# Chapter 9 Auction

## 9.1 General symmetric information model

### 9.1.1 Setting

- The argument of this section is based on Hendricks & Porter (2007).
- General symmetric information model (Milgrom & Weber, 1982).
- The seller has a single item.
- There are \(n\) potential risk neutral buyers/bidders.
- Each bidder \(i\) observes a real-valued private signal \(X_i\).
- There is a random variable or vector \(V\) that influences the value of the object to the bidders.
- The joint distribution of \((V, X_1, \cdots, X_n)\) is \(F\).
- Bidder \(i\)’s payoff is \(U_i = u_i(V, X_i, X_{-i})\) when the bidder \(i\) obtains the object being sold.
- The seller announces a reserve price \(r\).
- The primitives of the model, the number of bidders \(n\), the distribution \(F\), and the utility functions \(\{u_i\}_{i = 1}^n\), are common knowledge among buyers.
- Assumption: \(u_i\) is non-negative, continuous, increasing in each argument and symmetric in the components of \(X_{-i}\).
- Assumption: \((V, X_1, \cdots, X_n)\) are affiliated.
- The \(n\) random variables \(X = (X_1, \cdots, X_n)\) with joint density \(f(x)\) are affiliated if for all \(x\) and \(y\), \(f(x \wedge y) f(x \vee y) \ge f(x) f(y)\). If affiliated, they are non-negatively correlated.

- \(\to\) \(U_1, \cdots, U_n\) are affiliated.
- Let \(Y_1, \cdots, Y_{n - 1}\) is the ordering of the largest through the smallest signals from \(X_2, \cdots, X_n\).
- Then, \((V, X_1, Y_1, \cdots, Y_{n - 1})\) are also affiliated.

### 9.1.2 Discussion About the Asssumptions

- The private information is assumed to be single-dimensional.
- If the private information is multi-dimensional, the identification of type requires at least as many messages.
- Recently, there are few empirical studies of auctions of multi-dimensional private information such as Bajari, Houghton, & Tadelis (2014) and Takahashi (2018).

- The distribution of the signals is assumed to be symmetric across bidders.
- We can introduce asymmetric signals.
- A bidder may have more precise signals.

- The utility is independent of the winner’s identify when a bidder is not awarded the item.
- If the item is the valuable asset in the oligopoly industry, the implication to the profit can be different when the close competitor is awarded the asset.

- The number of potential bidders is assumed to be common knowledge.
- The participation to the auction may be endogenously determined.

### 9.1.3 Bidding Strategy

- A
**bidding strategy**for bidder \(i\) is a correspondence \(\beta_i: X_i \to \mathbb{R}_+\). - A mapping from the private signal into a non-negative real value.
- Under the stated assumptions, there exists a Bayesian Nash equilibrium with non-decreasing bid functions (Krishna, 2009).
- The following argument crucially depends on this property of the equilibrium bidding rule.

### 9.1.4 Special Cases

- Two special cases about the payoff functions:
**Private value (PV)**: \(u_i(v, x_i, x_{-i}) = x_i\): Bidder \(i\) knows his own valuation and is only uncertain about how much others value the item.**Pure common values (PCV)**: \(u_i(v, x_i, x_{-i}) = v\): All buyers have the same valuation, which is unknown to them when they bid, and only learned though the private signals.- A remark on the private value assumption:
- The assumption holds in more general settings than we may think.
- Suppose that \(u_k(v, x_i, x_{-i})\) depends on \(v, x_i, x_{-i}\).
- Suppose that bidder \(i\) is uncertain about \(V\) and the distribution of \(V\) conditional on \(x_i\) is independent of \(x_{-i}\).
- Suppose that \(u_i(v, x_i, x_{-i}) = u(v, x_i)\).
- Then, \(\mathbb{E}\{u(v, x_i)|X_i = x_i, X_{-i} = x_{-i}\} = f(x_i)\) for some monotone increasing function \(f\).
- This is still a private value model.
- If the distribution of \(V\) is not independent of some \(x_i\), it is no longer a private value model.
- The last case is referred to as the
**common values (CV)**model.

