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# System prepended metadata

title: Chapter 11. Network Models of Markets with Intermediaries

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# 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 $\leq 3.50$


![](https://i.imgur.com/RYASVF3.png)
:::

## 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 $v_j$
![](https://i.imgur.com/ON9TZxr.png)

---

### 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.
:::

- 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.

![](https://i.imgur.com/lgMfE2k.png)

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

---

### Payoffs.

:::danger
Noted that we now only consider cases where all the seller $v_i$'s are $0$.
:::

- 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.
![](https://i.imgur.com/GsQh3bc.png)

---

### 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
![](https://i.imgur.com/0b9NGxf.png)

### 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.
![](https://i.imgur.com/ZYOtjSV.png)
:::

:::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)
:::


## 11.3 Equilibria in Trading Networks

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

Example.
![](https://i.imgur.com/HQiJmzV.png)
***Perfect Competition***: Direct competition with another trader

![](https://i.imgur.com/wsFnDGr.png)
***Implicit Perfect Competition***: No direct competition between two traders.

![](https://i.imgur.com/brsD3m0.png)

:::


## 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
:::
Modeling a second-price auction with a trading network:
![](https://i.imgur.com/20KdiW8.jpg)
- 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
![](https://i.imgur.com/bcviYVh.jpg)
- 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
- ![](https://i.imgur.com/mKIbogl.jpg)Each good contributes
$$
(b_{ti}-v_i)+(a_{tj}-b_{ti})+(v_j-a_{tj})=v_j-v_i 
$$
to the social welfare

- Example:
    |![](https://i.imgur.com/NxfjMHx.jpg)|![](https://i.imgur.com/cmTPH1K.jpg)|
    |---|---|
    |$(1-0)+(2-0)+(4-0)=7$|$(2-0)+(3-0)+(4-0)=9$|

- 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:![](https://i.imgur.com/tbcvYKw.jpg)
    - 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:![](https://i.imgur.com/5WwhwwI.jpg)
    - 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**
        :::

- **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)
    :::

## 11.7 Reflections on Trade with Intermediaries
_(skip)_