---
# System prepended metadata

title: Chapter 9. Auctions

---

# Chapter 9. Auctions
[ToC]

## 9.1 Types of Auctions
- In this chapter, we'll consider the case of a seller auctioning one item to a set of buyers
- Underlying assumption: Each bidder has an **intrinsic value** for the item being auctioned
    - _the highest price she is willing to pay_

- 4 main types:
    - interactive
        1. Ascending-bid auctions
        2. Descending-bid auctions
    - simultaneous

        3. First-price sealed-bid auctions
        4. Second-price sealed-bid auctions

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## 9.2 When are Auctions Appropriate?
1. Sellers do not have a good estimate of the buyers’ intrinsic values for an item
2. Buyers do not know each other’s values (private, independent)

:::info
**If the values are known, auctions are not needed.**
Consider this:

Suppose that a seller values at $x$, and the maximum value held by a potential buyer of the item is some larger number $y$. Then what would happen is either

_1. <u>Seller announces that the item is for sale at a fixed price just below $y$</u>_

or

_2. <u>Buyer announces that she is willing to purchase the item for a price just above the larger of $x$ and the values held by all other buyers</u>_
:::

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## 9.3 Relationships between Different Auction Formats
This section gives some **informal** observations about different auction formats

### Descending-Bid and First-Price Auctions

- The seller is lowering the price from its high initial starting point

- During the auctions, bidders **learn nothing** except that no one has accepted the current price yet
- The process is equivalent to a **sealed-bid first-price auction**
---

### Ascending-Bid and Second-Price Auctions

- Bidders gradually drop out as the seller steadily raises the price

- The item goes to the highest bidder at a price equal to the second-highest bid
- ||Ascending-bid|Sealed-bid second-price|
    |---|---|---|
    |Difference|involve real-time interaction|purely through sealed bids|
- However, the latter can be viewed as a **simulation** of the former

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## 9.4 Second-Price Auctions
Let's formulate the second-price auction as a game
- Players: the bidders $i$
- Strategies: an amount to bid $b_i = f(v_i)$, where $v_i$ is bidder $i$'s true value
- Payoffs: $$ \begin{cases}0 & \text{, if $b_i$ is not the winning bid}\\ v_i-b_j & \text{, if $b_i$ is the winning bid, and $b_j$ is the second-place bid}\end{cases} $$
    > **Note**:
    > 1. If $b_i = v_i$, and $b_i$ and $b_j$ are tied for the largest, then the one summitted first wins, but with payoff of $b_i-b_j=0$
    > 2. $b_i$ only affects whether $i$ wins or loses, but never affects how much $i$ pays in the event that she wins

:::info
**Claim:** In a sealed-bid second-price auction, it is a _dominant strategy_ for each bidder $i$ to choose a bid $b_i = v_i$.
:::
:::success
**Proof**
We need to show that <u>no deviation from this bid would improve her payoff</u>, regardless of what strategy everyone else is using
1. If bidder $i$ chooses $b_i'>v_i$
    ![](https://i.imgur.com/PMKSFwd.jpg)
    - Payoff is affected only if $i$ would lose with bid $v_i$ but would win with bid $b_i'$
    - In such case, $i$ wins but pays more than her value $v_i$ after the change
    
2. If bidder $i$ chooses $b_i''<v_i$
    ![](https://i.imgur.com/BMjAGCX.jpg)
    - Payoff is affected only if $i$ would win with bid $v_i$ but would lose with bid $b_i''$

:::
**Conclusion:** In a second-price auction, it makes sense to <u>bid your true value</u> even if other bidders are overbidding, underbidding, colluding, or behaving in other unpredictable ways.

