of such games is motivated by the observation that many games of supply-side competition arising in practice fall in this category, where the price is identical for all players and it has a uniform influence on the players.

Our main thrust in this paper is that this class of games has a peculiar structure which one can exploit to provide results for a variety of settings concerning such games. To make our contributions precise, we introduce a game with following notation. Let be the set of players, , be the set of decisions of player , be the set of decision-tuples of all players, let be the decision of player , be the decisions of all players, and be the decisions of every player except player . Finally, let be solution of the optimization problem .

#### Player ’s problem in a price-coupling game

For a game , we assume player ’s problem denoted by which is given as follows:

Here represents the price. The objective function of player is a summation of two terms, which is dependent on and which is independent of . Also note that the part of the objective function which is dependent on has the same form in the objective function of all players. The other part which is independent of may have different forms for different players. The price in a price-coupling game is given as a solution of an optimization problem denoted by which is given below. We call this problem as the price-determining player problem.

#### Price-determining player

For a game , the price-determining player takes from the other players and determines a price for all players. The optimization problem of the price-determining player in its general form is given as follows:

Here the function is the cost function that the price-determining player minimizes over , the constraint on the price. This is an illustration of one kind of a price-coupling game. There are other price-coupling games with variations from this structure which will be discussed in later sections.

Guaranteeing the existence of an equilibrium is not straightforward in a price-coupling game. One reason is that unlike a standard game, the set of constraints of players are coupled, that is the decision of one player changes the feasible set of other players. Another reason is the non-convexity of the feasible region of a player, which occurs due to the appearance of in the constraints of .

The main contributions of the paper are as follows. A player’s problem in a price-coupling game is formulated as a price-anticipative formulation and price-taking formulation depending on how the price determined by the price-determining player influences the participants’ decision making. In the price-anticipative formulation, the players conjecture the solution of the price-determining player problem while making their decision. Thus, we model it as a leader-follower Stackelberg game with the players considered as the leaders and the price-determining player as the follower. It is known that there need not exist an equilibrium even for a simple leader-follower game as shown by Pang and Fukushima in [6]. In this formulation, we categorize and analyze the existence of equilibria of the price-coupling game in two different classes depending on the mathematical properties of the game. We provide conditions for the existence of equilibria in these classes of price-coupling games.

The second formulation is the price-taking formulation in which the price is taken as a given parameter for each player except the price-determining player. As pointed out in a recent paper [2] there has not been much progress regarding the existence of an equilibrium in price-taking formulation after the results by Arrow and Debreu in [3]. In our formulation an equilibrium in the price-taking formulation is given as an equilibrium of an player game with the additional player is the price-determining player. In this formulation, we use a potential function approach to provide conditions for the existence of an equilibrium. Finally, we consider a concrete application – the generation expansion planning problem and we apply our results to this application. The existence of equilibria in generation expansion planning problem presented in [7] is analyzed in both these formulations.

Our work follows a long line of work electricity markets and various related models. The main challenges in such problems have been discussed in [8], [9], [10], [11], and the recent works [12], [13]. The volume of work here is substantial and we refer the interested reader to the above works for details. The paper is organized as follows. The next section gives an example of price-coupling games from the electricity market. Section III elaborates the price-anticipative formulation of the price-coupling game. The price-taking formulation and the different cases are explained in Section IV. Section V elaborates the application of results to generation expansion planning problem. The paper ends with a conclusion in Section VI.

## Ii An Example: Cournot Model of Electricity Trading

We consider a well-known example of the Cournot model of electricity trading and show how it fits into our framework of a price-coupling game. In a Cournot model of competition [5], the players submit their bids as quantity which they are willing to supply. Let be the total quantity bids of the players. Suppose is the demand curve of that market which gives the relation of price that the consumers are willing to pay for a quantity of electricity. The ISO (Independent system operator) determines the market clearing price from bids by the players and the demand curve . Suppose . We denote the ISO’s problem in the Cournot model of competition as . Here the price is determined trivially from the demand curve. More generally the price may be a solution of some optimization problem denoted by as given in the introduction. Let be the cost for producing quantity of electricity for supplier and be the maximum production capacity of supplier . Given , the decision of rivals, the supplier ’s problem in the Cournot model of competition is given by the problem denoted by .

