Markets
Markets with Intermediaries
Where do the prices in a real market come from? Who sets prices in real markets and why are particular prices chosen over others?
Price Setting
Auctions provide a concrete example of price determination in a controlled setting. In a second price sealedbid auction buyers bid their true values. They choose prices (via their bids) in a procedure selected by the seller. But who sets the prices, who trades with whom if there are many buyers and many sellers?
Intermediaries. In many examples of markets, buyers and sellers do not interact directly with each other, but trade trough intermediaries, brokers, auctioneers (real estate agents), or other middlemen who set the prices. Such a market forms a network that connects buyers and sellers to different intermediaries. The network structure constrains who can trade with whom, and how prices are set.
Network Structure. What follows is a simple network model of trade between sellers and buyers through intermediate agents (traders).
Assume that there is a single type of good for trade that comes in indivisible units. Each seller $i$ initially holds one unit of the good and values it at $v_i$. This means he is willing to sell it at any price that is at least $v_i$. Each buyer $j$ values a unit of the good at $v_j$ and will try to obtain a unit by paying no more than $v_j$. All participants (buyers, sellers and traders) are assumed to know these valuations.
Prices. The flow of goods from sellers to buyers is determined by a game in which traders first set prices, then sellers and buyers react, as follows.

Each trader $t$ offers a bid price $b_{ti}$ to each seller $i$ that he is connected to as an offer to buy seller $i$’s unit at a price of $b_{ti}$.

At the same time, each trader $t$ offers an ask price $a_{tj}$ to each buyer $j$ to which he is connected as an offer to sell a unit to buyer $j$ at a price of $a_{tj}$.

When all traders have announced all their prices, each seller and each buyer chooses (at most) one trader to deal with. Each seller sells their unit to the chosen trader, and each buyer buys one unit from their chosen trader. This determines a flow of goods from the sellers, through the traders, to the buyers.
Example. The following diagram illustrates one set of trader’s prices (as labels on the edges) and one set of buyers’ and sellers’ choices (as arrows) and thus the resulting flow of goods.
Note that at most one unit moves along any edge of the network. And while there is no limit on the number of units a trader can trade, he can only sell as many units as he receives from sellers.
Payoffs. Specifying a game requires a description of strategies and payoffs. A trader’s strategy is a choice of bid and ask prices. A seller or buyer’s strategy is a choice of a trader to deal with (or not to deal at all).
The payoffs of the different kinds of participants are as follows.

A trader’s payoff is the profit he makes from all his transactions, that is the sum of the ask prices of his accepted offers to buyers, minus the sum of the bid prices of his accepted offers to sellers.

Seller $i$’s payoff is $b_{ti}$ if they choose trader $t$ or $v_i$ if they choose not to sell (and get to keep their unit of the good).

Buyer $j$’s payoff from selecting trader $t$ is $v_j  a_{tj}$ (as they receive the unit of the good but have to give up $a_{tj}$ in money), or $0$ if they choose not to buy.
Note that this game, in contrast to the ones studied earlier, proceeds in two stages: first the traders choose their strategies, and then the buyers and sellers choose theirs. However, in the second stage one can assume that the players will always choose an optimal strategy, their best repsonse is to choose the trader with the best offer.
Example. In the above example, the payoff for trader $T_1$ is $0.8  0.2 = 0.6$ and the payoff for trader $T_2$ is $1.70.3 = 1.4$. The payoffs for the three sellers are $0.2$, $0.3$, and $0$, respectively. And the payoffs for the three buyers are $1  0.8$, $1  0.7 = 0.3$, and $1  1 = 0$.
However, trader $T_1$ could improve his payoff in several ways. He could lower his bid price to seller $S_1$ and raise his ask price to buyer $B_1$, and both would still want to deal with $T_1$ as they have no other option. And he could raise his bid for seller $S_2$ to $0.4$, and lower his ask for buyer $B_2$ to $0.6$ and take trade away from trader $T_2$. For these reasons, the current choices do not constitute an equilibrium.
Equilibrium
In an equilibrium state, each player chooses a best response to the other players’ strategies.
For buyers and sellers this simply means to choose a trader with the best offer for them.
For traders, the situation is more complex, but it can be analyzed in some small, typical examples as follows.
Monopoly. A buyer, or a seller, is monopolized if they have only access to a single trader. In such a situation, the trader can dictate the price and receive all the profits.
This is the only equilibrium state, as the trader could always improve on any other bid or ask and increase their profit.
Perfect Competition. Two traders $T_1$ and $T_2$ are in perfect competition to buy a unit of goods from a seller $S_1$ to a buyer $B_1$ if both traders have access to the seller and the buyer.
In this case one can argue that in an equilibrium state the trader performing the trade must be doing this at $0$ profit, i.e., for an ask price $x$ that equals the bid price, for otherwise the other trader could outbid them with a more competitive bid and ask.
As the buyer must buy from the trader who obtained the unit from the seller, it can be argued that the other trader, in an equilibrium state, also offers an ask price equal to the bid price equal to $x$.
Any choice of a value for $x$ between $0$ and $1$ in the above network yields an equilibrium state. The tie between traders $T_1$ and $T_2$ for the transaction of the unit from seller $S_1$ to buyer $B_1$ has been broken in an arbitrary way. The choice of $x$ determines how the payoff is distributed between the buyer and the seller.
Example. Identifying monopolies and competitions in the example network we have been studying, reveals the equilibrium states of that network.
Any choice of a value for $x$ between $0$ and $1$ in the above network yields an equilibrium state. The tie between traders $T_1$ and $T_2$ for the transaction of the unit from seller $S_2$ to buyer $B_2$ has been broken in an arbitrary way.
Chapter 11 of the book contains more details and applications of this network model of markets with intermediate traders.