- Two special cases about the signals:
**Independent signals (IPV, ICV)**: Signals \(X_1, \cdots, X_n\) are independent.**Affiliated signals (APV, ACV)**: Signals \(X_1, \cdots, X_n\) are affiliated.- Example: Offshore oil and gas leases.
- \(V\): the size of oil or gas deposits under the tract.
- CV: bidders are uncertain about \(V\) and have different private information about the value of \(V\) because of the seismic data they obtain.
- PV: bidders are almost certain about \(V\) or have little discrepancy in private assessment of \(V\). But there is a heterogeneity in the costs of exploration and drilling and this information is private.

- In the following, we mostly consider a type of CV model such as \(u_i(v, x_i, x_{-i}) = u(v, x_i)\).

## 9.2 Second-Price Auctions

### 9.2.2 Equilibrium Bidding Strategies

- Case 1:
- A bidder cannot observe the prices at which the other bidders take fingers off.
- The equilibrium strategy is to take the fingers off at the threshold: \[ \beta(x) = w(x, x). \]
- Suppose that the other bidders follow the bidding strategy and the price level is \(b\) and the auction does not yet end.
- Suppose that I win at this moment.
- This means that there is at least one other bidder that has signal \(y = \beta^{-1}(b)\).
- Then, the payoff to me is \(w(x, y) - b = w(x, y) - w(y, y)\).
- It is positive if and only if \(x > y\), because affiliation implies that the expected payoff is increasing in the own signal.
- Thus, \(\beta(x) = w(x, x)\) is the best response.

- Case 2:
- Active bidders observe the prices at which rivals drop out.
- No bidder who drops out can become active again.
- The bidding strategy is the mapping from the number of rivals who dropped out and the prices at which they dropped out.
- Let \(\beta_k(x)\) be the price at which a bidder drops out when \(k\) rivals dropped out at prices \(b_1, \cdots, b_k\).
- At the equilibrium it should be: \[ \beta_k(x) = \mathbb{E}\{u(V, x)| X_1 = x, Y_1 = \cdots = Y_{n - k - 1} = x, Y_{n - k} = \beta_{k - 1}^{-1}(b_k), \cdots, Y_{n - 1} = \beta_0^{-1}(b_1)\} \]
- Suppose that the other bidders follow the bidding strategy and the price level is \(b\) and \(k\) bidders dropped at \(b_1, \cdots, b_k\).
- Suppose that I win at this moment.
- This means there are \(n - k - 1\) bidders with signals \(y = \beta_k^{-1}(b)\).
- Then, the payoff to me is: \[\begin{equation} \begin{split} &\mathbb{E}\{u(V, x)| X_1 = x, Y_1 = \cdots = Y_{n - k - 1} = y, Y_{n - k} = \beta_{k - 1}^{-1}(b_k), \cdots, Y_{n - 1} = \beta_0^{-1}(b_1)\}\\ &- \mathbb{E}\{u(V, x)| X_1 = y, Y_1 = \cdots = Y_{n - k - 1} = y, Y_{n - k} = \beta_{k - 1}^{-1}(b_k), \cdots, Y_{n - 1} = \beta_0^{-1}(b_1)\}. \end{split} \end{equation}\]
- This is positive if and only if \(x > y\), because affiliation implies that the expected payoff is increasing in the own signal.
- Thus, the above bidding strategy is the best response.

### 9.2.3 Reserve Price

- The seller does not sell the item if the winning bid is below \(r > 0\).
- The participation threshold is: \[ x^*(r) = \inf\left\{x: \mathbb{E}[w(x, Y_1)| X_1 = x, Y_1 < x] \ge r\right\}. \]
- That is, a bidder participates if and only if \(x \ge x^*(r)\).