It can be seen that this game has a structure of a price-coupling game. That is, the part of the objective function which is dependent on the price is the revenue for player and this part has the same form for all players and the price which is determined by the ISO is identical for all the players. Also, the term which is independent of price is the cost of generation which may vary for players. However this game has a structure which is distinct from the structure discussed in the introduction. These distinctions will matter in the results and are discussed more precisely in Section III.

## Iii Price-Anticipative Formulation

In this section we consider the price-anticipative formulation of a price-coupling game. As mentioned in Section I, in a price-anticipative formulation, the price-coupling game is modelled as a multi-leader single-follower game with players as the leaders and the price-determining player as the follower. For player , consider a price-coupling game with the player ’s problem denoted by . In the price-anticipative formulation, the player conjectures the price determined by the price-determining player. Hence, we denote the price as , which is dependent on player . We define the feasible set of player in price-anticipative formulation as With these considerations, the player ’s problem in price-anticipative formulation can be rewritten as .

We use in to identify that this is the first class of price-coupling games that we consider in the price-anticipative formulation. We use to denote the class of price-coupling games in the price-anticipative formulation with player ’s problem given by .

### Iii-a Existence of equilibria for game

Define We define an equilibrium for game as follows:

###### Definition III.1

A point is said to be an equilibrium for game , if ,

To show the existence of an equilibrium we require that the functions admit a potential function. We recall the definition of a potential function from [14] which is given as follows:

###### Definition III.2

The functions is said to admit a potential function if ,

In order to show the existence of equilibria for the class of games , we need a result which is given by the following lemma.

###### Lemma III.1

A point is feasible for player ’s problem if and only if . That is,

Proof: “” For a given , consider a point . That means, . Now by combining and we can rewrite the above equations as . Hence .

“” Suppose . That is, . For some , we separate the decisions of player and adversaries as and . For some fixed , the equations can be rewritten as . Hence, .

Now for the game , consider the following optimization problem denoted by .

Theorem III.1 gives a relation of an equilibrium of the game to the solution of the optimization problem . From this relation, we provide conditions for the existence of an equilibria for game in Corollary III.2.

###### Theorem III.1

Consider the game denoted by . Suppose the functions admit a potential function . For this game, consider an optimization problem denoted by . Suppose is a maximizer of the problem , then is also an equilibrium of the game .

Proof: Suppose is a maximizer of the problem . Then,

That is, for some ,

These inequalities hold true even if . Thus for we can rewrite the inequalities as,

By Lemma III.1, for ,

Since is a potential function for ,

Since we consider an arbitrary , this condition is valid for all . Hence is an equilibrium of the price-coupling game .

Since the maximizer of the problem is an equilibrium of game , condition for the existence of equilibria is given as the condition for the existence of a solution for the optimization problem which is given in the following corollary.

###### Corollary III.2

For a price-coupling game with admitting a potential function , admits an equilibrium if the function, is continuous and the set is non empty and compact. The set is compact if the set is compact, the function is continuous and the set valued function is continuous and uniformly bounded.

We omit the detailed proof, but we give an outline of the proof. The first part follows from Theorem III.1 and Weirstrass theorem. By Theorem in [15], since and are continuous, is closed. Also, is bounded since is uniformly bounded and is compact. Hence is compact. Thus, to guarantee the existence of an equilibrium for a price-coupling game , one can check the conditions given in the Corollary III.2.

Remark: Suppose the player dependent term in the objective function of a player is dependent only on , and is independent of , then a potential function of the terms is given by .

### Iii-B Shared constraints and existence of equilibria for game

In this section, we consider the price-coupling game in price-anticipative formulation with the player ’s problem given by . The difference of from is that the identical function part is dependent on and and is independent of . We use to denote this class of price-coupling games.

An equilibrium for the price-coupling game is defined as follows:

###### Definition III.3

A point is said to be an equilibrium for the game , if ,

Recall that we have considered as the price conjectured by player . Thus a player can make a conjecture independent of other players. To give a result for the existence of equilibria for the game , we modify the game with an additional constraint. The additional constraint in a player’s problem is that the price conjectured by the players need to be consistent, i.e. . We call this constraint as the consistent conjecture of the price. With this additional constraint, we define the feasible set of a player as the set denoted by .