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## 9.5 First-Price Auctions and Other Formats
Unlike second-price auctions, the value of your bid not only affects whether you win **but also how much you pay**. As a result, the conclusions are now different.
> **Game formulation**
> - Players: the bidders $i$
> - Strategies: an amount to bid $b_i = f(v_i)$, where $v_i$ is bidder $i$'s true value
> - Payoffs: $$ \begin{cases}0 & \text{, if $b_i$ is not the winning bid}\\ v_i-b_i & \text{, if $b_i$ is the winning bid}\end{cases} $$

- Bidding $v_i$ is no longer a dominant strategy
- The optimal way depends on knowledge of the other bidders and their distribution of possible values
- Intuition:
    - with many competing bidders => higher bid
    - with only a few competing bidders => lower bid

(More detail in Section 9.7)

---
### All-pay auctions
All bidders pay their bids, regardless of whether they win or lose
> **Game formulation**
> - Players: the bidders $i$
> - Strategies: an amount to bid $b_i = f(v_i)$, where $v_i$ is bidder $i$'s true value
> - Payoffs: $$ \begin{cases}-b_i & \text{, if $b_i$ is not the winning bid}\\ v_i-b_i & \text{, if $b_i$ is the winning bid}\end{cases} $$

- Like first-price auctions, one must balance the trade-off between bidding high and bidding low
- In general, bids will typically be shaded much lower than in a first-price auction
- Games of this type:
    - Political lobbying
    - Design competitions

(More detail in Section 9.7)

---

## 9.6 Common Values and The Winner's Curse
- In situations that the bidders intend to resell the object, there should be a <u>common eventual value</u> for the object

- Each bidder has an estimation $v_i$ of the common value $v$ based on her private information about it
$$
v_i = v + x_i
$$
, where $x_i$ is a random error with a mean of 0
- The winning bid is more likely to be an **over-estimate** of the common value, so the winner will lose money on the resale
    - The phenomenon is known as **winner's curse**
- Rational bidders who take the winner’s curse into account will <u>shade their bids downward</u> when the second-price format is use. _(Of course, bids will be reduced even further with first-price format.)_


## 9.7 Advanced Material: Bidding Stragegies in First-Price and All-Pay Auctions

### Equilibrium Bidding in First-Price Auctions

#### Private Value
The **private value** for bidders $1,\ldots,n$ are i.i.d. $X_1,\ldots,X_n$ with probability density function $f$, distribution function $F$ on $[0,1]$.

#### Strategy
A **strategy** for a bidder is a function $s:[0,1]\to[0,1]$ satisfy
- $s$ is continuous on $[0,1]$ and differentiable on $(0,1)$
- $s$ is stritly increasing
- $s(v)\leq v$

where $s(v)=b$, $v$ is the true value and $b$ is the bid

#### Bidders with the Same $s$
For bidder $i$ with his true value $v_i\in[0,1]$, bid price $v$, then the probability that $i$ win the item is
$$
\begin{split}
&P(X_1\leq v,X_2\leq v,\ldots,X_n\leq v)\\
=&\prod_{j=1,j\neq i}^n P(X_j\leq v)\\
=&P(X_1\leq v)^{n-1}\\
=&F(v)^{n-1}
\end{split}
$$
and thus the expect payoff is
$$
g(v)=F(v)^{n-1}\cdot(v_i-s(v))+(1-F(v)^{n-1})\cdot0=F(v)^{n-1}(v_i-s(v))
$$

We want $g(v)$ to be maximized when setting $v=v_i$. That is, $g'(v_i)=0$. Then we have
$$
\begin{split}
&g'(v)\mid_{v=v_i}\\
=&(n-1)F(v)^{n-2}f(v)(v_i-s(v))-F(v)^{n-1}s'(v)\mid_{v=v_i}\\
=&(n-1)F(v_i)^{n-2}f(v_i)(v_i-s(v_i))-F(v_i)^{n-1}s'(v_i)\\
=&0
\end{split}
$$
Now solve $s(v_i)$,
$$
s'(v_i)=(n-1)f(v_i)\left(\frac{v_i-s(v_i)}{F(v_i)}\right)
$$
which don't have explicit form of solution.