### 9.2.4 Variations

- If PV, i.e., \(w(x, y) = x\), then, the equilibrium strategy is to participate and drop at \(\beta(x) = x\) if \(x \ge r\) and not to participate if \(x < r\).
- This is the unique equilibrium with weakly dominant strategies.
- Under the common value assumption, there can be many asymmetric equillibria (Bikhchandani & Riley, 1991; Milgrom, 1981).
- If bidders can call their bids, the game becomes more complicated because bidders can announce a “jump” bid at any time to signal their valuations (Avery, 1998).
- Be aware of the complications arising in the versions of the English auction.

### 9.2.7 Optimal Reserve Price

- The expected revenue to the seller who values the item as \(x_0\) when setting the reserve price at \(r\) is: \[ R = x_0 F_X(r)^n + r n F_X(r)^{n - 1}[1 - F_X(r)] + \int_r^{\overline{x}} w n(n - 1)F_X(w)^{n - 2}[1 - F_X(w)] f_X(w) dw. \]
- The first-order condition is: \[ r = x_0 + \frac{1 - F_X(r)}{f_X(r)}. \]
- Thus, the identification of \(F_X\) allows the seller to set the revenue maximizing reserve price.

### 9.2.8 Likelihood of IPV English Auction with Bid Data

- The likelihood function is: \[ L = \prod_{t = 1}^T [1 - F_X(w_t)] \left[ \prod_{i = 2}^{m_t} f_X(b_{it}) \right] F_X^{n_t - m_t}(r_t). \]
- \(b_{1t} \ge b_{2t} \ge \cdots \ge b_{m_t} \ge r\).
- If \(n_t\) is not observed to econometrician, the econometrician can:
- assume \(n_t = n\) and estimate \(n\) as a parameter;
- assume \(n = \max_{t = 1, \cdots, T} \{m_t\}\);
- assume \(n_t\) is drawn from a parametric distribution and estimate the parameters.

### 9.2.9 Observed Heterogeneity

- Let \(z_{it}\) be the observed attribute of bidder \(i\) in auction \(t\).
- Assume that: \[ x_{it} = \alpha + \beta z_{it} + u_{it}. \]
- Then: \[ x_{it} \ge b_{it} \Leftrightarrow u_{it} \ge b_{it} - \alpha - z_{it} \beta := \tilde{b}_{it}. \]
- We can first regress \(b_{it}\) on \(z_{it}\) to estimate \(\alpha\) and \(\beta\) to compute \(\tilde{b}_{it}\).
- Then, the rest of the argument is the same as above by replacing \(b_{it}\) with \(\tilde{b}_{it}\).

### 9.2.10 Identification

- Athey & Haile (2002) and Athey & Haile (2007) synthesize and extend the identification arguments of various auction models.
- Button auction with the symmetric IPV framework is non-parametrically identified only by the winning bid data.
- Button auction with the asymmetric IPV framework is non-parameterically identified by the winning bid and winner’s identity data.
- The non-parametric identification can fail with a common value in general.
- The actual English auctions can be dirty and not easy to characterize the equilibrium: they are open cry auctions that signals their values, bidders may not indicate they are inactive at every highest bid, and there may be a minimum bid increment.
- Haile & Tamer (2003) considers a set identification of the signal distribution:
- signal is no less than the higher bid by the bidder: \(x_i \ge b_i\).
- signal is no greater than the winning bid plus the minimum bid increment: \(x_i \le w + \Delta\).

- Let \(F_{i:n}\) be the distribution of the \(i\)-th highest order statistics from \(F_X\).
- Let \(G_{i:n}\) be the empirical distribution of the \(i\)-th highest bids.
- By 1, we have \(F_{i:n}(x) \le G_{i:n}(x)\).
- By 2, we have \(F_{2:n}(x) \ge G_{1:n}(x + \Delta)\).
- These inequalities put the bounds on \(F_X\).

## 9.3 First-Price Auctions

### 9.3.1 First-Price Sealed Bid Auction

- Each bidder independently submit a bid to the auctioneer.
- The high bidder wins and pays his bid.