(1) |

Let With the additional constraint of the consistent conjecture of price, we redefine the player ’s problem in game as which is given as follows:

We call this modified game as a price-coupling game with consistent price conjectures and denote it by . For defining an equilibrium in the game , consider the set which is defined as follows.

Now we define an equilibrium in the game as follows.

###### Definition III.4

A point is said to be an equilibrium of the game if ,

The following lemma gives a relation between the original price-coupling game and the price-coupling game with consistent price conjecture .

###### Lemma III.2

Consider a game . Suppose is an equilibrium of this game. Then is also an equilibrium of the game [11].

Proof: Since at an equilibrium of the game , the price conjectured by the players are the same, an equilibrium of is also an equilibrium of .

To give conditions for the existence of an equilibria we use the shared constraint structure of the price-coupling game with consistent conjecture [16],[11]. In a coupled constraint game, the action of a player restricts the action set of other players. Moreover, if the coupled action set in a game is same for all players, then we call it as a shared constraint game. In our case, the set mapping is said to be a shared constraint mapping if there exists a set such that,

Shared constraint games were introduced by Rosen [17] and have attracted significant study over the last decade [18], [19], [9], [10]. The following lemma shows that for the game , the set mapping has a shared constraint structure.

###### Lemma III.3

Consider a price-coupling game with consistent price conjectures . There exists a shared constraint set such that a point is feasible for player ’s problem if and only if . That is, where, , where

Proof: Let be an arbitrary player for game . Consider the constraint set of player with the condition of consistent conjecture of price denoted by which is given in (1). By using the defined sets and , the set can be redefined as,

i.e., , which is independent of . Since the implication holds for each , as required for a shared constraint.

Thus is a shared constraint for the game . For the game , consider the following problem denoted by

The following theorem relates the solution of the problem to an equilibrium of the game .

###### Theorem III.3

Consider the game . Suppose admits a potential function . For this game, consider the optimization problem denoted by . Suppose is a maximizer of the problem , then is also an equilibrium of . There exists an equilibrium for this game if the functions and are continuous, the set is compact and the set valued function is continuous and uniformly bounded.

Proof: Let be a maximizer of the problem . Then, ,

That is, for some ,

These inequalities hold true even if and . Thus for we can rewrite the inequalities as,

By Lemma
and by cancelling equal terms on both sides, for ,
Since is a potential function for ,,
Since we considered an arbitrary , this condition is valid for all . Hence is an equilibrium of the game .

## Iv Price-taking Formulation

In this section, we consider the price-taking formulation of a price-coupling game. As discussed in Section I, in this formulation, a player makes the decision by considering the price as a given parameter. That is, given and , the player ’s problem in price-taking formulation has a structure of the problem which is given as follows:

Similar to the case of price-anticipative formulation, here we use in to denote that the problem is player ’s problem in the price-taking formulation. Even though the price is a given parameter for a player in the game, it is determined as a solution of the price-determining player problem . We use to denote this price-coupling game in price-taking formulation. We define a set as,

Now we define an equilibrium for the game as follows.

###### Definition IV.1

A point is said to be an equilibrium of the game if the following inequalities are satisfied ,

As seen from the above definition, an equilibrium of the game is defined an equilibrium of a game with players. Thus we call this formulation also as an player formulation. The additional player in this game is the price-determining player who decides the equilibrium price . In the following sections, we study the existence of equilibria for the game by categorizing into two classes depending on the presence or absence of the decision of players in the constraint set of the price-determining player. In the first class of games, we consider the case where the constraint set is independent of the decision of players. We denote this game as . In the second class of games, we consider the general case where the decision of players also constrains the feasible set of the price-determining player problem.

### Iv-a Game : is independent of .

Here we consider a price-coupling game in the price-taking formulation with the player ’s problem given by and the price-determining player’s problem is given by the following problem denoted by .