#### Uniform Distribution
If we consider the special case, the distribution is uniform $U(0,1)$, then $f(v)=1$, $F(v)=v$. We have
$$
s'(v_i)=(n-1)\left(\frac{v_i-s(v_i)}{v_i}\right)=(n-1)\left(1-\frac{s(v_i)}{v_i}\right)
$$
Observe that
$$
s(v_i)=\frac{n-1}{n}v_i
$$
is a solution.
:::warning
Let $s(x)=\sum_{k=0}^\infty a_kx^k$, then apply in $xs(x)=(n-1)x-(n-1)s(x)$, we have
$$
\sum_{k=0}^\infty a_kkx^{k}=(n-1)x-(n-1)\sum_{k=0}^\infty a_kx^k
$$
Thus,
$$
\begin{cases}
a_0=-(n-1)a_0\\
a_1=(n-1)-(n-1)a_1\\
a_k=-(n-1)a_k & ,k\geq 2
\end{cases}
$$
Hence
$$
\begin{cases}
a_1=(n-1)/n\\
a_k=0 & ,k\neq 1
\end{cases}
$$
and $s(x)=(n-1)x/n$
:::
Hence, every bidder bidding for a ratio $\frac{n-1}{n}$ of his/her true value is an equilibrium.

### Seller Revenue
:::info
**Theorem.**
Let $X_1,\ldots,X_n\stackrel{\small i.i.d}{\sim}U(0,1)$, $X_{(1)},\ldots,X_{(n)}$ be their order statistics, then for $1\leq k\leq n$,
$$
E(X_{(k)})=\frac{k}{n+1}
$$
:::

> For seller, which type of auctions should be used?

In a Second-Price Auction, the bidders follow their dominant strategies and bid truthfully, then the expected revenue is
$$
\frac{n-1}{n+1}
$$

By theorem and $\frac{n-1}{n}v_i$ being an equilibrium, in a First-Price Auction, the expected revenue is
$$
\frac{n}{n+1}\frac{n-1}{n}=\frac{n-1}{n+1}
$$

Observe that the expected revenue is the same.

#### Reserve Prices
The seller announce a **reserve price** $r$ before the auction. At the end of the auction, if the concluded price is smaller than $r$, then the item is reserved.

> What value of $r$ should the seller set?

Assume the seller values the item $u$, the reserve price $r$. Consider the special case: Second-Price Auction with one bidder with $v\sim U(0,1)$. In this case, we regard the seller as a bidder bidding with a constant price $r$. Then the expected revenue is
$$
g(v)=P(v\leq r)\cdot u+P(v> r)\cdot r=ru+(1-r)r
$$
which is maximized when setting $r=(1+u)/2$.

### Equilibrium Bidding in All-Pay Auctions
Similar to the way we find the equilibrium of first-price auction. Recall that the chance to win the item is $F(v)^{n-1}=v^{n-1}$ if the distribution is $U(0,1)$. Then now the expected payoff is
$$
\begin{split}
g(v)=&v^{n-1}\cdot(v_i-s(v))+(1-v^{n-1})\cdot(-s(v))\\
=&v_iv^{n-1}-s(v)
\end{split}
$$
Then
$$
g'(v)=(n-1)v_iv^{n-2}-s'(v)
$$
Consider $g'(v_i)=0$, we have
$$
s'(v_i)=(n-1)v_i^{n-1}
$$
solve that
$$
s(v_i)=\left(\frac{n-1}{n}\right)v_i^n
$$
Note that $v_i<1$, $s(v)$ decreases exponentially as $n$ increases. The expected revenue of one bidder is
$$
\int_0^1s(v)f(v)dv=\int_0^1\left(\frac{n-1}{n}\right)v^ndv=\left(\frac{n-1}{n}\right)\left(\frac{1}{n+1}\right)
$$
Thus the total revenue is 
$$
n\left(\frac{n-1}{n}\right)\left(\frac{1}{n+1}\right)=\frac{n-1}{n+1}
$$
is the same as before.