# 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.
- 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 \(u(v, x_i, x_{-i}) = u(x, x_i)\).
- Then, \(\mathbb{E}\{u_i(V, X_i)|X_i = x_i, X_{-i} = x_{-i}\} = \mathbb{E}\{u_i(V, X_i)|X_i = x_i\} := f(x_i)\) 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.

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