### 9.3.2 Equilibrium Bidding Strategies

Assume IPV.

Then \(x^*(r) = r\).

Let \(\beta\) be the bid function that is increasing in the signal and \(\eta\) is the inverse of \(\beta\).

Suppose that the other bidders follow strategy \(\beta\).

The expected profit when a bidder with signal \(x\) submits a bid \(b\) is: \[ \pi(b, x) = (x - b) F_X[\eta(b)]^{n - 1}. \]

The first-order condition is: \[ (x - b) (n - 1) F_X[\eta(b)]^{n - 2} f_X[\eta(b)] \eta'(b)- F_X[\eta(b)]^{n - 1} = 0. \]

If \(\beta\) is the equilibrium strategy, we have: \[ [x - \beta(x)] (n - 1) F_X(x)^{n - 2} f_X(x) - \beta'(x) F_X(x)^{n - 1} = 0. \]

Let \(G(x) = F_X(x)^{n - 1}\) for \(x \ge r\) and \(G(r) = 0\) and \(g(x) = G'(x)\).

Then, we have: \[ [x - \beta(x)] g(x) - \beta'(x) G(x) = 0. \]

This is a linear differential equation such that: \[ \beta'(x) + p(x) \beta(x) = q(x), \] with a boundary condition: \[ \beta(r) = r, \] where \[ p(x) = \frac{g(x)}{G(x)}, \] and \[ q(x) = x \frac{g(x)}{G(x)}. \]

Let \(\mu(x)\) be a function such that: \[ \mu(x) p(x) = \mu'(x). \]

Multiply \(\mu(x)\) to the both sides of the first-order condition to get: \[ \begin{split} &\mu(x) \beta'(x) + \mu(x) p(x) \beta(x) = \mu(x) q(x)\\ &\Leftrightarrow \mu(x) \beta'(x) + \mu'(x) \beta(x) = \mu(x) q(x)\\ &\Leftrightarrow [\mu(x) \beta(x)]' = \mu(x) q(x). \end{split} \]

Hence, \[ \mu(x) \beta(x) = \mu(r) \beta(r) + \int_{r}^x \mu(t) q(t) dt. \]

On the other hand, \[ [\ln \mu(x)]' = p(x). \]

Hence, \[ \mu(x) = \mu(r) \exp\left(\int_{r}^x p(t) dt \right) = \exp\left(\int_{r}^x p(t) dt \right), \] by setting \(\mu(r) = 1\).

Now, \[ \begin{split} \int_{r}^x p(t) dt &= \int_{r}^x \frac{g(t)}{G(t)} dt\\ &= [\ln G(t)]_r^x. \end{split} \]

Hence, \[ \mu(x) = G(x). \]

Inserting these results gives: \[ \begin{split} \beta(x) &= \frac{\int_r^x \mu(t) q(t) dt}{\mu(x)}\\ &= \frac{\int_r^x G(t) t \frac{g(t)}{G(t)} dt}{G(x)}\\ &= \frac{\int_r^x t g(t) dt}{G(x)}\\ &= \frac{[t G(t)]_r^t - \int_{r}^x G(t) dt }{G(x)}\\ &= x - \frac{\int_r^x G(t) dt}{G(x)}\\ &= x - \frac{\int_r^x F_X(t)^{n - 1} dt}{F_X(x)^{n - 1}}. \end{split} \]

The term \(- \frac{\int_r^x F_X(t)^{n - 1} dt}{F_X(x)^{n - 1}}\) is called the

**markdown factor**, which is decreasing in the number of bidders \(n\) and increasing in the dispersion of the value distribution.The assumption of a binding reserve price ensures that there is a unique symmetric equilibrium (Athey & Haile, 2007).

### 9.3.3 Maximum Likelihood Estimation of the IPV First-Price Auction

Donald & Paarsch (1993) proposed a maximum likelihood estimator.