Note that the constraint of the price-determining player problem is independent of . The Bertrand model of competition for suppliers in an electricity market is an example of such games. The suppliers’ bids are the price of electricity that they are willing to supply which do not appear as a constraint in the ISO’s problem. These games are easier to analyze because they do not have a coupled constraint structure as in the general case. The following lemma gives a result for the existence of an equilibrium in such class of games.

###### Proposition IV.1

Consider the game denoted by . Suppose the sets and the set valued function are non-empty, convex and compact subset of some Euclidean space. Let the functions be continuous and be convex for all and is concave for all , then there exists an equilibrium for the game .

Proof: This proposition can be verified using the results for the existence of an equilibrium in a classical - player game [20].

Thus by checking the conditions in Proposition IV.1, we can guarantee the existence of an equilibrium for the game . But this result cannot be extended for a general class of games when the constraint is coupled which is described in the next section.

### Iv-B Game : dependent on .

Now we consider the general class of price-coupling games in the price-taking formulation where , the constraint set of the price-determining player is dependent on , the decision of players. To provide a result for the existence of equilibria in such class of games, we consider a modification of the original game . The modification is that the constraint of the price-determining player problem is included as a constraint to the players’ problem. We denote the game with this modification as and the problem of player in as which is given as follows:

We denote the feasible set of the player ’s problem as which is given as follows:

(2) |

In this section, we show that an equilibrium of the original game is also an equilibrium of the modified game . We also show that the reverse relation of equilibria, that is an equilibrium of game is also an equilibrium of under certain conditions. The price-determining player problem is the same as the one defined in Section I. An equilibrium of the game is defined as follows.

###### Definition IV.2

Consider the game . A point is said to be an equilibrium of if ,

A relation between an equilibrium of the original game and an equilibrium of the modified game is given in the following propositions.

###### Proposition IV.2

Consider the game . Suppose is an equilibrium of the game , then is also an equilibrium of the game .

We omit the proof owing to space constraints. The following proposition relates an equilibrium of the modified game to that of the original game .

###### Proposition IV.3

Consider the game denoted by . Suppose is an equilibrium of the game . Suppose , then is also an equilibrium of the game .

Proof: Suppose is an equilibrium of the game . Since , is also feasible as an equilibrium of the game . Suppose is not an equilibrium of the game . Then either, for some , an such that,

or
a such that,
Since , . In either of these cases, cannot be an equilibrium of the game which is a contradiction to the assumption. Thus an equilibrium of the game is also an equilibrium of the game if .

Thus, an equilibrium of the game is an equilibrium of the game if , where and are the equilibrium price and rivals action at the equilibrium of .
In relation to an electricity market, the condition says that for player , given , the price that the ISO determines is valid for all the bids of player .
Now we use the relations between the games and to provide a condition for the existence of equilibria for a game having a potential function. To make the notations simple, we denote the objective function of player as .
For a game which has a potential function [14], consider the following optimization problem denoted by .

Lemma IV.1 gives a relation of the maximizer of the problem and an equilibrium of the game .

###### Lemma IV.1

Consider the game . Suppose is a potential function for this game. If is a maximizer of problem , then is an equilibrium of this game.

Proof: Suppose is a maximizer of the problem . Then,

(3) |

The inequality (3) still holds even if we replace by in the above condition. Hence, Since , the inequality can be rewritten as, Since is a potential function of the game ,

(4) |

Similarly, the inequality (3) holds even if we replace by and by . That is,

Since , the above condition can be rewritten as, Since is a potential function of the game , ,

(5) |

Thus, from the conditions and it can be seen that is an equilibrium of the game .

Thus a maximizer of the optimization problem is an equilibrium for the game . Now we use this optimization problem to provide conditions for the existence of an equilibrium in such games.

###### Corollary IV.4

Consider the game . Suppose this game admits a potential function . Then there exists an equilibrium if the function is continuous and the set is non empty and compact. The set is compact if the set is compact and is closed and uniformly bounded.

Now by using the relation of equilibria between the games and , we give conditions for the existence of an equilibrium in the original game .

###### Corollary IV.5

Consider the game which has potential function . Consider the optimization problem for such a game. Suppose is a maximizer of the problem and suppose Then is an equilibrium of the game .

Proof: The maximizer of the problem is an equilibrium of the game

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