# Chapter 11. Network Models of Markets with Intermediaries

[ToC]

## 11.1 Price-Setting in Markets

### Trade with Intermediaries

- Individual buyers and sellers do not interact directly with each other </br> but instead trade through intermediaries.

:::info
**Definitions.**
- ***order books***: A list of orders that buyers and sellers have submittted for that stock.
- ***limit orders***: Only commit to sell or buy once the price reaches some limit.
- ***bid***: The highest outstanding offer to **buy** the stock.
- ***ask***: The lowest outstanding offer to **sell** the stock.
- ***market orders***: Only commit to sell or buy at current bid and ask.

Example.
- 4 orders:
- sell 100 shares if the price $\geq 5.00$
- sell 100 shares if the price $\geq 5.50$
- buy 100 shares if the price $\leq 4.00$
- buy 100 shares if the price $\leq 3.50$

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## 11.2 A Model of Trade on Networks

### Network Structure.

- Each seller $i$ holds one unit of the good which he values at $v_i$
- Each buyers $j$ holds one unit of the good which he values at $v_j$

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### Prices and the Flow of Goods.

:::info
The flow of goods from sellers to buyers is determined by a game </br> in which traders first set prices, and then sellers and buyers react to these prices.
:::

:::danger
Note that a trader can only sell as many goods to buyers as he receicves from sellers.
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- Each trader $t$ offers a *bid price* $b_{ti}$ to each seller $i$ he connected to
- Each trader $t$ offers a *ask price* $a_{tj}$ to each buyer $j$ he connected to

example.


- $S3$ and $B3$ are called ***indifference*** between accepting and rejecting the offer

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### Payoffs.

:::danger
Noted that we now only consider cases where all the seller $v_i$'s are $0$.
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- A trader's strategies is a choice of bid and ask price to propose to each neighboring sellers and buyers
- A seller or buyer's strategies is a choice of neighboring trader to deal with
- Payoffs
- traders: $\Sigma{a_{ti}} - \Sigma{b_{tj}},$ for all buyer $i$ and seller $j$ accepted by trader $t$
- sellers: for seller $i$, the payoff from selecting trader $t$ is $b_{ti} - v_i = b_{ti}$ ;</br> selecting no trader is $v_i = 0$
- buyers: for buyer $j$, the payoff from selecting trader $t$ is $v_j - a_{tj} = 1-a_{tj}$ ;</br> selecting no trader is $0$

example.

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### Best Responses and Equilibrium.

- T1's best response : change the offers to maximize his payoff.
- If a minimum change of offer is $0.1$, then it's a Nash equilibrium

### Additional : Subgame Perfect Nash Equilibrium

:::info
Definition.
***Subgame*** meets the following criteria:
1. The initial node is in a singleton information set
2. If a node is contained in the subgame then so are all of its successors
3. If a node in a particular information set is in the subgame </br> then all members of that information set belong to the subgame.

Example.
:::

:::info
Definition.
***Subgame Perfect Nash Equilibrium***: A strategy combination is call a ***SPNE*** if
1. It is a Nash Equilibrium
2. Each subgame is also a Nash Equilibrium

[For further imformation](http://www.columbia.edu/~md3405/GT_Game_7_17.pdf)
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## 11.3 Equilibria in Trading Networks

:::info
**Definitions.**
***Monopoly***: Buyer or Seller are have only one access to one trader.

Example.
***Perfect Competition***: Direct competition with another trader

***Implicit Perfect Competition***: No direct competition between two traders.


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## 11.4 Further Equilibrium Phenomena
### Second-price auctions
:::info
**Review**
- Several buyers are bidding one item
- The buyer with the highest bid wins the item
- The winner pays the <u>second highest bid</u> to the seller
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Modeling a second-price auction with a trading network:
- Each buyer has an intermediary as its "proxy" for the transaction
- 4 buyers' values are $w$, $x$, $y$, $z$ respectively, where $w > x > y > z$
- The figure on the right shows an equilibrium of the network.</br>**Why is it an equilibrium?** => consider **_indifference(等優性)_**

### Ripple Effects
- Left figure:
- $0\leq x\leq 2$
- The market has a **bottleneck** that is restricting the flow of goods
- Right figure:
- $1\leq y\leq 2$
- $1\leq z\leq 3$
- $S2$ is now in a much more powerful position
- $B2$'s payoff is reduced

## 11.5 Social Welfare in Trading Networks
We say a solution is _socially optimal (efficient)_ if it maximizes the **social welfare**
- **Social welfare** = sellers' payoff + traders' payoff + buyers' payoff
- Each good contributes
$$
(b_{ti}-v_i)+(a_{tj}-b_{ti})+(v_j-a_{tj})=v_j-v_i
$$
to the social welfare

- Example:
|||
|---|---|
|$(1-0)+(2-0)+(4-0)=7$|$(2-0)+(3-0)+(4-0)=9$|

- Observation:
- Richly connected networks allow a flow of goods achieving a **higher social welfare**
- Sparsely connected networks contains **more bottlenecks** that prevent a desirable flow

- It can be shown that in every trading network, there is always at least one equilibrium, and every equilibrium produces a flow of goods that achieves the social optimum

## 11.6 Trader Profits
- As the network becomes more _richly connected_, individual traders have **less and less power**, and their payoffs go down
- **Suggestion:** in order to make a positive payoff, a trader must in some way be **essential** to the functioning of the trading network
-> _Not exactly_
- First example:
- Any choice of $x$ between 0 and 1 will result in an equilibrium
- When $x=0$, the social welfare is 3
- seller+buyer payoff = 2
- **trader payoff = 1** _(from T3)_
- When $x=1$, the social welfare is also 3, but
- seller+buyer payoff = 1
- **trader payoff = 2** _(from T1 and T5)_
- Another example:
- For each buyer, the two asks must be the same
- If either $x$ or $y$ is positive, then the trader left out of the trade on the higher one has a profitable deviation by slightly **undercutting this ask**, so $x=y=0$ in any equilibrium
:::danger
**Even though T1 is essential, its profit is $0$ in any equilibrium**
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- **Stronger condition:** There exists an equilibrium in which T receives a positive payoff when T has an **essential edge** to a seller or buyer
:::info
**For the proof of this statement, you can refer to this paper:**<br>[_Larry Blume, David Easley, Jon M. Kleinberg, and Eva Tardos. Trading networks with price-setting agents._](https://www.cs.cornell.edu/~eva/traders.pdf)
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## 11.7 Reflections on Trade with Intermediaries
_(skip)_