# Chapter 9 Auction

## 9.1 General symmetric information model

### 9.1.1 Setting

• The argument of this section is based on .
• General symmetric information model .
• 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 and .
• 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 .
• 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

### 9.2.1 The Button Auction

• Let: $w(x, y) := \mathbb{E}\{u(V, x)|X_1 = x_1, Y_1 = y_1\},$ be the expected payoff of the bidder when her signal is $$x$$ and the highest rival’s signals is $$y_1$$.
• Let $$\underline{x}$$ is the lower bound of the support of $$X$$.
• Button auction:
• The price rises continuously.
• Bidders stays active as long as they keep fingers on the button.
• A bidder wins once all other bidders take their fingers off.
• The price paid by the winner is the price level when the second last bidder takes the fingers off.
• The bidding strategy is the mapping from a price level to being active or not.

### 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 .
• 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 .
• Be aware of the complications arising in the versions of the English auction.

### 9.2.5 Estimation of a IPV Button Auction

• Parameters of interest is $$F$$, the distribution of private signals and the payoff relevant random variable $$V$$.
• The data consists of $$\{w_t, n_t, r_t\}_{t = 1}^T$$ if $$m_t \ge 1$$ for auction $$t = 1, \cdots, T$$:
• $$w_t$$: the winning bid;
• $$n_t$$: the number of potential bidders;
• $$r_t$$: the reserve price;
• $$m_t$$: the latent variable about the number of actual bidders.
• This is the case only the winning bid is observed but not the other bids.
• The winning bid is $$w_t = \max\{x_{2:n_t}, r_t\}$$, where $$x_{2:n_t}$$ is the second highest bid among $$n_t$$ bids.

### 9.2.6 Likelihood of IPV Button Auction

• estimate the model with a maximum likelihood estimator.
• Assume that $$F_X(\cdot) = F_X(\cdot; \theta)$$ with a finite dimensional parameter $$\theta$$.
• The likelihood is:
• If $$m_t = 0$$, the $$F_X(r_t)^{n_t}$$.
• If $$m_t = 1$$ then $$\mathbb{P}\{m_t = 1\} = n_t F_X(r_t)^{n_t - 1} [1 - F_X(r_t)]$$.
• If $$m_t > 1$$, then $$h_t(w_t) := n_t (n_t - 1) F_X(w_t)^{n_t - 2} [1 - F_X(w_t)] f_X(w_t)$$.
• The likelihood function is, if the data is only about the auctions with $$m_t \ge 1$$: $L = \prod_{t = 1}^T \frac{h_t(w_t)^{1\{m_t > 1\}} \mathbb{P}\{m_t = 1\}}{1 - \mathbb{P}\{m_t = 0\}}$
• The approach is still valid when the private signals are asymmetric and/or some bidders are not risk neutral, because $$b(x) = x$$ is still a dominant strategy.

### 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

• and 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.
• considers a set identification of the signal distribution:
1. signal is no less than the higher bid by the bidder: $$x_i \ge b_i$$.
2. 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 .

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

• 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

• 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.
1. 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 $$b > r$$.
2. Form the implied $$x_{it}$$ by: $\hat{x}_{it} = b_{it} + \frac{\widehat{H}(b_{it})}{\hat{h}(b_{it})}.$
3. 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.
• 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.