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 \(y\), \(f(x \wedge y) f(x \vee y) \ge f(x) f(y)\).
    • \(x \wedge y \equiv (\min\{x_1, y_1\}, \cdots, \min\{x_n, y_n\})\) and \(x \vee y \equiv (\max\{x_1, y_1\}, \cdots, \max\{x_n, y_n\})\).
    • 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.
    • Inversion from the message to the private information is then not straightforward.
    • 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.
    • The assumption holds in more general settings than we may think.
    • If \(\mathbb{E}\{u(v, x_i, x_{-i})|X_i = x_i, X_{-i} = x_{-i}\} = f(x_i)\) for some monotone increasing function \(f\), then we can redefine \(x'_i = f(x_i)\) as the private signal.
    • In general, \(\mathbb{E}\{u(v, x_i, x_{-i})|X_i = x_i, X_{-i} = x_{-i}\} \neq f(x_i)\) for any monnotone increasing function \(f\). Then, it is referred to as the common value (CV) model.
  • 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.
  • 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 \(y_1\).
  • Affilication implies that \(w(x, y)\) is increasing in \(x\).
  • 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.
    • We guess that the equilibrium strategy is to take the fingers off at the threshold: \[ \beta(x) = w(x, x) \] and verify this.
    • 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, \(b = \beta(x) = w(x, x)\) 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 \(b_1, \cdots, b_k\).
    • We guess that the equilibrium strategy is: \[ \beta_k(x) = \mathbb{E}\{u(V, x)| X_1 = x, Y_1 = \cdots = Y_{n - k - 1} = x, Y_{n - k} = \beta_{k - 1}^{-1}(b_k), \cdots, Y_{n - 1} = \beta_0^{-1}(b_1)\} \] and verify this.
    • 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: \[\begin{equation} \begin{split} &\mathbb{E}\{u(V, x)| X_1 = x, Y_1 = \cdots = Y_{n - k - 1} = y, Y_{n - k} = \beta_{k - 1}^{-1}(b_k), \cdots, Y_{n - 1} = \beta_0^{-1}(b_1)\}\\ &- b\\ &=\mathbb{E}\{u(V, x)| X_1 = x, Y_1 = \cdots = Y_{n - k - 1} = y, Y_{n - k} = \beta_{k - 1}^{-1}(b_k), \cdots, Y_{n - 1} = \beta_0^{-1}(b_1)\}\\ &- \mathbb{E}\{u(V, x)| X_1 = y, Y_1 = \cdots = Y_{n - k - 1} = y, Y_{n - k} = \beta_{k - 1}^{-1}(b_k), \cdots, Y_{n - 1} = \beta_0^{-1}(b_1)\}. \end{split} \end{equation}\]
    • 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.

9.2.5 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

  • Donald & Paarsch (1996) 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 \(n = \max_{t = 1, \cdots, T} \{m_t\}\);
    • assume \(n_t = n\) and estimate \(n\) as a parameter;
    • 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:
    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 \(\beta\).

  • Assume \(\beta(r) = r\) and \(\beta(x) = 0\) for \(x < r\) (this is a restriction).

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

  • 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 (\(\beta(r) = r\) and \(\beta(x) = 0\) for \(x < r\)) 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.
  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.
  • 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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