The data consists of \(\{w_t, r_t, n_t\}_{t = 1}^T\) for the sample where the number of actual bidders \(m_t \ge 1\).

The probability density of having \(w_t\) is: \[ \begin{split} h_t(w_t) &= n_t F_X[\eta_t(w_t)]^{n_t - 1} f_X[\eta_t(w_t)] \eta_t'(w_t)\\ &= \frac{n_t F_X[\eta_t(w_t)]^{n_t}}{(n_t - 1)[\eta_t(w_t) - w_t]}, \end{split} \] where the second equation is from the first-order condition.

Because the probability of \(m_t \ge 1\) is \(1 - F_X(r_t)^{n_t}\), the likelihood is: \[ L = \prod_{t = 1}^T \frac{h_t(w_t)}{1 - F_X(r_t)^{n_t}}. \]

To apply this approach, we may need to have a closed-form for \(\eta\), and this may require to assume a specific functional-form for \(F_X\).

### 9.3.4 Non-Parametric Approach

- Guerre, Perrigne, & Vuong (2000) proposed a non-parametric approach.
- The data consists of \(\{\{b_{it}\}_{i = 1}^{m_t}, n_t, r_t\}_{t = 1}^T\) and some observed covariates \(z_{it}\) for \(t\) with \(m_t \ge 1\).
- Assume \(n_t = n\), or in other words, focus on the data with the same number of potential bidders and estimate separately across different \(n\).
- Let \(H(b)\) be the distribution of the highest rival’s bid and \(h(b)\) be its density.
- Then, the expected payoff of bidding \(b\) when the signal is \(x\) is: \[ \pi(b, x) = (x - b) H(b). \]
- The first-order condition with respect to \(b\) is: \[ \begin{split} & (x - b) h(b) - H(b) = 0\\ &\Leftrightarrow x = b + \frac{H(b)}{h(b)} \end{split} \] where the right-hand side is actually \(\eta(b)\), the inverse of the bidding strategy \(\beta(x)\).
- The idea is that \(H(b)\) and \(h(b)\) are directly identified from the data, and so, the value \(\eta(b)\) can be computed for each bid.

### 9.3.5 Non-Parmaetric Approach: Estimation

- Note that: \[ H(b) = F_b(b)^{n - 1}, \] and \[ h(b) = (n - 1) f_b(b) F_b(b)^{n - 2}, \] where \(f_b\) and \(F_b\) are the density and distribution of the bids.

- Estimate \(f(b)\) non-parametrically, say, by a kernel regression: \[ \hat{f}_b(b) = \frac{1}{TN h_b}\sum_{t = 1}^T \sum_{i = 1}^n K\left(\frac{b_{it} - b}{h_b}\right) \] \[ \widehat{F}_b(b) = \frac{\#\{b_{it} = r\}}{NT} + \int_{r}^b \hat{f}_b(t) dt. \] for \(b > r\).
- Form the implied \(x_{it}\) by: \[ \hat{x}_{it} = b_{it} + \frac{\widehat{H}(b_{it})}{\hat{h}(b_{it})}. \]
- Estimate \(f_X\) non-parametrically, say, by a kernel regression: \[ \hat{f}_X(x) = \frac{1}{NT h_x} \sum_{t = 1}^T \sum_{i = 1}^n K\left(\frac{\hat{x}_{it} - x}{h_x}\right). \] and construct: \[ \widehat{F}_X(r) = \frac{\#\{\hat{x}_{it} = r\}}{NT} + \int_r^x \hat{f}_X(t) dt. \]

- For this argument to hold, it has to be that \(\eta(b)\) is strictly increasing in \(b\). Otherwise, for the same \(x\), multiple \(b\) can be associated.
- This approach can be extended to the symmetric IPV and affiliated values.
- Krasnokutskaya (2011) considered a model with unobserved heterogeneity, in which the bidder’s cost is \(c_i = x_i v\) and \(x_i\) is private and independent and \(v\) is known among bidders but not to econometrician.